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Lecture Slide 1
Econ 310
Princeton University
February 2012
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about the class
I 10 problem sets: do them and understand them.
I work them out by yourself; go over the answers to make sure
you understand them.
I read material before class, even if it is difficult to understand
I check blackboard regularly
I you are graded on the best 8 problem sets.
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what is (micro)economics
build models consisting oftrade-off talking rational economic agents(totreps)1
I totrep has preferences: for every pair of alternatives
(consumption goods, jobs,...), the preference tells us what totrep
would choose.I totreps preferences are rational that means there are no
inconsistencies.
I totrep always chooses his most preferred alternative among the
available options.
use these models to analyze economic institutions and policies.
1This term is taken from David Kreps, Notes on the theory of choice."
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Plan
I analyze totreps behavior.
I optimal consumption choices of a consumer.I optimal price and output decisions of a monopolist.I ...
I analyze the interaction of totreps (equilibrium analysis.)
I what happens when totreps interact in a competitive
market?I what happens when one totrep sells a used car to another?I
what happens when oligopolistic firms run by totrepscompete?
I ...
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Purpose:
I Develop an understanding of how economic institutions
(regulations, taxes, etc.) affect economic outcomes.
I When an economist says: this is what happens when we
institute a minimum wage, she really means: this is what
happens to the choices of a population of totreps if we institute a
minimum wage.
I Obvious question: are the choices of totreps a good
approximation of what really happens?
I Why this approach?
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Part 1: preference and utility
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Outline:
I define preferences
I introduce utility functions
I what does it mean for a utility function to represent a
preference?
I book: this part is a bit different from the book; read chapter 3.
Key Takeaway: if totreps preferences have no inconsistencies then we candescribe them with a utility function.
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preferences
I there is a (finite) set of objects X = {x,y, z,...}
I for example: X could be the occupations, or consumption plans.
I totrep makes pairwise comparisons, expressing a preference for
some objects over others.
The symbol denotes a binary relation on the set of choices X:
interpret x y as x is weakly preferred to y.
totrep is indifferent between x and y ifx y and y x: we write
x y.
totrep strictly prefers x over y ifx y but we dont have y x: we
write x y.
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rationality
Assumption 1: The preference is complete: for all x,y in the set X we
have x y or y x (possibly both).
Assumption 2: The preference is transitive: x y and y z implies
x z.
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utility
A utility function is a convenient way to describe totreps preferences.
A utility function u associates to each object in X a real number u(x).
The utility function u represents the preference if
x y implies u(x) u(y) and u(x) u(y) implies x y
TheoremThe preference can be represented by a utility function if and only if it
satisfies Assumptions 1 and 2.
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example
X = {MD,JD,MBA, PhD}
I MD JD;MD MBA;MD PhD
I JD MBA;JD PhD
I MBA JD;MBA PhD
complete? transitive?
MD JD MBA PhD
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example continued...
We have a complete and transitive preference
MD JD MBA PhD
Two (of many) possible utility functions:
I Utility function #1:
u1(MD) = 1, u1(JD) = u1(MBA) = 0, u1(PhD) = -.01
I Utility function #2:
u2(MD) = 1, u2(JD) = u2(MBA) = .99, u2(PhD) = -1000
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utility
TheoremThe preference can be represented by a utility function if and only if itsatisfies Assumptions 1 and 2.
We will prove this theorem. It has two parts:
I Only if part: If the preference can be represented by a utility
function, then the preference must be complete and transitive.
We will do this first (very easy).
I If part: If the preference is complete and transitive then we can
find a utility function that represents it. This is the more difficultpart. We will construct a particular utility function for the
preference that does the job.
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Only if part
I
We have a utility function u that assigns each element ofX a realnumber.
I The utility function represents a preference . That is: x y if
and only ifu(x) u(y).
First, we prove that the preference must be complete:
I take x,y then we have u(x) u(y) or u(y) u(x).
I In the first case, we have x y since u represents the preference.
I In the second case we have y x since u represents thepreference.
I So, must be complete. Easy!
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Only if part continued
Next, we prove that the preference must be transitive:
I take x,y, z with x y and y z. We must prove that x z.
I Since u represents the preference we know that u(x) u(y) and
u(y) u(z)
I But this implies that u(x) u(z).
I Since u represents the preference this, in turn, implies x z.
I Therefore, is transitive. Done!
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an intermediate step
if the preference is transitive then x z and y x imply y z.
I first, we show that y z: since x z we know that x z and
therefore transitivity implies y z.
I next, we show that we cannot have z y: ifz y then y x
and transitivity would imply z x. But x z implies that we
dont have z x and therefore z y is impossible.
I so: y z and we dont have z y. This means y z. Done!
f
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If partNow, we prove that a complete and transitive preference can be
represented by a utility function.
Define Z(x) to be all those elements ofX that are strictly worse than x
Z(x) = {z 2 X|x z}
Here is what we know about Z(x):
(i) Ifx y then Z(x) = Z(y). Why? ifx z then, since y x itfollows that y x. The intermediate step now implies that
y z. Conversely, ify z it follows that x z
(ii) Ifx y then Z(y) is a subset ofZ(x): every element ofZ(y) is in
Z(x). Why? ify z and x y then it follows from theintermediate step that x z.
(iii) Ifx y then Z(y) is a strict subset ofZ(x): at least one elementofZ(x) is not in Z(y). Why? y is in Z(x) because x y. But y isnot in Z(y).
if
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...if part...
Here is our utility function:
u(x) = number of elements in Z(x)
utility ofx: is the number of choices that are worse than x.
We must prove that for this choice ofu:
(i) x y implies u(x) u(y)
(ii) u(x) u(y) implies x y.
if
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...if part...
(i) x y implies u(x) u(y)
I either x y or x y.
I
ifx
y then (as we have shown) Z(x) = Z(y). But that meansu(x) = u(y). Good!
I ifx y then (as we have shown) Z(y) is a strict subset ofZ(x).But that means u(x) > u(y). Good!
if t
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...if part....
(ii) u(x) u(y) implies x y.
I we are trying to prove that u(x) u(y) implies x y.
I which is the same as: ifx is not preferred to y then u(x) is notgreater or equal to u(y).
I which is the same as: ifx is not preferred to y then u(y) > u(x)
I by completeness: ifx is not preferred to y then y x
I So, to complete the proof we must show that y x implies
u(y) > u(x).
I But we have already done that on the previous page! Done!
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Part 2: consumers
tli
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outline
I introduce the framework of consumer choice
I describe preferences by indifference curves
I compute marginal rates of substitution
I book: we follow the book. read chapters 3 and 4 in Varian.
consumption vectors
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consumption vectors
I in consumer theory, the alternatives are consumption vectors.
I 2 goods; a vector (x1, x2) 2 R2 specifies the quantities of good 1
and good 2.
I we call totrep a consumer if his preferences are over consumption
vectors
I set of alternatives: all non-negative vectors in R2.
utility functions
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utility functions
Maintained assumption: the consumer has complete and transitive
preferences on the non-negative vectors in R2 that can be represented
by a utility function.
The utility function u represents the consumers preference if
(x1, x2) (x01, x
02) implies u(x1, x2) u(x
01, x
02)
and, conversely,
u(x1, x2) u(x01, x
02) implies (x1, x2) (x
01, x
02)
transformations of utility functions
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transformations of utility functions
Fact: ifu represents a preference then so does any strictly increasing
function ofu
Why? Iff is a strictly increasing function then
u(x1, x2) u(x01, x
02) if and only iff(u(x1, x2)) f(u(x
01, x
02))
for example:
I u and 2 u represent the same preference
I u and u3 represent the same preference
I Ifu is always positive then u and ln u represent the same
preference
indifference curves: an example
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indifference curves: an example
utility function is
u(x1, x2) = x1 x2
Indifference curve: all the combinations ofx1 and x2 that give the
same utility:
u(x1, x2) = x1 x2 = utility level k
Therefore
x2 = g(x1) =k
x1
indifference curves are a family of hyperbolas
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indifference curves
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indifference curves
I for each consumption vector (x1, x2) the indifference curvethrough (x1, x2) divides R
2+ into those consumptions that are
better than (x1, x2) and those that are worse than (x1, x2).
averages are better
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averages are better
I the indifference curves in our
example have a particular shape: the
average of two points on the
indifference curve is better than the
original points.
I We say that the consumerspreferences are convex: the set of
vectors better than (x1, x2) is aconvex set.
I Can you think of situations where
convexity makes sense?
I Can you think of situations where it
does not make sense?
marginal rate of substitution
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marginal rate of substitution
I the marginal rate of substitution is the slope of the indifference
curve
I the slope of a curve is the derivative of the function thatdescribes the indifference curve.
marginal rate of substitution in the example
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marginal rate of substitution in the example
Recall
I u(x1, x2) = x1 x2.
I the function g : R+ ! R+ where g(x1) = k/x1 describes theindifference curve
the derivative ofg is
g 0(x1) = -k(x1)2
since k is the utility level, we have
k = x1 x2
substitute for k:
-k
(x1)2= -
x1 x2(x1)2
= -x2x1
= MRS
MRS: an alternative derivation
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MRS: an alternative derivation
I for u(x1, x2) = x1
x2 we found that
MRS = -x2x1
I Notice x2 =u(x1,x2)
x1 and x1 =u(x1,x2)
x2
Therefore:
MRS = -x2
x1= -
u(x1,x2)x1
u(
x1,x2)
x2
Is this true for all utility functions?
...yes... as long as the MRS is well defined
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y g
Indifference curve is a function g that solves the equation
u(x1,g(x1)) = k
The indifference curve g gives us for each x1 the corresponding
x2 = g(x1) so that utility is k.
Therefore: if we change x1, the utility must stay unchanged:
u(x1,g(x1))
x1+
u(x1,g(x1))
x2 g 0(x1) = 0
and solving for g 0 we get
g 0(x1) = -
u(x1,g(x1))x1
u(x1,g(x1))x2
= -
u(x1,x2)x1
u(x1,x2)x2
= MRS
other examples of utility functions...
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p yThe good x2 is money that is spent on many other goods. Utility is
linear in money:
u(x1, x2) = x2 + v(x1)
and v is a concave function.
MRS = -
u(x1,x2)x1
u(x1,x2)x2
= -v 0(x1)
MRS is independent ofx2.
Each indifference
curve is a verticallyshifted version of a
single indifference
curve
..examples contd...
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p
The goods are red and green pencils. The consumer cares about the
total number of pencils
u(x1, x2) = x1 + x2
MRS = -1
the indifference
curves are straight
lines with slope-
1
left shoe, right shoe
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g
the goods are perfect complements; the consumer needs one of each
u(x1
, x2
) = min{x1
, x2}
In this case, we cant use our formula to compute the MRS
I the indifference
curves are
L-
shaped.
I ifx1 x2 the
additional right
shoe delivers
no utility.I ifx2 x1 the
additional left
shoe delivers
no utility.
useless stuff
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the consumer has use for only one of the goods
u(x1, x2) = x1
Again, we cant use the formula to compute the MRS because the
marginal utility of good 2 is zero.
I indifference
curves are
vertical lines