Practice Exercises Advanced Micro

EconS 501: ADVANCED MICROECONOMIC THEORY – I

(Ph.D. program in Economics)

PRACTICE EXERCISES ON ADVANCED MICROECONOMIC THEORY - I

Felix Munoz-Garcia1 School of Economic Sciences Washington State University

1 103G Hulbert Hall, School of Economic Sciences, Washington State University. Pullman, WA 99164-6210, [email protected]. Tel. 509-335-8402.

Micro Theory IRecitation #1 - Preferences and Choice

1. MWG 1.B.2. Prove property (ii) of Proposition 1.B.1: If a preference relation % isrational then:

(a) � is both irre�exive (x � x never holds) and transitive (if x � y and y � z, thenx � z).

(b) � is re�exive (x � xfor all x), transitive (if x � y and y � z, then x � z), andsymmetric (if x � y, then y � x).

(c) if x � y % z, then x � z.

� Re�exivity: Since x % x for every x 2 X, x � x for every x 2 X as well. Thus � isre�exive.

� Transivity: Suppose that x � y and y � z. Then on one hand x % y and y % x, andon the other hand y % z and z % y. By transitivity of %, this implies that x % z andz % x. Thus x � z. Hence � is transitive.

� Symmetry: Suppose that x � y. Then x % y and y % x. Thus y % x and x % y. Hencey � x. Thus � is symmetric..

2. MWG 1.C.1. Consider the choice structure (B; C(�)) with B = (fx; yg; fx; y; zg) andC(fx; yg) = fxg. Show that if (B; C(�)) satis�es the weak axiom, then we must haveC(fx; y; zg) = fxg;= fzg; or = fx; zg.

� Recall that the choice structure (B; C(�)) satis�es the WARP if,

1. for some budget set B 2 B with x; y 2 B we have that element x is chosen, x 2 C(B),then

2. for any other budget set B0 2 B where alternatives x and y are also available, x; y 2 B0,and where alternative y is chosen, y 2 C(B0), then we must have that alternative x ischosen as well, x 2 C(B0).

� Then, if y 2 C(fx; y; zg), then the WARP would imply that y 2 C(fx; yg). Butcontradicts the equality C(fx; yg) = fxg. Hence y =2 C(fx; y; zg). Thus

either C (fx; y; zg) = fxg , orC (fx; y; zg) = fzg , orC (fx; y; zg) = fx; zg , butC (fx; y; zg) 6= fyg

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3. MWG 2.D.3(b). Consider an extension of the Walrasian budget set to an arbitraryconsumption set X such that

Bp;w = fx 2 X : p � x � wg:

Show that if X is a convex set, then Bp;w is convex as well.

� Let x 2 Bp;w and x0 2 Bp;w: Now consider the linear combination of these two bundlesx00 = �x+ (1� �)x0 where � 2 [0; 1]. Since X is convex, x00 2 X. Moreover,

p � x00 = �(p � x) + (1� �) (p � x0) � �w + (1� �)w = w

Thus x00 2 Bp;w.

4. MWG 2.D.4 Show that the budget set in Figure 2.D.4 is not convex.

� It follows from a direct calculation that consumption level M can be attained by8+ M�8s

s0 hours of labor. It follows from the de�nition that (24; 0) and�16� M�8s

s0 ;M�

are in the budget set (the latter is the kink where the worker obtains a wealth of M).But their convex combination of these two consumption vectors with ratio

M�8ss0

8 + M�8ss0

;8

8 + M�8ss0

!

is not in the budget set: the amount of leisure of this combination equals to 16 (so thelabor is eight hours), but the amount of the consumption good is

M8

8 + M�8ss

>8

8 + M�8ss0

=M8Ms

= 8s

5. MWG 3.B.1 Show the following:

(a) If % is strongly monotone, then it is monotone.(b) If % is monotone, then it is locally nonsatiated.

� Answer: (a) Assume that % is strongly monotone and x >> y, i.e., bundle x is higherthan bundle y in every component. Then x � y and x 6= y. Hence x � y. Thus % ismonotone.

� Answer (b). Assume that % is monotone, x 2 X, and � > 0. Let e = (1; : : : ; 1) 2 RLand y = x+ �p

Le. Then ky � xk � � and y � x. Thus % is locally nonsatiated.

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6. MWG 3.B.3 Draw a convex preference relation that is locally nonsatiated but is notmonotone.

� Following is an example of a convex, locally nonsatiated preference relation that is notmonotone i R2+. For example, x >> y but y � x.

7. [Checking properties of preference relations]. Check that the following prefer-ence relations de�ned in X = R2+ satisfy: (i) completeness, (ii) re�exivity, (iii) transi-tivity, (iv) monotonicity, and (v) weak convexity.

(a) (x1; x2) % (y1; y2) if x1 > y1 � 1.� Let us �rst build some intuition on this preference relation. Take a bundle(2; 1), you can take any other of course! Then, the upper contour set of thisbundle is given by

UCS%(2; 1) = f(x1; x2) % (2; 1)() x1 � 2� 1g = f(x1; x2) : x1 � 1g

while the lower contour set is de�ned as

LCS%(2; 1) = f(2; 1) % (x1; x2)() 2 � x1 � 1g = f(x1; x2) : x1 � 3g

Finally, the consumer is indi¤erent between bundle (2,1) and the set of bun-dles where

IND%(2; 1) = f(x1; x2) � (2; 1)() 1 � x1 � 3g

Graphically, all bundles in R2+ such that x1 � 1 belong to the set of bundlesweakly preferred to (2,1), all bundles such that x1 � 3 belong to the set ofbundles weakly preferred to (2,1), and those bundles in between, 1 � x1 � 3,are indi¤erent to (2,1).

� Completeness. We need that for any pair of bundles (x1; x2) and (y1; y2), ei-ther (x1; x2) % (y1; y2) or (y1; y2) % (x1; x2), or both (i.e., (x1; x2) � (y1; y2)).Since this preference relation only depends on the �rst component of everybundle, we have that, for every pair of bundles (x1; x2) and (y1; y2), either:

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1. x1 � y1 � 1, which implies that (x1; x2) % (y1; y2), or2. x1 < y1 � 1, which implies y1 � x1 � 1, and hence (y1; y2) % (x1; x2), or3. y1 � 1 > x1 � y1 � 1, which implies that (x1; x2) � (y1; y2). Hence, thispreference relation is complete.

� Re�exivity. We need to show that, for every bundle (x1; x2), (x1; x2) %(x1; x2). In this case, (x1; x2) % (x1; x2) implies that x1 � x1 � 1, whichis indeed true for any x1 2 R+.

� Transitivity. We need to show that, for any three bundles (x1; x2), (y1; y2)and (z1; z2) such that

if (x1; x2) % (y1; y2) and (y1; y2) % (z1; z2), then (x1; x2) % (z1; z2)

This property does not hold for this preference relation. In order to showthat, let us consider the following three bundles (that is, we are �nding acounterexample to show that transitivity does not hold):

(x1; x2) = (5; 4)

(y1; y2) = (6; 1)

(z1; z2) = (7; 2)

First, note that (x1; x2) % (y1; y2) since x1 � y1 � 1 (i.e., 5 � 6 � 1). Addi-tionally, (y1; y2) % (z1; z2) is also satis�ed since y1 � z1 � 1 (i.e., 6 � 7� 1).However, (x1; x2) � (z1; z2) since x1 � z1 � 1 (i.e., 5 � 7 � 1). Hence, thispreference relation does not satisfy Transivity.

� Continuity. Continuity requires that both the upper and the lower contoursets are closed, which is satis�ed given that they both contain their boundarypoints.

� Monotonicity. This property does not hold. For a small increase � > 0 in theamount of good 1, x1+�, we don�t necessarily have that (x1+�; x2) % (x1; x2)since for that we need x1 + � � x1 � 1, which is not true for any � smallerthan 1, � < 1:

� Weak Convexity. This property implies that the upper contour set must beconvex, that is, if bundle (x1; x2) is weakly preferred to (y1; y2), (x1; x2) %(y1; y2), then the linear combination of them is also weakly preferred to(y1; y2),

�(x1; x2) + (1� �) (y1; y2) % (y1; y2) for any � 2 [0; 1]

In this case, (x1; x2) % (y1; y2) implies that x1 � y1 � 1; whereas �(x1; x2) +(1� �) (y1; y2) % (y1; y2) implies

�x1 + (1� �) y1 � y1 � 1() x1 � y1 � 1

which is true given that (x1; x2) % (y1; y2). Hence, this preference relation isweakly convex.

(b) (x1; x2) % (y1; y2) if x1 � y1 � 1 and x2 � y2 + 1.

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� Let us �rst build some intuition on this preference relation. Take a bundle(2; 1). Then, the upper contour set of this bundle is given by

UCS%(2; 1) = f(x1; x2) % (2; 1)() x1 � 2� 1 and x2 � 1 + 1g= f(x1; x2) : x1 � 1 and x2 � 2g

which is graphically represented by all those bundles in R2+ in the lower right-hand corner (below x2 = 2 and to the right of x1 = 1). On the other hand,the lower contour set is de�ned as

LCS%(2; 1) = f(2; 1) % (x1; x2)() 2 � x1 � 1 and 1 � x2 + 1g= f(x1; x2) : x1 � 3 and x2 � 0g

which is graphically represented by all those bundles in R2+ in the left half ofthe positive quadrant (above x2 = 0 and to the left of x1 = 3).Finally, theconsumer is indi¤erent between bundle (2,1) and the set of bundles where

IND%(2; 1) = f(x1; x2) � (2; 1)() 1 � x1 � 3 and 0 � x2 � 2g

� Completeness. From the above analysis it is easy to note that this propertyis not satis�ed, since there are bundles in the area x1 > 3 and x2 � 2 whereour preference relation does not specify if they belong to the upper contourset, the lower contour set, or the indi¤erence set. Another way to prove thatcompleteness does not hold is by �nding a counterexample. In particular, wemust �nd an example of two bundles such that neither (x1; x2) % (y1; y2) nor(y1; y2) % (x1; x2). Let us take two bundles,

(x1; x2) = (1; 2) and (y1; y2) = (4; 6)

We have that:

1. (x1; x2) � (y1; y2) since 1 � 4 � 1 for the �rst component of the bundle,and

2. (y1; y2) � (x1; x2) since 6 � 2 + 1 for the second component of the bun-dle. Hence, there are two bundles for which neither (x1; x2) % (y1; y2)nor (y1; y2) % (x1; x2), which implies that this preference relation is notcomplete.

� Re�exivity. We need to show that, for every bundle (x1; x2), (x1; x2) %(x1; x2). In this case, (x1; x2) % (x1; x2) implies that x1 � x1 � 1, whichis indeed true for any x1 2 R+, and x2 � x2 + 1 which is also true for anyx2 2 R+.



This property does not hold for this preference relation. In order to showthat, let us consider the following three bundles (that is, we are �nding a

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counterexample to show that transitivity does not hold):

(x1; x2) = (2; 1)

(y1; y2) = (3; 4)

(z1; z2) = (4; 6)

First, note that (x1; x2) % (y1; y2) since x1 � y1 � 1 (i.e., 2 � 3 � 1), andx2 � y2 + 1 (i.e., 1 � 4 + 1). Additionally, (y1; y2) % (z1; z2) is also satis�edsince y1 � z1 � 1 (i.e., 3 � 4� 1), and y2 � z2 + 1 (i.e, 3 � 4 + 1). However,(x1; x2) � (z1; z2) since x1 � z1 � 1 (i.e., 2 � 4 � 1). Hence, this preferencerelation does not satisfy Transivity.

� Continuity. Continuity requires that both the upper and the lower contoursets are closed, which is satis�ed given that they both contain their boundarypoints.



�(x1; x2) + (1� �) (y1; y2) % (y1; y2) for any � 2 [0; 1]

In this case, (x1; x2) % (y1; y2) implies that x1 � y1 � 1 and x2 � y2 + 1;whereas �(x1; x2) + (1� �) (y1; y2) % (y1; y2) implies

�x1 + (1� �) y1 � y1 � 1 for the �rst component, and�x2 + (1� �) y2 � y2 + 1 for the second component.

which respectively imply

� (x1 � y1) � �1, and� (x2 � y2) � 1

and since (x1 � y1) � �1 and (x2 � y2) � 1 by assumption, i.e., (x1; x2) %(y1; y2), then both of the above conditions are true for any � 2 [0; 1]. Hence,this preference relation is weakly convex.

(c) (x1; x2) % (y1; y2) if min f3x1 + 2x2; 2x1 + 3x2g > min f3y1 + 2y2; 2y1 + 3y2g.� Let us �rst build some intuition on this preference relation. Take a bundle(2; 1). Then, the upper contour set of this bundle is given by

UCS%(2; 1) = f(x1; x2) % (2; 1)g= fmin f3x1 + 2x2; 2x1 + 3x2g > 7 � min f8; 7gg

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which is graphically represented by all those bundles in R2+ which are strictlyabove both lines 3x1 + 2x2 = 7 and 2x1 + 3x2 = 7. On the other hand, thelower contour set is de�ned as

LCS%(2; 1) = f(2; 1) % (x1; x2)g= f7 > min f3x1 + 2x2; 2x1 + 3x2gg

which is graphically represented by all those bundles in R2+ which are strictlybelow both lines 3x1 + 2x2 = 7 and 2x1 + 3x2 = 7..Finally, there are nobundles for which the consumer is just indi¤erent between bundle (2,1) andany other bundle (note that there are no bundles for which the upper andlower contour set coincide). Hence,

IND%(2; 1) = ?

� Completeness. For all (3x1 + 2x2) 2 R+ and (2x1 + 3x2) 2 R+, the minimummin f3x1 + 2x2; 2x1 + 3x2g = a exists and a 2 R+. Similarly, the minimummin f3y1 + 2y2; 2y1 + 3y2g = b exists and b 2 R+. Therefore, we can easilycompare a and b, obtaining that either a > b which implies (x1; x2) % (y1; y2),or a < b which implies (y1; y2) % (x1; x2):

� Re�exivity. We need to show that, for every bundle (x1; x2), (x1; x2) %(x1; x2). In this case, (x1; x2) % (x1; x2) implies that

min f3x1 + 2x2; 2x1 + 3x2g > min f3x1 + 2x2; 2x1 + 3x2g

holds strictly, which cannot be true, for any x1; x2 2 R+. Therefore this pref-erence relation does not satisfy Re�exivity. [Note that this preference relationasks for the above inequality to hold strictly; if the preference relation were in-stead (x1; x2) % (y1; y2) ifmin f3x1 + 2x2; 2x1 + 3x2g � min f3y1 + 2y2; 2y1 + 3y2gwith weak inequality, the preference relation would satisfy Re�exivity].



First, note that (x1; x2) % (y1; y2) implies

a � min f3x1 + 2x2; 2x1 + 3x2g > min f3y1 + 2y2; 2y1 + 3y2g � b

and (y1; y2) % (z1; z2) implies that

b � min f3y1 + 2y2; 2y1 + 3y2g > min f3z1 + 2z2; 2z1 + 3z2g � c

Combining both conditions we have that

min f3x1 + 2x2; 2x1 + 3x2g > min f3z1 + 2z2; 2z1 + 3z2g

and we can conclude that this preference relation is Transitive.

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� Continuity. Continuity requires that both the upper and the lower contoursets are closed, which is not satis�ed for this preference relation given thatneither of them contains their boundary points.

� Monotonicity. We need to show that, for a small increase � > 0 in the amountof good 1, x1 + �, we have that (x1 + �; x2) % (x1; x2). That is, we need toshow that

min f3 (x1 + �) + 2x2; 2 (x1 + �) + 3x2g > min f3x1 + 2x2; 2x1 + 3x2g

rewriting the above inequality as

min fa0; b0g > min fa; bg

we have that: (1) if x2 > x1 then b > a; or (2) if x2 � � > x1 then b0 > a0.Given that x1 2 (x2 � �; x2), then we have that

3x1 + 2x2 < 2(x1 + �) + 3x2

and if x1 < x2 � 2�, then min fa; a0g = a. On the other hand, if x1 2(x2 � 2�; x2), then min fa; b�g = a and if min fb; b0g = b.


�(x1; x2) + (1� �) (y1; y2) % (y1; y2) for any � 2 [0; 1]

In this case we must show that min fa; bg > min fc; dg implies

min f�a+ (1� �) c; �b+ (1� �) dg > min fc; dg

1. First case: min fa; bg � a, min fc; dg � c and without loss of generality,a > c.Therefore,

min f�a+ (1� �) c; �b+ (1� �) dg � �a+ (1� �) c

and �a+ (1� �) c > min fc; dg � c. For this case, convexity is satis�ed.2. Second case: min fa; bg � a, min fc; dg � d and without loss of generality,a > d.Therefore,

min f�a+ (1� �) c; �b+ (1� �) dg � �a+ (1� �) c

and �a+ (1� �) c > min fc; dg � d given that a > d and c > d. For thiscase, convexity is satis�ed as well. Analogously for the other two cases.

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[More preference relations properties]. [This is not an exercise] Let us �rst introducesome properties of the strict preference relation, �, de�ned in X = R2+:

1. (a) The strict preference relation, �, satis�es Asymmetry since there are no pair ofbundles (x1; x2) and (y1; y2) such that

(x1; x2) � (y1; y2) and (y1; y2) � (x1; x2):

(b) The strict preference relation, �, satis�es Negative Transitivity given that forevery pair of bundles (x1; x2) and (y1; y2) where (x1; x2) � (y1; y2), then for everyother bundle (z1; z2) we must have that either

1. (x1; x2) � (z1; z2) (without specifying whether (z1; z2) � (y1; y2) or (y1; y2) �(z1; z2)), or

2. (z1; z2) � (y1; y2) (without specifying whether (x1; x2) � (z1; z2) or (z1; z2) �(x1; x2)), or

3. both, i.e., (x1; x2) � (z1; z2) � (y1; y2).4. Intuition: if we can order two bundles such that one is preferred to other,(x1; x2) � (y1; y2), then a third bundle (z1; z2) can be ordered either: in be-tween of them (as in iii), or after one or both of them (as in i), or before oneor both of them (as in ii).

5. Note: Note that, despite the name, �Negative Transivity�is not a negationof Transivity.

(c) The strict preference relation, �, satis�es Irre�exivity since there is no bundle(x1; x2) for which (x1; x2) � (x1; x2).

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Micro Theory IRecitation #2 - Utility and Demand

1. Check if the utility function u(x1; x2) = x�1x�2 where �; � > 0 satis�es the following

properties:

(a) Local non-satiation;

(b) Decreasing marginal utility for both goods 1 and 2;

(c) Quasiconcavity.

(d) Homogeneous

(e) Homothetic.

1. (a) Local non-satiation (LNS). When working with a di¤erentiable utility function wecan check LNS by checking that the marginal utility from additional amounts ofgoods 1 and 2 are non-negative,

@u(x1; x2)

@x1= �x��11 x�2 = �

x�1x�2

x1> 0 if and only if � > 0

@u(x1; x2)

@x2= �x�1x

��12 = �

x�1x�2

x2> 0 if and only if � > 0

(b) Decreasing marginal utility. We need to show that the marginal utilities we foundabove are nonincreasing. That is,

@2u(x1; x2)

@x21= �(�� 1)x��21 x�2 � 0 if and only if � � 1

@2u(x1; x2)

@x22= �(� � 1)x�1x

��22 � 0 if and only if � � 1

(c) Quasiconcavity. Let us �rst simplify the expression of the utility function byapplying a monotonic transformation on u(x1; x2), since any monotonic transfor-mation of a utility function maintains the same preference ordering. That is, if autility function is quasiconcave, any monotonic transformation of it will also bequasiconcave. In this case, we apply

z1 = ln u(�) = � lnx1 + � lnx2

We now need to �nd the bordered Hessian matrix, then �nd its determinant, ifthis determinant is greater than (or equal to) zero, then this utility function is

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quasiconcave; otherwise it is quasiconvex (see Appendix M.D. in MWG, or Simonand Blume�s Mathematics for Economists). The bordered Hessian matrix is��

0 z1 z2z1 z11 z12z2 z21 z22

�� =��0 �

x1

�x2

�x1

� �x21

0�x2

0 � �x22

��and the determinant of this matrix is ��(�+�)

x21x22

� 0 for all x1; x2 2 R+, whichimplies that u(�) is quasiconcave.

(d) Homogeneous. It is easy to show that this Cobb-Douglas utility function repre-sents preferences which are homogeneous of degree �+ � in x1 and x2,

u(�x1; �x2) = (�x1)� (�x2)

� = ��+�x�1x�2 = �

�+�u(x1; x2)

(e) Homothetic preferences. First note that

MRS1;2 = ��x��11 x�2

�x�1x��12

scaling up all goods by a factor t, we obtain

MRS1;2 = �� (tx1)

��1 (tx2)�

� (tx1)� (tx2)

��1 = �t��1+�

t�+��1�x��11 x�2

�x�1x��12

= ��x��11 x�2

�x�1x��12

which shows that the MRS1;2 does not change when we scale up all goods by acommon factor t, i.e., the slope of the indi¤erence cuve at a given point is notchanged.

� A couple of remarks on Homothetic preferences. When preferences are homo-thetic, the MRS between the two goods is just a function of the consumptionratio between the goods, x1

x2, but it does not depend on the absolute amounts

consumed. As a consequence, if we double the amount of both goods, theMRS does not change.

� Recall that this type of preferences induce wealth expansion paths that arestraight lines from the origin, i.e., if we double the wealth level of the in-dividual, then his wealth expansion path (the line connecting his demandedbundles for the initial and the new wealth level) are straight lines. A corollaryof this property is that the demand function obtained from homothetic pref-erences must have an income-elasticity equal to 1, i.e., when the consumer�sincome is increased by 1%, the amount he purchases of any good k mustincrease by 1% as well.

� Examples of preference relations that are homothetic: Cobb-Douglas (as inthe previous example), preferences over goods that are considered substitutes,preferences over goods that are considered complements and CES preferences.In contrast, quasilinear preference relations are not homothetic.

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2. Let us check the quasiconcavity of a utility function that is not di¤erentiable, u(x1; x2) =max fx1; x2g :

� In order to check for quasiconcavity, we now use the de�nition: u(x1; x2) =max fx1; x2g is quasiconcave if, for every bundle (x1; x2), the set of bundles (y1; y2)such that the consumer obtains a higher utility level than from (x1; x2) is convex.That is, for every bundle (x1; x2), its upper contour set

f(y1; y2) : u(y1; y2) � u(x1; x2)g is convex

that is,f(y1; y2) : max fy1; y2g � max fx1; x2gg is convex

� As we can see from the �gure below representing this preference relation, we can�nd bundles, like x, for which its upper contour set is not convex. That is,

y % x but �x+ (1� �)y � x for all � 2 [0; 1]

x2

x1

x

y

That is, max fy1; y2g � max fx1; x2g (which implies y % x) [In this examplemax fy1; y2g = y1, and max fx1; x2g = x2, and y1 > x2]. However, constructing alinear combination of these two bundles �x+ (1� �)y we have that

max f�x1 + (1� �)y1; �x2 + (1� �)y2g < max fy1; y2g = y1

This inequality is indeed satis�ed because either:

(a) max f�x1 + (1� �)y1; �x2 + (1� �)y2g = �x1 + (1 � �)y1 (i.e., if the linearcombination of x and y is below the main diagonal), then �x1+(1��)y1 � y1for any � 2 [0; 1]; or

(b) If, instead, max f�x1 + (1� �)y1; �x2 + (1� �)y2g = �x2 + (1� �)y2 (i.e., ifthe linear combination of x and y is above the main diagonal), then we alsohave �x2 + (1� �)y2 � y1 for any � 2 [0; 1].

3

RECITATION #2

ECON 501

MWG 3.D.1. Verify that the Walrasian demand function generated by the Cobb-Douglas utility function satisfies:

(i) Homogeneity of degree zero in (p,w):

• x(αp,αw)=x(p,w) for any p,w and for any scalar α>0

(ii) Walras’ law:

• p·x=w for all x that belong to x(p,w)

(iii) Convexity/Uniqueness: If the preference relation is convex, so that u(·) is quasiconcave, then x(p,w) is a convex set. Moreover, if the preference relation is strictly quasiconvex, so that u(·) is strictly quasiconcave, then x(p,w) consists of a single point.

Solution:

① Assume the Cobb-Douglas utility function to be 11 2U Ax xα α−= , then UMP will be

1 2

11 2

,maxx x

Ax xα α− , subject to 1 1 2 2w p x p x− −

Take FOCs, we can solve for the Walrasian demand function, and get

( )1 1, /x p w w pα= , ( )2 2, (1 ) /x p w w pα= −

② Then check condition (i),

( ) ( ) ( ) ( )1 1 1, / / 1 , ,x p w w p w p x p wλ λ α λ λ α= = =

( ) ( )( ) ( ) ( ) ( )2 2, 1 / 1 / ,2 2 .x p w w p w p x p wλ λ α λ λ α= − = − =

③ check condition (ii),

( ) ( ) ( )1 1 2 2 1 1 2 2, , / 1 / .px p x p w p x p w p w p p w p wα α= + = + − =

1

④ Condition (iii) is obvious.

MWG 3.D.2. Verify that the indirect utility function

v(p,w)=[αlnα+(1-α)ln(1-α)]+lnw-αlnp1-(1-α)lnp2

satisfies the following properties:

(i) Homogeneous of degree zero in (p,w):

• v(αp,αw)=v(p,w) for any p,w and for any scalar α>0

(ii) Strictly increasing in w and nonincreasing in pl for any l.

(iii) Quasiconvex: the set {(p,w) : v(p,w)≤v} is convex for any v.

(iv) Continuous in p and w.

Solution.

① To check condition (i),

( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )

( )

1 2

1 2

1 2

,

ln 1 ln 1 ln ln 1 ln

ln 1 ln 1 ln ln ln ln 1 ln 1 ln

ln 1 ln 1 ln ln 1 ln

, .

v w

w

w

w

v w

λρ λ

α α α α λ α λρ α λρ

α α α α λ α λ α ρ α λ α

α α α α α ρ α ρ

ρ

= + − − + − − −

= + − − + + − − − − − −

= + − − + − − −

=

ρ

.<

② To check condition (ii),

( )( )( ) ( )

1 1

2 2

, / 1/ 0,

, / / 0,

, / 1 / 0

v w w w

v w

v w

ρ

ρ ρ α ρ

ρ ρ α ρ

∂ ∂ = >

∂ ∂ = − <

∂ ∂ = − −

③ We can prove that the set ( ){ }: ,L v w vρ ρ++∈ ≤ is convex.

Define , and assume ( ) ( )ln 1 ln 1 ln wα α α αΔ = + − − +

2

( ) ( )( ) ( )

( ) ( )

1 2

1 2

, ln 1 ln

, ln 1 ln

, (1 ) ,

v w V

v w

w w

ρ α ρ α ρ

ρ α ρ α ρ

ρ λρ λ ρ

⎧ = Δ − − − ≤ Δ +⎪⎪ ′ ′′ = Δ − − − ≤ Δ +⎨⎪ ′′ ′= + −⎪⎩

V

If we can prove that ( ),v wρ′′ ≤ Δ +V , then the set ( ){ }: ,L v w vρ ρ++∈ ≤ is convex.

( ) ( ) ( ) ( ) (1 2 1 1 2, ln 1 ln ln (1 ) 1 ln (1 )v w )2ρ α ρ α ρ α λρ λ ρ α λρ λ ρ′′ ′′′′ ′ ′= Δ − − − = Δ − + − − − + −

( ) ( )( ) ( )

( ) ( ) ( ) ( )( )

3

1 2

1 2

1 1 2 2

, ln 1 ln

(1 ) , (1 ) (1 ) ln 1 (1 ) ln (1 ) (1 )

(1 ) , ln (1 ) ln 1 ln (1 ) ln

v w V

v w V

v w V

λ ρ λ αλ ρ α λ ρ λ λ

λ ρ λ α λ ρ α λ ρ λ λ

λ ρ α λ ρ λ ρ α λ ρ λ ρ

⎧ = Δ − − − ≤ Δ +⎪⎨

′ ′′− = − Δ − − − − − ≤ − Δ + −⎪⎩

′ ′′+ − = Δ − + − − − + − ≤ Δ +,v wλ ρ⇒

Therefore,

( ) ( ) ( ) ( )

( ) ( )( )

( )

1 1 2

1 1 2

, ln (1 ) 1 ln (1 )

ln (1 ) ln 1 ln (1 ) ln

,

v w

Vv w V

ρ α λρ λ ρ α λρ λ

α λ ρ λ ρ α λ ρ λ ρ

ρ

′′ ′ ′⎡ ⎤= Δ − + − + − + −⎣ ⎦⎡ ⎤′ ′≤ Δ − + − + − + −⎢ ⎥⎣ ⎦

≤ Δ +

′′⇒ ≤ Δ +

2

2

ρ

( According to the property of ln(.), we have ( )1 2 1ln (1 ) ln (1 ) ln 2x x xλ λ λ λ+ − ≥ + − x . )

Therefore, the set ( ){ }: ,L v w vρ ρ++∈ ≤ is convex.

④ Condition (iv) follows the functional form of ( )v ⋅ .

MWG 3.D.6. Consider the three good setting in which the consumer has utility function

u(x)=(x1-b1)α(x2-b2)β(x3-b3)γ

a) Why can you assume that α+β+γ=1 without loss of generality? Do so for the rest of the problem.

• Solution. Define ( ) ( ) ( ) ( ) (1/1 1 2 2u x u x x b x b ) ( )3 3 ,x bα β γ α β γ′ ′ ′

with + += = − − −

( ) ( ) ( )/ , / , / .α α α β γ β β α β γ γ γ α β γ′ ′ ′= + + = + + = + + Then 1α β γ =′ ′+ + ′

and ( )u ⋅ represents the same preferences as ( )u ⋅ , because the function ( )1/u u α β γ+ +→ is a monotone transformation. Thus we can assume without loss of generality that

1.α β γ + + =

b) Write down the first-order conditions for the UMP, and derive the consumer’s Walrasian demand and indirect utility functions. [This system of demands is known as the linear expenditure system, and it is due to Stone (1954)].

• Solution. Use another monotone transformation of the given utility function,

( ) ( ) ( ) ( ) ( )1 1 2 2 3 3ln ln ln ln .u x u x x b x b x bα β γ= = − + − + −

( ) ( ) ( )1 1 2 2 3 3 1 1 2 2 3 3

11 1 1

1 1 1 1

22 2 2

2 2 2 2

33 3 3 3 3 3 3

1 1 2 2 3 3

1 1 2 2

ln ln ln ( )

0

0:

0

0

x b x b x b w p x p x p

px x b

p x p bp

x x bFOCs p x p bp

x x b p x p b

w p x p x p x

p x p x

α β γ λ

α λα

β λλβλγ λ γλ

λ

= − + − + − + − − −

∂⎧ = − =⎪∂ − ⎧⎪ = +⎪∂⎪= − = ⎪⎪∂ −⎪ ⎪⇒ = +⎨ ⎨

∂⎪ ⎪= − =⎪ ⎪∂ − = +⎪ ⎪⎩∂⎪ = − − − =⎪∂⎩

⇒ + +

x

3 3 1 1 2 2 3 3p x p b p b p b wα β γλ

+ += + + + =

Let 1 1 2 2 3 3.p b p b p b p b⋅ = + +

( )

1 1 11 1 1

2 2 22 2 2

3 3 33 3 3

( ) ( )

( ) ( )

( ) ( )

1

w pb

w pb1

2

3

x b b wp p p

w pb

pb b

x b bp p p

w pb

w pb b

x b b wp p p

pb b

α β γλ

α α αλ α β γβ β βλ α β γγ γ γλ α β γ

α β γ

+ +⇒ =

−

⎧ −= + = + = − +⎪ + +⎪

⎪ −⇒ = + = + = − +⎨ + +⎪

⎪ −= + = + = − +⎪

+ +⎩+ + =

Get demand function ( ) ( ) ( )( )1 2 3 1 2 3, , , / , / , /x p w b b b w p b p p pα β γ= + − ⋅

Plug into u(x)=(x1-b1)α(x2-b2)β(x3-b3)γ, then we obtain the indirect utility function

( ) ( )( ) ( ) ( )1 2 3, / /v p w w p b p p p/ .α β γα β γ= − ⋅

4

c) Verify that these demand functions satisfy the properties listed in Propositions 3.D.2 for the osition 3.D.3 for the indirect utility function. Walrasian demand function, and in Prop

① To check the homogeneity of the demand function,

( ) ( ) ( )( )

( ) ( )( ) ( )

1 2 3 1 2 3

1 2 3 1 2 3

, , , / , / , /

, , / , / , / ,

x w b b b w b

b b b w b x p w

λρ λ λ λρ α λρ β λρ γ λρ

ρ α ρ β ρ γ ρ

= + − ⋅

= + − ⋅ = .

To check Walras law,

( ) ( ) ( ) ( )( )( )

( )( )

1 1 2 2 3 3

1 1 2 2 3 3 1 1 2 2 3 3

, , , ,

/ / /

.

x w x w x w x w

b b b w b

b w b w

ρ ρ ρ ρ ρ ρ ρ ρ

ρ ρ ρ ρ ρ α ρ ρ β ρ ρ γ ρ

ρ ρ α β γ

⋅ = ⋅ + ⋅ + ⋅

= + + + − ⋅ + +

= ⋅ + − ⋅ + + =

The uniqueness is obvious.

② To check the homogeneity of the indirect utility function,

( ) ( )( ) ( ) ( )

( ) ( )( ) ( ) ( )

( )( ) ( ) ( ) ( )

1 2 3

11 2

1 2 3

, / / /

/ / /

/ / / ,

v w w b

w b

w b v p w

3

.

α β γ

α β γα β γ

α β γ

λρ λ λ λρ α λρ β λρ γ λρ

λ ρ α ρ β ρ γ

ρ α ρ β ρ γ ρ

− + +

= − ⋅

= − ⋅

= − ⋅ =

ρ

,>

To check the monotonicity,

( ) ( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

1 2 3

1 1

2 2

3 3

, / / / / 0

, / , / 0,

, / , / 0,

, / , / 0.

v w w

v w v w

v w v w

v w v w

α β γρ α ρ β ρ γ ρ

ρ ρ ρ α ρ

ρ ρ ρ β ρ

ρ ρ ρ γ ρ

∂ ∂ =

∂ ∂ = ⋅ − <

∂ ∂ = ⋅ − <

∂ ∂ = ⋅ − <

5 The continuity follows directly from the given functional form.

In order to prove the quasiconvexity, it is sufficient to prove that, for any v and the set ∈ 0,w >

( ){ }3 : ,v w vρ ρ∈ ≤ is convex. Consider

( ) ( ) 1 2ln , ln ln ln ln ln ln lnv w w b 3.ρ α α β β γ γ ρ α ρ β ρ γ ρ= + + + − ⋅ − − −

Since the logarithmic function is concave, the set

( ){ }31 2 3: ln ln ln lnw b vρ ρ α ρ β ρ γ ρ∈ − ⋅ − − − ≤

is convex for every Since the other terms, ln ln.v∈ ln ,α α β β γ γ+ + do not depend on ,ρ this

implies that the set ( ){ }3ρ : ln ,v w vρ∈ ≤ is convex. Hence so is ( ){ }3ρ : ,v w vρ∈ ≤ .

(Or follow the same step in MWG 3.D.2 ③. )

MWG 3.D.7. There are two commodities. We are given two budget sets Bp0

,w0 and Bp

1,w

1 described, respectively, by

p0=(1,1) and w0=8

p1=(1,4) and w1=26

The observed choice at (p0, w0) is x0=(4,4), and at (p1, w1) the consumer’s choice satisfies p·x1=w.

a) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximiz of preferences. ation

• Since 1 0 1 1 0 and ,x w x xρ ⋅ < ≠ the weak axiom implies 0 1 0.x wρ ⋅ > Thus 1x has to be

on the bold line in the following figure.

6

In the following four questions, we assume the given preference can be a differentiable utility function

( ).u ⋅

b) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences that are quasilinear with respect to the first good.

• If the preference is quasilinear with respect to the first good, then we can take a utility

function ( )u ⋅ so that ( ) 1/u x x∂ ∂

( ) /t tu x x

1= for every x (Exercise 3.C.5(b). Hence the first‐order

condition implies 2 2 1/t tρ ρ∂ ∂ = for each 0,1t = . Since 0 0 1 12 1 2 1/ /ρ ρ ρ ρ< and

( )u ⋅ is concave, 0 12 2x x> . Thus 1x has to be on the bold line in the following figure.

7

c) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences that are quasilinear with respect to the second good.

• If the preference is quasilinear with respect to the second good, then we can take a utility

function ( )u ⋅ so that ( ) 2/u x x∂ ∂

( ) /t tu x x

1= for every x (Exercise 3.C.5(b)). Hence the first‐order

condition implies 1 1 2/t tρ ρ∂ ∂ for each 1,0t= = . Since 0 0 1 11 2 1 2/ /ρ ρ ρ ρ> and

( )u ⋅ is concave, we must have 0 11 1x x< . Thus 1x has to be on the bold line in the following

figure.

d) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences for which both goods are normal.

• Since 1 0 1x wρ ⋅ < and the relative price of good 1 decreased, 1tx has to increase if good 1 is

normal. If good 2 is normal, then the wealth effect (positive) and the substitution effect

(negative) go in opposite direction which gives us no additional information about 2x . Thus

1x has to be on the bold line in the following figure.

8

e) Determine the region of permissible choices for x1, if the choices x0 and x1 are consistent with maximization of preferences when preferences are homothetic.

• If the preference is homothetic, the marginal rates of substitution at all vectors on a ray are

the same, and they become less steep as the ray becomes flatter. By the first‐order

conditions and 11 2 1 2/ / ,0 0 1 1 xρ ρ ρ ρ> has to be on the right side of the ray that goes

through 0x . Thus 1x has to be on the bold line in the following figure.

9

EconS 501 Fall 2008 Felix Munoz

Exercises – Recitation #3

Exercise 1. Find the demanded bundle for a consumer whose utility function is u(x1,x2)= x1

3/2x2 and her budget constraint is 3x1+4x2=100. Solution. Write the Lagrangian

1 2 1 23( , ) ln ln (3 4 100)x x x x xλ λ= + − + −2

(Be sure you understand why we can transform u this way.) Now, equating the derivatives with respect to x1, x2, andλ to zero, we get three equations in three unknowns

1

2

1 2

21 4 ,

3 4 100.

x

xx x

λ=

+ =

3 3 ,λ=

1(3,4,100) 20x

Solving, we get 2(3,4,100) 10x= = . , and

Note that if you are going to interpret the Lagrange multiplier as the marginal utility of income, you must be explicit as to which utility function you are referring to. Thus, the marginal utility of income can be measured in original ‘utils’ or in ‘ln utils’. Let u*=lnu and, correspondingly, v*=lnv; then

*( , )v p m

( , ) ,( , ) ( , )

v p m mm v p m v p m

μλ

∂∂ ∂= = =

∂

μ denotes he Lagrange multiplier in the Lagrangian Where 3/21 2 1 2( , ) (3 4 100).x x x x xμ μ= − + − 3/2204

μ =Check that in this problem we’d get , 1λ = 3/2(3, 4,100) 20 10v =

1/2 1/31 2 1 1 2 2( , ) ( ),

40, and .

Exercise 2. Use the utility function u(x1,x2)= x1

1/2x21/3 and the budget constraint m=p1x1+p2x2 to calculate

the Walrasian demand, the indirect utility function, the Hicksian demand, and the expenditure function. Solution. The Lagrangian for the utility maximization problem is

x x x p x p x mλ λ= − + − Taking derivatives,

1/2 1/31 2 1

1 ,

1/2 2/31 2 2

1 1 2 2

21 ,3

.

x x p

x x p

p x p x m

λ− =

+ =

λ− =

1


Solving, we get

1 21 2

3 2( , ) , ( , ) .5 5

m mx p m x p mp p

= =

Plugging these demands into the utility function, we get the indirect utility function

( )1/2 1/3 1/2 1/35/6

1 2 1 2

3 2 3 2( , ) ( , ) .5 5 5

m m mv p m U x p mp p p p

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞= = =⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

Rewrite the above expression replacing v(p, m) by u and m by e(p, u). Then solve it for e(.) to get 3/5 2/5

6/51 2( , ) 53 2p pe p u u⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

Finally, since , the Hicksian demands are /ih e p= ∂ ∂ i2/5 2/5

6/51 21

3/5 3/56/51 2

2

( , ) ,3 2

( , ) .3 2

p ph p u u

p ph p u u

−

−

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞ ⎛ ⎞= ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Exercise 3. Consider a two-period model with Dave’s utility given by ( )1 2,u x x where 1x represents

his consumption during the first period and 2x is his second period’s consumption. Dave is endowed

with ( 1 2, )x x which he could consume in each period, but he could also trade present consumption for future consumption and vice versa. Thus, his budget constraint is

1 1 2 2 1 1 2 2,p x p x p x p x+ = +

where 1p and 2p are the first and second period prices respectively.

a) Derive the Slutsky equation in this model. (Note that now Dave’s income depends on the value of his endowment which, in turn, depends on prices: 1 1 2 2m p x p x= + .)

Solution. Differentiate the identity ( )( , ) , ( , )j jh p u x p e p u≡ with respect to ip to get

( )( , ) ( , ) , ( , ) ( , )j j j

i i iWe must be careful with this last term. Look at the expenditure minimization problem

h p u x p m x p e p u e p up p m p

∂ ∂ ∂ ∂= +

∂ ∂ ∂ ∂

{ }( , ) min ( ) : ( )e p u p x x u x u= − = By the envelope theorem, we have

( )( , ) ( , ) , ( , )i i ii

e p u h p u x x p e p u xp

∂i= − = −

∂

Therefore, we have ( ) ( )

( , ) ( , ) , ( , )( , )j j j

i ii i

h p u x p m x p e p ux p m x

p p m∂ ∂ ∂

= +∂ ∂ ∂

−

And reorganizing we get the Slutsky equation

2


( ) ( )( , ) ( , ) , ( , )

( , )j j ji i

i i

x p m h p u x p e p ux x p m

p p m∂ ∂ ∂

= + −∂ ∂ ∂

b) Assume that Dave’s optimal choice is such that 1 .x x< If 1p goes down, will Dave be better off

or worse off? What if 2p goes down? Solution. Draw a diagram, play with it and verify that Dave is better off when p2 goes down and worse off when p1 goes down. Just look at the sets of allocations that are strictly better or worse than the original choice—i.e., the sets ( ) { : }SB x z z x= and ( ) { : }SW x z z x= ≺ . When p1 goes down the new budget set is contained in SW(x), while when p2 goes down there’s a region of the new budget set that lies in SB(x).

Exercise 4. The utility function is ( ) { }1 2 2 1 1 2, min 2 , 2u x x x x x x= + + .

a) Draw the indifference curve for ( )1 2, 2x = 0.u x Shade the area where ( )1 2, 20.u x x ≥

Solution. Draw the lines and 2 12 2x x+ = 0 01 22 2x x+ = . The indifference curve is the northeast boundary of this X.

b) For what values of 1 2/p p will the unique optimum be 1 0?x =

Solution. The slope of a budget line is 1 2/p p− . If the budget line is steeper than 2, 1 0x = .

Hence the condition is . 1 2/ 2p p >

c) For what values of 1 2/p p will the unique optimum 2 0?x =

Solution. Similarly, if the budget line is flatter than 1/2, 2x will equal 0, so the condition is

. 1 2/ 1/p p < 2d) If neither 1x nor 2x is equal to zero, and the optimum is unique, what must be the value of

1 2/ ?x x

Solution. If the optimum is unique, it must occur where 2 1 1 22 2 .x x x x− = − This implies that

1 2x x= , so that . 1 2/ 1x x = Exercise 5. Under current tax law some individuals can save up to $2,000 a year in an Individual Retirement Account (I.R.A.), a savings vehicle that has an especially favorable tax treatment. Consider an individual at a specific point in time who has income Y, which he or she wants to spend on consumption, C, I.R.S. savings, , or ordinary savings S . Suppose that the “reduced form” utility function is taken to be:

1S 2

( )1 2 1 2, , .U C S S S CSα β γ= (This is a reduced form since the parameters are not truly exogenous taste parameters, but also include the tax treatment of the assets, etc.) The budget constraint of the consumer is given by:

3


1 2 ,C S S Y+ + = and the limit that he or she can contribute to the I.R.A. is denoted by L.

a) Derive the demand functions for 1S and 2S for a consumer for whom the limit L is not binding.

Solution. This is an ordinary Cobb-Douglas demand: 1S Yαα β γ

=+ + and

2S Yβα β γ

=+ +

b) Derive the demand function 1S and 2S for a consumer for whom the limit L is binding.

Solution. In this case the utility function becomes 1 1( , , )U C S L S L Cα β γ= . The L term is just a

constant, so applying the standard Cobb-Douglas formula 1S Yαα γ

=+

.

Exercise 3.E.7. Show that if a preference relation is quasilinear with respect to good 1, the Hicksian demand functions for the remaining goods 2, 3, …, L do not depend on u. What is the form of the expenditure function in this case? Solution. Exercise 3.C.5(b) in MWG shows that every quasilinear preference with respect to good 1 can be represented by a utility function of the form ( ) (1 2, , Lu x x u x x= + … ) . Let

( )1 1,0, ,0 Le = … ∈ . We shall prove that for every with 0 1 1, ,p up ,α= ∈ ∈

if ( )and , Lx +∈ −∞ ∞ × 1,− ( ),x h p u= , then ( ), .h p u1x eα α++ = Note first that

( )1 ,u x e uα α+ ≥ + that is, 1x eα+ satisfies the constraint of the EMP for ( )., ,p u α+ Let

( ) and Ly u y u α+∈ ≥ + . Then ( )1eα .uu y − ≥ Hence ( )1 .p y e p xα⋅ − ≥ ⋅ Thus

( )1 .p y p x eα⋅ ≥ ⋅ + Hence ( ), .u1x e h pα α++ = Therefore, for every { } ( ) ( )2 , , , and , , ,L u u h p u h p u′ ′∈ ∈ ∈ =…

2, L…( ) ( ) 1en ,h p u h p ue= = +

. That is, the Hicksian

demand functions for goods are independent of utility levels. Thus, if we define

. ( ) ( ),0 , thh p h p

Since ( ) ( ) ( ) ( )1, , , we have , ,h p u h p u e e p u e p u .α α α+ = + + = +α Thus, if we define

( ) ( ) ( ) ( ), 0 , then e , .e p p u e p u= = +e p

4


Exericse 3.E.8. For the Cobb-Douglas utility function, verify that the following relationships in (3.E.1) and (3.E.3) respectively hold.

e(p,v(p,w))=w and v(p,e(p,u))=u, and h(p,u)=x(p,e(p,u)) and x(p,w)=h(p,v(p,w))

Note that the expenditure function can be derived by simply inverting the indirect utility function, and vice versa.

Solution. We use the utility function ( ) ( )11 2 .u x x x αα −= To prove (3.E.1),

( )( ) ( ) ( )( )( )( ) ( ) ( )( )

1 11 11 2 1 2

1 11 11 2 1 2

, , 1 1

, , 1 1 .

e p v p w p p p p w w

v p e p u p p p p u u

α αα α α α α α

α αα α α α α α

α α α α

α α α α

− −− − − −

− −− − − −

= − − =

= − − =

,

To prove (3.E.3),

( )( ) ( )( ) ( )( )

( )( ) ( )

( )( ) ( ) ( )( )

( )( ) ( )

1 11 2 1 2

112

1 2

11 11 2

1 21 2

1 2

, , 1 , 1

1, , ,

1

1, , 1 ,

1

, 1 , .

x p e p u p p u p p

pp u u h p up p

pph p v p w p p wp p

w p p x p w

αα α α

α α

α ααα α α

α α α α

ααα α

ααα αα α

α α

−− −

−

−− − −

= − −

⎛ ⎞⎛ ⎞ ⎛ ⎞−⎜ ⎟= =⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞ ⎛ ⎞−⎜ ⎟= − ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟− ⎝ ⎠⎝ ⎠⎝ ⎠

= − =

Exercise 3.E.9. Use the relations in 3.E.1:

e(p,v(p,w))=w and v(p,e(p,u))=u

to show that the properties of the indirect utility function v(p,w) identified in Proposition 3.D.3:

1. Homogeneity of degree zero.

2. Strictly increasing in w and nonincreasing in pk for any good k.

3. Quasiconvex; that is, the set {(p,w): v(p,w)≤v} is convex for any v.

4. Continuous in p and w.

imply the properties of the expenditure function e(p,u) identified in Proposition 3.E.2:

1. Homogeneous of degree one in prices.

2. Strictly increasing in u and nondecreasing in pk for any good k.

5


3. Concave in prices.

4. Continuous in p and w.

Likewise, use the relations e(p,v(p,w))=w and v(p,e(p,u))=u

to prove that the properties of e(p,u) identified in Proposition 3.E.2 imply the properties of v(p,w) identified in Proposition 3.D.3. Solution. First, we shall prove that Proposition 3.D.3 implies Proposition 3.E.2 via (3.E.1). Let

0, 0, , , and 0.p p u u α′ ′∈ ∈ ≥

(i) Homogeneity of degree one in p: Let 0.α > Define ( ) (, , then ,w e p u u v p w= = ) by the second relation of (3.E.1). Hence

( ) ( )( ) ( )( ) ( ), , , , ,e p u e p v p w e p v p w w e p uα α α α α α α= = = = , ,

where the second equality follows from the homogeneity of ( ),v ⋅ ⋅ and the third from the first relation of (3.E.1).

(ii) Monotonicity: Let .u u′ > Define ( ) ( ) (, and , , then ,w e p u w e p u u v p w′ ′= = = ) and

( ), .′ By the monotonicity of u v p w′ = ( ),v ⋅ ⋅ in w, we must have ,w w′ > that is,

( )u ( ), ,e p u e p′ > .

Next let .p p′ ≥ Define ( ),w e p u= and ( ),w e p u ,′ ′= then, by the second relation of

(3.E.1), By the monotonicity of ( ),p w= = (u v v p′ ), .w′ ( ),v ⋅ ⋅ , we must have ,w w′ ≥ that

is, ( ) ).u ≥ (, ,e p ue p′

(iii) Concavity: Let [ ]0,1 . Define α ∈ ( ),w e p u and ( ), , then

, . Define

= w e p u′ ′=

( ),u v p w v= = ( )p w′ ( ) pα1p pα′′ ′− ′ and = + ( )1 wα α .w w′′ ′= + − Then, by

the quasiconvexity of ( ),v ⋅ ⋅ , ( ), w u′′ ′′ .v p ≤ Hence, by the monotonicity of ( ),v ⋅ ⋅ in w and

the second relation of (3.E.1), ( ), .w e p u′′ ′′≤ That is,

( )( ) ( ) ( ) ( )1 , , 1 ,e p p u e p u e p uα α α α .′ ′+ − ≥ + −

(iv) Continuity: It is sufficient to prove the following statement: For any sequence ( ){ }1

,n nn

p u∞

=

with ),( ) (,n np u p→ u and any w, if ( ),n ne p u w≤ for every n, then ( ),e p u w≤ ; if

for every n, then ( ),n nu w≥e p ( ),p ue Suppose w≥ . ( )e ,n np u w≤ for every n. Then, by

6


the monotonicity of ( ),v ⋅ ⋅ in w, and the second relation of (3.E.1), we have n nu for

every n. By the continuity of

( ),v p w≤

( ) ( ), , , .v w⋅ By the second relation of (3.E.1) and the

monotonicity of (u v p⋅ ≤

),v ⋅ ⋅ in w, we must have ( ), .e p u w≤ The same argument can be applied

for the case with ( ,ne p u y n. ) w for ever

and 0.

n

w

≥

,

Let’s next prove that Proposition 3.E.2 implies Proposition 3.D.3 via (3.E.1). Let 0, 0,p p w , α′ ′∈ ∈ ≥

i. Homogeneity: Let 0.α > Define ( ), .p w Then, by the first relation of (3.E.1), u v=

( ),e p u .= Hence w

( ) ( )( ) (( ) () ), ,w=, , ,p e p w v pα α α= = ,e pα ,v p u u v pα α =w v

where the second equality follows from the homogeneity of ( ),e ⋅ ⋅ and the third from the second relation of (3.E.1).

ii. Monotonicity: Let .w w′ > Define ( ),u v p w and ( ), , then u v p w′ ′==

( ),e p u ( ), .u we p ′ ′=w= and By the monotonicity of ( ),e ⋅ ⋅ and w w′ > , we must

have u u′ > , that is, ( ) ( ), ,v p w v p w′ > .

. Define ( ),p wp p′ ≥ u v=Next, assume that and then

By the monotonicity of

( ), ,u v p w′ ′=

( )u (, e p= ), .w= ( ),e ⋅ ⋅ and pu′ ′ p≥e p ′ , we must have

, that is, u′ ≤ u ( ) ( ), ,w v p w′≥v p .

iii. Quasiconvexity: Let [ ]0,1α ∈ . Define ( ),u v p w and ( ), .u v p w Then

= and

= ′ ′ ′=

( ),e p u w ( ),e p u w′ = . Without loss of generality, assume that u u′ ≥ .

Define 1 pα ′− and ( )p pα= +′′ ( )1 wα ′w wα= +

( )

( ) ( )

( ) ( )

( )

,

, 1

, 1

1 ,

e p u

e p u e

e p u e

w w

α α

α α

α α

′′ ′

− . Then

(

(

,

,

p u

p u

w

)

)

′ ′ ′≥ + −

′ ′≥ + −

′ ′′= + − =

7


8

where the first inequality follows from the concavity of ( ),e u⋅ the second from the

monotonicity of ( ),e ⋅ ⋅ in u and .u u′ ≥ We must thus have ( ), .w u′′ ′≤v p′′

iv. Continuity: It is sufficient to prove the following statement. For any sequence

( ){ }1

,n nn

p w=

∞ with ( ) (,n n ),p w p→ w and any u, if ( ),n nv p w u≤ for every n,

then ( ),v p w u≤ ; if ( ),n nv p w u≥ for every n, then ( ),v p w u≥ . Suppose

for every n. Then, by the monotonicity of (( ),n nv p w u≤ ),e ⋅ ⋅ in u and the first

relation of (3.E.1), we have ( ),np u for every n. By the continuity of nw e≤

( ) ( ), ,e w⋅ ⋅ ≤ , .e p u We must thus have ( ), .w uv p ≤ The same argument can be

applied for the case with ( ),n nv p w u≥ for every n.

Alternative: An alternative, simpler way to show the equivalence on the concavity/quasiconvexity and the continuity uses what is sometimes called the epigraph.

For the concavity/quasiconvexity, the concavity of ( ),e u⋅ is equivalent to the convexity

of the set ( ) ( ){ }, : ,p w e p u w≥ and the quasi-convexity of ( )v ⋅ is the equivalent to the

convexity of the set ( ) ( ){ }, :p w ,v p w u≤ for every u. But (3.E.1) and the

monotonicity imply that ( ),v p w u≤ if and only if ( ),u ≥ .e p w Hence the two sets

coincide and the quasiconvexity of ( )v ⋅ is equivalent to the concavity of ( ),e u⋅ .

As for the continuity, the function ( )e ⋅ is continuous if and only if both

( ) ( ){ }, , : ,p w u e p u w≤ and ( ) ( ){ }, , ,p w u u w≥: e p are closed sets. The function

( )v ⋅ is continuous if and only if both ( ) ( ){ }, , : ,p w u v p w u≥ and

( ) ( ){ }, , : ,p w u v p w u≤ are closed sets. But, again by (3.E.1) and the monotonicity,

( ) ( ){ } ( ) ( ){ }

( ) ( ){ } ( ) ( ){ }

, , : , , , : , ;

, , : , , , : ,

p w u e p u w p w u v p w u

p w u e p u w p w u v p w u

≤ = ≥

≥ = ≤

Hence the continuity of ( )e ⋅ is equivalent to that of ( )v ⋅ .

Felix Munoz Fall 2008 EconS 501

Microeconomic Theory – R 1. Jan’s utility function for goods X and Y is .75 .257200 .U X Y= She must pay $90 for a

unit of good X and $3

ecitation #3 – Exercises.

0 for a unit of good Y. Jan’s income is $1200.

a. Determine the amounts of goods X and Y Jan purchases to maximize her utility given her budget constraint.

1


b. Determine the maximum amount of utility Jan receives.

c. Determine the value of *λ associated with this problem. d. Interpret the value of *λ you computed in part c. as it specifically applies to Jan.

2


2. a. Formulate the dual constrained expenditure minimization problem associated with 4.3 and determine the optimal amounts of goods X and Y Jan should purchase.

3


b. Determine the minimum amount of expenditure made by Jan.

c. Determine the optimal value of D

λ and provide a written interpretation of this value as it specifically applies to Jan in this problem.

d. Compare the optimal values of X, Y and λ you computed in exercise 4.3 with

those you computed in parts a. and c. of this exercise.

4


3. Raymond derives utility from consuming goods X and Y, where his utility function is

.25 .2580 .U X Y= He spends all of his income, I, on his purchases of goods X and Y, and he must pay prices of xP and yP for each unit of these goods, respectively. Assume that his income is $3200, the unit price of good X is $100, and the unit price of good Y is $100.

a. Determine the amounts of goods X and Y that Raymond should purchase to maximize his utility given his budget constraint.

5


b. Determine the maximum amount of utility Raymond can receive.

. fe4 Re r to your response to exercise 5.1.

a. Derive Raymond’s own‐price demand curve for good X.

b. Derive Raymond’s own‐price demand curve for good Y.

6


5. feRe r to your responses to exercise 5.1.

a. Derive Raymond’s Engel curve for good X.

b. Is good X a normal good or an inferior good? Justify your response mathematically.

7


c. Derive Raymond’s Engel curve for good Y.

d. Is good Y a normal good or an inferior good? Justify your response mathematically.

8


6. Assume an individual’s own‐price demand function for good X is ( ), , 200 4 1.5 0.008x y x YX X P P I= where of P P I = − − + xP and yP denote the unit

e. prices of goods X and Y, respectively, and I denotes the consumer’s money incom

a. Compute the individual’s cross‐price demand curve for good X when the unit price of good X is $2 and the consumer’s income is $40,000.

b. Are goods X and Y gross substitutes or gross complements? Justify your response mathematically.

9


7. Recall from exercise 5.1 Raymond’s utility function, when he consumes goods X and Y, is .25 .2580 .U X Y= Once again, assume the unit price of good X, xP , is $100, and the unit price of good Y, yP , is $100. Determine the quantities of goods X and Y Raymond should purchase that will minimize his expenditures on these goods and yield 320 units of utility to him.

10


8. feRe r to your response to exercise 5.5.

a. Determine Raymond’s compensated demand curve for good X.

b. Determine Raymond’s compensated demand curve for good Y.

11


. Is it possible for an individual’s demand curve for a good to be positively sloped?

Support your response with an appropriate graphical analysis. 9

12


13

EconS 501 Recitation #4 Fall 2010 Felix Munoz

1

Exercise 1. Prove that Proposition 3.G.1 in MWG is implied by Roy’s identity (Proposition 3.G.4). Note: we are always assuming that we are at an optimum.

Answer. Since the identity ( )( ), ,v p e p u u= holds for all p, differentiation with respect to p yields

( )( ) ( )( ) ( )

( ) ( )( )( , )

, ,, , , 0.

where we differentiate the first term , and the second term , ,

We apply the chain rule when differentiating the second term

p p

e p u

v p e p uv p e p u e p u

w

v p v e p u

∂∇ + ∇ =

∂

⋅ ⋅

By Roy’s identity,

( )( ) ( )( )

( )( ) ( )( ) ( )

( )( ) ( )

, , , ,( , )

, , , ,( , ) , 0

, ,( , ) , 0

ll

l p

l p

v p e p u v p e p ux p w

w p

v p e p u v p e p ux p w e p u

w wv p e p u

x p w e p uw

∂ ∂− ⋅ =

∂ ∂

∂ ∂− ⋅ + ∇ =

∂ ∂∂

⎡ ⎤− +∇ =⎣ ⎦∂

By ( )( )If we have more money to spend we can reach a greater utility

, , / 0v p e p u w∂ ∂ > and ( ) ( )( )considered at an optimum

, , , ,h p u x p e p u= we obtain ( ) ( ), , .ph p u e p u= ∇

Exercise 2. Verify for the case of a Cobb-Douglas utility function that all of the propositions in Section 3.G hold.

Answer. From examples 3.D.1 and 3.E.1, for the utility function ( ) 11 2u x x xα α−= , we obtain

( ) ( )

( ) ( )

( )( ) ( )

( ) ( )( )

( ) ( )

( ) ( )

1

2

21

22

11 21

2

21 211 2

1

21 2 2

, ,1

0

, ,1

0

, ,11

1 1

, ,1 11

w

p

p p

pD x p w

p

wp

D x p ww

p

pp pe p u u

p

p ppp pD e p u D h p u u

p p p

α

α

α

α

α

α

α

α

α

αα α

α α α α

α α α α αα

−

−

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥−⎢ ⎥⎣ ⎦⎡ ⎤−⎢ ⎥⎢ ⎥= ⎢ ⎥−

−⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎛ ⎞⎛ ⎞ ⎢ ⎥⎜ ⎟∇ = ⎜ ⎟ ⎢ ⎥⎜ ⎟ −⎝ ⎠ −⎝ ⎠ ⎢ ⎥⎣ ⎦

⎡ − −−⎢⎛ ⎞⎛ ⎞ ⎢⎜ ⎟= = ⎜ ⎟ ⎢⎜ ⎟ − −⎝ ⎠ −⎝ ⎠ ⎢ −⎢⎣

⎤⎥⎥⎥⎥⎥⎦


2

The indirect utility function for ( ) 11 2u x x xα α−= is

( ) ( )

11 2, .

1p pv p w w

αα

α α

−⎛ ⎞⎛ ⎞= ⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠

(Note here that the indirect utility function obtained in Example 3.D.2 is for the utility function

( ) ( )1 2ln 1 lnu x x xα α= + − .) Thus

( ) ( ) ( )( )

( ) ( )1 2, , / , 1 / ,

,, .

p

w

v p w v p w p p

v p wv p w

w

α α∇ = − − −

∇ =

Hence, ( ) ( ), , ,ph p u e p u= ∇ ( ) ( )2 , , ,p pD e p u D h p u= which is negative semidefinite and

symmetric,

( ), 0,pD h p u p = ( ) ( ) ( ) ( ), , , , ,Tp p wD h p u D x p w D x p w x p w= + and

( ) ( )( )( )( )

, /, .

, /v p u p

x p wv p u w

∂ ∂= −

∂ ∂

Exercise 3. A utility function ( )u x is additively separable if it has the form ( ) ( ).u x u x=∑

Show that the induced ordering on any group of commodities is independent of whatever fixed values we attach to the remaining ones. It turns out that this ordinal property is not only necessary but also sufficient for the existence of an additive separable representation.

Answer. Define { }1, ,S L= … and let T be a subset of S. The commodity vectors for those in S are

represented by { } #1

TTz z +∈= ∈ and the like, and the commodity vectors for those outside S are

represented by { } #2

L TTz z −

+∈= ∈ and the like. We shall prove that for every

# # #1 1 2, , ,T T L Tz z z −

+ + +′∈ ∈ ∈ and ( ) ( )#

2 1 2 1 2, , ,L Tz z z z z−+

′ ′∈ if and only if

( ) ( )1 2 1 2, , .z z z z′ ′ ′ In fact, since ( )u ⋅ represents , ( ) ( )1 2 1 2, ,z z z z′ if and only if

( ) ( ) ( ) ( ).T T T Tu z u z u z u z∈ ∉ ∈ ∉

′+ ≥ +∑ ∑ ∑ ∑

Likewise, ( ) ( )1 2 1 2, ,z z z z′ ′ ′ if and only if

( ) ( ) ( ) ( ).T T T Tu z u z u z u z∈ ∉ ∈ ∉

′ ′ ′+ ≥ +∑ ∑ ∑ ∑

But both of these two inequalities are equivalent to

( ) ( ).T Tu z u z∈ ∈

′≥∑ ∑


3

Hence they are equivalent to each other.

b)Show now that the Walrasian and Hicksian demand functions generated by an additively separable

utility function admit no inferior goods if the functions ( )u ⋅ are strictly concave. (You can

assume differentiability and interiority to answer this question.) Answer. Suppose that the wealth level w increases and all prices remain unchanged. Then the demand for at least one good (say, good ) has to increase by the Walras’ law. From (3.D.4) we

know that ( )( ) ( ) ( )( ), / ,k k ku x p w p p u x p w′ ′= for every 1,k L= … . Since ( ),x p w

increased and ( )u ⋅ is strictly concave, the right hand side will decrease. Hence, again since

( )ku ⋅ is strictly concave, ( ),kx p w will have to increase. Thus all goods are normal.

Exercise 4. If leisure is an inferior good, what is the slope of the supply function of labor?

Answer. Use Slutsky equation to write:

( )SL L L L L

w w m∂ ∂ ∂

= + −∂ ∂ ∂

, where L is leisure, w is wage rate, m is income.

Note that the substitution effect is always negative, i.e., SL

w∂∂

<0, term ( )L L− measures the amount

of working hours and it is always positive. Hence, if leisure is a normal good, 0Lm∂

>∂

, and the sign

of the total effect is negative and unambiguous, as the following expression illustrates.

( )( ) +! ( ) (+)

SL L L L Lw w m

∂ ∂ ∂= + −

∂ ∂ ∂− +

In contrast, if leisure is inferior, 0Lm∂

<∂

, the total effect, Lw∂∂

, is not necessarily negative. Indeed,

( )( ) ? ( ) (+)

SL L L L Lw w m

∂ ∂ ∂= + −

∂ ∂ ∂− −

In order to provide a more general analysis of this case, let us rearrange the equation above, solving for the total effect,


4

( )SL L L L L

w w m∂ ∂ ∂

= − −∂ ∂ ∂

Thus the slope of the labor supply curve depends on whether the total effect is positive or negative, which ultimately depends on whether the (negative) substitution effect dominates the (positive)

income effect. Comparing the Substitution and Income effects, and noting that ( )L L L

w w∂ − ∂

= −∂ ∂

,

then:

1. If SE>IE, then ( )0, 0L L Land

w w∂ ∂ −

> <∂ ∂

. This implies that the total effect is positive,

which implies that the slope of the leisure curve is positive. Therefore the slope of the labor supply curve must be negative.

2. If SE<IE, then ( )0, 0L L Land

w w∂ ∂ −

< >∂ ∂

. This implies that the total effect is negative,

which implies that the leisure curve is negatively sloped. As a consequence, the labor supply curve is positively sloped.

Micro Theory IRecitation #5

1. Exercise 3.I.7 MWG: There are three commodities (i.e., L=3) of which the third isa numeraire (let p3 = 1) the market demand function x(p; w) has

x1(p; w) = a+ bp1 + cp2

x2(p; w) = d+ ep1 + gp2

a) Give the parameter restrictions implied by utility maximization.

Intuitively, note that:

1. b � 0 for ULD to be satis�ced (" p1 )# x1)

2. g � 0 for ULD to be satis�ced (" p2 )# x2)

3. What about the sign of c (or e)?

(a) c > 0 :)" p2 )" x1 (x1 and x2 are substitutes)(b) c < 0 :)" p2 )# x1 (x1 and x2 are complements)

Let�s analyse this more formally. By applying Walras�law and the homogeneity of degreezero, we can obtain the demand functions for all three goods de�ned over the whole domainf(p; w) 2 R3 � R : p� 0g. Thus, we can obtain the whole 3� 3 Slutsky matrix as well fromthe demand functions. In particular, since there are no income e¤ects

�@xk(p;w)@w

= 0�for any

good k, we can express the Slutsky matrix as follows:

S(p; w) =

264@x1(p;w)@p1

@x1(p;w)@p2

@x1(p;w)@p3

@x2(p;w)@p1

@x2(p;w)@p2

@x2(p;w)@p3

@x3(p;w)@p1

@x3(p;w)@p2

@x3(p;w)@p3

375The 2� 2 submatrix of the Slutsky matrix that is obtained by deleting the last row and thelast column is equal to:

S(p; w) =

"@x1(p;w)@p1

@x1(p;w)@p2

@x2(p;w)@p1

@x2(p;w)@p2

#=

�b ce g

�The 3�3 Slutsky matrix is symmetric if and only if this 2�2 matrix is symmetric. Moreover,just as in the proof of Theorem M.D.4(iii), we can show that the 3 � 3 Slutsky matrix isnegative semide�nite on R3 if and only if the 2 � 2 matrix is negative semide�nite. Inparticular this matrix is symmetric if c = e, and negative semide�nite if b � 0, g � 0, andbg � c2 � 0 .

1

b) Estimate the Equivalent Variation for a change of prices from (p1; p2) = (1; 1)to (p1; p2) = (2; 2). Verify that without appropriate symmetry, there is nopath independence. Assume independence for the rest of the exercise.

We have to verify that:

1. the corresponding Hicksian demand functions for the �rst two commodities are inde-pendent of utility levels, hl(p; u) = hl(p; u0), and,

2. Hicksian demand functions coincide with the Walrasian demand functions.

Let p be any price vector and u, u0 be any two utility levels. By (3.E.4) in MWG we have:

hl(p; u) = xl(p; e(p; u)) and hl(p; u0) = xl(p; e(p; u

0)) for l = 1; 2

also, since the walrasian demands xl(�) do not depend on wealth, we can write

xl(p; e(p; u)) = xl(p; e(p; u0))

then we have hl(p; u) = hl(p; u0). Hence, the hicksiand demands hl(p; u) do not depend onutility level and they are the same as the xl(p; w) in this speci�c example.

Let us now examine how the path of price increases might a¤ect the size of the equivalentvariation (EV):Let us �rst assume that prices change following the path (1; 1) ! (2; 1) ! (2; 2): First, wemust �nd the EV of increasing in p1 from p1 = 1 to p1 = 2. Second, we must �nd the EV ofincreasing in p2 from p2 = 1 to p2 = 2.

EV =

2Z1

h1(p1; 1; u)dp1 +

2Z1

h2(2; p2; u)dp2

And since Hicksian and Walrasian demands coincide in this exercise,

EV =

2Z1

x1(p1; 1; w)dp1 +

2Z1

x2(2; p2; w)dp2

Replacing by the Walrasian demand functions,

EV =

2Z1

(a+ bp1 + c)dp1 +

2Z1

(d+ 2e+ gp2)dp2

Where we �xed p2 = 1 in the �rst term and p1 = 2 in the second term. Integrating,

EV = (a+3

2b+ c) + (d+ 2e+

3

2g) (1)

Let us now consider that prices change following following the path (1; 1)! (1; 2)! (2; 2).Note that using this path for increasing prices, we �rst raise p2 from p2 = 1to p2 = 2, andthen we raise p1 from p1 = 1 to p1 = 2. Hence, in order to �nd the EV of these price changes,

2

we must �rst �nd the EV of increasing p2 (p2 = 1to p2 = 2), and second, for a �xed level ofp2 = 2, we must �nd the EV of increasing p1 (from p1 = 1to p1 = 2).

EV =

2Z1

h2(1; p2; u)dp2 +

2Z1

h1(p1; 2; u)dp1

And since Hicksian and Walrasian demands coincide in this exercise,

EV =

2Z1

x2(1; p2; w)dp2 +

2Z1

x1(p1; 2; w)dp1

Replacing by the Walrasian demand function,

EV =

2Z1

(d+ e+ gp2)dp2 +

2Z1

(a+ bp1 + 2c)dp1

Where we �xed p1 = 1 in the �rst term and p2 = 2 in the second term. Integrating,

EV = (d+ e+3

2g) + (a+

3

2b+ 2c) (2)

Note that the equivalent variation following the �rst path (expression 1) and following thesecond path (expression 2) coincide if and only if c = e (which we required in order to havea symmetric Slustky matrix).

� Hence, when the Slustky matrix is symmetric we can guarantee that an increase in theprice of two goods is �path independent�.

3

c) Let EV1, EV2 and EV be the equivalent variations for a change of pricesfrom (p1; p2) = (1; 1) to respectively (2; 1); (1; 2); and (2; 2). Compare EV withEV1+EV2 as a function of the parameters of the problem. Interpret.

Let us be precise about the notation we will use in this part of the exercise.

� EV1 measures the EV for the price change (1,1) to (2,1) - Only p1 increases.

� EV2 measures the EV for the price change (1,1) to (1,2) - Only p2 increases.

� EV measures the EV for the price change (1,1) to (2,2) - Both prices increase.

(For a graphical representation, see the �gures of EV1, EV2 and EV at the end of thehandout)

As we calculated before:

EV1 =

2Z1

x1(p1; 1; w)dp1 = a+3

2b+ c

EV2 =

2Z1

x2(1; p2; w)dp2 = d+ e+3

2g

We now want to �nd the EV from an increase in the price of both goods. Remember fromexercise (b) that we can increase the price of both goods following two di¤erent paths. Letus �rst �nd the EV from increasing the price of both goods by following the �rst path:

EV =

2Z1

x1(p1; 1; w)dp1 +

2Z1

x2(2; p2; w)dp2

EV = (a+3

2b+ c) + (d+ 2e+

3

2g)

Let us now �nd the EV by following the second path:

EV =

2Z1

x2(1; p2; w)dp2 +

2Z1

x1(p2; 2; w)dp1

EV = (d+ e+3

2g) + (a+

3

2b+ 2c)

And in the case that the Slutsky matrix is symmetric, c = e, we have that the EV fromincreasing the price of both goods is �path independent�and takes the value:

EV = a+3

2b+ 3c+ d+

3

2g

Let us now �nd the di¤erence between EV (resulting from increasing the price of both goods)and the sum of EV1 and EV2.

4

EV � (EV1 + EV2) = (a+3

2b+ 3c+ d+

3

2g)� (a+ 3

2b+ 2c+ d+

3

2g) = c:

The sum EV1 + EV2 does not contain the e¤ect on equivalent variation due to the shiftof the graph of the demand function for the second commodity when p1 goes up to 2 (orequivalently, the shift of the graph of the demand function for the �rst commodity when p2goes up to 2). (See �gures at the end of the handout, for a graphical comparison betweenthe area EV1 + EV2 and the area EV .

5

d) Suppose that the prices increases in (c) are due to taxes. Denote the dead-weight losses for each of the three experiments by DW1, DW2; and DW .Compare DW with DW1 +DW2 as a function of parameters of the problem.

We �rst calculate the deadweight loss if the tax a¤ects the price of good 1 alone, DW1, raisingit from p1 = 1 to p1 = 2. First, note that the tax rate is $1. Hence, since x1(2; 1; w) =a+2b+ c, the tax revenue from the �rst good is equal to T1 = 1�x1(2; 1; w). (See the �gurerepresenting DW1 at the end of the handout, page 2 of �gures). Thus,

DW1 = T1 � EV1 = (a+ 2b+ c)� (a+3

2b+ c) =

b

2:

We secondly calculate the deadweight loss if the tax a¤ects the price of good 2 alone, DW2,raising it from p2 = 1 to p2 = 2. First, note that the tax rate is $1. Hence, since x2(1; 2; w) =d + e + 2g, the tax revenue from the second good is equal to T2 = 1 � x2(1; 2; w). (See the�gure representing DW2 at the end of the handout, page 2 of �gures). Thus,

DW2 = T2 � EV2 = (d+ e+ 2g)� (d+ e+3

2g) =

g

2:

Third, we now �nd the deadweight loss from a tax that a¤ect both the price of good 1 and theprice of good 2. First, note that since x1(2; 2; w) = a+2b+2c, and x2(2; 2; w) = d+2e+2g,the tax revenue from taxing both commodities is equal to:

T = 1� (a+ 2b+ 2c) + 1� (d+ 2e+ 2g) = a+ 2b+ 4c+ d+ 2gThen, the deadweight loss in this case is DW = T � EV

DW = T � EV = (a+ 2b+ 4c+ d+ 2g)� (a+ 32b+ 3c+ d+

3

2g) =

b

2+ c+

g

2

Let us �nally examine the di¤erence between calculating the deadweight loss of the tax thata¤ects the price of both commodities, and the sum of the deadweight loss of the tax a¤ectingthe price of each commodity separately. It is easy to check that

DW � (DW1 +DW2) = c

6

e) Suppose the initial tax situation has prices (p1; p2) = (1; 1). The governmentwants to raise a �xed (small) amount of revenue R through commoditytaxes. Call t1 and t2 the tax rates for the two commodities. Determinethe optimal tax rates as a function of the parameters of demand if theoptimality criterion is the minimization of the deadweight loss.

The government�s problem is:

min(t1;t2)

DW (t1; t2)

subject to2Xl=1

hl(1 + t1; 1 + t2; u)� tl � R

where DW (t1; t2) = TR(t1; t2)� EV (t1; t2). That is,

DW (t1; t2) =

2Xl=1

hl(1 + t1; 1 + t2; u)tl � e(1 + t1; 1 + t2; u) + e(1; 1; u)

Set up the Lagrangian by

L(t1; t2; �) = DW (t1; t2) + �(R� TR(t1; t2))

Then the �rst order condition with respect to tl is:

@DW (t1; t2)

@tl� �@TR(t1; t2)

@tl= 0

but,

@DW (t1; t2)

@tl=

2Xk=1

@hk(1 + t1; 1 + t2; u)

@tltk �

@e(1 + t1; 1 + t2; u)

@tl+ hl(1 + t1; 1 + t2; u)

since @e(1+t1;1+t2;u)@tl

= hl(1 + t1; 1 + t2; u). Then,@DW (t1;t2)

@tl=P2

k=1@hk(1+t1;1+t2;u)

@tltk, and

@TR(t1; t2)

@tl= hl(1 + t1; 1 + t2; u) +

2Xk=1

@hk(1 + t1; 1 + t2; u)

@tltk

Hence the above �rst order condition can be written as:

2Xk=1

@hk(1 + t1; 1 + t2; u)

@tltk � �

"hl(1 + t1; 1 + t2; u) +

2Xk=1

@hk(1 + t1; 1 + t2; u)

@tltk

#= 0

And rearranging,

2Xk=1

@hk(1 + t1; 1 + t2; u)

@tltk(1 + �)� �hl(1 + t1; 1 + t2; u) = 0 for all l = 1; 2:

7

From this expression and TR =P2

l=1 hl(1 + t1; 1 + t2; u)� tl we obtain

�� = bt1 + ct2a+ b(1 + 2t1) + c(1 + 2t2)

=ct1 + gt2

a+ c(1 + 2t1) + g(1 + 2t2)

and

R = (a+ b(1 + t1) + c(1 + t2))t1 + (d+ c(1 + t1) + g(1 + t2))t2

Therefore, any combination of tax rates (t1; t2) that satis�es the previous condition minimizesthe total deadweight loss of taxation, DW , and allows the tax authority to reach a minimaltax revenue of TR dollars.

8

Micro Theory IRecitation #6

1. Show that if the preferences are u(x1; x2) = � 1x1+ x2 with x1 > 0, then the following

relation holds:@x2@p1

=@x1@p2

� x�1@x2@m

[Note that in this relation only Walrasian demand functions are involved, no Hicksian de-mands]

Solution:The Marginal Rate of Substitution (MRS) is:

MRS1;2 =MU1MU2

=1

x21=p1p2

Using the budget set constraint, p1x1 + p2x2 = m, we have the following Walrasian demandfunctions:

x1(p; m) =

�p2p1

� 12

and x2(p; m) =m

p2��p1p2

� 12

We can now calculate the derivatives to show the relation: @x2@p1

= @x1@p2� x�1 @x2@m

,

� @x2@p1

= � 12(p1p2)1=2

� @x1@p2

= 12(p1p2)1=2

� @x2@m= 1

p2

Thus,@x2@p1

=@x1@p2

� x�1@x2@m

=1

2(p1p2)1=2��p2p1

� 12�1

p2

�= � 1

2(p1p2)1=2

Generally, we can show that this relation holds for any quasilinear preference relationu(x1; x2) = f(x1) + x2. Indeed, since the preferences are quasilinear, the income e¤ectis null, because the demand of the good 1 does not depend on income: @x1

@m= 0. Then, the

Slutsky equation of good 2 with respect to p1 is:

@x2@p1

=@h2@p1

� x�1@x2@m

(1)

The Slutsky equation of the good 1 with respect to p2 is:

@x1@p2

=@h1@p2

� x�2@x1@m

1

but since @x1@m= 0 thus @x1

@p2= @h1

@p2:

Also by the symmetry of the Slutsky matrix we know that @h1@p2

= @h2@p1. Then, we can rewrite

(1) as:@x2@p1

=@x1@p2

� x�1@x2@m

2. The preferences of some consumer can be represented as: u(x1; x2) = min fx1; x2g.We have been informed that only the price of the good 2 has changed, from p02 to p

12,

but we have not informed about by how much did it change. We know, however, thatthe amount of income that has to be transferred to the consumer in order to recoverhis initial utility level is:

p02m

p01 + p02

dollars

where m is the initial income, and p01 and p02 are prices of goods 1 and 2 respectively. Can

you provide some information about the size of the price change, i.e., the di¤erence betweenp02 and p

12?

Solution:According to the information,

�p02m

p01+p02

�is the amount of income that, at the new price ratio,

has to be transferred to the consumer in order to recover his initial utility level, which is thede�nition of CV. Then,

CV =

�p02m

p01 + p02

�Since we can calculate the CV as:

CV = v(P1; P0;m0)� v(P1; P1;m1)

Or using the expenditure function as follows:

CV = e(P1; u0)� e(P0; u0)

From the last homework assignment, we know that under this utility function u(x1; x2) =min fx1; x2g, Walrasian demands are:

x1(p; m) = x2(p; m) =m

p1 + p2

thus the Indirect Utility Function is:

v(p; m) = min

�m

p1 + p2; :

m

p1 + p2

�=

m

p1 + p2

Using the identity v(p; e(p; u0)) = u0 into the previous Indirect Utility Function we havethat

2

e(p; u0)

p1 + p2= u0;

and solving e(p; u0) = (p1 + p2)u0.

On the other hand, we also know that:

e(P; v(P0;m)) = e(P; u0 ) =(p1 + p2 )m

p01 + p02

We can use this expression to calculate CV:

CV = e(P1; u0)� e(P0; u0) = (p11 + p12)m

p01 + p02

� (p01 + p02)m

p01 + p02

=p02m

p01 + p02

where the last equality CV = p02m

p01+p02was given in the introduction of the exercise. Since we

know that the price of good one is not changing p11 = p01;

CV =p02m

p01 + p02

= (p01 + p12)

m

p01 + p02

�m

Rearranging,p02

p01 + p02

=p01 + p

12

p01 + p02

� 1

p02 = p01 + p

12 � p01 � p02

p02 = p12 � p02 =) p12 = 2p

02

Then we can conclude that the price of the good 2 has doubled.

3

3. Consider a consumer with regular preferences that consumes an n-dimensional basket:x = (x1; x2; :::; xn) 2 Rn+. The price vector isP0 = (po1; po2; :::; pon)� 0. The consumerswealth is m dollars. Assume that the prices of all of the goods change in the sameproportion � > 1. Calculate analytically the CV and the EV.

Solution:First note that all prices change from P0 to P1 = �P0 where � > 1 (proportional priceincrease).

� We can �rst �nd the CV using the expenditure function:

CV = e(P1; u0 )� e(P0; u0 )

According to the information we know P1 = �P0, then we can rewrite the CV as:

CV = e(�P0; u0 )� e(P0; u0 )

Since the Expenditure Function is homogeneous of degree one, e(�P0; u0 ) = �e(P0; u0 ),then:

CV = �e(P0; u0 )� e(P0; u0 )Thus, CV = �m�m. Then the CV can be expressed as: CV = m(� � 1)

� Second, to �nd the EV:EV = e(P1; u1)� e(P0; u1)

Since P1 = �P0 thus P0 = 1�P1 then we can rewrite EV as

EV = e(P1; u1)� e(1�P1; u1)

Since the Expenditure Function is homogeneous of degree one, e(1�P1; u1) = 1

�e(P1; u1),

then:EV = e(P1; u1)� 1

�e(P1; u1)

EV = m� 1�m

Then the EV can be expressed as: EV = m(1� 1�)

� And note that, for all � > 1, the CV > EV .

� Finally, note that these are the CV and EV measuring welfare change due to a pro-portional increase in the prices of all goods.

4

4. [Midterm #1, Fall 2008] An individual consumes only good 1 and 2, and his preferencesover these two goods can be represented by the utility function

u(x1; x2) = x�1x

�2 where �; � > 0 and �+ � ? 1

This individual currently works for a �rm in a city where initial prices are p0 = (p1; p2),and his wealth is w.

(a) Find the Walrasian demand for goods 1 and 2 of this individual, x1(p; w) andx2(p; w).

� Walrasian demands are

x1(p; w) =�w

(�+ �) p1and x2(p; w) =

�w

(�+ �) p2

b. Find his indirect utility function, and denote it as v(p0; w).

� Plugging the above Walrasian demand functions in the consumer�s utilityfunction, we obtain

v(p; w) =

��w

(�+ �) p1

�� w

(�+ �) p2

��=

�w

�+ �

��+� ��

p1

��

p2

��c. The �rm that this individual works for is considering moving its o¢ ce to a di¤erentcity, where good 1 has the same price, but good 2 is twice as expensive, i.e., thenew price vector is p0 = (p1; 2p2). Find the value of the indirect utility function inthe new location, i.e., when the price vector is p0 = (p1; 2p2). Let us denote thisindirect utility function v(p0; w).

v(p0; w) =

�w

�+ �

��+� ��

p1

��

2p2

��d. This individual�s expenditure function is

e(p; u) = (�+ �)�p1�

� ��+�

�p2�

� ��+�

u1

�+�

Find the value of this expenditure function in the following cases:

1. Under initial prices, p0, and maximal utility level u0 � v(p0; w), and denoteit by e(p0; u0).

e(p0; u0) = (�+ �)�p1�

� ��+�

�p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

p2

��#| {z }

u

1�+�

=

= (�+ �)�p1�

� ��+�

�p2�

� ��+�

�w

�+ �

��+��+�

��

p1

� ��+�

��

p2

� ��+�

= w

5

2. Under initial prices, p0, and maximal utility level u0 � v(p0; w), and denote itby e(p0; u0).

e(p0; u0) = (�+ �)�p1�

� ��+�

�p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

2p2

��# 1�+�

= (�+ �)�p1�

� ��+�

�p2�

� ��+�

�w

�+ �

��+��+�

��

p1

� ��+�

��

2p2

� ��+�

=1

2�

�+�

w

3. Under new prices, p0, and maximal utility level u0 � v(p0; w), and denote itby e(p1; u0).

e(p1; u0) = (�+ �)�p1�

� ��+�

�2p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

p2

��# 1�+�

= (�+ �)�p1�

� ��+�

�2p2�

� ��+�

�w

�+ �

��+��+�

��

p1

� ��+�

��

p2

� ��+�

= 2�

�+�w


e(p0; u0) = (�+ �)�p1�

� ��+�

�2p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

2p2

��# 1�+�

= (�+ �)�p1�

� ��+�

�2p2�

� ��+�

�w

�+ �

��+��+�

��

p1

� ��+�

��

2p2

� ��+�

= w

e. Find this individual�s equivalent variation due to the price change. Explain howyour result can be related with this statement from the individual to the media:�I really prefer to stay in this city. In fact, I would accept a reduction in mywealth if I could keep working for the �rm staying in this city, instead of movingto the new location�

� We know that

EV = e(p1; u1)� e(p0; u1) = m� 1

2�

�+�

w

That is, this individual would be willing to accept a reduction in his wealthof w� 1

2�

�+�

w in order to avoid moving to a di¤erent city. [Alternatively, the

individual is willing to accept a reduction of�1� 1

2�

�+�

�% of his weatlh ]

f. Find this individual�s compensating variation due to the price change. Explainhow your result can be related with this statement from the individual to themedia: �I really prefer to stay in this city. The only way I would accept to moveto the new location is if the �rm raises my salary�.

6

� We know that

CV = e(p1; u0)� e(p0; u0) = 2�

�+�w � w

That is, we would need to raise this individuals� salary by 2�

�+�w � w inorder to guarantee that his welfare level at the new city (with new prices)coincides with his welfare level at the initial city (at the initial price level).

[Alternatively, the individual must receive an increase of�2

��+� � 1

�of his

wealth]

g. Find this individual�s variation in his consumer surplus (also referred as areavariation). Explain.

� We know that area variation is given by the area below the Walrasian demandbetween the initial and �nal price level. That is,

AV =

Z 2p2

p2

x2(p; w)dp =

Z 2p2

p2

�

(�+ �) pw dp

=�

(�+ �)w

Z 2p2

p2

1

pdp =

�

(�+ �)w ln 2

Hence, moving to the new city would imply a reduction in this individual�swelfare of �

(�+�)w ln 2, or

��

(�+�)ln 2�% of his wealth.

h. Which of the previous welfare measures in questions (e), (f) and (g) coincide?Which of them do not coincide? Explain.

� None of them coincide, since this individual�s preferences are not quasilinealin any of the goods.

i. Consider how the welfare measures from questions (e), (f) and (g) would be mod-i�ed if this individual�s preferences were represented, instead, by the utility func-tion v(x1; x2) = � lnx1 + � lnx2:

� Since we have just applied a monotonic transformation to the initial utilityfunction, u(x1; x2), this new utility function represents the same preferencerelation than function v(x1; x2). Hence, the welfare results that we wouldobtain from function v(x1; x2) would be the same as those with utility functionu(x1; x2):

7

(b) @xz@ehz(w; p; v) is the income e¤ect:

1. if @xz@e> 0 then an increase in wages makes that worker richer, and he decides

to work more (this would be an upward bending supply curve), or2. if @xz

@e< 0 then an increase in wages makes that worker richer, and he decides

to work less (e.g., nurses in Massachussets).

5. [15 points]Measuring welfare changes through the expenditure function]. Aconsumer has a utility function u (x1; x2) = x

1=21 x

1=22 , where good x1 is the consumption

of alcoholic beverages, and x2 is her consumption of all other goods. The price of alcoholis p > 0, and the price of all other goods is normalized to 1.

(a) [2 points] Set the consumer�s expenditure minimization problem. Find �rst orderconditions, and �nd his optimal consumption of x1 and x2.

� The consumer�s minimization problem is

minx1;x2

p1x1 + p2x2 + �hU � x1=21 x

1=22

iAnd the �rst order conditions are

p1 =1

2x�1=21 x

1=22

p2 =1

2x1=21 x

�1=22

x1 =p2x2p1

Substituting in the constraint, we have

U =

�p2x2p1

�1=2(x2)

1=2

And solving for x2 in this expression, we �nd the Hicksian demand for good2,

x2 = U

�p1p2

�1=2And similarly for x1, we �nd the Hicksian demand for good 1,

x1 = U

�p2p1

�1=2(b) [4 points] Substituting your results from part (a) into your objective function,

�nd the expenditure function e(p1; p2; U) for this consumer.

� Substituting x2 and x1 into p1x1 + p2x2,

e(p1; p2; U) = p1U

�p2p1

�1=2+ p2U

�p1p2

�1=2= 2U (p1p2)

1=2

5

For convenience, we can denote p1 = p be the price of alcohol, and p2 = 1 bethe price of all other commodities. Hence, the expenditure function can berewritten as

e(p; U) = 2U (p)1=2

(c) [9 points] Let us now consider a proposal to reduce the price of alcohol fromp = $2 to p = $1 per unit. If the new utility enjoyed by the consumer after theprice change is U = 100,

1. [4 points] what is his minimum expenditure in order to reach U = 100 whenp = $2? And when p = $1? [Hint: Use the expression of the expenditurefunction e(p1; p2; U) you found in part (b)]� His minimum expenditure in order to reach U = 100 when p = $2 is

e($2; 100) = 2 � 100 (2)1=2 = 282:84

And when p = $1 is

e($1; 100) = 2 � 100 (1)1=2 = 200

2. [5 points] what is then the maximum amount that this consumer would bewilling to pay for this price reduction?� The maximum amount that this consumer would be willing to pay forthis price reduction is the di¤erence in the minimum expenditure he needto maintain the same utility level (U=100). This is the EV. That is,

e($2; 100)� e($1; 100) = 82:84

6. [20 points] [Measuring welfare changes when preferences are quasilinear] Showthat the compensating and the equivalent variation coincide when the utility function isquasilinear with respect to the �rst good (and we �x p1 = 1). [Hint: use the de�nitionsof the compensating and equivalent variations in terms of the expenditure function(not the hicksian demand). In addition, recall that if u(x) is quasilinear, then we canexpress it as u(x) = x1 + � (x�1), and rearranging x1 = u(x) � � (x�1) where x�1represents all the reamining goods, l = 2; 3; :::; L.]

� From the de�nition of the compensating and the equivalent variation, we knowthat

CV�p0; p1; w

�= e

�p1; u1

�� e

�p1; u0

�EV

�p0; p1; w

�= e

�p0; u1

�� e

�p0; u0

�Moreover, we know that if u(x) is quasilinear, then we can express it as

u(x) = x1 + � (x�1)() x1 = u(x)� � (x�1)

where x�1 represents all the reamining goods, l = 2; 3; :::; L. Therefore, theexpenditure function becomes

e (p; u) =LXi=1

pixi = p1|{z}$1

x1 +LXk=2

pkxk| {z }p�1�x�1

= x1 + p�1 � x�1

6

EconS 501 - Micro Theory IRecitation #5 - Production Theory

Exercise 1

1. Exercise 5.B.2 (MWG): Suppose that f (�) is the production function associatedwith a single-output technology, and let Y be the production set of this technology.Show that Y satis�es constant returns to scale if and only if f (�) is homogeneous ofdegree one.

� Suppose �rst that a production set Y exhibits constant returns to scale [See �gurecorresponding to Exercise 5.B.2 at the end of this handout]. Let z 2 RL�1+ and� > 0. Then (�z; f (z)) 2 Y . By constant returns to scale, (��z; �f (z)) 2 Y .Hence �f (z) � f (�z).

� By applying this inequality to �z in place of z and 1�in place of �, we obtain

1

�f (�z) � f

�1

�(�z)

�= f (z) ; or f (�z) � �f (z)

Hence f (�z) = �f (z). The homogeneity of degree one is thus obtained.

� Suppose conversely that f (�) is homogeneous of degree one. Let (�z; q) 2 Y and� � 0, then q � f (z) and hence �q � �f (z) = f (�z). Since (��z; f (�z)) 2 Y ,we obtain (��z; �q) 2 Y . The constant returns to scale is thus obtained.

Exercise 2

2. Exercise 5.B.3 (MWG): Show that for a single-output technology, the productionset Y is a convex if and only if the production function f (z) is concave.

� In order to prove this �if and only if statement�we need to show �rst that: if theproduction set Y is convex, then the production function f (z) is concave. Andsecond, we need to show the converse: that if the production function f (z) isconcave then the set Y must be convex.

� First, suppose that Y is convex. [See �gure corresponding to Exercise 5.B.3 atthe end of this handout]. Let z; z0 2 RL�1+ and � 2 [0; 1] ; then (�z; f (z)) 2 Yand (�z0; f (z0)) 2 Y . By the convexity,

(� (�z + (1� �) z) ; �f (z) + (1� �) f (z)) 2 Y .

Thus, �f (z) + (1� �) f (z) � f (�z + (1� �) z). Hence f (z) is concave.

1

� Let us now suppose that f (z) is concave. Let (q;�z) 2 Y; (q0;�z0) 2 Y , and� 2 [0; 1], then q � f (z) and q0 � f (z0). Hence

�q + (1� �) q0 � �f (z) + (1� �) f (z0)

By concavity,�f (z) + (1� �) f (z0) � f (�f + (1� �) z0)

Thus�q + (1� �) q0 � f (�z + (1� �) z0)

Hence

(� (�z + (1� �) z0) ; �q + (1� �) q0) = � (�z; q) + (1� �) (�z0; q0) 2 Y

Therefore Y is convex.

Exercise 3

3. Given a CES (Constant Elasticity of Substitution) production function:

q = f(z1; z2) = A [�z�1 + (1� �)z

�2 ]

1� , where A > 0 and 0 < � < 1

Calculate the Marginal Rate of Technical Substitution (MRTS) and the Elasticity ofSubstitution (�).

(a) Is it an homogeneous production function?

(b) Show, using MRTS and �, that:

1. when �! �1 the CES production function represents the Leontief produc-tion function;

2. when � = 1 the CES production function represents a perfect substitutestechnology; and

3. when � = 0 the CES production function represent a Cobb-Douglas technol-ogy.

Solution:

a) We can calculate the MRTS between the two factors as:

MRTS21 =

@q@z1@q@z2

=

�1�

�A [�z�1 + (1� �)z

�2 ]

1��1 ��z��11�

1�

�A [�z�1 + (1� �)z

�2 ]

1��1 �(1� �)z��12

=�z��11

(1� �)z��12

Using the de�nition of the Elasticity of Substitution we have:

� =d ln (z2=z1)

d lnMRTS=d ln (z2=z1)

d ln�@q=@z1@q=@z2

�2

to �nd this expression we can use the expression of the MRTS we just found:

MRTS =�

(1� �)

�z2z1

�1��and using a logarithmic transformation

ln(MRTS) = ln

��

1� �

�+ (1� �) ln

�z2z1

�solving for ln(z2=z1) we have:

ln

�z2z1

�=

�1

1� �

��ln(MRTS)� ln

��

1� �

��thus,

� =d ln

�z2z1

�d lnMRTS

=1

1� �As we can observe, the elasticity of substitution, �, is a constant value for any pro-duction process and any output value. This is the reason for the name of the CESfunction.

b) To verify that it is an homogeneous production function:

f(�z1; �z2) = A [� (�z1)� + (1� �) (�z2)�]

1� = A [�� (z1)

� + (1� �)�� (z2)�]1�

and rearranging

f(�z1; �z2) = �A [� (z1)� + (1� �) (z2)�]

1� = �f(z1; z2)

then the function is homogeneous of degree one.

c) Now we analyze what happens for di¤erent values of parameter � :

i) When �! �1: From part (a) we have MRTS21 =�z��11

(1��)z��12

, the limit of MRTS

when �! �1 is

lim�!�1

MRTS21 = lim�!�1

�z��11

(1� �)z��12

=�

(1� �)

�z2z1

�1As we can see, if z2 > z1 the MRTS goes to1, if z2 < z1 the MRTS goes to zero.Remember that this values of the MRTS are the same of the Leontief or FixedProportions production function.

ii) When � = 1:q = f(z1; z2) = A�z1 + A(1� �)z2

this production function is a perfect substitutes inputs technology.

iii) When � ! 0: MRTS = �(1��)

�z2z1

�; which corresponds to the Cobb-Douglas

production function MRTS. In the extreme case where � = 0; the elasticity ofsubstitution � =

�11��

�becomes � = 1.

3

Exercise 4

4. Assume a standard technology represented by the production function q = f(z1; z2),show:

(a) If the function represents always Constant Returns to Scale (CRS) , it is truethat marginal productivity of the factors are constant along the same productionprocess?

(b) What if the degree of the production function is di¤erent from one?

Solution:

a) If the production function has CRS, then the function is homogeneous of degree one,thus, the marginal productivity of the factors (�rst derivatives of the production func-tion) are also homogeneous, but one degree less than the original function (degreezero).

f1(�z1; �z2) = �0f1(z1; z2) = f1(z1; z2)

f2(�z1; �z2) = �0f2(z1; z2) = f2(z1; z2)

This result indicates that the marginal product of every input is constant along a givenray (i.e., for production plans using the same ratio of inputs z1

z2). Note that since the

marginal product of every input is constant along a given ratio of input combinationsz1z2, then we must have that the ratio of marginal products f1(z1; z2)

f2(z1; z2)is also constant

along a given ray z1z2. Finally, since

f1(z1; z2)

f2(z1; z2)=MRTS(z1; z2)

then the MRTS between inputs 1 and 2 is constant along a given ray z1z2. Therefore,

the production function is homothetic.

b) If the production function is homogeneous of degree k 6= 1, then by Euler�s theorem weknow that the marginal product of every input is homogeneous of degree k � 1. Thatis

f1(�z1; �z2) = �k�1f1(z1; z2) for the marginal product of input 1, and

f2(�z1; �z2) = �k�1f2(z1; z2) for the marginal product of input 2

In this case, the marginal product of every input is not constant along a given ray z1z2(in

which we increase both z1 and z2 keeping their proportion z1z2unmodi�ed). However,

the production function is still homothetic since:

MRTS(�z1; �z2) =�k�1f1(z1; z2)

�k�1f2(z1; z2)=f1(z1; z2)

f2(z1; z2)=MRTS(z1; z2)

4

Exercise 5

5. Assume a �rm with the production function:

q = f(z) = �(1 + z��1 z�"2 )

�1 where �; �; " > 0

Calculate the product-elasticities of the two inputs and comment the type of returnsto scale that the function represents.

Solution:The product-elasticity of the input i, �i, is de�ned as the percentage change in outputq = f(z) with respect to a percentage change in the amount used of the input i, zi. We cancalculate it as:

�i =@f(z)

@zi

zif(z)

For the case of this particular functional form, we have that the product elasticity of input1 is

�1 =@f(z)

@z1

z1f(z)

= ��(1+z��1 z�"2 )�2(��)(z��11 z�"2 )z1

�(1 + z��1 z�"2 )

�1= �(1+z��1 z

�"2 )

�1z��1 z�"2

and the product elasticity of input 2 is

�2 =@f(z)

@z2

z2f(z)

= ��(1+z��1 z�"2 )�2(�")(z��1 z�"�12 )z1

�(1 + z��1 z�"2 )

�1= "(1+z��1 z

�"2 )

�1z��1 z�"2

The elasticity of scale can help us �nd this production function�s returns to scale. In partic-ular, de�ne elasticity of scale, �, as the percentage change in total output as a consequenceof a percentage change in all inputs.

� =@f(tz)

@t

t

f(tz)

��t=1

Alternatively, the scale elasticity can be calculated as the sum of the product-elasticities forall inputs in the production process:

� =

nXi=1

�i

which in this case is

� =nXi=1

�i = �1 + �2 = (� + ")(1 + z��1 z

�"2 )

�1z��1 z�"2

This function does not represent global, but local, returns to scale. That is, the type ofreturns to scale depends on the production level (or the amount of inputs used). We can usethe scale elasticity to determine for which values of inputs z1 and z2 the production functionexhibits constant, increasing or decreasing returns to scale, as follows.

5

� For values of z1 and z2 for which � = 1,

(� + ")(1 + z��1 z�"2 )

�1z��1 z�"2 = 1

the production function has constant returns to scale for these levels of z1 and z2.

� For values of z1 and z2 for which � > 1,

(� + ")(1 + z��1 z�"2 )

�1z��1 z�"2 > 1

the production function has increasing returns to scale for these levels of z1 and z2.

� Finally, for values of z1 and z2 for which � < 1,

(� + ")(1 + z��1 z�"2 )

�1z��1 z�"2 < 1

the production function has decreasing returns to scale for these levels of z1 and z2.

Exercise 6

6. Obtain the conditional factor demand functions, the cost function, supply correspon-dence and pro�t function for the technology: q = z�1 z

�2 with �; � � 0.

Solution: The technology is a Cobb-Douglass production function, then, the conditionalfactor demand functions can be calculated as the solution of the costs minimization problem:

minz1;z2

w1z1 + w2z2

subject to z�1 z�2 � q

The �rst order conditions are:

MRTS21 =f1(z)

f2(z)=�z2�z1

=w1w2

and q = z�1 z�2 .

This is a system of two equations and two unknowns (z1 and z2) that can be solved for theconditional factor demand functions h1 and h2.From the MRTS we have z2 =

��w1w2z1, which we can replace into the constraint as follows

q = z�1 z�2 = z

�1

��

�

w1w2z1

��= z�+�1

��

�

w1w2

��and rearranging

z�+�1 = q

��w2�w1

��6

z1 = h1(w1; w2; q) =

��w2�w1

��=(�+�)q1=(�+�)

Now replacing z1 = h1(w1; w2; q) into z2 =��w1w2z1 we have:

z2 = h2(w1; w2; q) =

��w1�w2

��=(�+�)q1=(�+�)

Using these values we can �nd the cost function as:

C(w; q) = w1h1(w1; w2; q) + w2h2(w1; w2; q)

that is,

C(w; q) = w1

��w2�w1

��=(�+�)q1=(�+�)| {z }

h1(w1; w2; q)

+ w2

��w1�w2

��=(�+�)q1=(�+�)| {z }

h2(w1; w2; q)

Let � = �(�+�)

and K =h��

i� ��

�1��: We can rewrite the function as:

C(w; q) = Kq1=(�+�)w1��1 w�2

In order to �nd the supply correspondence and the pro�t function, we have to solve thepro�t maximization problem as follows:

maxq

�(q) = p � q � C(w; q)

maxq�(q) = p � q �Kq1=(�+�)w1��1 w�2

The �rst order conditions are:

@�(q)

@q= p�

�1

�+ �

�Kq

1�+�

�1w1��1 w�2 (1)

And the second order derivative must satisfy:

@2�(q)

@q2= �

�1

�+ �� 1��

1

�+ �

�Kq

1�+�

�2w1��1 w�2 < 0

Note that when (� + �) < 1 the above second order condition is satis�ed. Intuitively, thiscondition holds when the function shows decreasing returns to scale. Hence, only underdecreasing returns to scale for this Cobb-Douglas production function we can �nd supplycorrespondences that maximize the pro�ts and a supply function that is nondecreasing inprice (satisfying the law of supply). Solving for q from (1) we have:

p��

1

�+ �

�Kq

1�+�

�1w1��1 w�2 = 0

q(w; p) =

��+ �

K� p

w1��1 w�2

� �+�1��

7

and now using this expression we can obtain the conditional factor demand for factors z1andz2

z1(w1; w2; q) =

��w2�w1

��=(�+�)q1=(�+�)

=

��w2�w1

��=(�+�) "��+ �

K� p

w1��1 w�2

� �+�1��

#1=(�+�)

z2(w1; w2; q) =

��w1�w2

��=(�+�)q1=(�+�)

=

��w1�w2

��=(�+�) "��+ �

K� p

w1��1 w�2

� �+�1��

#1=(�+�)And rearranging,

z1(w1; w2; q) =

��w2�w1

��=(�+�) ��+ �

K� p

w1��1 w�2

� 11��

z2(w1; w2; q) =

��w1�w2

��=(�+�) ��+ �

K� p

w1��1 w�2

� 11��

Finally, we can calculate the pro�t function �(q) = p � q � w1z1 � w2z2 as:

�(q) = p ��+ �

K� p

w1��1 w�2

� �+�1��

�w1��w2�w1

��=(�+�) ��+ �

K� p

w1��1 w�2

� 11��

| {z }z1(w1; w2; q)

� w2��w1�w2

��=(�+�) ��+ �

K� p

w1��1 w�2

� 11��

| {z }z2(w1; w2; q)

Exercise 7

Consider the following production function:

q = min fz�1 ; �z2g

with �; � � 0, with q as the output and z1, z2 as the inputs.

1. Calculate the conditional factor demand functions and the cost function.

2. Assume that the �rm sells the output at a �xed price p, �nd the parameter values for� and � that verify the su¢ cient conditions (second order) for pro�t maximization.

8

3. Assume that � = 1. Obtain the expression for the supply function of the output andthe pro�t function.

Solution:

1. The function q = min fz�1 ; �z2g with �; � � 0, is a Leontief (or �xed proportions)production function, thus, the optimal amount of factors must verify z�1 = �z2, then

q = min fz�1 ; z�1 g = z�1

Thus, the conditional demand for factor 1 is h1(w; q) = q1=� and the conditional de-mand for factor 2 is h2(w; q) =

q�.

The costs function is then

C(w; q) = w1h1(w1; w2; q) + w2h2(w1; w2; q) = w1q1=� + w2

q

�

2. According to the cost function we just found in (a), the maximum pro�ts are:

maxq

�(q) = p � q � C(w; q)

= p � q � w1q1� � w2

q

�

and the �rst order conditions are:

@�(q)

@q= p� 1

�w1q

( 1�)�1 � w2

1

�= 0

And the second order conditions are:

@2�(q)

@q2= �

�1

�� 1�1

�w1q

( 1�)�2 < 0

therefore, for the S.O.C.s to hold we need that � < 1 for any value of �.

3. If � = 1 then the above second order conditions do not hold. Let us see what happensunder this assumption. First, we �nd the conditional factor demand correspondencefrom the production function under � = 1. This function is

q = min fz1; �z2g with � � 0

The optimal amount of factors must satisfy

z1 = �z2

Thenq = min fz1; �z2g = min fz1; z1g = z1

Thus, the conditional demand for factor 1 is

h1(w; q) = q;

9

Using z2 = z1�from above, and the result we just obtained that q = z1, we �nd that

the conditional demand for factor 2 is

h2(w; q) =q

�

The costs function is then

C(w; q) = w1h1(w1; w2; q) + w2h2(w1; w2; q) = w1q + w2q

�

The maximum pro�ts are:

maxq

�(q) = p � q � C(w; q)

= p � q � w1q � w2q

�

where the �rst order conditions are

@�(q)

@q= p� w1 � w2

1

�= 0; p = w1 + w2

1

�

Then, the supply function is perfectly elastic, thus we cannot determine the factordemands. This result is due to the fact that the production function shows constantreturns to scale (homogeneous of degree one). Note that at prices p = w1+w2 1� , �rm�spro�ts are zero.

max �(q) = p � q � C(w; q)

=

�w1 + w2

1

�

�� q � w1q1=� � w2

q

�= 0:

10

Micro Theory I

Recitation #6 - Production Theory-II

Exercise 1

Consider the following pro�t function that has been obtained form a technology that uses asingle input:

�(p; w) = p2w�

where p is the output price, w is the input price and � is a parameter value.

1. For which values of � the pro�ts function is a real pro�t function with all of theappropriate properties.

2. Calculate the supply function of the product and the demand for inputs.

Solution:

1. The pro�t function has to be homogeneous of degree one. Thus,

�(�p; �w) = ��(p; w)

In this case we have:

�(�p; �w) = (�p)2(�w)� = �2+�p2w� (1)

and, on the other hand,��(p; w) = �p2w� (2)

since, by homogeneity, expressions (2) and (3) must coincide. Then,

�2+�p2w� = �p2w�

which implies that 2 + � = 1. That is, we need � = �1. As a consequence, the pro�tfunction that we obtain is

�(p; w) =p2

w

In this case the pro�t function satis�es the following the properties:

(a) Continuous: this property holds for every value of w 6= 0.(b) Non decreasing in the output price, p: @�(p;w)

@p= 2p

w� 0

(c) Non increasing in the factor prices: @�(p;w)@w

= � p2

w2� 0

(d) Homogeneous of degree 1: if � = �1 then �(�p; �w) = ��(p; w)

1

(e) Convex in prices, factor prices and output prices:�� @2�(p;w)@p2

@2�(p;w)@p@w

@2�(p;w)@w@p

@2�(p;w)@w2

�� =�� 2

w� 2pw2

� 2pw2

2p2

w3

�� = 0Since the Hessian is a positive semi-de�nite matrix, the function�(p; w) is convex.

(f) Di¤erentiable: we checked this when checking for previous properties.

2. Using Hotelling�s Lemma we can �nd the supply function,

q(p; w) =@�(p; w)

@p=2p

w

and the conditional factor demand correspondence

z(p; w) = �@�(p; w)@w

=p2

w2

2

Exercise 2

Exercise 5.C.11 Show that @z`(w;q)@q

> 0 if and only if marginal cost at q is increasing inw`:

1. � Assume that c (�) is twice continuously di¤erentiable. By Proposition 5.C.2(vi),z (�) is continuously di¤erentiable and

@z` (w; q)

@q=

�@

@q

��@c (w; q)

@w`

�=

�@

@w`

��@c (w; q)

@q

�.

Hence@z` (w; q)

@q> 0

if and only if �@

@w`

��@C (w; q)

@q

�> 0,

that is, marginal cost is increasing in w`.

Exercise 3

Exercise 5.C.13 A price-taking �rm produces output q from inputs z1 and z2 according toa di¤erentiable concave production function f (z1; z2). The price of its output is p > 0, andthe prices of its inputs are (w1; w2) >> 0. However, there are two unusual things about this�rm. First, rather than maximizing pro�t, the �rm maximizes revenue (the manager wantsher �rm to have bigger dollar sales than any other). Second, the �rm is cash constrained.In particular, it has only C dollars on hand before production and, as a result, its totalexpenditures on inputs cannot exceed C.Suppose one of your econometrician friends tells you that she has used repeated observationsof the �rm�s revenues under various output prices, input prices, and levels of the �nancialconstraint and has determined that the �rm�s revenue level R can be expressed as the fol-lowing function of the variables (p; w1; w2; C):

R (p; w1; w2; C) = p [ + lnC � � lnw1 � (1� �) lnw2] .

( and � are scalars whose values she tells you.) What is the �rm�s use of input z1 whenprices are (p; w1; w2) and it has C dollars of cash on hand?

� Denote the production function of the �rm by f (�). Then its optimization problem is

max(z1;z2)�0

p � f (z1; z2) .

subject to w1z1 + w2z2 � C

3

This is analogous to the utility maximization problem in Section 3.D and the functionR (�) corresponds to the indirect utility function. Hence, analogously to Roy�s identity(Proposition 3.G.4), the input demands are obtained as

� 1

rCR (p; w; C)rwR (p; w; C) =

��C

w1;(1� �)Cw2

�.

Exercise 4

Exercise 5.D.4 Consider a �rm that has a distinct set of inputs and outputs. The �rmproduces M outputs; let q = (q1; :::; qM) denote a vector of its output levels. Holding factorprices �xed, C (q1; :::; qM) is the �rm�s cost function. We say that C (�) is subaddittive iffor all (q1; :::; qM), there is no way to break up the production of amounts (q1; :::; qM) amongseveral �rms, each with cost function C (�), and lower the costs of production. That is, thereis no set of, say J �rms and collection of production vectors fqj = (q1j; :::; qMj)gJj=1 such thatX

j

qj = q andXj

C (qj) < C (q) :

When C (�) is subadditive, it is usual to say that the industry is a natural monopoly becauseproduction is cheapest when it is done by only one �rm.

a. Consider the single-output case,M = 1. Show that if C (�) exhibits decreasing averagecosts, then C (�) is subadditive.

� Suppose that q =PJ

j=1 qj. By the decreasing average costs (and C (0) =

0);�qjq

�C (q) � C (qj). By summing over j, we obtain C (q) �

PJj=1C (qj).

Hence there is no way to break up the production of q among multiple �rms andlower the cost of production. Hence C (�) is subadditive.

b. Now consider the multiple-output case, M > 1. Show by example that the followingmultiple-output extension of the decreasing average cost assumption is not su¢ cientfor C (�) to be subadditive:

C (�) exhibits decreasing ray average cost if for any q 2 RM+ ;

C (q) >C (kq)

kfor all k > 1:

� Let M = 2 and de�ne C(q) =

pmin fq1; q2g

, then C (�) exhibits decreasing ray average cost. But let q1 = (1; 8), q2 = (8; 1), andq = q1 + q2 = (9; 9). Then C (q1) = C (q2) = 1 and C (q) = 3. Hence

C (q) > C (q1) + C (q2)

Therefore, C (�) is not subadditive.

4

Exercise 5

Exercise 5.E.5 (MWG) (M. Weitzman) Suppose that there are J single output plants.Plant j�s average cost is ACj(qj) = � + �jqj for qj � 0. Note that the coe¢ cient �jmay di¤er from plant to plant. Consider the problem of determining the cost minimizingaggregate production plan for producing a total output of q, where q < �

maxjj�j j:

a) If �j > 0 for all j, how should output be located among the J plants?

b) If �j < 0 for all j, how should output be located among the J plants?

c) If �j > 0 for some plants and �j < 0 for others?

Solution: Let us �rst solve for the speci�c case where j = 1; 2. The cost-minimizationproblem in which we �nd the optimal combination of output q1 and q2 that minimizes thetotal cost of production across �rms is

minq1;q2

TC1(q1) + TC2(q2)

subject to q1 + q2 = q

or equivalently, the maximization problem in which �rms choose the optimal combination ofoutput q1 and q2 that maximizes the total pro�ts across all �rms is

maxq1;q2

pq1 � TC1(q1)| {z }�1

+ pq2 � TC2(q2)| {z }�2


If the average cost is ACj(qj) = �+ �jqj then the total cost is TCj(qj) = (�+ �jqj)qj, thuswe can rewrite the PMP as:

maxq1;q2

pq1 � (�+ �1q1)q1 + pq2 � (�+ �2q2)q2


The F.O.C. are

@(�1+�2)@q1

= p� �� 2�1q1 = �

@(�1+�2)@q2

= p� �� 2�2q2 = �

@(�1+�2)@�

= q1 + q2 = q

Using the �rst two order conditions:

5

p� �� 2�1q1 = p� �� 2�2q2

�1q1 = �2q2 thus q1 =�2�1q2

Replacing this expression into the F.O.C. wrt � we have:

q1 + q2 = q then q1 +�1�2q1 = q

q1

�1 + �1

�2

�= q thus q1 =

�2�1+�2

q (1)

In a more general set up, with j number of �rms:If the average cost is ACj(qj) = �+ �jqj then the total cost must be TCj(qj) = (�+ �jqj)qjthen, plant j�s marginal cost isMCj(qj) = �+2�jqj. Since �j > 0 for every j, the �rst ordernecessary and su¢ cient conditions for cost minimization are that:

1. �P

j qj = q

� MCj(qj) =MCj�(qj�) for all j and j�.

From these conditions we obtain qj =q�jPh

1�h

.

Note that this expression coincides with (1) for N=2 �rms.

q�1

1�1+ 1

�2

=�2

�1 + �2q

b. and c. In both cases it is cost-minimizing to concentrate on plants with the smallest�j < 0, because the average cost is decreasing at the highest rate at such plants.

6

Exercise 6

Assume there is a �rm that has a regular production function given by q = f(z1; :::; zn) withConstant Returns to Scale (CRS). Show that if the �rm pays the use of each input accordingto its exact marginal productivity, then pro�ts are equal to zero.

Solution: Since the production function shows CRS we know that it is homogeneous ofdegree one, then, according to Euler�s theorem for homogeneous functions:

@f(z)

@z1z1 +

@f(z)

@z2z2 + :::+

@f(z)

@znzn = f(z1; :::; zn) (3)

Then, if the �rm pays each input according to the market value of the marginal productivityof each input:

p@f(z)@zi

= wi :): @f(z)@zi= wi

p, for every input i

Using this result on (1),

w1pz1 +

w2pz2 + :::+

wnpzn = f(z1; :::; zn)

w1z1 + w2z2 + :::+ wnzn = pf(z1; :::; zn)

pf(z1; :::; zn)� (w1z1 + w2z2 + :::+ wnzn) = 0

Since pf(z1; :::; zn) is the total revenue of the �rm and (w1z1+w2z2+ :::+wnzn) is the totalcost, then this di¤erence is the total pro�t of the �rm, which is zero, as we wanted to show.

7

Exercise 7

Derive an expression for the cost function given the following information about the pro�tfunction:

�(p; w) = p2

4w1+ p2

4w2

Solution: Recall that using Hotelling�s Lemma we can �nd the input demand correspond-ence as follows:

z1(p; w) = �@�(p;w)@w1

= p2

4w21

z2(p; w) = �@�(p;w)@w2

= p2

4w22

Which do not depend on q (they are one of the results from the PMP).

In a similar manner we can �nd the total output:

q(w; p) = @�(p;w)@p

= 2p4w1

+ 2p4w2

= p(w1+w2)2w1w2

From this equation we can solve for p as:

p = 2w1w2q(w1+w2)

In order to obtain the conditional factor demand functions hi(w; q), we can replace the pricewe just found into the input demands z1(p; w) and z2(p; w) as:

h1(w; q) =14w21(p)2 = 1

4w21

�2w1w2q(w1+w2)

�2=

w22(w1+w2)2

q2

h2(w; q) =14w22(p)2 = 1

4w22

�2w1w2q(w1+w2)

�2=

w21(w1+w2)2

q2

Which depend on q (they are a result from the CMP).

Finally, using the conditional factor demand functions we can �nd the cost function:

C(w; q) = w1h1(w; q) + w2h2(w; q) = w1w22

(w1+w2)2q2 + w2

w21(w1+w2)2

q2 = w1w2w1+w2

q2

8

Exercise 8

Given a technology with just two variable inputs, �nd the production function that originatesthe following pro�t function:

�(p; w) = p2

8(w1w2)1=2

Solution: Using Hotelling�s Lemma we can �nd the demand for inputs as:

z1(p; w) = �@�(p;w)@w1

= p2

16(w1w2)1=2w1

z2(p; w) = �@�(p;w)@w2

= p2

16(w1w2)1=2w2

We can normalize input prices by using the output price as the numeraire:hwk =

wkp; where k = 1; 2

i,

then

z1(p; w) =1

16(w1w2)1=2w1

z2(p; w) =1

16(w1w2)1=2w2

From this system, we can obtain the values for w1 and w2 in terms of z1 and z2,

w1(p; z) =1

4z1=21 (z1=z2)1=4

w2(p; z) =1

4z1=22 (z2=z1)1=4

In a similar manner we can �nd output function:

q(w; p) = @�(p;w)@p

= p4(w1w2)1=2

Assuming the normalization of input prices and replacing the values of w1(p; z) and w2(p; z)into the above output function then:

q(w; p) = 14(w1w2)1=2

= 1

4

" 1

4z1=21 (z1=z2)

1=4

! 1

4z1=22 (z2=z1)

1=4

!#1=2

q(w; p) = 14

h�4z1=21 (z1=z2)

1=4��4z1=22 (z2=z1)

1=4�i1=2

Thus, the production function is:

q(w; p) = z1=41 z

1=22 = f(z1; z2)

9

Micro Theory IRecitation #7 - Competitive Markets

Exercise 1

1. Exercise 12.5, NS: Suppose that the demand for stilts is given by Q = 1; 500� 50Pand that the long-run total operating costs of each stilt-making �rm in a competitiveindustry are given by C(q) = 0:5q2 � 10q.

Entrepreneurial talent for stilt making is scarce. The supply curve for entrepreneurs is givenbyQs = 0:25w where w is the annual wage paid. Suppose also that each stilt-making �rmrequires one (and only one) entrepreneur (hence, the quantity of entrepreneurs hired is equalto the number of �rms). Long-run total costs for each �rm are then given by:

C(q; w) = 0:5q2 � 10q + w

(a) What is the long-run equilibrium quantity of stilts produced? How many stilts areproduced by each �rm? What is the long-run equilibrium price of stilts? How many �rmswill there be? How many entrepreneurs will be hired, and what is their wage?

(b) Suppose that the demand for stilts shifts outward to

Q = 2; 428� 50P

How would you know answer the questions posed in part a.

(c) Because stilt-making entrepreneurs are the cause of the upward-sloping long-run supplycurve in this problem, they will receive all rents generated as industry output expands.Calculate the increase in rents between parts (a) and (b). Show that this value is identicalto the change in long-run producer surplus as measured along the stilt supply curve.

Solution:

This problem introduces the concept of increasing input costs into long-run analysis byassuming that entrepreneurial wages are bid up as the industry expands. Solving part (b)can be a bit tricky; perhaps an educated guess is the best way to proceed.

(a) The equilibrium in the entrepreneur market requires Qs = 0:25w = QD = n or w = 4n.

Hence, given C(q; w) = 0:5q2 � 10q + w, the MC = q � 10 and AC = 0:5q � 10 + wq=

0:5q � 10 + 4nq

In long run equilibrium the MC = AC, thus:

q � 10 = 0:5q � 10 + 4nq

1

q � 0:5q = 4nq

0:5q = 4nq

q2 = 8n then q =p8n

Total output is given in terms of the number of �rms by

Q = nq = np8n

Now in terms of supply-demand equilibrium,

Q = 1500� 50P and P =MC = q � 10 or q = P + 10

Hence QS = nq = n(P + 10)

Have 3 equations in Q, n, P. Since Q = np8n and Q = n(P + 10), we have

np8n = n(P + 10)

P =p8n� 10

QD = 1500� 50P = 1500� 50(p8n� 10) = 2000� 50

p8n

Then since QD = QS

2000� 50p8n =

p8n thus n = 50 entrepreneurs

Finally, we can also calculate:

Q = np8n = 1000

q = Qn= 20

P = q � 10 = 10

w = 4n = 200.

(b) Using the results of the previous part and if Q = 2; 428� 50P then,

2

(np8n)� 50(

p8n� 10) = 2; 428

(np8n)� 50

p8n = 2; 928

(n� 50)p8n = 2; 928, therefore n = 72

and, we can then recalculate:

Q = np8n = 1728

q = Qn= 24

P = q � 10 = 14

w = 4n = 288.

So, as the demand shifts outward, the number of �rms in the industry increases, the totalproduction and �rm production increases, the price increases and the wages increase.

(c) The long-run supply curve is upward sloping because as new �rms enter the industry thecost curves shift up:

AC = 0:5q � 10 + 4nq

as n increases the average cost also increases.

Using linear approximations, the increase in the producers surplus (PS) from the supplycurve is given by 4 � 1000 + 0:5 � 728 � 4 = 5456. If we use instead the supply curve forentrepreneurs the area is 88 � 50 + 0:5 � 88 � 22 = 5368. These two numbers agree roughly.To get exact agreement would require recognizing that the long-run supply curve here is notlinear �it is slightly concave.

3

Exercise 2

2. Exercise 12.9, NS: Given an ad valorem tax rate (ad valorem tax is a tax on thevalue of transaction or a proportional tax on price) of t (t = 0:05 for a 5% tax), the gapbetween the price demanders pay and what suppliers receive is given by PD = (1+t)PS.

(a) Show that, for an ad valorem tax,d lnPDdt

= eSeS�eD and

d lnPSdt

= eDeS�eD

(b) Show that the excess burden of a small tax isDW = �0:5 eDeS

eS�eD t2P0Q0

(c) Compare these results to those for the case of a unit tax.

Solution:

This problem shows that the comparative statics results for ad valorem taxes are very similarto the results for per-unit taxes

(a) Given that the gap between the price demanders pay and what suppliers receive isPD = (1 + t)PS

Then, the introduction of a tax implies a small price change, i.e., dPD = (1 + t)dPS + dtPS,where we can evaluate this expression at t = 0 since the tax is impossed before any tax waspresent. Hence, the previous expression collapses to dPD = (dPS + dtPS.

We know also that

eD =@QD@P

� PQD

and eS =@QS@P

� PQS

In equilibrium with a tax rate of t, we will have

QD(PD) = QS(PS)

@QD@PD

dPD =@QS@PSdPS

but since dPD = dPS + dtPS then,@QD@PD

(dPS + dtPS) =@QS@PSdPS

@QD@PD

dPS +@QD@PD

dtPS =@QS@PSdPS

rearranging

@QD@PD

dtPS =@QS@PSdPS � @QD

@PDdPS

@QD@PD

dtPS =�@QS@PS

� @QD@PD

�dPS

@QD@PD

@QS@PS

� @QD@PD

= dPSdt� 1PS(*)

4

using ln we can have

ln

�@QD@PD

@QS@PS

� @QD@PD

�= ln

�dPSdt

�+ ln

�1PS

�or ln

�dPSdt

�+ ln

�1PS

�= ln

�@QD@PD

@QS@PS

� @QD@PD

�Thus,

dln�dPSdt

�= dln

�@QD@PD

@QS@PS

� @QD@PD

�=�

eDeS�eD

�

dln�dPSdt

�= eD

eS�eD

Similarly if we use PS = (1 + t)PD we can obtain

dln�dPDdt

�= eS

eD�eS

(b) A linear approximation of the DWL accompanying a small tax dt is given by:

DWL = 0:5(P0dt)(dQ)

Since eD =@QD@P

� PQD= dlnQ

dlnP

then dQ = eDQ0PdP and substituting into DWL

DWL = 0:5(P0dt)�eD

Q0PdP�

DWL = 0:5 � P0 � (dt)2 � eD �Q0 � dPdt1P

but from (*) we know dPDdt� 1PD= eS

eD�eS then

DWL = 0:5 � P0 � (dt)2 � eD �Q0 � eSeD�eS

DWL = 0:5�eDeSeD�eS

�(dt)2Q0P0

We can now generalize this result for any small t:

DWL = 0:5�eDeSeD�eS

�t2Q0P0

(c) The unit tax described in this chapter is equivalent to the value of the ad-valorem tax.In other words, the unit tax is equal to the ad-valorem tax multiplied by Ps. Therefore, theresults obtained using the ad-valorem tax are equivalent to the ones obtained using the unittax.

5

Exercise 3

3. Consider the utility function U = �log(x1) + �log(x2)� l and budget constraint wl =q1x1 + q2x2.

(a) Show that the price elasticity of demand for both commodities is equal to -1.

(b) Setting producer prices at p1 = p2 = 1, show that the inverse elasticity rule impliest1t2= q1

q2.

(c) Letting w = 100 and �+� = 1, calculate the tax rates required to achieve revenueof R = 10.

Solution:

(a) The consumer�s demands is solve

maxfx1;x2;lg

�log(x1) + �log(x2)� ls.t. wl = q1x1 + q2x2

or equivalently

maxfx1;x2;lg

�log(x1) + �log(x2)� ( q1w x1 +q2wx2)

The F.O.C.s are then

�x1= q1

wand �

x2= q2

w

Then, the demands are

x1 =�wq1and x2 =

�wq2

The elasticity of demand is de�ned by

"di =dxidqi

qixi

Calculating this for good 1 obtains

"d1 = ��wq21

q1(�wq1)= �1

Calculating this for good 2 obtains

"d2 = ��wq22

q2(�wq2)= �1

6

(b) The inverse elasticity rule states that

ti1+ti

= ��

1"di, i = 1; 2.

Hence

t11+t1

"d1 =��= t2

1+t2"d2

But "di = �1 and 1 + ti = qi, so

t1q1= t2

q2and rearranging we have t1

t2= q1

q2.

(c) Revenue is de�ned by

R = t1x1 + t2x2

Using the solutions for the demands x1 = �wq1and x2 =

�wq2we have

R = t1(�wq1) + t2(

�wq2)

Using the relation t1t2= q1

q2we just found in part b as t1 =

q1q2t2

R = ( q1q2t2)(

�wq1) + t2(

�wq2) = w[( q1

q2)( �q1) + �

q2]t2 =

wq2(�+ �)t2

Finally, since 1 + ti = qi,�+ � = 1, R = 10 and w = 100, the optimal tax on good 2 solves

10 = 1001+t2

t2

which has solution t2 = 19, and hence t1 = 1

9.

7

Exercise 4

4. Let the consumer have the utility function U = x�11 + x�22 � l.

(a) Show that the utility maximizing demands are x1 =h�1wq1

i1=[1��1]and x2 =h

�2wq2

i1=[1��2].

(b) Letting p1 = p2 = 1, use the inverse elasticity rule to show that the optimal tax

rates are related by 1t2=h�2��11��2

i+h1��11��2

i1t1.

(c) Setting w = 100, �1 = 0:75, �2 = 0:5, �nd the tax rates required to achieverevenue of R = 10 and R = 300.

(d) Calculate the proportional reduction in demand for the two goods comparing theno-tax position with the position after introduction of the optimal taxes for bothrevenue levels. Comment on the results.

Solution:

(a) If the consumer maximization problem is max U = x�11 + x�22 � l s.t. q1x1 + q2x2 = wl

Thus we can rewrite the budget constraint l = q1x1+q2x2w

and we can replace into the utilityfunction for an unconstrained optimization problem as:

max U = x�11 + x

�22 � q1x1

w� q2x2

w

F.O.C.

@U@xi= �ix

�i�1i � qi

w= 0

�ix�i�1i = qi

w

Solving for xi we get the utility maximizing demands as required.

xi =�w�iqi

� 11��i

(b) The �rst step is to calculate the price elasticity using the demand function we just found:

"di = � 11��i

With p1 = p2 = 1 the inverse elasticity rule states that (see previous exercise):

t11+t1

"d1 =t21+t2

"d2 or1+t2t2"d1 =

1+t1t1"d2

8

Substituting for the elasticities

1+t2t2

�� 11��1

�= 1+t1

t1

�� 11��2

�

1t2

�1+t21��1

�= 1

t1

�1+t11��2

�1��2t2(1 + t2) =

1��1t1(1 + t1)

1��2t2+ 1��2

t2t2 =

1��1t1+ 1��1

t1t1

1��2t2= 1��1

t1+ 1��1

t1t1 � 1��2

t2t2

1��2t2= 1��1

t1+ (1� �1)� (1� �2)

�nally

1t2= 1��1

1��2� 1t1+ �2��1

1��2or 1

t2=h�2��11��2

i+h1��11��2

i1t1.

(c) Using the parameter values gives

1t2= �0:5 + 0:5 1

t1

so

t2 =2

1t1�1

Then, given the revenue constraint

R = t1x1 + t2x2

but we know the optimal values for the demand and using the fact that 1 + ti = qi , then

R = t1

�w�1q1

�1=(1��1)+ t2

�w�2q2

�1=(1��2)R = t1

�w�11+t1

�1=(1��1)+ t2

�w�21+t2

�1=(1��2)and t2 is also known, then

R = t1

�w�11+t1

�1=(1��1)+

�2

1t1�1

�0B@ w�2

1+

2

1t1�1

!1CA1=(1��2)

9

Replacing the values of the parameters

R = t1

�751+t1

�4+

�2

1t1�1

�0B@ 25

1+

2

1t1�1

!1CA2

The revenue curve has a maximum level of revenue around t1 = 0:4, this is known as La¤erproperty. For R = 10 the solution is t1 = 0:0031 and t2 = 0:0062. In the case R = 300 thesolution is t1 = 0:1814 and t2 = 0:4431.

(d) The proportional reduction in demand for the two goods comparing the no-tax positionwith the position after introduction of the optimal taxes for both revenue levels is in thenext table.

R x1 % x2 %0 3; 164 � 25 �10 3; 125 1:23 24:69 1:24300 1; 624 48:67 12:00 52:00

As we can see, the optimal taxes do reduce demand in approximately the same proportionfor both commodities. In this case the interpretation of the Ramsey rule is applicable evenwhen the tax intervention has a signi�cant e¤ect on the level of demand.

Exercise 5

5. (Ramsey rule) Consider a three-good economy (k = 1; 2; 3) in which every consumerhas preferences represented by the utility function U = x1 + g(x2) + h(x3), where thefunctions g(�) and h(�) are increasing and strictly concave. Suppose that each goodis produced with constant returns to scale from good 1, using one unit of good 1 perunit of good k 6= 1. Let good 1 be the numeraire and normalize the price of good 1 toequal 1. Let tk denote the (speci�c) commodity tax on good k so the consumer priceis qk = (1 + tk).

(a) Consider two commodity tax schemes t = (t1; t2; t3) and t = (t01; t02; t

03). Show

that if (1 + t0k) = �(1 + tk) for k = 1; 2; 3 for some scalar � > 0, then the two taxschemes raise the same amount of tax revenue.

(b) Argue from part a that the government can without cost restrict tax schemes toleave one good untaxed.

(c) Set t1 = 0, and suppose that the government must raise revenue of R. What arethe tax rates on goods 2 and 3 that minimize the welfare loss from taxation?

(d) Show that the optimal tax rates are inversely proportional to the elasticity of thedemand for each good. Discuss this tax rule.

(e) When should both goods be taxed equally? Which good should be taxed more?

Solution:

10

(a) The budget constraint for the consumer with tax scheme t = (t1; t2; t3) is:

(1 + t1)x1 + (1 + t2)x2 + (1 + t3)x3 = 0

Hence tax revenue R is:

R � t1x1 + t2x2 + t3x3 = �(x1 + x2 + x3)

Similar reasoning shows that with tax scheme t0 = (t01; t02; t

03) the tax revenue R

0 is:

R0 � t01x01 + t02x02 + t03x03 = �(x01 + x02 + x03)

But the demand for each commodity is homogeneous of degree zero so that

xi = xi(1 + t1; 1 + t2; 1 + t3)

xi = xi(�(1 + t1); �(1 + t2); �(1 + t3))

xi = xi(1 + t01; 1 + t

02; 1 + t

03) = x

0i

Therefore

(x1 + x2 + x3) = (x01 + x

02 + x

03)

and R = R0.

(b) The value for � can be chosen arbitrarily. In particular, a tax system with a tax tk ongood k can be shown to be equivalent to one with no tax on good k by choosing

� = 11+tk

(c) The optimization decision for the consumer is

maxfx1;x2;x3g

U = x1 + g(x2) + h(x3)

s.t. x1 + (1 + t2)x2 + (1 + t3)x3 = 0

Substituting the constraint into the objective function for x1 reduce the F.O.C. to

g0(x2)� (1 + t2) = 0 and h0(x3)� (1 + t3) = 0

11

These necessary conditions result in demand functions x2 = x2(1 + t2) and x3 = x3(1 + t3)so

x1 = �(1 + t2)x2 � (1 + t3)x3.

The optimization of the government can now be written as

maxft2;t3g

U = �(1 + t2)x2 � (1 + t3)x3 + g(x2) + h(x3)

s.t. R = t2x2 + t3x3

where x2 and x3 are, in their turn, functions of (1 + t2) and (1 + t3) respectively.

The solution to this problem provides the tax rates that minimize welfare loss. The necessaryconditions are:

g0x02 � x2 � (1 + t2)x02 � �(x2 + t2x02) = 0

h0x03 � x3 � (1 + t3)x03 � �(x3 + t3x03) = 0.

From the consumer�s choice problem g0 = 1 + t2 and h0 = 1 + t3. These allow the implicitsolutions

t2 = �x2x02

1+��and t3 = �x3

x03

1+��

(d) The elasticity of demand for good k is de�ned as

"dk =(1+tk)x

0k

xk

by this de�nition,

t21+t2

= � 1"d2

1+��

and

t31+t3

= � 1"d3

1+��.

The tax rate on good k is therefore inversely proportional to the elasticity of demand forthat good. Setting the relative taxes in this way minimizes the excess burden resulting fromthe need to raise the revenue R.

(e) This tax rule implies that the good with the lower elasticity of demand should havehigher tax rate. The two goods should be taxed at the same rate only if they have the sameelasticity of demand.

12

Micro Theory IRecitation #8 - Uncertainty-I

Exercise 1

1. Exercise 6.B.2, MWG: Show that if the preference relation % on L is representedby a utility function U (�) that has the expected utility form, then % satis�es theindependence axiom.

� Assume that the preference relation % is represented by an v:N �M expectedutility function U (L) =

Pn unpn for every L = (p1; :::; pN) 2 L. Let

L = (p1; :::; pN) 2 L; L0 =�p0

1; :::; p0

N

�2 L; L00 =

�p00

1 ; :::; p00

N

�2 L,

and � 2 (0; 1). Then L % L0 if and only ifP

n unpn �P

n unp0n. This inequality

is equivalent to

�

Xn

unpn

!+ (1� �)

Xn

unp00

n

!� �

Xn

unp0

n

!+ (1� �)

Xn

unp00

n

!.

This latter inequality holds if and only if

�L+ (1� �)L00 % �L0 + (1� �)L00.

Hence L % L0 if and only if

�L+ (1� �)L00 % �L0 + (1� �)L00.

Thus the independence axiom holds.

Exercise 2

2. Exercise 6.B.5, MWG: The purpose of this exercise is to show that the Allaisparadox is compatible with a weaker version of the independence axiom. We considerthe following axiom, known as the betweenness axiom [see Dekel (1986)]:

For all L; L0 and � 2 (0; 1) ; if L � L0, then �L+ (1� �)L0 � L.

Suppose that there are three possible outcomes.

a. Show that a preference relation on lotteries satisfying the independence axiom alsosatis�es the betweenness axiom.

1

� Answer: This follows from Exercise 6.B.1 (in your homework assignment). Fol-lowing the independence axiom we can state that

if L � L0 then (1� �)L+ �L| {z }L

� �L+ (1� �)L0

Thus L � �L+ (1� �)L0. This means that if the preference relation satis�es theindependence axiom it then also satis�es the betweenness axiom.

b. Using a simplex representation for lotteries similar to the one in Figure 6.B.1, showthat if the continuity and betweenness axioms are satis�ed, then the indi¤erence curvesof a preference relation on lotteries are straight lines. Conversely, show that if theindi¤erence curves are straight lines, then the betweenness axiom is satis�ed. Do thesestraight lines need to be parallel?

� Answer: Indi¤erence courves are straight lines if for every pair of lotteries L,L0, we have that L � L0 implies �L+ (1� �)L0 � L for all � 2 (0; 1). That is, ifdecision maker is indiferent between the compond lottery �L+(1��)L0 (the linearcombination of two simple lotteries) and either of the simple lotteries L or L0 thatgenerated such compound lottery. (See �gure representing the betweenness axiomat the end of the handout). The independence axiom guarantees that indi¤erencecurves over lotteries must be not only straight lines but also parallel (See �gurerepresenting the independence axiom at the end of the handout).

c. Using (b), show that the betweenness axiom is weaker (less restrictive) than the inde-pendence axiom.

� Answer: Any preference represented by straight, but not parallel indi¤erencecurves, satis�es the betweenness axiom but does not satisfy the independenceaxiom. Hence the betweenness axiom is weaker than the independence axiom.In other words, the IA =) BA, but IA:BA. (See �gure 3 at the end of thehandout, illustrating an example of indi¤erence curves that satisfy the BA but donot satisfy the IA).

Exercise 3

3. Suppose that all individuals have a Bernoulli utility function u (x) =px.

a. Calculate the Arrow-Prat coe¢ cients of absolute and relative risk aversion at the levelof wealth w = 5.

u (x) =px = x

12

u0 (x) =1

2x�

12 =

1

2px

u00 (x) =

��12

��1

2

�x�

32 =

1

4x�

32

2

� Arrow-Pratt coe¢ cient of absolute risk aversion:

rA (x) = �u00 (x)

u0 (x)= ��14x�

32

12x�

12

=1

2� x(�

32�(� 1

2)) =

=1

2� x� 3

2+ 12 =

1

2� x�1 = 1

2x

when x = 5rA (5) =

1

2 � 5 =1

10= 0:1

� Arrow-Pratt coe¢ cient of relative risk aversion:

rR (x; u) = �x � u00 (x)

u0 (x)= x � rA (x) =

= x � 12x=1

2= 0:5

regardless of the speci�c value of x (but this is just for this case).

b. Calculate the certainty equivalent and the Probability Premium for a gamble�16; 4;

1

2;1

2

�

u (x) =px

u (�) =p16 � 1

2+p4 � 12= 4 � 1

2+ 2 � 1

2= 2 + 1 = 3

u (�) = 3 =) x = 9

c (F; u) = 9 Certainty Equivalent

�1

2+ �

�u (x+ ") +

�1

2� �

�u (x� ") = u (x)�

1

2+ �

� p16|{z}

utility derived from the prize 16.

+

�1

2� �

�p4 =

p10|{z}

u(EV )

since EV =1

2� 16 + 1

2� 4 = 1

2(16 + 4) =

1

2� 20 = 10

p16

2+p16� +

p4

2�p4� =

p10�p

16�p4�� =

p10�

p16

2�p4

2

2� =p10� 3

� =

p10� 32

Probability Premium

3

Calculate the Certainty Equivalent and the Probability Premium for the gamble�36; 16; 1

2; 12

�:

Compare this result with the one in part (b) and interpret.

u (x) =px

u (�) =p36 � 1

2+p16 � 1

2= 6 � 1

2+ 4 � 1

2= 3 + 2 = 5

u (�) = 5 =) x = 25


�1

2+ �

�u (x+ ") +

�1

2� �

�u (x� ") = u (x)�

1

2+ �

�p36 +

�1

2� �

�p16 =

p26|{z}

u(EV )

since EV =1

2� 36 + 1

2� 16 = 36 + 16

2=52

2= 26

1

2

p36 +

p36� +

1

2

p16�

p16� =

p26�p

36�p16�� =

p26�

p36

2�p16

2

(6� 4)� =p26� 6

2� 42

2� =p26� 5

� =

p26� 52

� Probability Premium

The di¤erence between the mean and the c (F; u) is equal to 1 for both of them.However, the �rst lottery has a higher � (x; "; u) where rR (x; u) is constant in w andrA (x; u) is decreasing in w.

Exercise 4

4. A security agency with vNM utility function u evaluates two disaster plans for theevacuation of an area prone to �ooding. The probability of �ooding is 1%. There arefour possible outcomes: 8>><>>:

a1 : no evacuation, no �ooding,a2 : no evacuation, but �ooding,a3 : evacuation, no �ooding,a4 : evacuation, �ooding.

The agency is indi¤erent between the sure outcome a3 and the lottery of a1 withprobability p 2 (0; 1) and a2 with probability 1 � p and between the sure outcome a4

4

and the lottery of a1 with probability q 2 (0; 1) and a2 with probability 1�q. Further,u (a1) = 1 and u (a2) = 0. Moreover,

a3 � (a1; a2; p; 1� p)a4 � (a1; a2; q; 1� q)

u (a1) = 1; u (a2) = 0

a. Express u (a3) and u (a4) in terms of p and q.

� Answer: Given % on L can be represented by a utility function u (�)

u (a3) = pu (a1) + (1� p)u (a2) = pu (a4) = qu (a1) + (1� q)u (a2) = q

The two disaster plans are summarized as follows:

� Plan 1: results in an evacuation in 90% of the cases where a �ooding does occur andin 10% of the cases where no �ooding occurs.

� Plan 2: results in an evacuation in 95% of the cases where a �ooding does occur andin 15% of the cases where no �ooding occurs.

b. For each of these two plans, compute the probability distribution over the four outcomesfa1; a2; a3; a4g.

Insert Figure here (to be discussed during the Review Session)

c. Compute the expected utility of each of the two plans. When is plan 1 strictly preferredover plan 2?

� Answer:

u (a1) = 1

u (a2) = 0

u (a3) = p

u (a4) = q

�

u (Plan1) = 0:891 � u (a1) + 0:001 � u (a2) + 0:099 � u (a3) + 0:009 � u (a4)= 0:891 + 0:099p+ 0:009q

u (Plan2) = 0:8415 � u (a1) + 0:0005 � u (a2) + 0:1485 � u (a3) + 0:00954 � u (a4)= 0:8415 + 0:1485p+ 0:0095q

Hence, Plan 1 is strictly preferred to Plan 2 if and only if

u (Plan1) > u (Plan2)

5

() 0:891 + 0:099p+ 0:009q > 0:8415 + 0:1485p+ 0:0095q

() 0:0495p+ 0:0005q < 0:0495

() q < 99 (1� p)

But given that q 2 (0; 1) ; this condition can always be satis�ed when,

1 < 99 (1� p)() 98

99> p

i.e., for almost all possible values of p, Plan 1 will always be strictly preferred to Plan 2.

Exercise 5

5. Assume there is an apartment complex with 10 units, each of the houses has a marketvalue of $100.000. Each of the neighbors that lives in this houses has a total wealth of$200.000 dollars. Assume that a piromaniac is surrounding the area and is intendingto burn just one of the houses. The neighbors are trying to set a private security found,in which each of them will deposit $10.000. All of the money collected in the foundwill be given to the neighbor that will su¤er the lost. If you were one of the neighborsand your utility function is a Bernulli type such as:

u(x) =px

Will you take the insurance?

Answer: We can start calculating the expected wealth in the case the neighbor takes theinsurance and in the case he doesn�t.

� In the case he does not take the insurance, the neighbor is subject to a lottery wherex0 represents the real part of the wealth:

GRAPH 1

In the linear form we can express: xF =�910; 300:000; 200:000

�The expected value of wealth is: x = E(xF ) =

�910

�� 300:000 +

�110

�� 200:000 = 290:000

� In the case he pays the insurance, the neighbor is subject to a lottery where x0 repres-ents the real part of the wealth:

GRAPH 2

6

In the linear form we can express: x0F =�910; 290:000; 290:000

�The expected value of wealth is in this case: x0 = E(x0F ) =

�910

�� 290:000+

�110

�� 290:000 =

290:000

In both states of nature the payment of $10.000 dollars is satis�ed, but in the case when thehouse is burned the amount received is $100.000.

Note that, in the case where the neighbor takes the insurance, the wealth is no longer random,since in both states of the nature the total wealth is $290.000. Evidently, there is no variancein this measure.

To �nd the most convenient option for the neighbor, lets calculate the certain equivalencein both cases:

No-insurance:

pc(F; u(xF )) = 0; 9 �

p300:000 + 0; 1 �

p200:000 = 17:00267

This is:

c(F; u(xF )) = (17:00267)2 = 289:090; 81

Insurance:

pc(F; u(x0F )) = 0; 9 �

p290:000 + 0; 1 �

p290:000 = 17:02939

This is:

c(F; u(x0F )) = (17:02939)2 = 290:000

Then, insurance is the best option.

7

Micro Theory IRecitation #9 - Monopoly

Exercise 1

A monopolist faces a market demand curve given by: Q = 70� p.(a) If the monopolist can produce at constant average and marginal costs of AC =MC = 6,what output level will the monopolist choose in order to maximize pro�ts? What is the priceat this output level? What are the monopolist�s pro�ts?

(b) Assume instead that the monopolist has a cost structure where the total costs are de-scribed by:

C(Q) = 0:25Q2 � 5Q+ 300

With the monopolist facing the same market demand and marginal revenue, what price-quantity combination will be chosen now to maximize pro�ts? What will pro�ts be?

(c) Assume now that a third cost structure explains the monopolist�s position, with totalcosts given by:

C(Q) = 0:0133Q3 � 5Q+ 250

Again, calculate the monopolist�s price-quantity combination that maximizes pro�ts. Whatwill pro�t be? Hint : Set MC = MR as usual and use the quadratic formula to solve thesecond order equation for Q.

(d) Graph the market demand curve, theMR curve, and the three marginal cost curves fromparts a, b and c. Notice that the monopolist�s pro�t-making ability is constrained by (1) thedemand curve (along with its associated MR curve) and (2) the cost structure underlyingproduction.

Solution:

(a) We have that the demand is given by Q = 70� p or p = 70�Q thus the total revenue isTR = p �Q = (70�Q)Q, then the marginal revenue for the monopolist is MR = 70� 2Q.We know that the monopolist pro�t maximization condition is MR = MC and by theinformation we know that MC = 6 then we can set 70 � 2Q = 6 and solving for thequantities we have that Q = 32, P = 38 and � = (p�MC)Q = (38� 6)32 = 1024.(b) If the total costs are described by C(Q) = 0:25Q2 � 5Q + 300 then the marginal costsare MC = 0:5Q� 5. Thus, equalizing again MR =MC we have that 70� 2Q = 0:5Q� 5.In this case Q = 30, P = 40 and � = pQ�TC = (40 � 30)� (0:25(30)2� 5(30)+ 300) = 825.As we can see, the change in the costs structure reduces the total production, increases theprice and reduce the pro�ts of the �rm.

(c) If the total costs are described by C(Q) = 0:0133Q3 � 5Q+ 250 then the marginal costsare MC = 0:0399Q2 � 5. Thus, equalizing again MR = MC we have that 70 � 2Q =

1

0:0399Q2� 5. In this case the positive solution of the quadratic equation is Q = 25, P = 45and � = pQ � TC = (45 � 25) � (0:0133(25)3 � 5(25) + 250) = 792:2. The new change inthe costs structure reduces the total production, increases the price and reduce the pro�tsof the �rm.

Exercise 2

Suppose a government wishes to combat the undesirable allocational e¤ects of a monopolythrough the use of a subsidy.

(a) Why would a lump-sum subsidy not achieve the government�s goal?

(b) Use a graphical proof to show how a per-unit-of-output subsidy might achieve the gov-ernment�s goal.

(c) Suppose the government wants its subsidy to maximize the di¤erence between the totalvalue of the good to consumers and the good�s total cost. Show that, in order to achievethis goal, the government should set:

tP= � 1

eQ;P,

where t is the per-unit subsidy and P is the competitive price. Explain your result intuitively.

Solution:

(a) The government wishes the monopoly to expand output toward P =MC. A lump-sumsubsidy (T ) will have no e¤ect on the monopolist�s pro�t maximizing choice, so this will notachieve the goal. If the monopoly maximizes � = p�Q�TC+T then the pro�t maximizationcondition is MR =MC.

(b) A subsidy per unit of output (t) will e¤ectively shift the MC curve downward. If themonopoly maximizes � = p � Q � (TC � t � Q) then the pro�t maximization condition isMR = MC � t, thus if the marginal cost curve shifts to the right (or downward if themonopoly has constant marginal costs), then the monopoly will produce more units at alower price.

(c) A subsidy (t) must be chosen so that the monopoly chooses the socially optimal quantity,given t. Since the social optimality requires P =MC and pro�t maximization requires thatMR = MC � t = P

�1 + 1

e

�, substitution yields P � t = P

�1 + 1

e

�thus 1� t

P= 1 + 1

eand

tP= �1

eas was to be shown.

Intuitively, the monopoly creates a gap between price and marginal cost and the optimalsubsidy is chosen to equal that gap expressed as a ratio to price.

2

Exercise 3

The taxation of monopoly can sometimes produce results di¤erent from those that arise inthe competitive case. This problem looks at some of those cases. Most of these can beanalyzed by using the inverse elasticity rule.

(a) Consider �rst an ad valorem tax on the price of a monopoly�s good. This tax reducesthe net price received by the monopoly from P to P (1 � t) where t is the proportional taxrate. Show that, with a linear demand curve and constant marginal cost, the imposition ofsuch a tax causes price to rise by less than the full excent of the tax.

(b) Suppose that the demand curve in part a were a constant elasticity curve. Show thatthe price would now increase by precisely the full extent of the tax. Explain the di¤erencebetween these two cases.

(c) Describe a case where the imposition of an ad valorem tax on a monopoly would causethe price to rise by more than the tax.

(d) A speci�c tax is a �xed amount per unit of output. If the tax rate is � per unit, totaltax collections are �Q. Show that the imposition of a speci�c tax on a monopoly will reduceoutput more (and increase price more) than will the imposition of an ad valorem tax thatcollects the same tax revenue.

Solution:

(a) Recall that the Inverse Elasticity Rule is P = MC1+ 1

e

when the monopoly is subject to an

ad valorem tax of t, this becomes P = MC(1�t) �

11+ 1

e

.

With linear demand, e falls (becomes more elastic) as prices rises. Hence,

Paftertax =MC(1�t) �

11+ 1

eaftertax

< MC(1�t) �

11+ 1

epretax

= Ppretax(1�t)

(b) With constant elasticity demand eaftertax = epretax, thus the inequality in part a becomesan equality so Paftertax =

Ppretax(1�t) .

(c) If the monopoly operates on a negatively sloped portion of its marginal cost curve wehave (in the constant elasticity case)

Paftertax =MCaftertax

(1�t) � 11+ 1

e

> MCpretax(1�t) � 1

1+ 1e

= Ppretax(1�t)

(d) The key part of this question is the requirement of equal tax revenues. That is tPaQa =�Qs where the subscripts refer to the monopoly�s choices under the two tax regimes. Supposethat the tax rates were chosen so as to raise the same revenue for a given output level, say Q.Then � = tPa hence � > tMRa. But in general under an ad valorem taxMRa = (1�t)MR =MR� tMR whereas under a speci�c tax,MRs =MR�� . Hence, for a given Q, the speci�ctax that raises the same revenue reduces MR by more than does the ad valorem tax. Withan upward sloping MC, less would be produced under the speci�c tax, thereby dictating aneven higher tax rate. In all, a lower output would be produced, at a higher price than underthe ad valorem tax. Under perfect competition, the two equal-revenue taxes would haveequivalent e¤ects.

3

Exercise 4

Consider the market for the G-Jeans (the latest fashion among people in their late thirties).G-Jeans are sold by a single �rm that carries the patent for the design. On the demandside, there are nH = 200 high-income consumers who are willing to pay a maximum amountof V H = $20 for a pair of G-Jeans, and nL = 300 low-income consumers who are willing topay a maximum amount of V L = $10 for a pair of G-Jeans. Each consumer chooses whetherto buy one pair of jeans or not to buy at all.

(a) Draw the market aggregate-demand curve facing the monopoly.

� The aggregate demand curve should be drawn according to the following formula:

Q (p) =

8<:0 if p > $20200 if $10 < p � $20200 + 300 if p � $10.

(b) The monopoly can produce each unit at a cost of c = $5. Suppose that the G-Jeans monopoly cannot price discriminate and is therefore constrained to set a uniformmarket price. Find the pro�t-maximizing price set by G-Jeans, and the pro�t earnedby this monopoly.

� Setting a high price, p = $20 generates Q = 200 consumers and a pro�t of �H =(20� 5) 200 = $3000:

� Setting a low price, p = $10 generates Q = 200 + 300 consumers and a pro�t of�H = (10� 5) 500 = $2500 < $3000. Hence, p = $20 is the pro�t-maximizingprice. Type L consumers will not buy under these prices.

(c) Compute the pro�t level made by this monopoly assuming now that this monopolycan price discriminate between the two consumer populations. Does the monopolybene�t from price discrimination. Prove your result!

�The monopoly will change p = $20 in market H and p = $10 in market L: Hence,total pro�t is given by

� = �H + �L = (20� 5) 200 + (10� 5) 300 = 3000 + 1500 = $4500 > $3000.

Clearly, the ability to price discriminate cannot reduce the monopoly pro�t sinceeven with this ability, the monopoly can always set equal prices in both mar-kets. The fact that the monopoly chooses di¤erent prices implies that pro�tcan only increase beyond the pro�t earned when the monopoly is unable to pricediscriminate.

4

Exercise 5

The demand function for concert tickets to be played by the Pittsburgh symphony orchestravaries between nonstudents (N) and students (S). Formally, the two demand functions ofthe two consumer groups are given by

qN = 2401

p2Nand qS = 540

1

p3S.

Assume that the orchestra�s total cost function is C (Q) = 2Q where Q = qN + qS is to totalnumber of tickets sold. Compute the concert ticket prices set by this monopoly orchestra,and the resulting ticket sales, assuming that the orchestra can price discriminate betweenthe two consumer groups.

� The demand price elasticity is �2 in the nonstudents�market, and �3 in the students�market. In the nonstudents�market, the monopoly sets pN to solve

pN

�1 +

1

�2

�= $2 yielding pN = $4 and hence qN =

240

42= 15.

In the students�market, the monopoly sets pS to solve

pS

�1 +

1

�3

�= $2 yielding pS = $3 and hence qS =

540

33= 20.

5

Micro Theory IRecitation #10 - Externalities

Exercise 1

If the two consumers in the economy have preferences U1 = [x11x12]�[x21x

22]1�� and U2 =

[x21x22]�[x11x

12]1��, show that the equilibrium is e¢ cient despite the externality. Explain this

conclusion.

Solution:

The marginal utility of good 1 for consumer 1 is:

@U1@x11

= �[x11x12]��1(x12)[x

21x22]1�� = �

[x11x12]�[x21x

22]1��

x11

and the marginal utility of good 2 is:

@U1@x21

= (1� �)[x11x12]�[x21x22]��(x22) = (1� �)[x11x

12]�[x21x

22]1��

x21.

From these the marginal rate of substitution for consumer 1 can be calculated as:

MRS11;2 =��1�� x21x11

Similar calculations for consumer 2 give

MRS21;2 =��1�� x12x22

Notice that each of the marginal rates of substitution is independent of the externality e¤ect.Each consumer equates his or her MRS to the price ratio that ensures that the marginalrates of substitution are equal. Therefore the externality does not a¤ect the fact that theequilibrium is e¢ cient.This conclusion holds because the externality does not a¤ect the proportions in which thetwo consumers purchase the goods. (Observe that the externality e¤ect can be factored outof the utility functions as a constant.) The same equilibrium is reached with the externalityas it is without. Such externalities are called �Pareto irrelevant�.

1

Exercise 2

There is a large number of commuters who decide to use either their car or the tube. Com-muting by train takes 70 minutes whatever the number of commuters taking the train.Commuting by car takes C(x) = 20 + 60x minutes, where x is the proportion of commuterstaking their car, 0 � x � 1.(a) Plot the curves of the commuting time by car and the commuting time by train as afunction of the proportion of cars users.

(b) What is the proportion of commuters who will take their car if everyone is taking herdecision freely and independently so as to minimize her oun commuting time?

(c) What is the proportion of car users that minimizes the total commuting time?

(d) Compare this with your answer given in part b. Interpret the di¤erence. How large isthe deadweight loss from the externality?

(e) Explain how a toll could achieve the e¢ cient allocation of commuters between train andcar and the bene�cial for everyone.

Solution:

(a) The commuting times are shown in the �gure. The time by tube is constant, but the timetaken by car increases as car use increases. Every traveller�s decision d(x) can be expressedas

d(x) =

�car if C(x) � 70, ortube if C(x) > 70

(b) The proportion of car users , if independent choices are made, will be such thet the timesof travel by tube and by car are equated.Thus, 70 = 20 + 60xm solving for xm gives xm = 5=6 = 0:833. This solution corresponds tothe intersection point of the two commuting time courves.

(c) The total commuting time is (20 + 60x)x + 70(1 � x), where x is the proportion of carusers. Setting the derivative with respect to x equal to zero gives: 20 + 120x � 70 = 0 or120xo � 50 = 0 thus xo = 5=12 = 0:416 is the time-minimizing car use.(d) The free-market outcome for the proportion of car users is greater than the sociallyoptimal outcome because the individual commuters do not take into account the negativeexternality generated by car travel, meaning the tra¢ c congestion. The deadweight lossfrom the externality is the di¤erence between the total commuting times. Using the earlierresults obtains

Tm =

�20 + 60

�5

6

��5

6

�| {z }

car

+ 70

�1

6

�| {z }tube

= 4206= 70

and

To =

�20 + 60

�5

12

��5

12

�| {z }

car

+ 70

�7

12

�| {z }

tube

= 71512= 59:58

2

The di¤erence is Tm � To = 70� 59:58 = 10:41.(e) Suppose that the commuters attach monetary value to their travel time. It takes 45minutes per car user, and 70 per train user. Then a toll may induce car users to switchfrom car to tube if the amount of the toll exceeds the bene�ts of shorter travel time. Giveninformation on the monetary value of travel time, the amount of the toll can be computedso that the proportion of commuters that still �nd it bene�cial to travel by car is exactlyequal to the socially optimal level.

Exercise 3

On the island of Pago Pago there are two lakes and 20 anglers. Each angler can �sh on eitherlake and keep the average catch on his particular lake. On lake X, the total number of �shcaught is given by

F x = 10lx � 12l2x

where lx is the number of people �shing on the lake. For lake y the relationship is

F y = 5ly

(a) Under this organization of society, what will be the total number of �sh caught?

(b) The chief of Pago Pago, having once read an economics book, believes it is posible toraise the total number of �sh caught by restricting the number of people allowed to �sh onlake X. What number should be allowed to �sh on lake x in order to maximize the totalcatch of �sh? What is the number of �sh caught in this situation?

(c) Being opposed to coercion, the chief decides to require a �shing license for lake x. If thelicensing procedure is to bring about the optimal allocation of labor, what should the costof a license be (in terms of �sh)?

(d) Explain how this example sheds light on the connection between property rights andexternalities.

Solution:

(a) F x = 10lx � 0:5l2x and F y = 5lyFirst, show how total catch depends on the allocation of labor.

lx + ly = 20 thus ly = 20� lxF T = F x + F y

F T = (10lx � 0:5l2x) + (5ly) = (10lx � 0:5l2x) + (5 (20� lx))F T = 5lx � 0:5l2x + 100

Equating the average catch on each lake gives

3

Fx

lx= F y

ly

10� 0:5lx = 5

then lx = 10 and ly = 10and

F T = 5(10)� 0:5(10)2 + 100F T = 100

(b) The problem is to maxF T = 5lx � 0:5l2x + 100thus the FOC wrt lx is dF

T

dlx= 5� lx = 0 then lx = 5, ly = 15 and then F T = 112:5

(c) F xcase 1 = 10(10)� 0:5(10)2 = 50 average catch is F xcase 1 = 50=10 = 5F xcase 2 = 10(5)� 0:5(5)2 = 37:5 average catch is F xcase 2 = 37:5=5 = 7:5thus the license fee on lake X should be equal to 2.5.

(d) The arrival of a new �sher on lake X imposes an externality on the �shers already therein terms of a reduced average catch. Lake X is treated as a common property here. If thelake were private property, its owner would choose lx to maximize the total catch less theopportunity cost of each �sher (the 5 �sh he can catch on lake Y ). So the problem is tomaximize F x � 5lx which yields lx = 5 as in the optimal allocation case.

Exercise 4

Suppose the oil industry in Utopia is perfectly competitive and that all �rms draw oil froma single (and practically inexhaustable) pool. Assume that each competitor belives that itcan sell all the oil it can produce at a stable world price of $10 per barrel and that the costof operating a well for one year is $1,000. Total output per year (Q) of the oil �eld is afunction of the number of wells (n) operating in the �eld. In particular,

Q = 500n� n2

and the amount of oil produced by each well (q) is given by:

q = Qn= 500� n.

(a) Describe the equilibrium output and the equilibrium number of wells in this perfectlycompetitive case. Is there a divergence between private and social marginal cost in theindustry?

(b) Suppose now that the government nationalizes the oil �eld. How many oil wells shouldit operate? What will total output be? What will the output per well be?

(c) As an alternative to nationalization, the Utopian gevernment is considering an annuallicense fee per well to discourage overdrilling. How large should this license fee be if it is toprompt the industry to drill the optimal number of wells?

4

Solution:

(a) Every �rm increases q until � = pq� 1000 = 0. That is, p(500�n)� 1000 = 0, implyingn = 400.In addition, note that revenue per well is revenue

well= 5000�10n, which declines in the number

of wells being drilled. There is hence an externality here because drilling another well reducesoutput in all other wells.

(b) The social planner chooses the number of �rms n in order to maximize aggregate pro�ts

maxn

pQ� 1000n = 5000n� 10n2 � 1000n

Taking FOCs with respect to n, we obtain

5000� 20n� 1000 = 0

solving for n, n = 200. Hence, total output is Q = 200� (500� 200) = 60; 000. So individualproduction is q = 300.Alternatively, the social planner chooses Q where MV P =MC of well. Total value:

5000n� 10n2. MV P = 5000� 20n = 1000. Thus n = 200.

(c) Let tax = x. Want revenuewell

� x = 1000 when n = 200. At n = 200 the averagerevenuewell

= 3000.So, charge x = 2000.

5

FIGURE – EXERCISE 2

Commuting cost (in minutes):

1. Cost of commuting by train = 70 (flat horizontal line)

2. Cost of commuting by car = 20+60x (positively sloped line)

Micro Theory IRecitation #12 - Public Goods

Exercise 1

[1.] Take an economy with 2 consumers, 1 private good, and 1 public good. Let eachconsumer have an income of M . The prices of public and private good are both 1. Let theconsumers have a utility functions:

UA = log(xA) + log(G) and UB = log(xB) + log(G)

(a) Assume that the public good is privately provided, so G = gA+gB. Eliminating xA fromthe utility function using the budget constraint, show that along an indi¤erence curve:

dgAh

1gA+gB

� 1M�gA

i+ dgB

h1

gA+gB

i= 0

and hence that:

dgB

dgA= gA+gB

M�gA � 1

Solve the last equation to �nd the locus of points along which the indi¤erence curve of A ishorizontal and use this to sketch the indi¤erence curves of A.

(b) Consider A choosing gA to maximize utility. Show that the optimal choice satis�es:

gA = M2� gB

2

(c) Repeat part (b) for consumer B, and calculate the level of private provision for the welfarefunction W = UA + UB. Contrast this with the private provision level.

Solution:

(a) The utility of consumer A is given by UA = log(xA) + log(G), which can be written asUA = log(M � gA) + log(gA + gB). Totally di¤erentiating gives:

dUA =�� 1M�gA +

1gA+gB

�dgA +

�1

gA+gB

�dgB

we know that along the indi¤erence curve the chance in utility is zero, then, dUA = 0, thus:�1

gA+gB� 1

M�gA

�dgA +

�1

gA+gB

�dgB = 0

(First result)

1

�1

gA+gB

�dgB = �

�1

gA+gB� 1

M�gA

�dgA

dgB

dgA= �

�1

gA+gB� 1

M�gA

��

1

gA+gB

� = gA+gB

M�gA � 1

dgB

dgA= gA+gB

M�gA � 1(Second result)

The indi¤erence curve of A is horizontal when dgB

dgA= 0. Hence the locus of points where the

indi¤erence curves of A are horizontal is the solution to:

0 = gA+gB

M�gA � 1 orgA+gB

M�gA = 1

gA + gB =M � gA

2gA =M � gB

gA = M�gB2

= M2� gB

2

GRAPH (Best response function)

(b) The utility maximization decision of A is:

MaxfgAg

UA = log(M � gA) + log(gA + gB)

which has a necessary condition:

� 1M�gA +

1gA+gB

= 0

solving for gA we have:

gA = M2� gB

2

(c) The utility maximization decision of B is:

MaxfgBg

UB = log(M � gB) + log(gA + gB)

which has a necessary condition:

� 1M�gB +

1gA+gB

= 0

solving for gA we have:

gB = M2� gA

2

2

The consumers are identical, so the equilibrium will be symmetric with gA = gB = g. As aresult, the necessary condition gives:

g = M2� g

2

g = M3

with total provision

G = gA + gB = 2g = 2M3

(d) The e¢ cient level of provision will have the cost equally allocated between the consumers.Recall W = UA + UB, it therefore solves:

MaxfGg

UA + UB = log�M � gA

�+ log(gA + gB) + log

�M � gB

�+ log(gA + gB)

MaxfGg

UA + UB = log (M � g) + log(g + g) + log (M � g) + log(g + g)

MaxfGg

UA + UB = 2log (M � g) + 2log(2g)

MaxfGg

UA + UB = 2log�M � G

2

�+ 2log(G)

The necessary condition is:

�12

2M�G

2

+ 2G= 0

Solving for G we have:

eG =MAs we can see comparing eG with G shows that provision at the Nash equilibrium is belowwhat is optimal.

3

Exercise 2

[2.] Consider two consumers (1; 2), each with incomeM to allocate between two goods. Good1 provides 1 unit of consumption to its purchaser and �, 0 � � � 1, units of consumption tothe other consumer. Each consumer i, i = 1; 2, has the utility function U i = log (xi1) + x

i2,

where xi1 is the consumption of good 1 and xi2 is the consumption of good 2.

(a) Provide an interpretation of �.

(b) Assume that good 2 is a private good. Find the Nash equilibrium levels of consumptionwhen both goods have a price of 1.

(c) By maximizing the sum of utilities, show that the equilibrium is Pareto-e¢ cient if � = 0but ine¢ cient for all other values of �.

(d) Now assume that good 2 also provides 1 unit of consumption to its purchaser and �,0 � � � 1, units of consumption to the other consumer. For the same preferences, �nd theNash equilibrium and show that it is e¢ cient for all values of �.

(e) Explain the conclusion in part d.

Solution:

(a) The parameter � measures the degree of publicness of the good.

(b) U1 = log (y11 + �y21) + x

12 where y

i1 is the purchase of good 1 by i. Using the budget

constraint (and assuming both goods have unit price) obtains

U1 = log (y11 + �y21) +M � y11.

the choice of y11 satis�es:1

y11+�y21� 1 = 0

The game is symmetric. So the solution is y11 = y21 = y1 =

11+�: Hence the consumption level

in equilibrium is:

x11 = x21 = x1 = [1 + �]y1 = 1.

(c) The level of social welfare is:

W = log(y11 + �y21) +M � y11 + log(y21 + �y21) +M � y21

Applying symmetry obtains:

W = 2log((1 + �)y1) + 2[M � y1]so,@W@y1

= 2y1� 2 = 0.

Hence, y1 = 1 and x1 = 1 + �. The two outcomes are the same if � = 0.

(d) Utility now becomes

U1 = log (y11 + �y21) +M � y11 + �(M � y21).

The Nash equilibrium remains at y11 = y21 = y1 =

11+�: With symmetry the level of welfare

is:

4

W = 2log((1 + �)y1) + 2(1 + �)[M � y1],so y1 = 1

1+�. The two outcomes are identical for all �.

(e) In part b there is one private good and one public good when � 6= 0. So free riding takesplace when � 6= 0. With � = 0, there are two private goods, so the outcome is e¢ cient. Inpart d both goods have an identical degree of publicness so the consumption externalitiesare balanced. It is possible to free-ride on both goods, so e¢ ciency results.

Exercise 3

[3.] Suppose the production possibility frontier for an economy that produces one publicgood (y) and one private good (x) is given by:

x2 + 100y2 = 5000

This economy is populated by 100 identical individuals, each with a utility function of theform

utility =pxiy

where xi is the individual�s share of private good production (= x=100). Notice that thepublic good is nonexclusive and that everyone bene�ts equally from its level of production.

(a) If the market for x and y were perfectly competitive, what levels of those goods wouldbe produced? What would the typical individual�s utility be in this situation?

(b) What are the optimal production levels for x and y? What would the typical individual�sutility level be? How should consumption of good x be taxed to achieve this result? Hint :The numbers in this problem do not come out evently, and some approximations shouldsu¢ ce.

Solution:

(a) The solution here requires some assumption about how individuals form their expecta-tions about what will be purchased by others. If each assumes he or she can be a free rider,y will be zero as will be each person�s utility.

(b) Taking total di¤erential of production possibility frontier.

2xdx+ 200ydy = 0

RPT = �dxdy= 200y

2x= 100 y

x

Individual MRSi =MUyMUx

=0:5pxi=y

0:5py=xi

= xiy= x=100

y= 1

100� xy

For e¢ ciency require that the sum of MRS should equal RPTPi

MRSi =xy. Hence x

y= 100 y

xthus, x = 10y.

Using production possibility frontier yields:

200y2 = 5000 then y = 5, x = 50, xi = 0:5 and utility =p2:5.

Ratio of per-unit tax share of y to the market price of x should be equal to the MRS =xiy= 1

10:

5

Exercise 4

[4.] (M.W.G. 11.D.4) Reconsider the nondepletable externality example discussed in section11.D, but now assume that the externalities produced by the J �rms are not homogeneous.In particular, suppose that if h1; h2; :::; hJ are the �rms�externality levels, then consumer i�sderived utility is given by �i(h1; h2; :::; hJ)+wi for each i = 1; :::; I. Compare the equilibriumand e¢ cient levels of h1; h2; :::; hJ . What tax/subsidy scheme can restore e¢ ciency? Underwhat condition should each �rm face the same tax/subsidy rate?

Solution:

For the Pareto optimal outcome we solve:

Maxfhig

IPi=1

�i(h1; h2; :::; hJ) +JPj=1

�j(hj)

which yields the F.O.C.s

IPi=1

�@�i(h

o1; h

o2; :::; h

oJ )

@hj

�� 0j(hoj)

with equality if hoj > 0 for all j = 1; :::; J .

On the other hand, in a competitive equilibrium each �rm maximizes pro�ts individually,and we get the FOC:

�j(h�j) � 0, with equality if h�j > 0.

To restore Pareto-optimal outcome in a competitive equilibrium, we must set an individualtax for each j of

tj = �IPi=1

�@�i(h

o1; h

o2; :::; h

oJ )

@hj

�

Each �rmwill face the same tax rate if and only if we haveIPi=1

�@�i(h

o1; h

o2; :::; h

oJ )

@hj

�=

IPi=1

�@�i(h

o1; h

o2; :::; h

oJ )

@hk

�for all j; k.

6

Exercise 5

[4.] (M.W.G. 11.D.7) A continuoum of individuals can build their houses in one of twoneighborhoods, A or B. It costs cAto build a house in neighborhood A and cb < cA tobuild in neighborhood B. Individuals care about the prestige of the people living in theirneighborhood. Individuals have varying levels of prestige, denoted by the parameter �.Prestige varies between 0 and 1 and is uniformly distributed across the population. Theprestige of neighborhood k (k = A; B) is a function of the average value of � in thatneighborhood, denoted by �k. If individual i has prestige parameter � and builds her housein neighborhood k, her derived utility net of building costs is (1 + �)(1 + �k) � ck. Thus,individuals with more prestige value a prestigious neighborhood more. Assume that cA andcB are less than 1 and that (cA � cB) 2 (12 ; 1).(a) Show that in any building-choice equilibrium (technically, the Nash equilibrium of thesimultaneous-move game in which individuals simultaneously choose where to build theirhouse) both neighborhoods must be occupied.

(b) Show that in any equilibrium in which the prestige levels of the two neighborhoods di¤er,every resident of neighborhood A must have at least as high a prestige level as every residentof neighborhood B; that is, there is a cuto¤ level of �, say �, such that all types � � � buildin neighborhood A and all � < � build in neighborhood B. Characterize this cuto¤ level.

(c) Show that in any equilibrium of the type identi�ed in (b), a Pareto improvement can beachieved by altering the cuto¤ value of � slightly and allowing transfers between individuals.

Solution:

(a) Assume in negation that only one neighborhood is occupied. First assume it is B, andconsider the most prestigious individual with � = 1. Since �B = 1

2, then this individual�s

utility from staying in neighborhood B is (1+1)(1+ 12)� cB = 3� cB � 3. If he would move

to neighborhood A his utility would be (1 + 1)(1 + 1)� cA = 4� cA > 3, so all individualsin neighborhood B cannot be an equilibrium. Now assume that only A is occupied andagain consider the most prestigious individual with � = 1. His utility from staying in theneighborhood A is (1+1)(1+ 1

2)� cA = 3� cA, and his utility from moving to neighborhood

B is (1 + 1)(1 + 1) � cB = 4 � cB > 3 � cA so all individuals in neighborhood A cannot bean equilibrium - contradiction.

(b) Let an equilibrium be a pair (�A; �B), where�B � f� : type � locates in neighborhood ig,and let �A; �B be the average prestige levels associated with such an equilibrium.

Claim: �A must take the on the form [�; 1] for some �.

Proof: Assume �0 prefers A to B: (1 + �0)(1 + �A)� cA > (1 + �0)(1 + �B)� cBRearranging gives us: (1 + �0) � cA�cB

�A��B, which implies that all types locates in which

neighborhood, and it is calculated by solving:

(1 + �)�1 + 1+�

2

�� cA = (1 + �)

�1 + �

2

�� cB

which yields, � = 2(cA � cB)� 1

7

(c) Starting at the equilibrium with � as given above, if a small group of individuals fromthe lower end of neighborhood A move to neighborhood B, then the average prestige in bothneighborhoods will rise. In particular, if for some " > 0 the segment [�; �+ "] moved from Ato B, the average prestige in both neighborhoods would rise by "

2. So, in both neighborhoods,

an individual of type � who did not move will have a positive change in utility of (1 + �) "2.

For a type � individual who moved from A to B, there will be a negative change in utilityequal to (1 + �)

�1 + �

2+ "

2

�� cB � [(1 + �)

�1 + 1

2+ �

2

�� cA] = (1 + �)

�"�12

�+ (cA + cB).

We denote the total bene�t from such a change as B, and the total cost as C, so that wehave:

B(") =R �0(1 + �)

�"2

�d� +

R 1�+"(1 + �)

�"2

�d�

C(") =R �+"0

�(1 + �)

�"�12

�(cA � cB)

�d�,

and we can evaluate the e¤ect of such a change when " = 0:

dB(")d"

j"=0=R �0(1 + �)

�12

�d� +

R 1�+"(1 + �)

�12

�d� � (1 + � + ")

�"2

�dB(")d"

j"=0= �2+ �

2

2+ 1

2+ 1

4� �+"

2� (�+")

2

2� (1 + � + ")

�"2

�= 3

4

anddC(")d"

j"=0=R �+"0(1 + �)

�12

�d� +

�(1 + � + ")

�"2

�+ cA � cB

�dC(")d"

j"=0= �+"2� �

2+(�+")

2

4� �

2

4+ [2(cA � cB) + "]

�"�12

�+ cA � cB = 0

Note that the last equality is true since from the conclusion of part (b)

8

1

Review Session #12 - Exercise 11.E.1 MWG

Consider the setting studied in Section 11.E (where only the firm knows its type θ and only the consumer knows his type η ). Suppose that

( , ) /h h bhπ θ β θ∂ ∂ = − + and ( , ) /h h chφ η γ η∂ ∂ = − + ,

where θ and η are random variables with expectation [ ] [ ] [ ] 0E E Eθ η θη= = = , that all take strictly

positive values ( , , ) 0b cβ , and 0γ > . Denote 2 2[ ]E θθ σ= and 2 2[ ]E ηη σ= .

(a) Identify the best quota *h for a planner who wants to maximize the expected value of aggregate surplus. (Assume the firm must produce an amount exactly equal to the quota.) SOLUTION:

Firm must produce exactly equal to the quota *h . The social planner determines the optimal

quantity *h by choosing the value of h that maximizes expected value of aggregate surplus (since the social planner does not know the precise realization of parameters η and θ ),

ˆmax [ ( , )] [ ( , )]

hE h E hη θφ η π θ+

And taking FOC with respect to h, we obtain * *ˆ ˆ( , ) ( , ) 0h hE Eh hη θ

φ η π θ⎡ ⎤ ⎡ ⎤∂ ∂+ ≤⎢ ⎥ ⎢ ⎥∂ ∂⎣ ⎦ ⎣ ⎦

We can now substitute the functional forms for the marginal benefit for consumers, ( , )hh

φ η∂∂

,

and the marginal profits for the firm, ( , )hh

π θ∂∂

, we obtain

* *ˆ ˆ[ ] [ ] 0ch E bh Eγ η β θ− + + − + ≤ ,

from which we can solve for *h to have

* *ˆ ˆ with equality for 0h hc bγ β+

≥ ≥+

.

(b) Identify the best tax *t for this same planner.

SOLUTION: Given a tax *t , the firm will maximize profits and will choose h that maximizes its profits (reduced in tax payments), that is

max ( , )h

h thπ θ −

The firm hence takes FOC with respect to h, obtaining

2

( , ) 0h th

π θ∂− =

∂

And since we know that ( , )h bhh

π θ β θ∂= − +

∂by definition, the above FOC becomes

0bh tβ θ− + − = . Solving for h, we obtain the firm’s profit-maximizing externality ( , )h t θ , as

a function of the tax rate t and its “type” θ , as follows

( , ) th tb

θ βθ + +=

Importantly, note that ( , )h t θ describes the firm’s “reaction function” (or “best response function”) after observing that the regulator imposes a particular tax rate t. Provided this best response function, we can now find what is the optimal tax that the social planner imposes, anticipating the firm’s best response function, as follows

*max [ ( ( , ), )] [ ( ( , ), )]

tE h t E h tφ θ η π θ θ+

(where note that, rather than writing any general level of h, we wrote the level of h that the firm optimally chooses in the second stage, after observing the tax rate t imposed by the regulator in the first stage). Taking FOC with respect to h, we obtain

( ( , ), ) ( , ) ( ( , ), ) ( , ) 0h t h t h t h tE Eh t h t

φ θ η θ π θ θ θ∂ ∂ ∂ ∂⎡ ⎤ ⎡ ⎤⋅ + ⋅ =⎢ ⎥ ⎢ ⎥∂ ∂ ∂ ∂⎣ ⎦ ⎣ ⎦

(note that we needed to use the chain rule in this FOC).

And since ( , ) th tb

θ βθ + += then

( , ) 1h tt bθ∂

= −∂

is a constant, that can be taken out of the

expectation operator. Therefore, we can cancel out the ( , )h ttθ∂

∂ from the FOC.

Substituting the functional form of our marginal benefit and marginal profit functions, the above FOC becomes:

[ ] [ ][ ] [ ] 0E t E tc E b Eb b

θ β θ βγ η β θ+ − + −− ⋅ + + − ⋅ + =

Rearranging, and solving for t, we obtain

*

( [ ] ) ( [ ] ) ( [ ] [ ] )[ ] [ ] [ ] [ ]

( ) [ ] [ ] [ ] [ ]Recall [ ] 0 and [ ] 0( )

( )

c E t b E t b E EcE c ct bE b bt bE b bE b

t c b bE b bE b cE c bE bE E

t c b b b c bc bt

c b

θ β θ β η β θ γθ β θ β η β θ γ

η β θ γ θ β θ βη θ

β γ β ββ γ

− + − − + − = + + −− − + − − + = − − +

+ = + + − + + + += =

+ = − − + +−

=+

(c) Compare the two instruments: Which is better and when?

SOLUTION:

3

We need to compare the expected difference in losses in order to determine when a tax or a quota instrument is better. Figure 11.E.1 illustrates the choices of *h and *t

In figure 11.E.1 the intersection of the expected marginal profits and marginal utility curves determine *h and *t . Consider a realization of θ and η that results in the curves intersecting at

point x . The optimal level of the externality would then be *h . If we use a quota instrument *hthe loss is the shaded triangle xuv . If we use a tax instrument *t then the firm will choose

*( , )h t θ and the loss is the shaded triangle xyz . Thus in the case pictured the tax instrument is better. (This is, of course, not a proof, but an introduction to the proof). Let’s consider when each instrument will be best. First we must introduce a non-standard way of calculating the area of a triangle. Area is normally calculated as: Area=1/2* base *height. In figure 11.E.1(b) below this would be 1

2A ed= .

4

We can divide the edge e into 1e and 2e , and we can then write 1edb

= , where b is the slope of

the top edge of the triangle. We can also write 1be e

b c= ⋅

+. Combining

edb c

=+

. Next we

plug d back into our normal area calculation to get: 21

2eA

b c= ⋅

+.

We apply this non-standard area calculation to determine the area of triangle xuv . In words, the calculation is the edge of uv squared, divided by twice the sum of the slopes of both marginal curves. The height of the edge uv is

* *( , ) ( , )h hh h

b cc b c b

π θ φ η

γ β γ ββ θ γ η

θ η

∂ ∂−

∂ ∂+ +⎛ ⎞ ⎛ ⎞− ⋅ + + − ⋅ +⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

+

We may therefore calculate the loss from quantity regulation (quota) as 2( )

2( )hLb c

θ η+=

+ . Next we

calculate the area of the loss from taxation. The height of the edge yz is

( )

* *

* *

* *

* *

* *

( ( , ), ) ( ( , ), )

We know ( ( , ) )

h t h th h

t tc bb b

c t tb

t c hc h t h

φ θ η π θ θ

θ β θ βγ η β θ

γ θ β η

γ

θ η

∂ ∂−

∂ ∂⎛ ⎞ ⎛ ⎞+ − + −

− − ⋅ + − − ⋅ +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

− + + − − −

= ⋅ −

− −

5

Next calculate * *( , )h t hbθθ − = and plug back in

cbθ η− . We may therefore calculate the loss

from taxation as 2

2( )t

cbLb c

θ η⎛ ⎞−⎜ ⎟⎝ ⎠=

+

Comparison of losses. Last we calculate the expected differences in losses

22

2

2

1( ) ( )2( )

( ) 2

h tcE L L E

b c b

b cb

θ

θθ η η

σ

⎡ ⎤⎛ ⎞− = ⋅ + − −⎢ ⎥⎜ ⎟+ ⎝ ⎠⎢ ⎥⎣ ⎦−

=

To conclude, we have just found that the optimal choice of quantity or tax instrument depends on the sign of ( )b c− :

1. When this term is positive, b c> , the loss from the quota system is greater so the tax instrument is preferred.

2. When instead c b> , the reverse is true and the quota instrument is preferred. Recall that under the tax system the level of the externality is changed depending on the firm’s realized marginal profits.

EconS 501

Recitation #13 – Imperfect Competition

Exercise 15.3 (NS). [ON YOUR OWN] This exercise analyzes Cournot competition when firms have different marginal costs. This departure from identical firms allows the student to shift around firm’s best-responses independently on a diagram.

Let ic be the constant marginal and average cost for firm i (so that firms may have different marginal costs). Suppose demand is given by 1P Q= − .

a. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry profits, consumer surplus, and total welfare. ANSWER:

11 1 2 1 1

1 2 11

2 11

1 22

max (1 )

:1 2

1 best response for consumer 12

Likewise 1 best response for consumer 2

2

qq q q c q

q q cq

q cq

q cq

− − −

∂− − =

∂− −

=

− −=

Solving simultaneously,

1 21

1

1 1 21

1122

4 112 2 2 2

q c cq

q q c c

− −− −

=

− = − + + −

3/)21( 211 ccqc +−=

and 3/)21( 122 ccqc +−= .

2

Further, 1 2(2 )3

c c cQ − −= , 1 2(1 )

3c c cP + += ,

21 2(1 2 )9

ci

c cπ − += , ccc

21 ππ +=Π ,

21 2(2 )18

c c cCS − −= , and ccc CSW +Π= .

b. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1’s cost would change the equilibrium. Draw a representative isoprofit for firm 1. ANSWER: Point E in Figure 15.3 represents the Nash equilibrium. The curved line represents firm 1’s isoprofit.

The reduction in firm 1’s marginal cost shifts its best response out and shifts the equilibrium from E to E’. Firm 1 will produce more for any given 2q .

Exercise 15.7 (NS). This exercise analyzes the Stackelberg game both with and without the possibility of entry-deterring investment.

Assume as in Problem 15.1 that two firms with no production costs, facing demand 150Q P= − , choose quantities 1q and 2q .

a. Compute the subgame-perfect equilibrium of the Stackelberg version of the game in which firm 1 chooses 1q first and then firm 2 chooses 2q . ANSWER:

BR2(q1

)

●

q

q

BR1(q2

)

E

E’●

3

We solve the game using backward induction starting with firm 2’s action. Firm 2 moves second and best responds to firm 1’s choice. We saw from Problem 15.1 (b) that firm 2’s best-response function is 2/75 12 qq −= . We substitute this back into firm 1’s profit function so that firm 1 is making its optimal choice given what it expects firm 2 to do

11 1 1 2 1 1

1 11

1

[150 ( )] 150 752

:150 2 75

75

qq q q q q

q qq

q

π ⎡ ⎤⎛ ⎞= − + = − − −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦∂

− − = −∂=

.

Taking the first-order condition with respect to 1q and solving yields 75*1 =q .

Substituting this back into firm 2’s best-response function yields

75

2 2*2

75

37.5

q

q

= −

=.

b. Now add an entry stage after firm 1 chooses 1q . In this stage, firm 2 decides whether or not to enter. If it enters it must sink cost 2K , after which it is allowed to choose 2q . Compute the threshold value of 2K above which firm 1 prefers to deter firm 2’s entry. ANSWER:

If firm 1 accommodates 2’s entry, the outcome in part (a) arises. P=150 – Q. P = 37.5. There are no production costs, thus profit for firm 1= P*Q=37.5*75= 2,812.5.

When firm 1 produces 1q if firm 2 best responds to 1q , and enters it will generate

profit 22

1 4/)150( Kq −− . In order to deter entry this profit must be less than or equal to 0. Setting firm 2’s profit function equal to zero

21

2

1 2

(150 ) 04150 2

q K

q K

−− =

= −

The threshold value of 21 2150 Kq −= . Firm 1’s profit from operating alone in

the market and producing this output is Q*P = )2)(2150( 22 KK− , which

4

exceeds 2,812.5 if 6.1202 ≥K . (as can be shown by graphing both sides of the inequality)

Exercise 15.9 (NS). This exercise examines the “Herfindahl index” of market concentration. Many economists subscribe to the conventional wisdom that increases in concentration are bad for social welfare. This problem leads students through a series of calculations showing that that the relationship between welfare and concentration is not this straightforward.

One way of measuring market concentration is through the use of the Herfindahl index, which is defined as:

2

1 where

ni

i ii

qH s sQ=

= =∑

Where is is firm i’s market share. The higher is H, the more concentrated the industry is said to be. Intuitively, more concentrated markets are thought to be less competitive because dominant firms in concentrated markets face little competitive pressure. We will assess the validity of this intuition using several models.

a. If you have not already done so, answer Problem 15.2(d) by computing the Nash equilibrium of this n-firm Cournot game. Also compute market output, market price, consumer surplus, industry profit, and total welfare. Compute the Herfindahl index for this equilibrium. ANSWER: Firm i’s profit is )( cbQbqaq iii −−− − with associated first-order condition

02 =−−− − cbQba i . This is the same for every n firm so we may impose symmetry [ ** )1( ii qnQ −=− ] Plugging in

*

*

2 ( 1) 0( )( 1)

i

i

a b b n q ca cqn b

− − − − =−

=+

.Further,

* ( )( 1)n a cQn b

−=

+, * ( )

( 1)a ncPn+

=+

,2

* * ( )( 1)i

n a cnb n

π⎡ ⎤−

Π = = ⎢ ⎥+⎣ ⎦,

22* ( )

( 1)n a cCSb n⎡ ⎤−

= ⎢ ⎥+⎣ ⎦

2* ( )

( 1)n a cW

n b⎡ ⎤−

= ⎢ ⎥+ ⎣ ⎦.

5

Because firms are symmetric, nsi /1= , thus we can solve for the Herfindahl index 21 1( )n nH n= = .

b. Suppose two of the n firms merge, leaving the market with n-1 firms. Recalculate the Nash equilibrium and the rest of the items requested in part (a). How does the merger affect price, output, profit, and total welfare. Compute the Herfindahl index for this equilibrium. ANSWER:

We can obtain a rough idea of the effect of merger by seeing how the variables in part (a) change with a reduction in n . Per-firm output, price, industry profit, and the Herfindahl index increase with a reduction in n, caused by the merger. Total output, consumer surplus, and welfare decrease with a reduction in n, caused by the merger.

c. Put the model used in parts (a) and (b) aside and turn to a different setup: that of Problem 15.3, where Cournot duopolists face different marginal costs. Use your answer to Problem 15.3(a) to compute equilibrium firm outputs, market output, price, consumer surplus, industry profit, and total welfare, substituting the particular cost parameters 1

1 2 4c c= = . Also compute the Herfindahl index. ANSWER:

Substituting 4/121 == cc into the answers for 15.3, we have 4/1* =iq ,

2/1* =Q , 2/1* =P , 8/1* =Π , 8/1* =CS , and 4/1* =W . Also, 2/1=H .

d. Repeat your calculations in part (c) while assuming that firm 1’s marginal cost 1c falls to 0 but 2c stays at 1

4 . How does the merger affect price, output, profit, consumer surplus, total welfare, and the Herfindahl index. ANSWER:

Substituting 01 =c and 4/12 =c into the answers for 15.3, we have

12/5*1 =q , 12/2*

1 =q , 12/7* =Q , 12/5* =P , 144/29* =Π ,

288/49* =CS , and 288/107* =W . Also, 49/29=H .

6

e. Given your results from parts (a)-(d), can we draw any general conclusions about the relationship between market concentration on the one hand and price, profit, or total welfare on the other? ANSWER:

Comparing part (a) with (b) suggests that increases in the Herfindahl index are associated with lower welfare. The opposite is evidenced in the comparison of part (c) to (d): welfare and the Herfindahl increase together. General conclusions are thus hard to reach.

Exercise 15.10 (NS). [ON YOUR OWN] This exercise extends the Inverse Elasticity Pricing Rule (IEPR) from a market structure with only one firm (monopoly) to market structures with more than one firm. It derives an alternatives form of the IEPR we know under monopoly that we can apply into a Cournot model of quantity competition.

a. Use the first-order condition (Equation 15.2) for a Cournot firm to show that the usual inverse elasticity rule form Chapter 11 holds under Cournot competition (where the elasticity is associated with an individual firm’s residual demand, the demand left after all rivals sell their output on the market). Manipulate Equation 15.2 in a different way to obtain an equivalent version of the inverse elasticity rule:

,

where ii

Q P i

sP MC QsP e q−

= − =

Where is is firm i’s market share and ,Q Pe is the elasticity of market demand. Compare this version of the inverse elasticity rule to that for a monopolist from the previous chapter.

ANSWER:

Equation 15.2 can be rearranged as follows:

,

Equation 15.2

( ) ( ) ( ) 0

' / 1| |

i

ii i i

i

i

i i i

q P

P Q P Q q C qq

P qP CP P

P q dP dq qP CP P P

π

ε

∂ ′ ′= + − =∂

′′ −−=

′ − − ⋅−= = =

,

7

where Pqi ,ε is the elasticity of demand with respect to firm i ’s output. The

second equality uses the fact that idqdPdQdPP // ==′ . Multiplying numerator and denominator by Q , we can also rearrange Equation 15.2 as

,

/| |

i i

Q P

q sdP dQ QP Q ε

⎛ ⎞− ⋅⎛ ⎞ =⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

.

MWG 12.C.10. Consider a J-firm Cournot model in which firms’ costs differ. Let ( ) ( )j j j jc q c qα= denote firm j’s cost function, and assume that ( )c ⋅ is strictly increasing

and convex. Assume that 1 ... jα α> > .

(a) Show that if more than one firm is making positive sales in a Nash equilibrium of this model, then we cannot have productive efficiency; that is, the equilibrium aggregate output *Q is produced inefficiently. ANSWER: Each firm i chooses its output 0iq ≥ to maximize its profits

( ) ( )i i i i ip Q q q c qπ α−= + − FOC, assuming a positive solution

: ( ) ( ) ( )i i ii

p Q c q p Q qq

α∂ ′ ′= −∂

Where Q is the total output. Since ( )c ⋅ is increasing and convex, the right-hand side of the FOC is increasing in iα and decreasing in iq . Since the FOC holds for every firm, we must have i jq q> whenever j iα α> . The marginal cost of production for each firm i is ( )i ic qα ′ . Imagine the case where firm i differs from firm j we may calculate the difference in marginal cost between the two firms as

( ) ( )j j i ic q c qα α′ ′− From our FOCs we know that after canceling out the ( )p Q terms this is equivalent to

( )( )i jp Q q q′− − Imagine the particular case where j iα α> so that also i jq q> . Because ( )p Q′ < 0 the entire term ( )( )i jp Q q q′− − is positive. This implies that ( ) ( )j j i ic q c qα α′ ′> . Thus the marginal cost for firm j is greater in this case, therefore marginal costs across firms are not necessarily equalized. Likewise aggregate output is not necessarily produced efficiently.

(b) If so, what is the correct measure of welfare loss relative to a fully efficient (competitive) outcome? [Hint: Reconsider the discussion in Section 10.E] ANSWER:

8

The correct measure of welfare loss relative to a fully efficient outcome in this case is equal to the loss of consumer surplus due to non-competitive pricing plus the higher production cost due to productive inefficiency. This productive inefficiency was not considered in section 10.E.

(c) Provide an example in which welfare decreases when a firm becomes more productive (i.e., when jα falls for some j). [Hint: Consider an improvement in cost for firm 1 in the model of Exercise 12.C.9.] Why can this happen? ANSWER: Let’s use our results from Exercise 12.C.9 in order to provide an example in which welfare decreases when a firm becomes more productive. From 12.C.9 the Cournot equilibrium output and price levels are

1 21

2 12

1 2

2323

3

a c cqb

a c cqb

a c cp

− +=

− +=

+ +=

Total profits of the two firms can now be computed as 2 2 2

1 1 2 2 1 2 1 2 1 2( ) ( ) (2 5 5 2 ( ) 8 ) / 9p c q p c q a c c a c c c c b− + − = + + − + − Consumer surplus can be computed as

1 2 1 221 2 1 200

( ) ( / 2) ( )(5 ) /18q q q q

p q dq aq bq a c c a c c b+ +

= − = − − + +∫

Adding up total profits and consumer surplus, and differentiating with respect to 1c , we obtain

1 2

1

9 9 49

c c aSurplusc b

− −∂=

∂

This derivative is positive when 41 2 9( )c c a> + . This will occur when firm 1’s

costs are much greater than firm 2’s costs. In this case a decrease in 1c reduces social welfare. The reason is that when 1c slightly falls, firm 1 steals more business from firm 2, which raises production inefficiency. When 1c is substantially larger than 2c , this effect actually dominates the increase in consumer surplus due to a lower price.

MWG 12.C.12. Consider two strictly concave and differentiable profit functions ( , ), 1, 2,j j kq q jπ = defined on [0, ]jq q∈ .

(a) Give sufficient conditions for the best-response functions ( )j jb q to be increasing or decreasing. ANSWER:

9

Assume that 11( , ) 0 for 1,2ii jq q iπ < = . Where the subscript ii means differentiate

twice with respect to the first element. Each firm will maximize profit.

FOC: 11

12

( ) ( ( ), ):

( ( ), )

ii j i j j

ij j i j j

b q b q qq q b q q

ππ

∂∂= −

∂ ∂

Therefore, the sign of ( )i j

j

b qq

∂

∂ is the same as the sign of 12 ( ( ), )i

i j jb q qπ .

This means firm i’s best-response function is increasing when 12iπ is positive.

Firm i’s best-response function is decreasing when 12iπ is negative.

(b) Specialize to the Cournot model. Argue that a decreasing (downward-sloping)

best-response function is the “normal” case. ANSWER: In the Cournot model

1( , ) ( ) ( )i j i j i iq q p q q q c qπ = + − Then differentiating further w.r.t. jq

12 ( , ) ( ) ( )ii j i j i i jq q p q q q p q qπ ′′ ′= + + +

Which is negative if ( )p ⋅ is downward sloping and not to convex. This seems likely for the inverse demand function, thus the “normal” slope of the best response functions in the Cournot model is negative.

MWG 12.D.1. Consider an infinitely repeated Bertrand duopoly with discount factor 1δ < . Determine the conditions under which strategies of the form in (12.D.1) sustain the

monopoly price in each of the following cases:

From page 401 (12.D.1):

11

if all elements of equal ( , ) or 1( )

otherwise

m m mt

jt tp H p p t

P Hc

−−

⎧ == ⎨⎩

(a) Market demand in period t is ( ) ( )ttx p x pγ= where 0γ > is the rate of growth of

demand across periods. ANSWER: Monopoly profit in period t is

max ( )( ) max ( )( )t t t m

p px p p c x p p cγ γ γ π− = − =

If a firm deviates at t τ= , it can obtain mτγ π in that period, and it will get zero forever after. If it does not deviate, its payoff is

0

1( )2 (1 ) 2

m mt

tτ τπ πγ γδ γ

γδ∞

==

−∑

10

Monopoly price can be sustained when deviation from the strategy is not profitable. Deviation is not profitable if and only if

1 1 or (1 ) 2 2

mmτ τπγ γ π δ

γδ γ≥ ≥

−

Hence, the minimal discount factor supporting cooperation decreases in the rate of growth of demand, i.e., cooperation can be sustained under a larger set of discount factors as demand grows faster across periods. [See figure].

γ

δ

12

1416

1

1 2 3 4

Cooperation

Cheating

(b) At the end of each period, the market continues to exist with probability [0,1]γ ∈ . ANSWER: If a firm deviates, it can obtain mπ in that period, and it will get zero forever after. If it does not deviate, its payoff is

0

1( )2 (1 ) 2

m mt

t

π πγδγδ

∞

=

=−∑

Therefore, deviation is not profitable if and only if 1 1 or

(1 ) 2 2

mmπ π δ

γδ γ≥ ≥

−

Thus, cooperation cannot be sustained under any discount factor (between zero and one) when the probability that demand continues existing is relatively low, but can be sustained when the probability that demand continues existing is sufficiently high (and decreases as this probability gets closer to 100%). [See figure].

11

γ

δ

12

23

1

1

Cooperation

Cheating

34

12

14

Prob of demand continuing

(c) It takes K periods to detect and respond to a deviation from the collusive agreement. ANSWER: If a firm deviates, it can obtain

1

0

(1 )(1 )

KK t m mt

δδ π πδ

−

=

−=

−∑

In the next K periods, and it will get zero forever after. If it does not deviate, its payoff is

0

12 (1 ) 2

m mt

t

π πδδ

∞

==

−∑

Therefore, deviation is not profitable if and only if 1

1 (1 ) 1 or (1 ) 2 (1 ) 2

m K Kmπ δ π δδ δ

− ⎛ ⎞≥ ≥ ⎜ ⎟− − ⎝ ⎠

Hence, the more periods of time K that a cheating firm remains undetected by its colluding partners, the more attractive cheating becomes. Cooperation therefore can only be sustained under more restrictive sets of parameter values. [See figure].

12

K

δ

12

0.707

1

1 2 3 4

Cooperation

Cheating

Micro Theory I - EconS 501Midterm #1 - Answer key

1. [20 points] [Checking properties of preference relations]. Let us consider thefollowing preference relations de�ned in X = R2+. First, de�ne the upper countourset, the lower contour set and the indi¤erence set for every preference relation. Then,check if they satisfy: (i) completeness, (ii) transitivity, (iii) monotonicity, and (v) weakconvexity. [Answer only one of the following 2 questions]

(a) [20 points] (x1; x2) % (y1; y2) if and only if x1 � y1 � 1 and x2 � y2 + 1.� Let us �rst build some intuition on this preference relation. Take a bundle(2; 1). Then, the upper contour set of this bundle is given by

UCS%(2; 1) = f(x1; x2) % (2; 1)() x1 � 2� 1 and x2 � 1 + 1g= f(x1; x2) : x1 � 1 and x2 � 2g

which is graphically represented by all those bundles in R2+ in the lower right-hand corner (below x2 = 2 and to the right of x1 = 1). On the other hand,the lower contour set is de�ned as

LCS%(2; 1) = f(2; 1) % (x1; x2)() 2 � x1 � 1 and 1 � x2 + 1g= f(x1; x2) : x1 � 3 and x2 � 0g

which is graphically represented by all those bundles in R2+ in the left half ofthe positive quadrant (above x2 = 0 and to the left of x1 = 3).Finally, theconsumer is indi¤erent between bundle (2,1) and the set of bundles where

IND%(2; 1) = f(x1; x2) � (2; 1)() 1 � x1 � 3 and 0 � x2 � 2g

� Completeness. From the above analysis it is easy to note that this propertyis not satis�ed, since there are bundles in the area x1 > 3 and x2 � 2 whereour preference relation does not specify if they belong to the upper contourset, the lower contour set, or the indi¤erence set. Another way to prove thatcompleteness does not hold is by �nding a counterexample. In particular, wemust �nd an example of two bundles such that neither (x1; x2) % (y1; y2) nor(y1; y2) % (x1; x2). Let us take two bundles,

(x1; x2) = (1; 2) and (y1; y2) = (4; 6)

We have that:

1. (x1; x2) � (y1; y2) since 1 � 4 � 1 for the �rst component of the bundle,and

2. (y1; y2) � (x1; x2) since 6 � 2 + 1 for the second component of the bun-dle. Hence, there are two bundles for which neither (x1; x2) % (y1; y2)nor (y1; y2) % (x1; x2), which implies that this preference relation is notcomplete.

1



This property does not hold for this preference relation. In order to showthat, let us consider the following three bundles (that is, we are �nding acounterexample to show that transitivity does not hold):

(x1; x2) = (2; 1)

(y1; y2) = (3; 4)

(z1; z2) = (4; 6)

First, note that (x1; x2) % (y1; y2) since x1 � y1 � 1 (i.e., 2 � 3 � 1), andx2 � y2 + 1 (i.e., 1 � 4 + 1). Additionally, (y1; y2) % (z1; z2) is also satis�edsince y1 � z1 � 1 (i.e., 3 � 4� 1), and y2 � z2 + 1 (i.e, 3 � 4 + 1). However,(x1; x2) � (z1; z2) since x1 � z1 � 1 (i.e., 2 � 4 � 1). Hence, this preferencerelation does not satisfy Transivity.



�(x1; x2) + (1� �) (y1; y2) % (y1; y2) for any � 2 [0; 1]

In this case, (x1; x2) % (y1; y2) implies that x1 � y1 � 1 and x2 � y2 + 1;whereas �(x1; x2) + (1� �) (y1; y2) % (y1; y2) implies

�x1 + (1� �) y1 � y1 � 1 for the �rst component, and�x2 + (1� �) y2 � y2 + 1 for the second component.

which respectively imply

� (x1 � y1) � �1, and� (x2 � y2) � 1

and since (x1 � y1) � �1 and (x2 � y2) � 1 by assumption, i.e., (x1; x2) %(y1; y2), then both of the above conditions are true for any � 2 [0; 1]. Hence,this preference relation is weakly convex.

(b) [20 points] (x1; x2) % (y1; y2) if and only if max fx1; x2g > max fy1; y2g.1. Completeness. For all (x1; x2),(y1; y2) 2 R2, eithermax fx1; x2g >max fy1; y2g,or max fy1; y2g > max fx1; x2g, or both. It follows that either (x1; x2) �(y1; y2), or (x1; x2) � (y1; y2), or both. Hence, this preference relation iscomplete.

2

2. Transitivity. Take some (x1; x2),(y1; y2) and (z1; z2) 2 R2 with (x1; x2) %(y1; y2) and (y1; y2) % (z1; z2). Then, max fx1; x2g > max fy1; y2g, andmax fy1; y2g > max fz1; z2g. Therefore, max fx1; x2g > max fz1; z2g, andso (x1; x2) % (z1; z2). Hence the preference relation is transitive. Since, it isalso complete, this preference relation is rational.

3. Monotonicity. Take any (x1; x2) and (y1; y2) 2 R2 with x1 > y1 and x2 > y2.Then, max fx1; x2g > max fy1; y2g with strict inequality, and it follows that(x1; x2) % (y1; y2). In addition, since max fx1; x2g > max fy1; y2g, we canalso say that (y1; y2) � (x1; x2). It follows that (x1; x2) � (y1; y2), and hencethe preference relation is monotone.

4. Weak Convexity. Take some (x1; x2),(y1; y2) and (z1; z2) 2 R2 with (y1; y2) %(x1; x2) and (z1; z2) % (x1; x2). Therefore, max fy1; y2g > max fx1; x2g, andmax fz1; z2g > max fx1; x2g. However, the convex combination of (y1; y2) and(z1; z2) with �,

max f�y1 + (1� �)z1; �y2 + (1� �)z2g

is not necessarily higher than max fx1; x2g. In order to see that, consider anexample in whichmax fy1; y2g >max fx1; x2g, andmax fz1; z2g >max fx1; x2g,such as (y1; y2) = (0; 4) and (x1; x2) = (3; 3) and (z1; z2) = (4; 0). Now, notethat the convex combination of (y1; y2) and (z1; z2) with �, will give us valuesbetween 0 and 4. For intermediate values of � (such as � = 1

2) we have that

max

�1

20 +

1

24;1

24 +

1

20

�= max f2; 2g < max f3; 3g

Hence, the preference relation is not convex.

2. [15 points] [Lexicographic preference relations are rational]. Let us de�ne alexicographic preference relation in a continuouous consumption set X � Y , where forsimplicity both X = [0; 1] and Y = [0; 1], as follows:

(x1; x2) % (y1; y2) if and only if�

x1 > y1, or ifx1 = y1 and x2 > y2

� Show that % is a rational preference relation (i.e., it is complete and transitive).� Answer :(a) Completeness. For all (x1; x2),(y1; y2) 2 R2, either (x1; x2) % (y1; y2) or

(y1; y2) % (x1; x2), or both.Hence, we need to show that(x1; x2) � (y1; y2) =) (y1; y2) % (x1; x2)

Indeed, note that

(x1; x2) � (y1; y2) if�

y1 > x1, and ifx1 6= y1 or y2 > x2

(1)

First, note that we changed �or� for �and�, and viceversa, in order to con-struct a full negation of the argument in (1). Now, note that (2) implies(y1; y2) % (x1; x2). Therefore, we have shown that (x1; x2) � (y1; y2) implies(y1; y2) % (x1; x2). Hence, the preference relation is complete.

3

(b) Transitivity. Let us take some (x1; x2),(y1; y2) and (z1; z2) 2 R2 with (x1; x2) %(y1; y2):

(x1; x2) % (y1; y2) if�

x1 > y1, or ifx1 = y1 and x2 > y2

and (y1; y2) % (z1; z2), that is

(y1; y2) % (z1; z2) if�

y1 > z1, or ify1 = z1 and y2 > z2

Hence, we need to check for transitivity in the four possible cases in which(x1; x2) % (y1; y2) and (y1; y2) % (z1; z2).

� 1. If x1 > y1, and y1 > z1, then x1 > z1. As we know that x1 > z1 implies(x1; x2) % (z1; z2). Hence, transitivity is checked in this case.

2. If (x1 = y1 and x2 > y2) and (y1 = z1 and y2 > z2), then (x1 = z1 and x2 > z2).And we know that (x1 = z1 and x2 > z2) implies (x1; x2) % (z1; z2), whichvalidates transitivity.

3. If x1 > y1, and (y1 = z1 and y2 > z2), then x1 > z1. As we know that x1 > z1implies (x1; x2) % (z1; z2). Hence, transitivity is checked in this case.

4. If y1 > z1 and (x1 = y1 and x2 > y2), then x1 > z1. As we know that x1 > z1implies (x1; x2) % (z1; z2), which validates transitivity. We have then checked allfour cases under which (x1; x2) % (y1; y2) and (y1; y2) % (z1; z2) may occur, and inall of them we obtained (x1; x2) % (z1; z2), con�rming that this preference relationis transitive. Therefore, since the preference relation is complete and transitive,we can conclude that it is rational.

1. [15 points] [CheckingWARP].Check whether the following demand function satis�esthe weak axiom of revealed preference (WARP). You can use �gures to help yourdiscussion, but your �nal reasoning must be in terms of the de�nition of the WARP:

� �Average demand�: The consumer�s walrasian demand is the expected value ofa uniform randomization over all points on her budget frontier, for any (strictlypositive) prices p1, p2 and wealth w.

�Answer : First, note that if the consumer randomizes uniformly over all pointsin her budget line, then the expected random demand is allocated at themidpoint of the budget line.

x2

Bp,w

x1

x(p,w)

0.5w/p2

0.5w/p1

w/p2

w/p1

Average demand

4

Let us now prove that WARP is satis�ed for average demand. Let us workby contradiction, by assuming that average demand violates WARP. Thereare two possibilities in which this violation might take place, as the followingtwo �gures illustrate.

x2

Bp,w

x1

x(p,w)

w/p2

w/p1

Bp’,w’

x(p’,w’)

w’/p1'

x2

Bp,w

x1

x(p,w)

w/p2

w/p1

Bp’,w’

x(p’,w’)

w’/p1'x1' x1

Let us compute point x1 and x01. Recall that these points have to be allocatedat the midpoint of the budget line. Hence,

x1 =1

2

w

p1and x01 =

1

2

w0

p01

therefore 2x1 = wp1and 2x01 =

w0

p01. Moreover, we can see in both �gures that

x01 < x1. Therefore, 2x01 < 2x1, which implies

w0

p01<w

p1

But in both �gures we in fact see that w0

p01> w

p1. Hence, we have reached a

contradiction, and average demand cannot violate WARP.

4. [5 points] [Concavity of the support function] We know that, given a non-empty,closed set K, its support function, �K (p), is de�ned by

�K (p) = inf fp � xg for all x 2 K and p 2 RL

Hence, the value of this support function, �K , satis�es �K � p � x for every element xin the set K. Given this de�nition, prove the concavity of the support function. Thatis, show that

�K (�p+ (1� �) p0) > ��K (p) + (1� �)�K (p0)for every p; p0 2 RL and for any � 2 [0; 1].

� First, from the de�nition of the support function we know that, for a given pricevector p, and for every element x in the set K,

�K(p) � p � x, then ��K(p) � �p � x, for all � 2 [0; 1] (2)

And similarly for any other price vector p0,

�K(p0) � p0 � x, then ��K(p0) � �p0 � x, for all � 2 [0; 1] (3)

5

Similarly,

�K(p) � p � x, then (1� �)�K(p) � (1� �) p � x, for all � 2 [0; 1] (4)

�K(p0) � p0 � x, then (1� �)�K(p0) � (1� �) p0 � x, for all � 2 [0; 1] (5)

Summing up expressions (1) and (4), we have

��K(p) + (1� �)�K(p) � �p � x+ (1� �) p0 � x

which can be simpli�ed to

��K(p) + (1� �)�K(p) � [�p+ (1� �) p0] � x

and by the de�nition of the support function, we know that �K(�p+(1� �) p0) =[�p+ (1� �) p0] � x. Therefore,

��K(p) + (1� �)�K(p) � �K(�p+ (1� �) p0)

and hence the support function �K(p) is concave.

5. [25 points] [Compensating and Equivalent variation] An individual consumesonly good 1 and 2, and his preferences over these two goods can be represented by theutility function

u(x1; x2) = x�1x

�2 where �; � > 0 and �+ � ? 1


(a) [1 point] Find the Walrasian demand for goods 1 and 2 of this individual, x1(p; w)and x2(p; w).

� Walrasian demands are

x1(p; w) =�w

(�+ �) p1and x2(p; w) =

�w

(�+ �) p2

(b) [1 point] Find his indirect utility function, and denote it as v(p0; w).

� Plugging the above Walrasian demand functions in the consumer�s utilityfunction, we obtain

v(p; w) =

��w

(�+ �) p1

�� w

(�+ �) p2

��=

�w

�+ �

��+� ��

p1

��

p2

��(c) [1 point] The �rm that this individual works for is considering moving its o¢ ce to

a di¤erent city, where good 1 has the same price, but good 2 is twice as expensive,i.e., the new price vector is p0 = (p1; 2p2). Find the value of the indirect utilityfunction in the new location, i.e., when the price vector is p0 = (p1; 2p2). Let usdenote this indirect utility function v(p0; w).

v(p0; w) =

�w

�+ �

��+� ��

p1

��

2p2

��6

(d) [4 points] This individual�s expenditure function is

e(p; u) = (�+ �)�p1�

� ��+�

�p2�

� ��+�

u1

�+�



e(p0; u0) = (�+ �)�p1�

� ��+�

�p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

p2

��#| {z }

u

1�+�

= w


e(p0; u0) = (�+ �)�p1�

� ��+�

�p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

2p2

��# 1�+�

=1

2�

�+�

w


e(p1; u0) = (�+ �)�p1�

� ��+�

�2p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

p2

��# 1�+�

= 2�

�+�w


e(p0; u0) = (�+ �)�p1�

� ��+�

�2p2�

� ��+�

"�w

�+ �

��+� ��

p1

��

2p2

��# 1�+�

= w

(e) [4 points] Find this individual�s equivalent variation due to the price change.Explain how your result can be related with this statement from the individual tothe media: �I really prefer to stay in this city. In fact, I would accept a reductionin my wealth if I could keep working for the �rm staying in this city, instead ofmoving to the new location�

� We know that

EV = e(p1; u1)� e(p0; u1) = m� 1

2�

�+�

w

That is, this individual would be willing to accept a reduction in his wealthof w� 1

2�

�+�

w in order to avoid moving to a di¤erent city. [Alternatively, the

individual is willing to accept a reduction of�1� 1

2�

�+�

�% of his weatlh ]

7

(f) [4 points] Find this individual�s compensating variation due to the price change.Explain how your result can be related with this statement from the individualto the media: �I really prefer to stay in this city. The only way I would accept tomove to the new location is if the �rm raises my salary�.

� We know that

CV = e(p1; u0)� e(p0; u0) = 2�

�+�w � w

That is, we would need to raise this individuals� salary by 2�

�+�w � w inorder to guarantee that his welfare level at the new city (with new prices)coincides with his welfare level at the initial city (at the initial price level).

[Alternatively, the individual must receive an increase of�2

��+� � 1

�of his

wealth]

(g) [4 points] Find this individual�s variation in his consumer surplus (also referredas area variation). Explain.

� We know that area variation is given by the area below the Walrasian demandbetween the initial and �nal price level. That is,

AV =

Z 2p2

p2

x2(p; w)dp =

Z 2p2

p2

�

(�+ �) pw dp

=�

(�+ �)w

Z 2p2

p2

1

pdp =

�

(�+ �)w ln 2

Hence, moving to the new city would imply a reduction in this individual�swelfare of �

(�+�)w ln 2, or

��

(�+�)ln 2�% of his wealth.

(h) [4 points] Which of the previous welfare measures in questions (e), (f) and (g)coincide? Which of them do not coincide? Explain.

� None of them coincide, since this individual�s preferences are not quasilinealin any of the goods.

(i) [2 points] Consider how the welfare measures from questions (e), (f) and (g) wouldbe modi�ed if this individual�s preferences were represented, instead, by the utilityfunction v(x1; x2) = � lnx1 + � lnx2:

� Since we have just applied a monotonic transformation to the initial utilityfunction, u(x1; x2), this new utility function represents the same preferencerelation than function v(x1; x2). Hence, the welfare results that we wouldobtain from function v(x1; x2) would be the same as those with utility functionu(x1; x2):

6. [10 points] [Slutsky equation in labor markets]. Explain the income and substitu-tion e¤ect in the labor market. Help your discussion with a �gure, but you must relateyour �gure with the Slutsky equation in labor economics.

8

(a) � We know that the worker�s problem can be written as a EMP

miny;zM = py � wz subject to v(y; z) = v

where y is the composite commodity, z is the number of working hours, pis the price of the composite commodity, and w is the wage. Finally, notethat v(y; z) = v represents the utility level v that this worker wants to reach.From this EMP we can �nd the optimal hicksian demands, hy(w; p; v) andhz(w; p; v), and inserting them into the objective function, we obtain the valuefunction of this EMP (the expenditure function):

e(w; p; v) = phy(w; p; v) + whz(w; p; v)

We know thatxz(w; p; e(w; p; v)) = hz(w; p; v)

Di¤erentiating on both sides and using the chain rule

@xz@w

+@xz@e

@e

@w=@hz@w

() @xz@w

=@hz@w

� @xz@e

@e

@w

and since we know that @e(w;p;v)@w

= �hz(w; p; v), then

@xz@w

=@hz@w

+@xz@ehz(w; p; v)

Using the Slutsky equation (SE and IE) in the analysis of labor markets:

@xz@w

=@hz@w

+@xz@ehz(w; p; v)

where:

1. @hz@w> 0 is the substitution e¤ect:

(a) an increase in wages increases the worker�s supply of labor, if we make his wealthlevel constant; and

2. @xz@ehz(w; p; v) is the income e¤ect:

(a) if @xz@e> 0 then an increase in wages makes that worker richer, and he decides to

work more (this would be an upward bending supply curve), or

(b) if @xz@e< 0 then an increase in wages makes that worker richer, and he decides to

work less (e.g., nurses in Massachussets).

7. [10 points] [Aggregate demand]. Answer only one of the following 2 questions:

9

(a) [10 points] We know that aggregate demand can be expressed as a function ofaggregate wealth, i.e.,

IXi=1

xi(p; wi) = x

p;

IXi=1

wi

!

if the following condition is satis�ed for any two individuals i and j, for a givengood k, and for any wealth of these two individuals, wi and wj.

@xki(p; wi)

@wi=@xkj(p; wj)

@wj

Explain what this condition implies in terms of these individuals�wealth expansionpaths (you can use a �gure to help your discussion). Can you give an example ofa preference relation satisfying this condition?

� This condition states that: for any �xed price vector p, for any good k,and for any wealth level of any two individuals i and j, the wealth e¤ectis the same across individuals. In other words, the wealth e¤ects arisingfrom the distribution of wealth across consumers cancel out. Graphically,this condition is equivalent to say that all consumers exhibit parallel, straightlines:

� Straight, because the coincidence in wealth e¤ects do not depend on theindividuals�wealth level.

� Parallel, because individuals�wealth e¤ects must coincide (and recall thatwealth expansion paths just represent how an individual demand changes ashe becomes richer). (See �gure from Handout #8).

� Examples of parallel, straigh wealth expansion paths? Homothetic prefer-ences, and Quasilinear preferences (all consumers with respect to the samegood).

� Recall that we can embody all these cases as special cases of a particular typeof preferences? If every consumer�s indirect utility function can be expressedas

vi(p; wi) = ai(p) + b(p)wi (Gorman form)

with the same b(p)�s for all consumers, then their wealth expansion paths are parallel, straightlines. And as a consequence, aggregate demand can be represented as a function of aggregatewealth.

1. b. [10 points] Show that if an individual�s preference relation is homothetic, thenthis individual�s Walrasian demand satis�es the Uncompensated Law of Demand(ULD). [Hint: instead of showing ULD, you can alternatively show thatDpxi(p; wi)is negative semide�nite, since we know that both properties are equivalent. Inorder to show the latter, �rst use the Slustsky equation, then use homotheticity,and �nally pre- and post-multiply all elements by dp]

10

� As we observed in Homework #3, we �rst write the Slutsky equation

Si(p; wi) = Dpxi(p; wi) +Dwxi(p; wi)xi(p; wi)T

and for homothetic preference relations, xi(p; wi) = �iwi, (or alternatively,�i =

xi(p;wi)wi

), we have thatDwxi(p; wi) = �i, with we can write asDwxi(p; wi) =xi(p;wi)wi

. Plugging and rearranging,

Dpxi(p; wi) = Si(p; wi)�xi(p; wi)

wixi(p; wi)

T

� Now we pre- and post-multiply all elements by dp,

dp �Dpxi(p; wi) � dp = dp � Si(p; wi) � dp| {z }< 0 if dp 6= �p= 0 if dp = �p

�dp�xi(p; wi)wi

xi(p; wi)T| {z }

> 0 if xi> 0= 0 if xi= 0

�dp

Either way, dp �Dpxi(p; wi) � dp < 0, except when zero consumption and thechange in prices is proportional to the initial price level, i.e., dp = �p. SinceDpxi(p; wi) is then negative semide�nite, and a few minutes ago we saw that

ULD () Dpxi(p; wi) is negative semide�nite

Hence, xi(p; wi) satis�es ULD.

11

Micro Theory I - EconS 501Midterm #2 - Answer Key

Instructions: Please read the questions carefully, answer them in a formal and concisemanner, but include all your steps, this will allow you to obtain partial credit. You haveuntil 10:25a.m. to complete the exam. Good luck!!

1. [25 points] [True or false?] Identify which of the following statements are true, andwhich are false, and provide a very short explanation of why this is the case.

(a) [3 points] All preference relations are rational.

� False. See recitation #1 for examples of preference relations which do notsatisfy rationality.

(b) [3 points] If a preference relation is rational (satis�es completeness and transitiv-ity), it can be represented by a utility function.

� True. The Lexicographic preference relation satis�es rationality, and it cannotbe represented by a utility function when de�ned over continuous consump-tion sets, e.g., X 2 [0; 1]; since this preference relation is not continuous.However, when the lexicographic preference relation is de�ned over discreteconsumption sets, X = fx11;x12 ; :::; x1Ng, this preference relation can be rep-resented by a utility function (see Homework #1 for an example). If thestatement was �If a preference relation is rational (satis�es completeness andtransitivity), it can be represented by a utility function for any consumptionset�then it would have been false, since it cannot be represented when theconsumption set is continuous.

(c) [3 points] If a preference relation is quasilinear, the substitution e¤ect is zero, andthe income e¤ect is positive.

� False. If a preference relation is quasilinear, the income e¤ect is zero.(d) [3 points] Gi¤en goods does not need to be inferior.

� False. Every Gi¤en good must be inferior. The opposite direction is not true,however: every inferior good does not need to be Gi¤en.

(e) [3 points] The Walrasian demand is negatively sloped, for any preferences of theconsumer.

� False. The Walrasian demand of Gi¤en goods is positively sloped (quantitydemanded increases when prices increase, or alternatively, quantity demandeddecreases when prices decrease).

(f) [3 points] The area variation (change in consumer surplus) is never a good ap-proximation of the change in consumer welfare resulting from price changes (orequivalently, from tax changes), for any type of consumer preferences.

1

� False. The area variation is a good approximation of the change in con-sumer welfare for quasilinear preferences, and more generally, for goods withrelatively small income e¤ects.

(g) [7 points] If a production function satis�es increasing average product, it mustalso satisfy increasing marginal product.

� False. Recall that the average product can be represented through the slopeof the rays from the origin to the production function, and the slope is simplythe slope of the production function at a given point. A counterexample ofthe above statement is represented below:

f(z)

z

f’(z)

2. [20 points] [Properties of the pro�t function] The pro�t function, �(p), is de�nedas

�(p) = max fp � y j y 2 Y gor alternatively, �(p) > p � y for every y 2 Y .

(a) [10 points] Show that the pro�t function �(p) is convex in prices.

� We need to show that, for any � 2 [0; 1] and p 2 RL++,

�(�p+ (1� �)p0) � �� (p) + (1� �)�(p0)

Proof. From the de�nition of the pro�t function we know that

� (p) � p � y, for any y 2 Y and p >> 0, and� (p0) � p0 � y, for any y 2 Y and p0 >> 0

And similarly, using that the pro�t function is homogeneous of degree 1 inprices, we can take any � 2 [0; 1],

�� (p) � �p � y, for any y 2 Y and p >> 0, and(1� �)� (p0) � (1� �) p0 � y, for any y 2 Y and p0 >> 0

Adding up the two previous inequalities,

�� (p) + (1� �)� (p0) � �p � y + (1� �) p0 � y

2

where the right-hand side �p�y+(1� �) p0�y coincides with the pro�t functionfor price level �p+ (1� �) p0, i.e., � (�p+ (1� �) p0). Hence,

�� (p) + (1� �)� (p0) � � (�p+ (1� �) p0)

and therefore we can conclude that the pro�t function is convex in prices.

(b) [10 points] Prove the Hotelling�s lemma using the Duality theorem. [Hint: easy,just rewrite]

� Hotelling�s lemma states that if the output function evaluated at prices c,y(p), consists of a single point, then the pro�t function � (p) is di¤erentiableat the price level p; and moreover such derivative is rp.� (p) = y(p).

� Let us �rst express the pro�t function as the support function that, for everyprice vector p, chooses the in�mum of p � (�y), i.e., instead of choosing themax of p � y, we rede�ne it as the inf of p � (�y).

� (p) = inf fp � (�y) jy 2 Y g

� In order to emphasize the similarities with the Duality Theorem, we reproduceit here again: Let K be a nonempty and closed set, and let us �K(p) be itssupport function. Then, there exists a unique element x in the set K suchthat �K (p) = p � x if and only if �K (p) is di¤erentiable at p. Moreover,rp�K (p) = x.

� Therefore, given that we have noticed that the pro�t function can be ex-pressed as a support function, we can rewrite Hotelling�s lemma as a directapplication of the Duality Theorem.(just changing labels!): Let �Y be a non-empty and closed (production) set, and let us ��Y (p) be its support function.Then, there exists a unique production function y (p) in the set �Y suchthat � (p) = p � y (p) if and only if � (p) is di¤erentiable at p. Moreover, thisderivative is rp� (p) = y (p).

3. [8 points] [Independence axiom and convexity]. Show that the independenceaxiom implies convexity, i.e., for three di¤erent lotteries L, L0 and L00, if L % L0 andL % L00�then L % �L0 + (1� �)L00:

� From L % L0 we can apply the independence axiom, and obtain

�L+ (1� �)L � �L0 + (1� �)L

Similarly, from L % L00 we can apply the independence axiom, and obtain

(1� �)L+ �L0 � (1� �)L00 + �L0

And by transitivity (from the two previous expressions), we have

�L+ (1� �)L � (1� �)L00 + �L0

and rearrangingL � �L0 + (1� �)L00

3

4. [18 points] [Purchasing health insurance] Consider an individual with the followingBernouilli utility function

u(C;H) = lnC � �

Hwhere C is his expenditure in consumption goods and H is his expenditure on healthinsurance. Parameter � denotes his losses if he becomes sick, where for simplicity

� =

�1 if he is sick, and0 if he is healthy

Note that this utility function implies that, when getting sick, this individual�s disu-tility is decreasing in the amount of health insurance that he purchased (e.g., he canhave access to better doctors and care facilities, and the negative e¤ects of the illnessare reduced). Finally, the probability of getting sick is given by 2 [0; 1], and thisindividual�s wealth is given by m > 0, where m = C +H.

(a) [3 points] What is this individual utility maximization problem? [Hint: it is easierto choose C as your choice variable. You can �nd the optimal amount of H lateron]

maxC;H

(1� ) lnC + �lnC � 1

H

�where we subsitute � = 0 when the individual is healthy (which occurs with prob-ability ), and � = 1 when the individual is sick (which occurs with probability1� ). And since m = C +H, then H = m� C, hence,

maxC

(1� ) lnC + �lnC � 1

m� C

�which reduces the choice variables of this maximization problem to only one: C.

(b) [3 points] Find the �rst order conditions associated to the previous maximizationproblem.

1

C�

(m� C)2 = 0

(c) [6 points] Determine the optimal amount of consumption goods, C�, and healthinsurance, H�.

� Rearranging,C2 � (2m+ )C +m2 = 0

with solutions

C =2m+ +

p 2 + 4m

2and C =

2m+ �p 2 + 4m

2

but given that the amount spent on consumption cannot exceed the indi-

vidual�s wealth, C � m, the only feasible solution is C� =2m+ �

p 2+4m

2.

Therefore, the optimal amount of health insurance that this individual buysis

H� = m� C� = m� 2m+ �p 2 + 4m

2=

p 2 + 4m �

2

4

(d) [6 points] Determine if the optimal amount of health insurance, H�, is increasing,decreasing, or constant in m. Interpret.

� Di¤erentiating H� with respect to m,

@H�

@m=

p + 4m

> 0 for all parameter values

That is, the optimal amount of health insurance, H�, is increasing in theindividual�s wealth level, m.

5. [15 points] [Nonconstant coe¢ cient of absolute risk aversion]. Suppose that theutility function is given by u(W ) = aW � bW 2, where a; b > 0.

(a) [2 points] First, �nd the coe¢ cient of absolute risk-aversion. Does it increases ordecreases in wealth? Interpret.

� First, note that u0 = a�2bW and u00 = �2b. Hence, the coe¢ cient of absoluterisk-aversion is

rA(x; u) = �u00(x)

u0(x)=

2b

a� 2bWNote that as W rises, the denominator decreases, and as a consequencerA(x; u) rises, i.e., the decision maker becomes more risk averse as he wealthincreases.

(b) [3 points] Let us now consider that this decision maker is deciding how much toinvest in a risky asset. This risky asset is a random variable R, with mean R > 0and variance �2R. Assuming that his initial wealth isW , state the decision maker�s(expected) utility maximization problem, and �nd �rst order conditions. [Hint:First, note that the decision maker�s wealth (W in his utility function) is now arandom variable W + xR, where x is the amount of risky asset that he acquires.You must insert this expression in the decision maker�s utility function, for everyW . Then, we must take expectations over the entire expression, since the riskyasset is a random variable.]

� The choice problem of the decision maker is

maxxE�a (W + xR)� b (W + xR)2

�And the associated �rst order condition is

E [aR� 2bR (W + x�R)] = 0

(c) [4 points] Simplify the �rst order condition you found before. [Hint: Note thatyou must use the property that E[R2] = R

2+ �2R].

� Simplifying the above �rst order condition,

E [aR� 2bR (W + x�R)] = aR� 2bRW � E�2bR2x�

�=

= aR� 2bRW � 2bx��R2+ �2R

�= 0

5

(d) [4 points] What is the optimal amount of investment in risky assets?

� Solving for x� in the above expression,

x� =(a� 2bW )R2b�R2+ �2R

�(e) [2 points] Show that the optimal amount of investment in risky assets (the expres-

sion you found in the previous part) is a decreasing function in wealth. Interpret.

� Di¤erentiating x� with respect to wealth,

@x�

@W= � R�

R2+ �2R

�which is negative, since R; �2R > 0. Intuitively, the larger the decision maker�swealth, the lower is the amount of risky assets he wants to hold. This expla-nation is consistent with his coe¢ cient of absolute risk aversion found at thebeginning of the exercise, where we showed that the individual becomes morerisk averse as his wealth increases.

6. [12 points] [Concavity and Coe¢ cient of risk aversion] Let u and v be two utilityfunctions, where v(W ) = f (u(W )), and f(�) is a concave function, i.e., v is moreconcave than u.

(a) [4 points] Find the coe¢ cient of absolute risk-aversion for v.

� First, note that v0 = f 0u0 and v00 = f 00 [u0]2+f 0u00. Therefore, the Arrow-Prattcoe¢ cient of absolute risk aversion of v is

rA(x; v) = �v00(x)

v0(x)= �f

00 [u0]2 + f 0u00

f 0u0

= �u00

u0� f

00u0

f 0

(b) [8 points] Prove that the coe¢ cient of absolute risk-aversion for v is greater thanfor u.

� We want to compare the Arrow-Pratt coe¢ cient of absolute risk aversion of v,�u00

u0 �f 00u0

f 0 , with respect to that of u, �u00

u0 . Hence, we want to know the sign

of f00u0

f 0 , First, note that u (�) is increasing in wealth, and then u0 > 0. Second,note that function f(�) is a concave, i.e., f 0 > 0 and f 00 < 0. Summarizing,the ratio f 00u0

f 0 is negative, which implies

�u00

u0� f

00u0

f 0> �u

00

u0

which implies that rA(x; v) > rA(x; u). Intuitively, the coe¢ cient of absoluterisk aversion is higher the more concave is the utility function.

6

Micro Theory I - EconS 501Final Exam - Answer Key.


(a) [3 points] Homothetic preferences have non-straight wealth expansion paths.

� False. Recall that this type of preferences induce wealth expansion pathsthat are straight lines from the origin, i.e., if we double the wealth levelof the individual, then his wealth expansion path (the line connecting hisdemanded bundles for the initial and the new wealth level) are straight lines.

(b) [7 points] Homothetic preferences induce a demand function with non-constantincome elasticity.

� False. As a corollary of the straight wealth expansion paths, we can concludethat the demand function obtained from homothetic preferences must havean income-elasticity equal to 1, i.e., when the consumer�s income is increasedby 1%, the amount he purchases of any good k must increase by 1% as well.

(c) [2 points] Consider the demand function of an individual with homothetic prefer-ences. The marginal rate of substitution resulting from this individual�s demandfunction does not vary if we increase his consumption of one of the goods

� False. When preferences are homothetic, the MRS between the two goods isjust a function of the consumption ratio between the goods, x1

x2, but it does

not depend on the absolute amounts consumed. As a consequence, if wedouble the amount of both goods, the MRS does not change.� Let us consider an example of a Cobb-Douglas utility function, u(x1; x2) =x�1x

�2 . First note that

MRS1;2 = ��x��11 x�2

�x�1x��12

scaling up all goods by a factor t, we obtain

MRS1;2 = �� (tx1)

��1 (tx2)�

� (tx1)� (tx2)

��1 = �t��1+�

t�+��1�x��11 x�2

�x�1x��12

= ��x��11 x�2

�x�1x��12

which shows that the MRS1;2 does not change when we scale up all goodsby a common factor t, i.e., the slope of the indi¤erence cuve at a given point(x1; x2) is not changed.

(d) [3 points] The Weak Axiom of Revealed Preference (WARP) states that, for anytwo price-wealth situations (p; w) and (p0; w0),

if p � x(p0; w0) � w and x(p0; w0) 6= x(p; w), then p0 � x(p; w) > w0

1

� False. WARP states that, for any two price-wealth situations (p; w) and(p0; w0),


(e) [4 points] The utility function u(x1; x2) = max fx1; x2g is quasiconcave (i.e., theupper contour set of any indi¤erence curve is convex).

� False. In order to check for quasiconcavity, we now use the de�nition: u(x1; x2) =max fx1; x2g is quasiconcave if, for every bundle (x1; x2), the set of bundles(y1; y2) such that the consumer obtains a higher utility level than from (x1; x2)is convex. That is, for every bundle (x1; x2), its upper contour set

f(y1; y2) : u(y1; y2) � u(x1; x2)g is convex

that is,f(y1; y2) : max fy1; y2g � max fx1; x2gg is convex

� As we can see from the �gure below representing this preference relation, wecan �nd bundles, like x, for which its upper contour set is not convex. Thatis,

y % x but �x+ (1� �)y � x for all � 2 [0; 1]x2

x1

x

y

That is, max fy1; y2g � max fx1; x2g (which implies y % x) [In this examplemax fy1; y2g = y1, and max fx1; x2g = x2, and y1 > x2]. However, construct-ing a linear combination of these two bundles �x+ (1� �)y we have that

max f�x1 + (1� �)y1; �x2 + (1� �)y2g < max fy1; y2g = y1

This inequality is indeed satis�ed because either:

�max f�x1 + (1� �)y1; �x2 + (1� �)y2g = �x1+(1��)y1 (i.e., if the linearcombination of x and y is below the main diagonal), then �x1+(1��)y1 �y1 for any � 2 [0; 1]; or

� If, instead, max f�x1 + (1� �)y1; �x2 + (1� �)y2g = �x2+(1��)y2 (i.e.,if the linear combination of x and y is above the main diagonal), then wealso have �x2 + (1� �)y2 � y1 for any � 2 [0; 1].

2

(f) [2 points] One of your friends studying intermediate microeconomics meets youand starts explaining how excited he is about all the concepts he is learningin his micro course. At one point he says: I particularly enjoyed the secondfundamental welfare theorem. You know, that theorem showing that any Paretooptimal allocation can be implemented by a central authority who transfers moneyamong consumers, and then allows the market work. It is fascinating that thecompetitive equilibrium resulting from allowing the market work can induce theutility levels of the Pareto optimal allocation, for all types of consumers and �rms!In other words, this theorem says that by redistributing money among people wecan achieve Pareto improvements. As a consequence, we would increase the utilityof at least some people, while not reducing the utility level of anybody else.�Yourfriend is de�nitely excited, but where is the �aw in his statement?

� False. The second fundamental welfare theorem cannot be generalized toall types of consumers and �rms. It can only be applied when consumers�preferences are convex, and �rms�production function is convex.

(g) [3 points] Consider an individual with Bernouilli utility function u(x) =px.

When facing the gamble�36; 16; 1

2:12

�his certain equivalent is c(F; u) =

p26 and

his probability premium is � =p262.

� False. The certain equivalent c (F; u) of this gamble isp36 � 1

2+p16 � 1

2= 6 � 1

2+ 4 � 1

2= 3 + 2 = 5

u (c(F; u)) = 5


which implies that the decision maker must be given an amount of moneythat provides him with a utility level of 5. Therefore, u (c(F; u)) = 5, andc (F; u) = 25.� False too. The probability premium � of this gamble is�

1

2+ �

�u (x+ ") +

�1

2� �

�u (x� ") = u (x)�

1

2+ �

�p36 +

�1

2� �

�p16 =

p26|{z}

u(EV )

since EV =1

2� 36 + 1

2� 16 = 36 + 16

2=52

2= 26

1

2

p36 +

p36� +

1

2

p16�

p16� =

p26�p

36�p16�� =

p26�

p36

2�p16

2

(6� 4)� =p26� 6

2� 42

2� =p26� 5

� =

p26� 52

� Probability Premium

3

1. h. [2 points] The independence axiom states that, for all three lotteries L, L0 andL00, and � 2 (0; 1) we have

L % L0 () �L+ (1� �)L0 % �L0 + (1� �)L00

� False. The independence axiom states that, for all three lotteries L, L0 andL00, and � 2 (0; 1) we have

L % L0 () �L+ (1� �)L00 % �L0 + (1� �)L00

i. [6 points] Consider the social welfare maximization problem

maxw1;:::;wI

W (u1 (x1) ; u2 (x2) ; :::; uI (xI))

subject to p � X

i

xi

!� w

And denote by v (p; w) the optimal solution of this problem, usually referred asthe indirect utility function. The indirect utility function v (p; w) is increasing inprices.

� False. Let us take p0 � p. Let us (w1; w2; :::; wI) be the solution to themaximization problem given (p0; w). Hence,

v (p0; w) =W (v1 (p01; w1) ; v2 (p

01; w2) ; :::; vI (p

01; wI))

As p0 � p, we have that

vi (p0; wi) � vi (p; wi) for all i

Since W (�) is increasing in the utility levels of every individual,

vi (p0; wi) � vi (p; wi)() W (vi (p

0; wi) ; :::; vI (p0; wI)) � w (vI (p; wI) ; :::; vI (p; wI))

Then, by the de�nition of v (p; w) we have,

W (v1 (p0; w1) ; v2 (p

0; w2) ; :::; vI (p0; wI)) � v (p; w)

Therefore,v (p0; w) � v (p; w) for all p0 � p

Hence, the indirect utility function v (p; w) is weakly decreasing in prices, oralternatively non-increasing in prices.

j. [5 points] If a lottery F �rst order stochastic dominates another lottery G, thenthe mean value of F must be higher than that of G, and viceversa.

� False. This statement says F %FOSD G =) E(xF ) � E(xG), and thecontrary F %FOSD G(= E(xF ) � E(xG):

4

� The �rst direction of the implication is true. Indeed, we know that distribu-tion function F (x) �rst-order stochastically dominates G(x) ifZ

u (x) dF (x) >Zu (x) dG(x)

Using the fact that the utility function is weakly increasing, and using u(x) =x, we have Z

xdF (x) >ZxdG(x)

� However, the second implication F %FOSD G(= E(xF ) � E(xG) is false. Itcan easily proved by providing any counterexample. Consider for instance theexample of a mean-preserving spread discussed in class. The mean of bothdistribution functions F (x) and G(x) was 5

2. However, neither F (x) FOSD

G(x), nor G(x) FOSD F (x), for all x.

k. [3 points] If a monopolist can perfectly identify the consumers who belong totwo di¤erent segments of the market, he will set prices to each segment such thecorresponding marginal revenue coincides with the monopolist marginal cost, forany cost structure and quantities.

� False. This pricing rule is only valid when the monopolist does not facecapacity constraints. When the monopoly faces capacity constraints, themonopolist sets the output produced in each of the N segments of the market(q1; q2; :::; qN) so that they satisfy

MRi(qi) =MRj(qj) for every segment i = f1; 2; :::; Ng , where i 6= j

.

2. [14 points] [Expected utility] Consider an individual with Bernouilli utility functionu(x) = �x2+ x. Show that his expected utility for any given lottery F (x) is determinedby the mean and variance alone. [Hint: recall that V ar(x) = E(x2)� E(x)2].

� We know that the expression of the expected utility function for any lottery F (x)is EU =

Zu(x) dF (x)

� In this case, then, EU =Z ��x2 + x

�| {z }u(x)

dF (x). And expanding it,

�

Zx2dF (x)| {z }E(x2)

+

ZxdF (x)| {z }E(x)

and on the other hand, we know that V ar(x) = E(x2)� E(x)2. Hence, E(x2) =V ar(x) + E(x)2. Substituing E(x2) in the above expression,

�V ar(x) + �E(x)2 + E(x)

and as a consequence, the EU for the Bernouilli utility function u(x) = �x2+ xis determined by the mean and the variance alone

5

3. [10 points] [Second-degree price discrimination] Let us consider two consumerswith the following quasilinear utility functions

u1(x1; y1) = �1x1 � y1

u2(x2; y2) = �2x2 � y2The price of the composite commodity y is 1, and each consumer has a large initialwealth. We know that �2 > �1. Both goods can only be consumed in weakly positiveamounts, xi � 0 and yi � 0. A monopolist supplies teh x-good. It has zero marginalcosts, but has a capacity constraint: it can supply at most 10 units of the x-good. Themonopolist will o¤er at msot two price-quantity packages (r1; x1) and (r2; x2), whereri is the cost of purchasing xi units of the good (total revenue for the monopolist fromselling xi units to consumer i).

(a) [1 point] Check if the single-crossing property is satis�ed. Interpret.

� Indeed, the single-crossing property holds, since

@u2(x; y)

@x>@u1(x; y)

@x() �2 > �1

Intuitively, this property implies that the marginal utility that consumer 2obtains from consuming additional amounts of good x is strictly higher thanthat of consumer 1, for all x (irrespective of the amount of good x they areconsuming).

(b) [1 point] Write down the monopolist�s pro�t maximization problem. You shouldhave 4 constraints (two participation constraints and two incentive compatibilityconstraints), plus the capacity constraint x1 + x2 � 10.�

maxr1;r2

r1 + r2

subject to �1x1 � r1 � 0 (PC1)

�2x2 � r2 � 0 (PC2)

�1x1 � r1 � �1x2 � r2 (IC1)

�2x2 � r2 � �2x1 � r1 (IC2)

and x1 + x2 � 10 (Capacity constraint)

(c) [2 points] Which constraints will be binding in the optimal solution?

� We know that PC1 is binding and IC2 is binding

�1x1 = r1 � PC1�2x2 � r2 = �2x1 � r1 � IC2

and that the capacity constraint is binding as well,

x1 + x2 = 10.

6

From IC2,

r2 = �2 (x2 � x1) + r1

r2 = �2

x2 � x1 +

r1z}|{�1x1

!

= �2

0@10� x1| {z }x2

� x1

1A+ �1x1 = 10�2 � 2x1�2 + �1x1Hence, IC2 can be expressed as r2 = 10�2 � 2x1�2 + �1x1

(d) [3 points] Substitute these binding constraints into the objective function. Whatis the resulting expression?

� The objective function r1 + r2 can be rewritten using PC1 and IC2 fromabove,

r1z}|{�1x1 +

r2z }| {10x2 � 2�2x1 + �1x1

Simplifying,

2�1x1 + 10�2 � 2�2x1 = 10�2 + 2 (�1 � �1)x1

(e) [3 points] What are the optimal values of the packages (r1; x1) and (r2; x2)?

� Since the objective function (simpli�ed in part c) is a function of x1 only, wejust have to take FOC with respect to x1,

2�1 � 2�2 � 0() �1 � �2

since �1 < �2 by de�nition, then 2�1 � 2�2 < 0 strictly, which implies acorner solution in the variable we were di¤erentiating, x1, where

x�1 = 0

Since x�1 = 0; then x�2 = 10:

� Additionally, from PC1,

r�1 = �1 x�1 = �1 � 0 = 0

And from IC2,r�2 = 10�2 � 2x�1�2| {z }

0

+ �1x�1|{z}

0

= 10�2

4. [17 points] [Distribution of tax burden] An ad valorem tax of � is to be levied onconsumers in a competitive market with aggregate demand curve x(p) = Ap", whereA > 0 and " < 0, and aggregate supply curve q(p) = �p , where � > 0 and > 0.Calculate the percentage change in consumer cost and producer receipts per unit soldfor a small (marginal) tax. Denote � = (1 + �). Assume that a partial equilibriumanalysis is valid.

7

(a) [10 points] Compute the elasticity of the equilibrium price with respect to �.

� To compute the price received by producers, we can use equation (10.C.8) inthe textbook:

p�0(0) = � x0 (p�)

x0 (p�)� q0 (p�)= � A"p"�1�

A"p"�1� � � p �1�= � A"p"�

A"p"� � � p �=

= � "x (p�)

"x (p�)� q (p�) = �"

"� .

(We have multiplied both the numerator and the denominator by p� and usedthe fact that p� is an equilibrium price, therefore x (p�) = q (p�).) The pricepaid by consumers is (p�) + t, and its derivative with respect to t at t = 0 is

p0 (0) + 1 = � "

"� + 1 = �

"� .

(b) [2 points] Argue that when = 0 producers bear the full e¤ect of the tax whileconsumers�total costs of purchase are una¤ected.

� From the above expression,

p0 (0) + 1 = � "

"� + 1 = �

"� .

we can see that when = 0 (supply is perfectly inelastic),

lim !0

�

"� = 0.

and the price paid by consumers is unchanged, and the price received byproducers

lim !0

� "

"� = 1.

decreases by the full amount of the tax.

(c) [3 points] Argue that when " = 0 consumers bear the full burden of the tax.

� When " = 0 (demand is perfectly inelastic),

lim"!0� "

"� = 0.

and the price received by producers is unchanged and the price paid by con-sumers

lim"!0�

"� = 1.

increases by the full amount of the tax.

(d) [2 points] What happens when each of these elastiticities approaches1 in absolutevalue?

8

� When "! �1 (demand is perfectly elastic), the price paid by consumers isunchanged,

lim"!�1

�

"� = 0.

and the price received by producers decreases by the amount of the tax. Incontrast, when !1 (supply is perfectly elastic),

lim !1

� "

"� = 0.

and the price received by producers is unchanged and the price paid by con-sumers increases by the amount of the tax.

5. [19 points] [Cost reducing investment]. Consider a situation in which there is amonopolist in a market with inverse demand function p(q). The monopolist makestwo choices: How much to invest in cost reduction, I, and how much to sell, q. If themonopolist invests I units in cost reduction, his (constant) per-unit cost of productionis c(I). Asume that c0(I) < 0 and that c00(I) > 0, i.e., investing in cost reductionreduces the monopolist�s per-unit cost of production, but at a decreasing rate. Assumethroughout that the monopolist�s objective function is concave in q and I.

(a) [7 points] Derive the �rst-order conditions for the monopolist�s choices.

� he monopolist will solve

maxq;I

p (q) � q � c (I) q � I

which yields the FOCs are

(i) p0 (qm) � qm + p (qm) = c (Im) ,(ii) �c0 (Im) qm = 1.

(b) [6 points] Compare the monopolist�s choices with those of a benevolent socialplanner who can control both q and I (a ��rst-best�comparison). Interpret yourresults.

� The social planner will maximize total surplus,

maxq;I

Z q

0

p (x) dx� c (I) q � I,

and the FOCs are,(iii) p (q�) = c (I�) ,(iv) �c0 (I�) q� = 1.

The monopolist produces less output than is socially optimal, qm < q�, andprice is above marginal cost. Given this, equations (ii) and (iv) imply that�c0 (I�) < �c0 (Im), which in turn implies that I� > Im (given that c0(I) < 0)That is, the monopolist invests less in cost-reducing technologies than thesocial planner would.

9

(c) [6 points] Compare the monopolist�s choices with those of a benevolent socialplanner who can control for I but not for q (a �second-best� comparison). Inparticular, suppose that the social planner chooses I and then the monopolistchooses q.

� Given a level bI set by the government, the monopolist will set q to maximizeits pro�ts, i.e., it will set q to equateMR =MC. Therefore, the governmentsproblem is to maximize social surplus subject to the monopolists�s behavior.That is,

maxq;I

Z q

0

p (x) dx� c (I) q � I

subject to p0 (q) � q + p (q) = c (I)The Lagrangian is

L =

Z q

0

p (x) dx� c (I) q � I � � [p0 (q) q + p (q)� c (I)] ,

which yields the FOCs,

(v) p (bq)� c�bI�� [p00 (bq) bq + 2p0 (bq)] = 0,(vi) �c0

�bI� (bq � �) = 1.When comparing (ii) and (vi) we can see that

�c0 (Im) qm = �c0�bI� (bq � �) , where � > 0

Hence, the social planner�s investment, bI is greater than the optimal invest-ment level I�, found in the previous exercise The intuition is that in thissecond-best environment, the social planner chooses an investment level bIlarger than optimal for the given level of output in order to induce the mo-nopolist to produce more.

10

Micro Theory I - EconS 501Midterm #1 - October 5th, 2011.

Instructions: Please read the questions carefully, answer them in a formal and concisemanner, but include all your steps, this will allow you to obtain partial credit. You haveuntil 5:30p.m. to complete the exam. Good luck!!

Short exercises: The following two exercises are relatively short. Please do not use morethan approximately half a page to answer each of their sections.

1. Consider the preference relation de�ned on a consumption set X � R2+: for all anytwo bundles x, x0 2 X, bundle x is weakly preferred to x0, i.e., x % x0, if and only if

x1 � x01 � 1 and x2 � x02 + 1:

Check if this preference relation satis�es completeness and transitivity.

2. Check if the following function is a proper cost function:

c(w; q) = 4w21w22q1=4;

where the vector of input prices w 2 R2++, and output level q 2 R+. [Hint: check theproperties of the cost function associated to the cost-minimization problem (CMP). Ifone of the properties is violated, then this function cannot represent a cost function.]

3. Consider a pro�t-maximization problem (PMP) that produces the following pro�t func-tion

�(p; w; r) =p2

4w+p2

4r,

where w 2 R+ denotes the wage rate, r 2 R+ represents the interest rate and p 2 R+denotes the price of the single output that the �rm produces. Obtain the expressionof its associated cost function c(w; r; q).

Long exercises:

4. Consider the utility functionu(x) =

Yn

i=1x�ii ;

where x denotes a vector of n di¤erent goods x 2 Rn+, and �i > 0. Check if u(x)satis�es: (1) additivity, (2) homegeneity of degree k, and (3) homotheticity.

5. Consider a function F (p; w) representing the inverse of the indirect utility functionv(p; w), that is

F (p; w) � 1

v(p; w)

where the indirect utility function satis�es the usual properties, and v(p; w) 6= 0.

1

(a) Use function F (p; w) to �nd the Walrasian demand of good i, i.e., xi(p; w).

(b) Let sj denote the share of income that the consumer spends on good j. Showthat sj can be expressed as follows

sj = �@F (p;w)@pj

pjF (p;w)

@F (p;w)@w

wF (p;w)

6. Consider a consumer with the following expenditure function

e(p; u0) = g(p) +�u0 � f(p)

�where functions g(p) and f(p) depend on the price vector p alone. Show that theincome elasticity of any good i converges to one when the consumer�s wealth leveltends to in�nity, i.e., lim

w!1"xi;w = 1.

7. Consider a consumer who, facing a initial price vector p0 2 Rn++, purchases an n-dimensional bundle x 2 Rn+ with an income of w dollars. Assume that the price of allgoods experience a common increase measured by factor � > 1.

(a) Compute the compensating variation (CV) of this price increase.

(b) Compute the equivalent variation (EV) of this price increase.

2

[You can use the following pages as scratch paper.]

Name: ________________________________

3

Name: ________________________________

4

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Micro Theory I - EconS 501Midterm #2 - First part, November 16th, 2011.

Instructions: Please read the questions carefully, answer them in a formal and concisemanner, but include all your steps, this will allow you to obtain partial credit. Good luck!!

1. [25 points] [Preference relations] Let % be a preference relation de�ned over theconsumption set X = RL+. We de�ne that preference relation % is monotone if x >> yimplies that x � y. In addition, we de�ne that preference relation% is weakly monotoneif x � y implies that x % y. Show that if the preference relation % is strictly convexand weakly monotone, then it is monotone.

2. [30 points] [WARP] Let (B; C(�)) be a choice structure where B includes all nonemptysubsets of X, i.e., C(B) 6= ? for all sets B 2 B. We de�ne the choice rule C(�) to bedistributive if, for any two sets B and B0 in B,

C(B) \ C(B0) 6= ? implies that C(B) \ C(B0) = C(B \B0)

In words, the elements that choice rule C(�) selects both when facing set B and whenfacing set B0, C(B) \ C(B0), coincide with the elements that choice rule would selectwhen confronted with the elements that belong to both sets B \ B0, i.e., C(B \ B0).Show that, if choice rule C(�) is distributive, then choice structure (B; C(�)) does notnecessarily satisfy the weak axiom of revealed preference. (A counterexample su¢ ces.)

3. [40 points] [Production theory] Consider an economy with only one input (labor, l)and one output (chairs, y). A �rm has access to the following three technologies (A,B and C) to produce chairs:

A. Each unit of labor leads to one chair.

B. 10 units of labor are required to buy a machine. After acquiring the machine,each extra unit of labor produces 1.5 chairs.

C. 5 units of labor are required to build a machine. After that, each extra unit oflabor leads to 2 chairs.

Assume that free disposal holds. Then,

(a) [5 points] Graph the technology set Y , with l in the horizontal axis, where negativenumbers indicate inputs, and y in the vertical axis. Show feasible combinationsof inputs and output when less than 20 workers are used.

(b) [5 points] Is Y convex? Does it satisfy non-increasing returns to scale? Does itsatisfy non-decreasing returns to scale? Does it satisfy additivity? Justify.

(c) [12 points] Let the price of output be p = 1 and the price of labor be w = 4.What is the quantity that this �rm (university) will produce? What will be the�rm�s pro�ts?

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(d) [5 points] Now, assume that the �rm can never use more than 20 units of labor.Let the price of output be p = 1 and the price of labor be w = 1. What is thequantity a �rm will produce? What will be the �rm�s pro�ts?

(e) [13 points] Now, in order to use technology C, the �rm has to pay a �xed cost ofK. Find the supply and pro�t functions for any vector of prices (p; w) and anyvalue of K. You can normalize output prices to be p = 1.

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Micro Theory I - EconS 501Midterm #2 - Second part, November 18th, 2011.


1. [20 points] [Consumer theory]Consider an individual with utlity function u(x1; x2; x3)for three goods, where the cross-price elasticities are null, i.e., "ij = 0 for any two goods

i and j. Show that the ratio of substitution e¤ects s23s13is equal to

@x2@w@x1@w

.

2. [40 points] [Uncertainty.] Given an individual with utility function u(y) = log y,where y denotes this individual�s income level,

(a) [10 points] Show that his Arrow-Pratt coe¢ cient of relative risk aversion, rR(y),is constant in income y.

(b) [15 points] This individual declares an amount of money x to the IRS, wherex � y. With probability 1 � p he is not audited and hence his income level isy � tx. With probability p, he is audited and his income decreases to

y � ty � Ft(y � x)

That is, after an audit the IRS taxes this individual for his real income y, reducinghis income in ty, but in addition, applies a �ne F for the amount of evaded taxes,t (y � x), further reducing his income in Ft (y � x). Write down the expectedutility maximization problem for this individual. Then take �rst-order conditionswith respect to this individual�s choice of declared income, x.

(c) [15 points] Using the �rst-order condition you found in part (b), show that theproportion of income this individual does not declare, x

y, is independent of his

income level y. [Hint: Since x � y, let x = �y in the �rst-order condition andshow that your results are independent on y.]

3. [40 points] [Monopoly] Consider the following two-period monopoly model: A �rmis a monopolist in a market with an inverse demand function (in each period) ofp(q) = a� bq. The cost per unit in period 1 is c1. In period 2, however, the monopolisthas �learned by doing,�and so its marginal costs decrease to c2 = c1 �mq1, where q1is the monopolist�s period 1 output level. Assume that a > c > 0 and b > m > 0. Alsoassume that the monopolist does not discount future earnings, i.e., the discount factor� = 1.

(a) [5 points] What is the monopolist�s output level in each of the periods, q1 and q2?Denote them by qM1 and qM2 .

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(b) [14 points] Consider a benevolent social planner with social welfare function Wgiven by

W = (CS1 + �1) + (CS2 + �2)

where CSt and �t represent respectively consumer surplus and pro�ts duringa given period t = f1; 2g. What output levels would be implemented by thebenevolent social planner? Denote them by qSP1 and qSP2 .

(c) [10 points] Can you interpret the choice of qSP1 as being selected according to the�price equal to marginal cost�rule?

(d) [11 points] Given that the monopolist will be selecting the period 2 output level,qM2 , would the social planner like the monopolist to slightly increase the level ofperiod 1 output above that identi�ed in part (a), qM1 ? Can you given any intuitionfor this?

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Micro Theory I - EconS 501Final Exam - December 13th 2011.

Instructions: Please read the questions carefully, answer them in a formal and concisemanner. Include all your steps, since this will allow you to obtain partial credit. You haveuntil 10:00a.m. to complete the exam. Good luck!!

1. [Consumer Theory] [20 points] The preferences of some consumer can be representedas: u(x1; x2) = min fx1; x2g. We have been informed that only the price of the good2 has changed, from p02 to p

12, but we have not informed about by how much did it

change. We know, however, that the amount of income that has to be transferred tothe consumer in order for him to recover his initial utility level is:

p02w

p01 + p02

dollars

where w is the initial income, and p01 and p02 are the initial prices of goods 1 and 2,

respectively. The above information should allow you to measure the exact size of theprice change. What is the di¤erence between p02 and p

12?

2. [Monopoly: leasing vs. selling] [27 points] Consider a two-period game where amonopolistic �rm wants to sell its durable good. The durable good will last only twoperiods, and after that it will become obsolete. There is no depreciation of the goodbetween the two periods. The discount factor � is identical for all consumers and the�rm. Demand for the ulization of the good is given by p = 1 � Q, where Q denotesthe aggregate quantity. Production is assumed to be costless. A resale market exists,since consumers who buy the good in one period might want to re-sell (or lease) it inthe second period.

(a) [14 points] Consider �rst the case where the �rm sells in each period.

1. [3 points] Starting from the second period, set up the pro�t-maximizationproblem for the monopolist, where it selects a production level q2 given ademand function p2 = 1 � q1 � q2. Determine q2, p2, and pro�ts �2 duringthis second period.

2. [8 points] Given the equilibrium price you found for the second-period monopoly,p2, the �rst-period demand is p1 = (1� q1)��p2, which intuitively representsthat the willingness to pay for the good in the �rst period is given by thecurrent value that the consumer assigns to this good (given by the demandfunction, 1�q1), plus the discounted value of the good tomorrow (which arisesif the current consumer leases the good in the second period at a price p2).Given this �rst-period demand, set up the monopolist�s pro�t-maximizationproblem, where its choice variable is now q1, and its objective function con-siders not only �rst-period but also the discounted value of second-periodpro�ts, ��2. Determine q1, p1, and overall pro�ts across both periods.

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3. [3 points] Show that equilibrium prices decline over time, i.e., p1 > p2.

(b) [8 points] Consider now that the monopolist leases (i.e., rents) the good in eachperiod, and �nd equilibrium prices and output. [Hint : when leasing its goods, themonopolist�s pro�ts on a given time period t become independent on other periodprices, pk, where k 6= t]

(c) [5 points] Find the monopolist�s equlibrium pro�ts from leasing. Are they higheror lower than the pro�t the monopolist makes from selling the good, i.e., yourresult from part a(3)?

3. [Mergers in a Cournot market] [28 points] Consider Cournot competition withn identi�cal �rms. Suppose that the inverse demand function is linear, with p(X) =a � bX, where X is total industry output, and a; b > 0. Each �rm has a linear costfunction of the form C(x) = cx, where x stands for individual output, and c denotesthe marginal cost of production, where a > c.

(a) [6 points] At the symmetric equilibrium of the Cournot model of quantity com-petition,

1. what are the industry output and the price level?2. what is the equilibrium social welfare?

(b) [12 points] Now let m 2 n �rms merge. Show that the merger is pro�table for them merged �rms if and only if it involves a pre-merger market share of 80 percent.Otherwise, the merger is unprotable.

(c) [6 points] Show that each of the remaining (n�m) nonmerged �rms is better o¤after the merger.

(d) [4 points] Show that the merger ofm �rms increases industry price and also lowersconsumer surplus.

4. [Externalities] [25 points] Consider two consumers with utility functions

uA = log(xA1 ) + xA2 �

1

2log(xB1 ) for consumer A, and

uB = log(xB1 ) + xB2 �

1

2log(xA1 ) for consumer B.

where the consumption of good 1 by individual i = fA;Bg creates a negative external-ity on individual j 6= i. For simplicity, consider that both individuals have the samewealth, m, and that the price for both goods is 1.

(a) [6 points] Equilibrium. Set up consumer A�s utility maximization problem, anddetermine his demand for goods 1 and 2, i.e., xA1 and x

A2 . Then operate similarly

to �nd consumer B�s demand for good 1 and 2, i.e., xB1 and xB2 .

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(b) [8 points] Social optimum. Calculate the social optimum amounts of xA1 , xA2 , x

B1

and xB2 , considering that the social planner maximizes a utilitarian social welfarefunction, i.e., W = uA + uB.

(c) [11 points] Restoring e¢ ciency. Show that the social optimum you found in (b)can be sustained by a tax on good 1 (so the after-tax price becomes 1 + t) withthe revenue returned equally to both consumers in a lump-sum transfer.

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EconS 501 – Microeconomic Theory I

Midterm Exam #1 – October 5th 2010

INSTRUCTIONS: Please read the questions carefully, answer them in a formal and concise manner, but

include all your steps, this will allow you to obtain partial credit. You have until 5:30p.m. to complete the

exam. Good luck!!

SHORT EXERCISES: The following two exercises are relatively short. Please do not use more than half a

page to answer each of their sections.

Exercise 1. [5 points] Show that if f: → is a strictly increasing function, and u:X→ is a utility function representing a rational preference relation , then the function v:X→ defined by v(x)=f(u(x)) is also a utility function representing the same rational preference relation .

Exercise 2. [10 points] Let us define the income-elasticity of the demand for good j as ( , )

( , )( , )

jj

j

x p w wp ww x p w

η∂

=∂

, where p denotes the price vector of J goods, w represents the individual’s

wealth level, and ( , )jx p w denotes this individual’s Walrasian demand for good j. a) [3 points] Show that if '( , ) ( , )j jp w p wη η= for any two different goods j and j’, then it must be that

'

'

( , ) ( , )j j

j j

x p w x p wp p

∂ ∂=

∂ ∂. Explain.

Prove the following two properties [3.5 points each]:

b) 1

( , )0

Jj

jjj

h p up

p=

∂=

∂∑ , where hj(p,u) denotes the Hicksian demand for good j, where j is one of J

goods in the economy, i.e., j={1,2,…,J}. c) Use your previous result to show that not all J goods can be net complements.

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LONG EXERCISES:

Exercise 3. [27 points] Consider a household that is seen to purchase quantities of just two goods, bread and cheese. Denote quantities of bread by x and quantities of cheese by y, with corresponding prices px and py. The household comprises two individuals; Andrew, whose preference relation can be represented by the utility function ( , )Au x y x= and Brenda, whose preference relation can be represented by the utility

function ( , )Bu x y y= . Denote the wealth of Andrew by wA and that of Brenda by wB.

a) [3 points] Derive the uncompensated demand functions for both Andrew and Brenda and their indirect utility functions.

b) [4 points] The households’ wealth w is divided evenly between Andrew and Brenda. Suppose that you observe the aggregate demands of this household and you interpret it as if it came from just a single consumer. Find the demands of the supposed single consumer.

Recall from the lectures that the equivalent variation of a change in prices and income from 0 0( , )p w to 1 1( , )p w can be defined as:

0 1 1 0 0 0( , ( , )) ( , ( , ))EV e p v p w e p v p w= − . If 0 1w w= and the change in prices are caused by the imposition of commodity taxes then the deadweight loss (DWL) or excess burden of the taxes is given by:

1 0

1( , )

L

l ll

DWL EV t x p w=

= − −∑ ,

where 1 0l l lt p p= − .

c) [5 points] Briefly explain why this measure may be viewed as a deadweight loss to (social) economic efficiency.

d) [6 points] Suppose that the household initially faces prices 0 (1,2)p = and has wealth 0 300w = . Then a specific tax of 2 is imposed on bread (i.e. good x) that leads to its price rising to 3 (with the price of cheese, i.e. good y, and the households’ wealth both remaining unchanged). Calculate the DWL under the false assumption that the household demands come from just one consumer.

e) [7 points] Using the individuals’ indirect utility functions derived in part (a) calculate the two individual DWLs, and A BDWL DWL . Explain why A BDWL DWL+ does or does not equal DWL.

Exercise 4. [20 points] Consider an individual with Cobb-Douglas preferences u(x1,x2)=(x1x2)0.5, where x1 and x2 denote the amounts consumed of goods 1 and 2, respectively. The prices of these goods are p1>0 and p2>0, respectively, and this individual’s wealth is w>0. The government needs to collect a large amount of money to finance a new Health Care plan, and contemplates two options:

1. Introduce an income tax equivalent to 40% of individuals’ wealth; or 2. Charge a sales tax over the price of good 1 (e.g., fuels) which would imply an increase in the

price of good 1 from p1 to p1(1+t), collecting the same dollar amount as with the income tax. Using the indirect utility function of this individual under option 1 (income tax) and option 2 (sales tax), explain which tax produces a smaller utility reduction to this individual (i.e., which tax is preferred by this individual). You can accompany your discussion with an intuitive explanation and/or a figure if necessary.

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Exercise 5. [18 points] Consider the three good setting in which the consumer has utility function

where b₁,b₂,b₃>0 represent the minimal amounts of goods 1, 2 and 3 that this individual must consume at every period in order to remain alive (e.g., calories, water and shelter).

a) [3 points] Why can you assume that α+β+γ=1 without loss of generality? Do so for the rest of the problem

b) [4 points] Write down the first-order conditions for the UMP, and derive the consumer's Walrasian demand and indirect utility functions. [This system of demands is known as the “linear expenditure system,” and it is due to Stone (1954), and the above utility function is usually referred as the Stone-Geary utility function.]

[Hint: Use another monotone transformation, v(x)=ln u(x) of the given utility function u(x)].

c) [2 points] Verify that the Walrasian demand functions x(p,w) obtained in part (b) satisfy homogeneity of degree zero.

d) [4 points] Verify that the indirect utility function v(p,w) obtained in part (b) satisfies: a. Homogeneity of degree zero, b. Increasing in wealth, c. Decreasing in prices, and d. Quasiconvex in prices.

e) [5 points] Let us now restrict our analysis to a utility function with only two goods, , where α+β=1. Are the preferences represented by this utility function

homothetic? [Hint: find the share of income spent on each good (i.e., budget shares)] Exercise 6. [20 points] A firm can produce one output q using two inputs called z1 and z2 by means of two different technologies. Technology 1 is represented by the production function q=min{z1,z2} for z1,z2≥0.

Technology 2 is represented by the production function q= 1 2

3 3z z+ for z1,z2≥0. Prices of the inputs are

w1,w2≥0. a) [4 points] Does Technology 1 exhibit constant returns to scale? What about Technology 2? b) [8 points] Derive the cost function for both of them. (Hint: You do not need to set up the

Lagrangian, using a nice picture and/or explanation is enough). c) [8 points] Suppose that w1<w2. Suppose also that the firm wants to produce some amount of output

q . For which values of w1 will the firm use technology 1, and for which values of w1 will the firm use technology 2?

Micro Theory I - EconS 501Midterm #2 (First part) - November 17th, 2010.


1. [60 points] [Production: Derived input demands] The demand for any inputultimately depends on the demand for the good that input produces. This can beshown more explicitly by deriving an entire industry�s demand for inputs. To do so, weassume that an industry produces a homogeneous good, Q, under constant returns toscale using only capital and labor. The demand function for Q is given by Q = D(P ),where P is the market price of the good being produced. Because of the constantreturns to scale assumption, P = MC = AC. Throughout the problem let C(r; w; 1)be the �rm�s unit cost function, where r > 0 denotes the price of a unit of capital andw > 0 represents the price of a unit of labor.

� [IMPORTANT : each of the following questions informs you about the result youare supposed to �nd for that question. This will allow you to move forwardfrom one question to the next, using the information provided to you in previousquestions, even if some of your previous results are not completely right.]

(a) [8 points] Explain why the total industry demands for capital and labor are givenby K = QCr and L = QCw, where Cr = @C

@rand Cw = @C

@w.

(b) [10 points] Using @K@r= QCrr +D

0C2r and@L@w= QCww +D

0C2w, prove that

Crr = �w

rCrw and Cww = �

r

wCwr

(c) [10 points] Use the results from part (b) together with the expression of theelasticity of substitution between labor and capital described in class, � = CCrw

CrCw,

to show that

@K

@r=wL

Q

�K

rC+D0K2

Q2and

@L

@w=rK

Q

�L

wC+D0L2

Q2

(d) [15 points] Convert the derivatives in part (c) into elasticities to show that

"K;r = �sL� + sK"Q;P and "L;w = �sK� + sL"Q;P

where sj is the share of input j on total cost, sL = wLQC

and sK = rKQC

(recall thatsince C denotes unit cost, QC represents total cost), and "Q;P denotes the priceelasticity of demand for the product being produced.

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(e) [17 points] Use your results in part (d) to identify the substitution and outpute¤ects in production. Discuss their sign as well as what factors (sL, sK , �, and"Q;P ) produce an increase or decrease in their relative values.

2. [40 points] [Cost reducing investment in monopoly]. Consider a situation in whichthere is a monopolist in a market with inverse demand function p(q). The monopolistmakes two choices: How much to invest in cost reduction, I, and how much to sell,q. If the monopolist invests I units in cost reduction, his (constant) per-unit cost ofproduction is c(I). Asume that c0(I) < 0 and that c00(I) > 0, i.e., investing in costreduction reduces the monopolist�s per-unit cost of production, but at a decreasingrate. Assume throughout that the monopolist�s objective function is concave in q andI.




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Micro Theory I - EconS 501Midterm #2 (2nd part) - November 19th, 2010.


1. [18 points] [Inverse elasticity rule for taxation in perfectly competitive mar-kets] Consider the utility function U = �log(x1) + �log(x2)� l and budget constraintwl = q1x1 + q2x2, where qi denotes the price that consumers pay for good i = f1; 2g,xi represents the number of units of good i, w is the wage rate per hour, and l denotesthe amount of hours that this individual works.

(a) [8 points] Show that the price elasticity of demand for both commodities is equalto �1.

(b) [10 points] Setting producer prices at p1 = p2 = 1, show that the inverse elasticityrule implies t1

t2= q1

q2.

2. [37 points] [Uncertainty about being audited by the IRS] Consider a taxpayerwith exogenous income y > 0 who faces a tax rate t, where 0 < t < 1. She is askedto report a number x to the government and pays taxes tx. If the taxpayer is honestshe will report x = y, but she may cheat by reporting a lower income 0 � x < y. Letz = y�x represent the amount by which income is understated. The government doesnot know the true income y and must enforce compliance through a system of auditsand penalties. Assume that the enforcement policy, known by the taxpayer, is to auditreports with probability p, where 0 < p < 1. Assume that p is constant and henceindependent of x. If there is an audit, we assume that the government always learnsthe true income y. If the taxpayer is caught cheating she must pay a penalty � on eachdollar of income evaded, �z, in addition to the evaded tax. Assume that the taxpayeris risk averse and maximizes expected utility.

(a) [8 points] For any z, where 0 � z � y, write the income the consumer will enjoyin each one of the two possible situations, i.e., if there is an audit and if there isnot an audit (notice that the choice variable for the consumer is z).

(b) [14 points] Calculate the minimum value of t such that she will choose to cheat[Hint: You just have to provide a condition under which z� > 0).

(c) [15 points] Assume that the consumer chooses z� > 0. Prove that the optimalvalue of z� decreases in the probability of being audited p and in the �ne �. [Hint:use the implicit function theorem].

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3. [45 points] [Regulating a natural monopoly] A water supply company provideswater to Pullman. The demand for water in Pullman is p(q) = 10 � q, and thiscompany�s costs are c(q) = 1 + 2q.

(a) [5 points] Depict in a �gure: the demand curve p(q), the associated marginalrevenue MR(q), the marginal cost of production MC(q) and the average cost ofproduction AC(q). Discuss why this situation illustrates a �natural monopoly.�

(b) [6 points] Determine the amount of water qm that this �rm will produce if leftunregulated as a monopolist. Determine the corresponding prices and pro�ts forthe �rm.

(c) [6 points] Determine the amount of water that this �rm will produce if a regulatoryagency in Pullman forces the �rm to produce an amount of output q� that solvesp(q�) =MC(q�). Determine the corresponding prices and pro�ts for the �rm.

(d) [28 points] Consider now that the regulatory agency allows the monopoly tocharge two di¤erent prices: a high price p1 for the �rst q1 units, and a low pricep(q�) for the remaining (q�� q1), i.e., the units from q1 until the output level youfound in part (b), q�. In addition, the regulatory agency imposes the conditionthat the �rm cannot make any pro�ts, � = 0, when charging these two prices.

1. [20 points] Find the value of q1, and the associated price p1. Using thisinformation, determine the number of units (q��q1) sold at a low price p(q�).

2. [8 points] Depict these two prices and quantities in a �gure, and shade thearea of bene�ts and losses for the �rm.

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Micro Theory I - EconS 501Final Exam - December 16th, 2010.


1. Exercise 1. [Consumer theory] [25 points]

An individual consumes only goods 1 and 2, and his indirect utility function, v(p1; p2; w), isgiven by the following expression

v(p1; p2; w) =w

p1 + �p2where � > 0, and p1, p2, w > 0

a. [8 points] Find this individual�s Walrasian demand for good 1, x1(p; w), and good 2,x2(p; w), where p denotes the price vector p � (p1; p2) [Hint : you should use someequivalence in order to go from indirect utility function to Walrasian demand. (Onestep)] Then, �nd the ratio

x2(p; w)

x1(p; w)

Explain the intuition behind your result.

b. [9 points] Find this individual�s Hicksian demand for good 1, h1(p; u0), and good 2,h2(p; u

0).[Hint : you should use some equivalences here as well, one to go from indi-rect utility function to expenditure function, and another one to go from expenditurefunction to hicksian demand] Then, �nd the ratio

h2(p; u0)

h1(p; u0)

Explain the intuition behind your result.

c. [8 points] Using Walrasian and Hicksian demands you found in parts (a) and (b), �ndthe Slustky equation for goods 1 and 2. Explain your result, and connect it with someof your intuitions in parts (a) and (b).

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Exercise 2. [Regulating externalities under incomplete information.] [25 points]Consider the setting studied in class where a regulator observes neither the type of the �rmemitting pollution (i.e., the realization of parameter �) nor the type of the consumers beinga¤ected by such pollution (the realization of parameter �). Suppose that the �rm�s marginalbene�t from an addition unit of pollution is

@�(h; �)

@h= � � bh+ �,

and that the marginal utility from an additional unit of pollution for the consumer is

@�(h; �)

@�= � ch+ �,

where � and � are random variables with expectation E[�] = E[�] = E[��] = 0, and alltake strictly positive values, i.e., �; � > 0. Parameters b; c and are also strictly positive byde�nition, i.e., b; c; > 0. Finally, denote E[�2] = �2� and E[�

2] = �2�.

a. [10 points] Identify the best quota bh� that a social planner selects to maximize theexpected value of aggregate surplus. (Assume that the �rm must produce an amountexactly equal to the quota.)

b. [10 points] Identify the best tax t� for this same planner.

c. [5 points] Compare the two instruments in terms of their associated deadweight loss.Two �gures are enough: one where the quota performs better and another where thetax performs better. [Note that I am not asking you to �nd the precise parameterconditions under which one instrument performs better than the other. A graphicalrepresentation is ok.]

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Exercise 3 [Tax Evasion under Uncertainty.] [25 points]Given an individual with utility function u(y) = log y, where y denotes this individual�sincome level.

a. [5 points] Show that his Arrow-Pratt coe¢ cient of relative risk aversion, rR(y), isconstant in y, where u0 and u00 denote the �rst-order (second-order) derivative of utilityfunction u(y) with respect to y, respectively. [Hint: just apply the formula of rR(y)]

b. [8 points] This individual declares an amount of money x to the IRS, where x � y.With probability 1 � p he is not audited and hence his income level is y � tx. Withprobability p, he is audited and his income decreases to

y � ty � Ft(y � x)

That is, after an audit the IRS taxes this individual for his real income y, reducinghis income in ty, but in addition, applies a �ne F for the amount of evaded taxes,t (y � x), further reducing his income in Ft (y � x). Write down the expected utilitymaximization problem for this individual. Then take �rst-order conditions with respectto this individual�s choice of declared income, x.

c. [12 points] Using the �rst-order condition you found in part (c), show that the pro-portion of income this individual does not declare, x

y, is independent of income level

y. [Hint: Since x � y, let x = �y in the �rst-order condition and show that y caneliminated, so all your results are independent on y.]

3

Exercise 4 - Mixed Oligopoly [25 points]Consider a market with one public �rm, denoted by 0, and one private �rm, denoted by 1.Both �rms produce a homogeneous good with identical and constant marginal cost c > 0per unit of output, and face the same inverse linear demand function p(X) = a � bX withaggregate output X = x0 + x1. It is assumed that a > c. The private �rm maximizes pro�t

�1 = p(X)x1 � c� x1 ,

and the public �rm maximizes a combination of social welfare and pro�ts

V0 = �W + (1� �)�0

where social welfare (W ) is given by W =

XZ0

p(y)dy � c � (x0 + x1), and pro�ts are �0 =

p(X)x0 � c� x0.Both �rms choose output as their choice variable in a simultaneous-move game (as in theCournot model of quantity competition).

a. [7 points] Calculate the best-response functions of the private �rm, x1(x2), and of thepublic �rm, x2(x1).

b. [8 points] Use a �gure of the best-response functions to illustrate how they are a¤ectedwhen � increases. [Hint : you can restrict your analysis to � = 0, � = 1=2 and � = 1]

c. [5 points] Calculate the equilibrium quantities for the private and public �rms. Findthe aggregate output in equilibrium as a function of �.

d. [5 points] Calculate the socially optimal output level (by using the marginal costpricing rule, p(X) = c) and compare it with the equilibrium outcome you obtained inpart (c).

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Micro Theory I - EconS 501Midterm #1 - October 5th, 2009.


1. [15 points] [True or false?] Identify which of the following statements are true, whichare false, and provide a very short explanation of why this is the case.

(a) [6 points] If a preference relation satis�es local non-satiation, then it must alsosatisfy monotonicity.

(b) [8 points] If two goods are gross complements, then they must be net substitutes.

2. [15 points] [Checking properties of preference relations]. Consider the followingpreference relation de�ned in X = R2+: (x1; x2) % (y1; y2) if and only if

min f3x1 + 2x2; 2x1 + 3x2g > min f3y1 + 2y2; 2y1 + 3y2g

(a) [5 points] For any given bundle (y1; y2), draw the upper contour set, the lowercontour set, and the indi¤erence set of this preference relation.

(b) [10 points] Check if this preference relation satis�es: (i) completeness, (ii) transi-tivity, (iii) monotonicity, and (iv) weak convexity.

3. [20 points] [Relationship between WARP and CLD] The �gure at the end ofthe exam illustrates the change in budget line Bp;w to Bp0;w (pivoting outwards), as aresult of the decrease in the price of good 1, maintaining both the price of good 2 andwealth constant. Then, the consumer receives a wealth compensation (changing hiswealth level from w to w0) that guarantees he can still a¤ord his initial consumptionbundle, x(p; w). Show that, if Walrasian demand satis�es the Weak Axiom of RevealedPreference (WARP), then

(a) [12 points] x(p0; w0) cannot lie on segment A, but it must lie on segment B.

(b) [8 points] What conclusions can you infer from your results in part (a) about theslope of the Walrasian demand function? What about the slope of the Hicksiandemand function?

1

4. [15 points] [Slutsky equation in labor markets]. Explain the income and substitu-tion e¤ect in the labor market. Help your discussion with a �gure, but you must relateyour �gure with the Slutsky equation in labor economics.

� [Hint : �rst, write the worker�s expenditure minimization problem, where theworker minimizes py � wz, subject to the constraint v(y; z) = v, where y isthe composite commodity, p is its price, z is the number of working hours,and w is the current wage. Then write the (general) expression of the hick-sian demands that you would obtain from this EMP, and use the duality propertyxz(w; p; e(w; p; v)) = hz(w; p; v). Then di¤erentiate both sides, use the chain rule,and the property that @e(w;p;v)

@w= �hz(w; p; v), you should obtain the expression

of the Slutsky equation for labor economics. Explain]

(a) Explain under which conditions can we observe that the uncompensated laborsupply of a certain individual is negatively sloped (the individual decreases herworking hours as the wage per hour increases), but her compensated labor supplyis positively sloped (the individual increases her working hours as the wage perhour increases).

5. [15 points]Measuring welfare changes through the expenditure function]. Aconsumer has a utility function u (x1; x2) = x

1=21 x

1=22 , where good x1 is the consumption

of alcoholic beverages, and x2 is her consumption of all other goods. The price of alcoholis p > 0, and the price of all other goods is normalized to 1.

(a) [2 points] Set the consumer�s expenditure minimization problem. Find �rst orderconditions, and �nd his optimal consumption of x1 and x2.

(b) [4 points] Substituting your results from part (a) into your objective function,�nd the expenditure function e(p1; p2; U) for this consumer.

(c) [9 points] Let us now consider a proposal to reduce the price of alcohol from p = $2to p = $1 per unit. If the current utility enjoyed by the consumer is U = 100,

1. [4 points] what is his minimum expenditure in order to reach U = 100 whenp = $2? And when p = $1? [Hint: Use the expression of the expenditurefunction e(p1; p2; U) you found in part (b)]

2. [5 points] what is then the maximum amount that this consumer would bewilling to pay for this price reduction?

6. [20 points] [Measuring welfare changes when preferences are quasilinear] Showthat the compensating and the equivalent variation coincide when the utility function isquasilinear with respect to the �rst good (and we �x p1 = 1). [Hint: use the de�nitionsof the compensating and equivalent variations in terms of the expenditure function(not the hicksian demand). In addition, recall that if u(x) is quasilinear, then we canexpress it as u(x) = x1 + � (x�1), and rearranging x1 = u(x) � � (x�1) where x�1represents all the reamining goods, l = 2; 3; :::; L.]

(a) Explain why the compensating and equivalent variations coincide under theseconditions.

2

Figure - Exercise 2

x1

2

wp

x2

1

wp 1'

wp

Bp’,w (uncompensated)

Bp,w

Bp’,w’(compensated)

1

''

wp

Compensation in wealth,from w to w’

x(p,w)

2

'wp

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Micro Theory I - EconS 501Midterm #2 - November 16th, 2009.


1. [25 points] [Short questions].

(a) [4 points] The betweeness axiom implies the independence axiom. True or false?

(b) [5 points] A production function q = f(z1; z2) satisfying constant returns to scale,does not need to be homothetic. True or false?

(c) [6 points] Consider the �extreme preference for certainty�preference relation overlotteries we discussed in class, where an individual weakly prefers lottery L =(p1; p2; :::; pN) to lottery L0 = (p01; p

02; :::; p

0N), L % L0, if and only if

maxn2N

pn > maxn2N

p0n

A friend (especially interested in macro) asks you the following: �I am interested inusing this type of preference relation for my research. In particular, I want to testif people select their investment portfolio nowadays according to this preferencerelation. Before proceeding, however, I must be sure that I can use the standardvNM expected utility functions in my research. I read section 6.B in MWGagain and I found that, according to the expected utility theorem, I need thispreference relation over lotteries to satisfy the independence axiom in order tobe able to use vNM expected utility functions. Can you please let me know ifthis particular preference relation satis�es the independence axiom?�You havegood and bad news for your friend: the good news is that he asked you (almostan expert in micro by now!) before starting his research using that particularpreference relation over lotteries. The bad news is that this preference relationdoes not satisfy the independence axiom. Justify.

(d) [4 points] Another crazy friend knocks on your door: this time he is an econome-trician! It seems that he received very good comments about you from your macrofriend, and he is now coming to you for a question about micro. He is consideringthe following Walrasian market demands for goods 1 and 2, respectively.

x1 = a� bp1 � cp2 and x2 = d� ep1 � gp2

According to his data, the market equilibrium price vector at year 2000 wasp0 = (15; 3), while that in year 2008 was p1 = (17; 5). He wants to measure thechange in consumer welfare due to this price change using the equivalent variation.He �rst found the equivalent variation from the change in both goods�prices, by

1

following the path (15; 3) ! (17; 3) ! (17; 5). He then computed the equivalentvariation again, but using the path (15; 3) ! (15; 5) ! (17; 5). He realizedthat the equivalent variation he found using the �rst path does not necessarilycoincide with that using the second path. Which assumption can you recommendhim in order to guarantee that the computation of the equivalent variation isindependent on the path he uses to raise prices, i.e., that the equivalent variationis �path independent�?

(e) [6 points] You are grading a term paper for a (bad) undergraduate student atWSU, and read the following statement: �The inverse elasticity rule used by taxauthorities to determine the proportional rate of tax on a given good i, ti

pi+ti, does

not have any relationship with the deadweight loss of taxation of the tax on goodi.�Explain why this statement is wrong.

2. [8 points] Consider a production function q = f(z1; :::; zn) satisfying Constant Returnsto Scale (CRS). Show that if the �rm pays each input according to its exact marginalproductivity, then pro�ts are equal to zero.

3. [15 points] The preferences of some consumer can be represented as: u(x1; x2) =min fx1; x2g. We have been informed that only the price of the good 2 has changed,from p02 to p

12, but we have not informed about by how much did it change. We know,

however, that the amount of income that has to be transferred to the consumer inorder for him to recover his initial utility level is:

p02w

p01 + p02

dollars

where w is the initial income, and p01 and p02 are the initial prices of goods 1 and 2,

respectively. Can you provide some information about the size of the price change, i.e.,the di¤erence between p02 and p

12?

4. [15 points] Consider the following pro�t function that has been obtained form a tech-nology that uses a single input:

�(p; w) = p2w�

where p is the output price, w is the input price and � is a parameter value.

(a) [10 points] Check if the pro�t function satis�es: (1) non-decreasing in output pricep, (2) non-decreasing in input prices w, (3) homogeneous of degree one, (4) convexin prices p and w. In particular, determine for which values of � these propertiesare satis�ed (some properties might be satis�ed for all values of �, while othersmight hold only for certain values of �).

(b) [5 points] Calculate the supply function of the product, q(p; w), and the demandfor inputs, z(p; w).

2

5. [25 points] Let the consumer have the utility function U = x�11 + x

�22 � l, where x1

and x2 are consumption goods, and l are labor hours (which creates a disutility to theindividual).

(a) [5 points] Show that the utility maximizing demands are x1 =h�1wq1

i1=[1��1]and

x2 =h�2wq2

i1=[1��2].

(b) [7 points] Letting p1 = p2 = 1, use the inverse elasticity rule to show that the

optimal tax rates are related by 1t2=h�2��11��2

i+h1��11��2

i1t1.

(c) [6 points] Setting w = 100, �1 = 0:75, �2 = 0:5, �nd the tax rates required toachieve revenue of R = 10 and R = 300.

(d) [7 points] Calculate the proportional reduction in demand for the two goods com-paring the no-tax position with the position after introduction of the optimaltaxes for both revenue levels. Comment on the results.


u(C;H) = lnC � �

H

where C is his expenditure in consumption goods and H is his expenditure on healthinsurance. Parameter � denotes his losses if he becomes sick, where for simplicity

� =







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1. [10 points] [Gorman form for vi(p; wi)] Prove that if the indirect utility functionvi(p; wi) admits the Gorman form (i.e., it can be represented as a linear combinationof the individuals�wealth, as follows),

vi(p; wi) = ai(p) + b(p)wi

then all consumers exhibit parallel, straight wealth expansion paths at any price vectorp. [Hint: Use Roy�s identity].

(a) Show also that, if preferences admit Gorman-form indirect utility functions, vi(p; wi) =ai(p) + b(p)wi, with the same b(p) for all individuals, then preferences admit ex-penditure function, ei(p; ui), of the form

ei(p; ui) = c(p)ui + di(p)

2. [15 points] [Concavity and Coe¢ cient of risk aversion] Let u and v be two utilityfunctions, where v(W ) = f (u(W )), and f(�) is a concave function, i.e. v is moreconcave than u.



1

3. [20 points] [�Learning by doing�in monopoly] Consider the following two-periodmodel: A �rm is a monopolist in a market with an inverse demand function (in eachperiod) of p(q) = a � bq. The cost per unit in period 1 is c1. In period 2, however,the monopolist has �learned by doing,� and so its constant cost per unit of outputis c2 = c1 � mq1, where q1 is the monopolist�s period 1 output level. Assume thata > c > 0 and b > m > 0. Also assume that the monopolist does not discount futureearnings.

(a) [5 points] What is the monpolist�s level of output in each of the periods, q1 andq2? Denote them by qM1 and qM2 .

(b) [5 points] Consider a benevolent social planner with social welfare function Wgiven by

W = (CS1 + �1) + (CS2 + �2)

where CSt and �t represent respectively consumer surplus and pro�ts duringa given period t = f1; 2g. What output levels would be implemented by thebenevolent social planner? Denote them by qSP1 and qSP2 .

(c) [2 points] Is there any sense in which qSP1 is selected according to the �price equalto marginal cost�rule?

(d) [8 points] Given that the monopolist will be selecting the period 2 output level,qM2 , would the social planner like the monopolist to slightly increase the level ofperiod 1 output above that identi�ed in part (a), qM1 ? Can you given any intuitionfor this?

4. [20 points] [The problem of the commons] Lake Ec can be freely accessed by�shermen. The cost of sending a boat out on the lake is r > 0. When b boats are sentout onto the lake, f(b) �sh are caught in total (so each boat catches f(b)

b�sh), where

f 0(b) > 0 and f 00(b) < 0 at all b � 0. The price of �sh is p � 0, which is una¤ected bythe level of catch from Lake Ec.

(a) [5 points] Characterize the equilibrium number of boats that are sent out on thelake.

(b) [5 points] Characterize the optimal number of boats that should be sent out onthe lake.

(c) [5 points] Compare your answers in parts (a) and (b). Explain.

(d) [5 points] Suppose that the lake is instead owned by a single individual who canchoose how many boats to send out. What number of boats would this ownerchoose?

2

5. [35 points] [Private contributions to a public good] Take an economy with 2consumers i = A; B, 1 private good x, and 1 public good G. Let each consumer havean income of M . The prices of public and private good are both 1. Let the consumershave a utility functions:

UA = log(xA) + log(G), for individual A, and

UB = log(xB) + log(G), for individual B

Assume that the public good is privately provided, so total contributions to the publicgood are G = gA + gB. Note that you can eliminate xi from the utility function usingthe budget constraint M = xi + gi.

(a) [7 points] Consider individual A choosing his contribution to the public good gA

to maximize utility. Show that the optimal choice satis�es:

gA =M

2� g

B

2

(b) [1 points] Repeat part (a) for consumer B.

(c) [5 points] Find the competitive (Nash) equililibrium contributions to the publicgood by consumer A and B.

(d) [8 points] Show that along an indi¤erence curve the following property must besatis�ed:

dgA�

1

gA + gB� 1

M � gA

�+ dgB

�1

gA + gB

�= 0

and hence that:dgB

dgA=gA + gB

M � gA � 1:

(e) [7 points] Solve the last equation to �nd the locus of points along which the indif-ference curve of individual A is horizontal and use this to sketch the indi¤erencecurves of the individual A.

(f) [7 points] Calculate the Pareto e¢ cient level of private provision for the welfarefunction W = UA +UB. Contrast this with the private provision level you foundin section (c).

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Micro Theory I - EconS 501Midterm #1 - October 2nd, 2008.


1. [20 points] [Checking properties of preference relations]. Let us consider thefollowing preference relations de�ned in X = R2+. First, de�ne the upper countourset, the lower contour set and the indi¤erence set for every preference relation. Then,check if they satisfy: (i) completeness, (ii) transitivity, (iii) monotonicity, and (v) weakconvexity. [Answer only one of the following 2 questions].

(a) [20 points] (x1; x2) % (y1; y2) if and only if x1 � y1 � 1 and x2 � y2 + 1.(b) [20 points] (x1; x2) % (y1; y2) if and only if max fx1; x2g > max fy1; y2g.

2. [15 points] [Lexicographic preference relations are rational]. Let us de�ne alexicographic preference relation in a continuouous consumption set X � Y , where forsimplicity both X = [0; 1] and Y = [0; 1], as follows:

(x1; x2) % (y1; y2) if and only if�

x1 > y1, or ifx1 = y1 and x2 > y2

� Show that % is a rational preference relation (i.e., it is complete and transitive).

3. [15 points] [CheckingWARP].Check whether the following demand function satis�esthe weak axiom of revealed preference (WARP). You can use �gures to help yourdiscussion, but your �nal reasoning must be in terms of the de�nition of the WARP:

� �Average demand�: The consumer�s walrasian demand is the expected value ofa uniform randomization over all points on her budget frontier, for any (strictlypositive) prices p1, p2 and wealth w.

4. [5 points] [Concavity of the support function] We know that, given a non-empty,closed set K, its support function, �K (p), is de�ned by

�K (p) = inf fp � xg for all x 2 K and p 2 RL

Hence, the value of this support function, �K , satis�es �K � p � x for every element xin the set K. Given this de�nition, prove the concavity of the support function. Thatis, show that

�K (�p+ (1� �) p0) > ��K (p) + (1� �)�K (p0)for every p; p0 2 RL and for any � 2 [0; 1].

1

5. [25 points] [Compensating and Equivalent variation] An individual consumesonly good 1 and 2, and his preferences over these two goods can be represented by theutility function

u(x1; x2) = x�1x

�2 where �; � > 0 and �+ � ? 1


(a) [1 point] Find the Walrasian demand for goods 1 and 2 of this individual, x1(p; w)and x2(p; w).

(b) [1 point] Find his indirect utility function, and denote it as v(p0; w).

(c) [1 point] The �rm that this individual works for is considering moving its o¢ ce toa di¤erent city, where good 1 has the same price, but good 2 is twice as expensive,i.e., the new price vector is p0 = (p1; 2p2). Find the value of the indirect utilityfunction in the new location, i.e., when the price vector is p0 = (p1; 2p2). Let usdenote this indirect utility function v(p0; w).

(d) [4 points] This individual�s expenditure function is

e(p; u) = (�+ �)�p1�

� ��+�

�p2�

� ��+�

u1

�+�






(e) [4 points] Find this individual�s equivalent variation due to the price change.Explain how your result can be related with this statement from the individual tothe media: �I really prefer to stay in this city. In fact, I would accept a reductionin my wealth if I could keep working for the �rm staying in this city, instead ofmoving to the new location�

(f) [4 points] Find this individual�s compensating variation due to the price change.Explain how your result can be related with this statement from the individualto the media: �I really prefer to stay in this city. The only way I would accept tomove to the new location is if the �rm raises my salary�.

(g) [4 points] Find this individual�s variation in his consumer surplus (also referredas area variation). Explain.

(h) [4 points] Which of the previous welfare measures in questions (e), (f) and (g)coincide? Which of them do not coincide? Explain.

2

(i) [2 points] Consider how the welfare measures from questions (e), (f) and (g) wouldbe modi�ed if this individual�s preferences were represented, instead, by the utilityfunction v(x1; x2) = � lnx1 + � lnx2:

6. [10 points] [Slutsky equation in labor markets]. Explain the income and substi-tution e¤ect in the labor market. Help your discussion with a �gure, but you mustrelate your �gure with the Slutsky equation in labor economics. [Hint: �rst, writethe worker�s expenditure minimization problem, where the worker minimizes py�wz,subject to the constraint v(y; z) = v, where y is the composite commodity, p is itsprice, z is the number of working hours, and w is the current wage. Then write the(general) expression of the hicksian demands that you would obtain from this EMP,and use the duality property xz(w; p; e(w; p; v)) = hz(w; p; v). Then di¤erentiate bothsides, use the chain rule, and the property that @e(w;p;v)

@w= �hz(w; p; v), you should

obtain the expression of the Slutsky equation for labor economics. Explain]

7. [10 points] [Aggregate demand]. Answer only one of the following 2 questions:

(a) [10 points] We know that aggregate demand can be expressed as a function ofaggregate wealth, i.e.,

IXi=1

xi(p; wi) = x

p;

IXi=1

wi

!

if the following condition is satis�ed for any two individuals i and j, for a givengood k, and for any wealth of these two individuals, wi and wj.

@xki(p; wi)

@wi=@xkj(p; wj)

@wj

Explain what this condition implies in terms of these individuals�wealth expansionpaths (you can use a �gure to help your discussion). Can you give an example ofa preference relation satisfying this condition?

(b) [10 points] Show that if an individual�s preference relation is homothetic, thenthis individual�s Walrasian demand satis�es the Uncompensated Law of Demand(ULD). [Hint: instead of showing ULD, you can alternatively show thatDpxi(p; wi)is negative semide�nite, since we know that both properties are equivalent. Inorder to show the latter, �rst use the Slustsky equation, then use homotheticity,and �nally pre- and post-multiply all elements by dp]

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Micro Theory I - EconS 501Midterm #2 - November 18th, 2008.



(a) [3 points] All preference relations are rational.

(b) [3 points] If a preference relation is rational (satis�es completeness and transitiv-ity), it can be represented by a utility function.

(c) [3 points] If a preference relation is quasilinear, the substitution e¤ect is zero, andthe income e¤ect is positive.

(d) [3 points] Gi¤en goods does not need to be inferior.

(e) [3 points] The Walrasian demand is negatively sloped, for any preferences of theconsumer.

(f) [3 points] The area variation (change in consumer surplus) is never a good ap-proximation of the change in consumer welfare resulting from price changes (orequivalently, from tax changes), for any type of consumer preferences.

(g) [7 points] If a production function satis�es increasing average product, it mustalso satisfy increasing marginal product.

2. [20 points] [Properties of the pro�t function] The pro�t function, �(p), is de�nedas

�(p) = max fp � y j y 2 Y gor alternatively, �(p) > p � y for every y 2 Y .

(a) [10 points] Show that the pro�t function �(p) is convex in prices.

(b) [10 points] Prove the Hotelling�s lemma using the Duality theorem. [Hint: easy,just rewrite]

3. [8 points] [Independence axiom and convexity]. Show that the independenceaxiom implies convexity, i.e., for three di¤erent lotteries L, L0 and L00, if L % L0 andL % L00�then L % �L0 + (1� �)L00:

1


u(C;H) = lnC � �

H

where C is his expenditure in consumption goods and H is his expenditure on healthinsurance. Parameter � denotes his losses if he becomes sick, where for simplicity

� =







5. [17 points] [Nonconstant coe¢ cient of absolute risk aversion]. Suppose that theutility function is given by u(W ) = aW � bW 2, where a; b > 0.

(a) [2 points] First, �nd the coe¢ cient of absolute risk-aversion. Does it increases ordecreases in wealth? Interpret.

(b) [3 points] Let us now consider that this decision maker is deciding how much toinvest in a risky asset. This risky asset is a random variable R, with mean R > 0and variance �2R. Assuming that his initial wealth isW , state the decision maker�s(expected) utility maximization problem, and �nd �rst order conditions. [Hint:First, note that the decision maker�s wealth (W in his utility function) is now arandom variable W + xR, where x is the amount of risky asset that he acquires.You must insert this expression in the decision maker�s utility function, for everyW . Then, we must take expectations over the entire expression, since the riskyasset is a random variable.]

(c) [4 points] Simplify the �rst order condition you found before. [Hint: Note thatyou must use the property that E[R2] = R + �2R].

(d) [4 points] What is the optimal amount of investment in risky assets?

(e) [2 points] Show that the optimal amount of investment in risky assets (the expres-sion you found in the previous part) is a decreasing function in wealth. Interpret.

2

6. [12 points] [Concavity and Coe¢ cient of risk aversion] Let u and v be two utilityfunctions, where v(W ) = f (u(W )), and f(�) is a concave function, i.e., v is moreconcave than u.



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(a) [3 points] Homothetic preferences have non-straight wealth expansion paths.

(b) [7 points] Homothetic preferences induce a demand function with non-constantincome elasticity.

(c) [2 points] Consider the demand function of an individual with homothetic prefer-ences. The marginal rate of substitution resulting from this individual�s demandfunction does not vary if we increase his consumption of one of the goods.

(d) [3 points] The Weak Axiom of Revealed Preference (WARP) states that, for anytwo price-wealth situations (p; w) and (p0; w0),


(e) [4 points] The utility function u(x1; x2) = max fx1; x2g is quasiconcave (i.e., theupper contour set of any indi¤erence curve is convex). [Hint: draw some indi¤er-ence curves]

(f) [2 points] One of your friends studying intermediate microeconomics meets youand starts explaining how excited he is about all the concepts he is learningin his micro course. At one point he says: �I particularly enjoyed the secondfundamental welfare theorem. You know, that theorem showing that any Paretooptimal allocation can be implemented by a central authority who transfers moneyamong consumers, and then allows the market work. It is fascinating that thecompetitive equilibrium resulting from allowing the market work can induce theutility levels of the Pareto optimal allocation, for all types of consumers and �rms!In other words, this theorem says that by redistributing money among people wecan achieve Pareto improvements. As a consequence, we would increase the utilityof at least some people, while not reducing the utility level of anybody else.�Yourfriend is de�nitely excited, but where is the �aw in his statement?

(g) [3 points] Consider an individual with Bernouilli utility function u(x) =px.

When facing the gamble�36; 16; 1

2:12

�his certain equivalent is c(F; u) =

p26 and

his probability premium is � =p262.

(h) [2 points] The independence axiom states that, for all three lotteries L, L0 andL00, and � 2 (0; 1) we have

L % L0 () �L+ (1� �)L0 % �L0 + (1� �)L00

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(i) [6 points] Consider the social welfare maximization problem:

maxw1;:::;wI

W (u1 (x1) ; u2 (x2) ; :::; uI (xI))

subject to p � X

i

xi

!� w

And denote by v (p; w) the optimal solution of this problem, usually referred asthe indirect utility function. The indirect utility function v (p; w) is increasing inprices..

(j) [5 points] If a lottery F �rst order stochastic dominates another lottery G, thenthe mean value of F must be higher than that of G, and viceversa.

(k) [3 points] If a monopolist can perfectly identify the consumers who belong totwo di¤erent segments of the market, he will set prices to each segment such thecorresponding marginal revenue coincides with the monopolist marginal cost, forany cost structure and quantities.

2. [14 points] [Expected utility] Consider an individual with Bernouilli utility functionu(x) = �x2+ x. Show that his expected utility for any given lottery F (x) is determinedby the mean and variance alone. [Hint: recall that V ar(x) = E(x2) + E(x)2].

3. [10 points] [Second-degree price discrimination] Let us consider two consumerswith the following quasilinear utility functions

u1(x1; y1) = �1x1 � y1

u2(x2; y2) = �2x2 � y2The price of the composite commodity y is 1, and each consumer has a large initialwealth. We know that �2 > �1. Both goods can only be consumed in weakly positiveamounts, xi � 0 and yi � 0. A monopolist supplies teh x-good. It has zero marginalcosts, but has a capacity constraint: it can supply at most 10 units of the x-good. Themonopolist will o¤er at msot two price-quantity packages (r1; x1) and (r2; x2), whereri is the cost of purchasing xi units of the good (total revenue for the monopolist fromselling xi units to consumer i).

(a) [1 points] Check if the single-crossing property is satis�ed. Interpret.

(b) [1 points] Write down the monopolist�s pro�t maximization problem.

(c) [2 points] Which constraints will be binding in the optimal solution?

(d) [3 points] Substitute these binding constraints into the objective function. Whatis the resulting expression?

(e) [3 points] What are the optimal values of the packages (r1; x1) and (r2; x2)?

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4. [17 points] [Distribution of tax burden] An ad valorem tax of � is to be levied onconsumers in a competitive market with aggregate demand curve x(p) = Ap", whereA > 0 and " < 0, and aggregate supply curve q(p) = �p , where � > 0 and > 0.Calculate the percentage change in consumer cost and producer receipts per unit soldfor a small (marginal) tax. Denote � = (1 + �). Assume that a partial equilibriumanalysis is valid.

(a) [10 points] Compute the elasticity of the equilibrium price with respect to �.

(b) [2 points] Argue that when = 0 producers bear the full e¤ect of the tax whileconsumers�total costs of purchase are una¤ected.

(c) [3 points] Argue that when " = 0 consumers bear the full burden of the tax.

(d) [2 points] What happens when each of these elastiticities approaches1 in absolutevalue?

5. [19 points] [Cost reducing investment]. Consider a situation in which there is amonopolist in a market with inverse demand function p(q). The monopolist makestwo choices: How much to invest in cost reduction, I, and how much to sell, q. If themonopolist invests I units in cost reduction, his (constant) per-unit cost of productionis c(I). Asume that c0(I) < 0 and that c00(I) > 0, i.e., investing in cost reductionreduces the monopolist�s per-unit cost of production, but at a decreasing rate. Assumethroughout that the monopolist�s objective function is concave in q and I.




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Practice Exercises Advanced Micro

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