ECON2001 Microeconomics Lecture Notes Term 1uctp100/2001/2001notes.pdfECON2001 Lecture Notes...

ECON2001 Lecture Notes

ECON2001 MicroeconomicsLecture Notes

Term 1Ian Preston

Budget constraint

Consumers purchase goods q from within a budget set B of affordable bun-dles. In the standard model, prices p are constant and total spending has toremain within budget p′q ≤ y where y is total budget. Maximum affordablequantity of any commodity is y/pi and slope ∂qi/∂qj |B = −pj/pi is constantand independent of total budget.

In practical applications budget constraints are frequently kinked or discon-tinuous as a consequence for example of taxation or non-linear pricing. If theprice of a good rises with the quantity purchased (say because of taxation abovea threshold) then the budget set is convex whereas if it falls (say because of abulk buying discount) then the budget set is not convex.

Marshallian demands

The consumer’s chosen quantities written as a function of y and p are theMarshallian or uncompensated demands q = f(y, p)

Consider the effects of changes in y and p on demand for, say, the ith good:

• total budget y

– the path traced out by demands as y increases is called the incomeexpansion path whereas the graph of fi(y, p) as a function of y iscalled the Engel curve

– we can summarise dependence in the total budget elasticity

εi =y

qi

∂qi

∂y=

∂ ln qi

∂ ln y

– if demand for a good rises with total budget, εi > 0, then we say itis a normal good and if it falls, εi < 0, we say it is an inferior good

– if budget share of a good, wi = piqi/y, rises with total budget, εi > 1,then we say it is a luxury or income elastic and if it falls, εi < 1, wesay it is a necessity or income inelastic

• own price pi

– the path traced out by demands as pi increases is called the offercurve whereas the graph of fi(y, p) as a function of pi is called thedemand curve

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– we can summarise dependence in the (uncompensated) own priceelasticity

ηii =pi

qi

∂qi

∂pi=

∂ ln qi

∂ ln pi

– if uncompensated demand for a good rises with own price, ηii > 0,then we say it is a Giffen good

– if budget share of a good rises with price, ηii > −1, then we say it isprice inelastic and if it falls, ηii < −1, we say it is price elastic

• other price pj , j 6= i

– we can summarise dependence in the (uncompensated) cross priceelasticity

ηij =pj

qi

∂qi

∂pj=

∂ ln qi

∂ ln pj

– if uncompensated demand for a good rises with the price of another,ηij > 0, then we can say it is an (uncompensated) substitute whereasif it falls with the price of another, ηij < 0, then we can say it is an(uncompensated) complement. These are not the best definitions ofcomplementarity and substitutability however since they may not besymmetric ie qi could be a substitute for qj while qj is a complementfor qj . A better definition, guaranteed to be symmetric, is one basedon the concept of compensated demands to be introduced below.

Adding up

We know that demands must lie within the budget set:

p′f(y, p) ≤ y.

If consumer spending exhausts the total budget then this holds as an equality,

p′f(y, p) = y,

which is known as adding upBy adding up

• not all goods can be inferior

• not all goods can be luxuries

• not all goods can be necessities

Also certain specifications are ruled out for demand systems. It is not possi-ble, for example, for all goods to have constant income elasticities unless theseelasticities are all 1. Otherwise piqi = Aiy

αi , say, and 1 =∑

i Aiαiyαi−1 for all

budgets y which is impossible unless all αi = 1. This does not rule out constantelasticities for individual goods.

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There are also restrictions on price effects - for example, if price of somegood goes up then purchases of some good must be reduced so no good can bea Giffen good unless it has strong complements.

Homogeneity

Multiplying y and p by the same factor does not affect preferences or thebudget constraint so choices should not be affected either, assuming that y andp influence choice only through the budget constraint.

Marshallian demands should therefore be homogeneous of degree zero:

f(λy, λp) = f(y, p)

for any λ > 0.

Preferences

Suppose the consumer has a preference relation % where qA % qB means“qA is at least as good as qB”. For the purpose of modelling demand this canbe construed as an inclination to choose the bundle qA over the bundle qB . Formodelling welfare effects the interpretation needs to be strengthened to includea link to consumer wellbeing.

A weak preference relation suffices to define strict preference � and indiffer-ence ∼ if we let

• % and - be equivalent to ∼,

• % and � be equivalent to �.

We want the preference relation to provide a basis to consistently identify aset of most preferred elements in any possible budget set and for this we needassumptions.

• Completeness Either qA % qB or qB % qA. This ensures that choice ispossible in any budget set.

• Transitivity qA % qB and qB % qC implies qA % qC . This ensures thatthere are no cycles in preferences within any budget set.

Together these ensure that the preference relation is a preference ordering.

Indifference curves

For any bundle qA define

• the weakly preferred set R(qA) as all bundles qB such that qB % qA

• the indifferent set I(qA) as all bundles qB such that qB ∼ qA

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To make these sets well behaved we make the technical assumption:

• Continuity If qA % qB and qB % qC then there is a bundle indifferent toqB on any path joining qA to qC . This rules out discontinuous jumps inpreferences.

Continuity is violated by the example of lexicographic preferences. Say thatthere are two goods q1 and q2 and that the consumer prefers one bundle toanother if and only if it either has more of q1 or the same amount of q1 andmore of q2. Such preferences do not satisfy continuity and indifferent sets aresingle points.

A further assumption rules out consumers ever being fully satisfied:

• Nonsatiation Given any bundle there is always some direction in whichchanging the bundle will make the consumer better off.

If this is true then indifferent sets have no “thick” regions to them and we canvisualise them as indifference curves.

Utility functions

A utility function u(q) is a representation of preferences such that qA % qB

if and only if u(qA) ≥ u(qB). A utility function exists if preferences give acontinuous ordering.

Utility functions are not however unique since if u(q) represents certain pref-erences then any increasing transformation φ(u(q)) also represents the samepreferences. We say that utility functions are ordinal.

Shape of indifference curves

We now consider assumptions which put some actual shape on indifferencecurves. For example:

• Monotonicity Larger bundles are preferred to smaller bundles.

Given monotonicity, indifference curves must slope down. This slope isknown as the marginal rate of substitution (MRS).

Monotonicity corresponds to increasingness of the utility function u(q). Anindifference curve is defined by u(q) being constant and therefore the MRS isgiven by

MRS =dq2

dq1

∣∣∣u

= −∂u/∂q1

∂u/∂q2

which is obviously negative if ∂u/∂q1, ∂u/∂q2 > 0.

• Convexity λqA + (1 − λ)qB % qB if qA % qB and 1 ≥ λ ≥ 0. This saysthat weakly preferred sets are convex or, equivalently, MRS is diminishing.

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Convexity can be interpreted as capturing taste for variety. It says that aconsumer will always prefer to mix any two bundles between which they areindifferent. The corresponding property of the utility function is known asquasiconcavity.

Homotheticity

Preferences are said to be homothetic if qA ∼ qB implies that λqA ∼ λqB forany λ > 0. Graphically this means that higher indifference curves are magnifiedversions of lower ones from the origin. This is a strong restriction that wouldrarely be made in practice but it is useful to consider as a reference case. It is nota restriction on the shape of any one indifference curve but on the relationshipbetween indifference curves within an indifference map.

If preferences are homothetic then marginal rates of substitution are constantalong rays through the origin. This is only true for homothetic preferences andthis is usually an easy way to check whether given preferences are homothetic.

If there exists a homogeneous utility representation u(q) where u(λq) =λu(q) then preferences can be seen to be homothetic. Since increasing transfor-mations preserve the properties of preferences, then any utility function whichis an increasing function of a homogeneous utility function also represents ho-mothetic preferences.

Quasilinearity

Quasilinearity is another strong restriction based on a similar idea. Prefer-

ences are quasilinear if

qA1

qA2

qA3...

∼

qB1

qB2

qB3...

implies

qA1 + λqA2

qA3...

∼

qB1 + λqB2

qB3...

.

In other words adding the same amount to one particular good preserves indif-ference. This means that higher indifference curves are parallel translations oflower ones. In this case, marginal rates of substitution are constant along linesparallel to axes.

If there exists a utility representation u(q) such that u(q1, q2, . . . ) = q1 +F (q2, q3, . . . ), say, then preferences are quasilinear. This is also true of anyutility functions which are increasing transformations of functions with thisproperty.

Some Examples

• Perfect substitutes u(q1, q2) = aq1 + bq2: The MRS is −a/b and isconstant. Indifference curves are parallel straight lines. These are theonly preferences which are homothetic and quasilinear.

• Perfect complements u(q1, q2) = min[aq1, bq2]: Indifference curves areL-shaped with the kinks lying on a ray through the origin of slope a/b.These preferences are homothetic but not quasilinear.

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• Cobb-Douglas: u(q1, q2) = a ln q1 + b ln q2: Preferences are homothetic,indifference curves are smooth and MRS aq2/bq2 is diminishing

Revealed preference

Suppose the consumer chooses qA at prices pA when qB was cheaper:

pA′qA > pA′

qB .

We say that qA is (directly) revealed preferred to qB . The Weak Axiom ofRevealed Preference (WARP) says that the consumer would never then chooseqB at prices pB when qA was affordable:

pB ′qA ≤ pB ′

qB .

This is an implication of consumer optimisation.The Strong Axiom of Revealed Preference (SARP) says that there should be

no cycles in revealed preference eg we should never find qA revealed preferred toqB , qB revealed preferred to qC and qC revealed preferred to qA (or any longercycle). This is equivalent to consumer optimisation.

Negativity

Suppose that as prices change from pA to pB the consumer is compensatedin the sense that their total budget is adjusted to maintain affordability of theoriginal bundle. this is known as Slutsky compensation. Then choices before andafter satisfy pB ′

qA = pB ′qB . But the later choice cannot then have been cheaper

at the initial prices or the change would violate WARP. Hence pA′qA ≤ pA′

qB .By subtraction we get negativity :

(pB − pA)′(qB − qA) ≤ 0.

This shows a sense in which price changes and quantity changes must move, onaverage, in opposite directions if the consumer is compensated.

If we consider the case where the price of only one good changes then wesee that this implies that Slutsky compensated effects of own price rises mustbe negative. In other words, Slutsky compensated demand curves necessarilyslope down.

Note that convexity was not assumed anywhere in the argument.

Slutsky equation

The Slutsky compensated demand function given initial bundle qA is definedby

g(qA, p) = f(p′qA, p)

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- that is to say it is the amount demand if budget is constantly adjusted tokeep qA affordable. Differentiating establishes a relationship between Slutskycompensated and Marshallian price effects

∂gi

∂pj= qA

j

∂fi

∂y+

∂fi

∂pj.

The difference is the income effect qAj ∂fi/∂yand it is positive if the good is

normal. Hence, since the Slutsky compensated effect has been shown to benegative, so must be the Marshallian effect for normal goods. This is the Lawof Demand.

The Slutsky equation is highly useful. Its importance is that it allows testingof restrictions regarding compensated demands since it shows how to calculatecompensated effects from the sort of uncompensated effects estimated in applieddemand analysis.

Tangency condition

If all goods are chosen in positive quantities and preferences are convex thenthe solution to the consumer’s optimisation problem is at a tangency betweenan indifference curve and the boundary of the budget set.

This is true even for non-linear budget sets. However if the budget set islinear (or indeed simply convex) then we know that such a tangency is uniqueso finding one guarantees that we have found the best choice for the consumer.If the budget set is not convex, on the other hand, then there can be multipletangencies and the optimum can typically be found only by comparing the levelof utility at each of them.

The nature of Marshallian demands can then be inferred by moving thebudget constraint to capture changes in y and p and tracing out movement ofthe tangency.

Demand under homotheticity and quasilinearity

As income increases. slopes of budget constraints do not change. Income ex-penditure paths traced out by the tangencies as incomes are increased thereforeall occur at points with the same MRS.

Homotheticity and quasilinearity are each characterised by the nature ofpaths along which MRS is constant and therefore each give rise to income ex-pansion paths of particular shapes.

• Homotheticity MRS is constant along rays through the origin so incomeexpansion paths are rays through the origin. Quantities consumed are pro-portional to total budget y given prices and budget shares are independentof y.

• Quasilinearity MRS is constant along lines parallel to one of the axesso income expansion paths are parallel to one of the axes. Quantitiesdemanded of all but one of the goods are independent of y (for interiorsolutions).

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Constrained optimisation

Mathematically, Marshallian demands solve

maxq

u(q) s.t. p′q ≤ y

This can be solved by finding stationary points of the Lagrangean

u(q)− λ(p′q − y).

For interior solutions, first order conditions require

∂u

∂qi= λpi

which imply

MRS = − ∂u/∂qi

∂u/∂qj= − pi

pj.

This is a confirmation of the tangency condition - the slope of indifference curveand budget constraint are equal at interior solutions.

Duality

In comparison with the primal problem

maxq


consider now the dual problem of minimising the expenditure necessary to reacha given utility

minq

p′q s.t. u(q) ≥ υ

Solutions to this problem are Hicksian or compensated demands q = g(υ, p).The problem can be solved by finding stationary points of the Lagrangean

p′q − µ(u(q)− υ).

which, for interior solutions, gives first order conditions requiring

pi = µ∂u

∂qi

and therefore

MRS = − ∂u/∂qi

∂u/∂qj= − pi

pj.

Note that this is exactly the same tangency condition encountered in solvingthe primal problem.

We can define important functions giving the values of the primal and dualproblems. The value of the maximised utility function as a function of y and

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p can be found by substituting the Marshallian demands back into the directutility function u(q). We call this the indirect utility function

v(y, p) = u(f(y, p)) = maxq


The value of the minimised cost in the dual problem as a function of υ andp can be found by costing the Hicksian demands. We call this the expenditurefunction or cost function

c(υ, p) = p′g(υ, p) = minq

p′q s.t. u(q) ≥ υ.

The duality between the two problems can be expressed by noting the equal-ity of the quantities solving the two problems

f(c(υ, p), p) = g(υ, p) f(y, p) = g(v(y, p), p)

or noting that v(y, p) and c(υ, p) are inverses of each other in their first argu-ments

v(c(υ, p), p) = υ c(v(y, p), p) = y.

Expenditure function

The expenditure function c(υ, p) has the properties that

• it is increasing in each price in p and in υ

• it is homogeneous of degree one in prices p, c(υ, λp) = λc(υ, p). TheHicksian demands are homogeneous of degree zero so the total cost ofpurchasing them must be homogeneous of degree one

c(υ, λp) = λp′g(υ, λp) = λp′g(υ, p) = λc(υ, p)

Indirect utility function

The properties of the indirect utility function v(y, p) correspond exactly tothose of the expenditure function given that the two are inverses of each other.In particular it is

• increasing in y and decreasing in each element of p

• homogeneous of degree zero in y and p:

v(λy, λp) = v(y, p)

This should be apparent also from the homogeneity properties of Marshal-lian demands.

Shephard’s Lemma

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Among the most useful features of these functions are their simple links tothe associated demands. For example, it is possible to get from the expenditurefunction to the Hicksian demands simply by differentiating.

Since c(υ, p) = p′g(υ, p)

∂c(υ, p)∂pi

= gi(υ, p) + p′∂g(υ, p)

∂pi

= gi(υ, p) + µ∑

i

∂u

∂qi

∂g(υ, p)∂pi

= gi(υ, p)

using the first order condition for solving the cost minimisation problem andthe fact that utility is held constant in that problem.

This is known as Shephard’s Lemma. Its importance is that it allows com-pensated demands to be deduced simply from the expenditure function.

Roy’s Identity

Since v(c(υ, p), p) = υ

∂v(y, p)∂pi

+∂v(y, p)

∂y

∂c(u, p)∂pi

= 0

⇒ −∂v(y, p)/∂pi

∂v(y, p)/∂y= gi(v(y, p), p)

= fi(y, p)

using Shephard’s Lemma.This is Roy’s identity and shows that uncompensated demands can be de-

duced simply from the indirect utility function by differentiation.In many ways it is easier to derive a system of demands by beginning with

well specified indirect utility function v(y, p) or expenditure function c(υ, p) anddifferentiating than by solving a consumer problem directly given a direct utilityfunction u(q).

Slutsky equation, again

Since g(υ, p) = f(c(υ, p), p)

∂gi(υ, p)∂pj

=∂fi(y, p)

∂pj+

∂fi(y, p)∂y

∂c(υ, p)∂pj

=∂fi(y, p)

∂pj+

∂fi(y, p)∂y

fj(y, p)

Notice that this is the same as the Slutsky equation derived earlier forSlutsky-compensated demands. Hicks-compensated price derivatives are the

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same as Slutsky-compensated price derivatives since the two notions of com-pensation coincide at the margin.

This means the results derived earlier can simply be carried over. Hicksiandemands therefore also satisfy negativity at the margin. In particular

∂gi(υ, p)∂pi

≤ 0.

(Negativity can actually be stated slightly more strongly than this, involving alsorestrictions on cross-price effects, but this is the most important implication).

Slutsky symmetry

There is one more property of Hicksian demands that can now be deduced.From Shephard’s Lemma

∂gi(υ, p)∂pj

=∂2c(υ, p)

∂pipj=

∂gj(υ, p)∂pi

Compensated cross-price derivatives are therefore also symmetric. Holdingutility constant, the effect of increasing the price of one good on the quantitychosen of another is numerically identical to the effect of increasing the price ofthe other good on the quantity chosen of the first good.

This shows that notions of complementarity and substitutability are con-sistent between demand equations if using compensated demands and providesa strong argument for defining complementarity and substitutability in suchterms. This would not be true if using uncompensated demands because in-come effects are not symmetric.

Demand restrictions

To summarise, if demands are consistent with utility maximising behaviourthen they have the following properties

• Adding up: Demands must lie on the budget constraint and therefore

p′f(y, p) = y

p′g(υ, p) = c(υ, p)

• Homogeneity: Increasing all incomes and prices in proportion leaves thebudget constraint and therefore demands unaffected

fi(y, p) = fi(λy, λp)gi(υ, p) = gi(υ, λp)

• Negativity of compensated own price effects: In particular, a compensatedincrease in any good’s price can only reduce demand for that good

∂gi/∂pi = ∂fi/∂pi + fi∂fi/∂y ≤ 0



• Symmetry of compensated cross price effects:

∂gi/∂pj = ∂gj/∂pi

If demands satisfy these restrictions then there is a utility function u(q)which they maximise subject to the budget constraint. We say demands areintegrable. These are all the restrictions required by consumer optimisation.

We know a system of demands is integrable if any of the following hold

• They were derived as solutions to the dual or primal problem given a wellspecified direct utility function

• They were derived by Shephard’s Lemma from a well specified cost func-tion or they were derived by Roy’s identity from a well specified indirectutility function

• They satisfy adding up, homogeneity, symmetry and negativity.

The connections between the concepts discussed can be summarised in thediagram below:

Utility maximisation problem Cost minimisation problemmaxq u(q) s.t. p′q ≤ y ←→ minq

∑p′q s.t. u(q) ≥ υ

↓ ↓Uncompensated demands Compensated demandsqi = fi(y, p) i = 1, 2, ... ←→ qi = gi(υ, p) i = 1, 2, ...

l lIndirect utility function Expenditure function

v(y, p) ←→ c(υ, p)

Some worked examples of consumer choiceproblems

• Example 1:

The direct utility function is

u(q1, q2) = (q1 − a1)(q2 − a2)

The tangency condition defining optimum consumer choice is

MRS = −∂u/∂q1

∂u/∂q2= −q2 − a2

q1 − a1= −p1

p2

⇒ p1q1 − p1a1 = p2q2 − p2a2



Substituting into the budget constraint

y = 2p1q1 − p1a1 + p2a2

and thus uncompensated demands are

f1(y, p1, p2) = (y + p1a1 − p2a2)/2p1 = a1 + (y − p1a1 − p2a2)/2p1

f2(y, p1, p2) = (y − p1a1 + p2a2)/2p2 = a2 + (y − p1a1 − p2a2)/2p2

Substituting into the direct utility function gives the indirect utility func-tion

v(y, p1, p2) = (f1(y, p1, p2)− a1)(f2(y, p1, p2)− a2)= (y − p1a1 − p2a2)2/4p1p2

Inverting v(y, p1, p2) in y then gives the expenditure function

c(υ, p1, p2) = 2√

υp1p2 + p1a1 + p2a2

The compensated demands are then most easily found by differentiatingc(υ, p1, p2) (using Shephard’s Lemma) or by substituting c(υ, p1, p2) intothe uncompensated demands

g1(υ, p1, p2) =∂c(υ, p1, p2)

∂p1= f1(c(υ, p1, p2), p1, p2) = a1 +

√υ

p2

p1

g2(υ, p1, p2) =∂c(υ, p1, p2)

∂p2= f2(c(υ, p1, p2), p1, p2) = a2 +

√υ

p1

p2

• Example 2:

Direct utility isu(q1, q2) = ln q1 + q2

Preferences are quasilinear.

The tangency condition is

MRS = −∂u/∂q1

∂u/∂q2= − 1

q1= −p1

p2

⇒ p1q1 = p2

This defines an interior optimum assuming y > p2.

Hence, directly and by substituting into the budget constraint, uncom-pensated demands are

f1(y, p1, p2) = p2/p1

f2(y, p1, p2) = (y/p2)− 1



The uncompensated demand for the first good is independent of totalbudget y.


v(y, p1, p2) = ln f1(y, p1, p2) + f2(y, p1, p2)= ln(p2/p1) + (y/p2)− 1

Inverting in y gives the expenditure function

c(υ, p1, p2) = p2(υ − ln(p2/p1) + 1)

Differentiating or substituting then gives the compensated demands

g1(υ, p1, p2) =∂c(υ, p1, p2)

∂p1= f1(c(υ, p1, p2), p1, p2) = p2/p1

g2(υ, p1, p2) =∂c(υ, p1, p2)

∂p2= f2(c(υ, p1, p2), p1, p2) = υ − ln(p2/p1)

The compensated demand for the first good is independent of υ.

• Example 3:

Direct utilityu(q1, q2) = min[q1, q2]

Goods are perfect complements and at the optimum

q1 = q2

⇒ f1(y, p1, p2) = f2(y, p1, p2) =y

p1 + p2


v(y, p1, p2) = min[f1(y, p1, p2), f2(y, p1, p2)]

=y

p1 + p2

Inverting in y gives the expenditure function

e(υ, p1, p2) = υ(p1 + p2)

Differentiating or substituting gives the compensated demands

g1(υ, p1, p2) =∂c(υ, p1, p2)

∂p1= υ =

∂c(υ, p1, p2)∂p2

= g2(υ, p1, p2)



Consumer surplus

By Roy’s identity the effect of a small change in price pi on utility is pro-portional to the quantity consumed

∂v

∂pi=

∂v

∂yqi.

The horizontal distance of the demand curve from the vertical axis is thereforean indicator of the marginal welfare cost of increasing price. If ∂v/∂y is constantthen the effect of increasing the price to the point where none of the good isdemanded is therefore a triangular area underneath a demand curve. We callthis consumer surplus.

What sort of demand curve do we need to use to keep ∂v/∂y constant?If preferences are quasilinear then this will be true of the Hicksian demandcurve. Even if preferences are not quasilinear the general idea behind calculatingconsumer surplus as the area under the compensated demand curve usually stillgives a reasonable approximation to a good measure of welfare.

Cost of living indices

The expenditure function is the ideal concept for comparing cost of living.We can define a cost of living index as the ratio of the minimum cost of reachinga given utility in two periods. Say that we are comparing current prices pA

with prices in a base period pB . Then the cost of living index is the ratio ofexpenditure functions

T (υ, pA, pB) =c(υ, pA)c(υ, pB)

Notice that such a cost of living index depends on the utility level υ at whichwe make the comparison. Must this be so? There are only two case in whichnot:

• If prices are proportional pA = λpB then the cost of living index is equalto λ at all υ, whatever preferences,

• If preferences are homothetic then the cost of living index is equal at allυ, whatever prices.

Two common approximations to the true index are used. Both compare thecost of purchasing a fixed bundle of goods.

The Laspeyres index is the ratio of the costs of purchasing the base periodbundle qB

L(pA, pB) =pA′

qB

pB ′qB

.

If L(pA, pB) < 1 and consumer’s total expenditure is unchanged then theconsumer can afford the base bundle and cannot be worse off.



We also know that pB ′qB = c(υB , pB) where υB is the base period utility

and also that pA′qB cannot be less than the minimum cost of attaining uB in

the current period (since qB gives utility υB but not necessary most cheaplyat current prices). Hence the Laspeyres index is greater than the true cost ofliving index at base period utility,

L(pA, pB) ≥ T (υB , pA, pB).

This is so because consumers are free to substitute away from goods whichbecome more expensive and therefore evaluating the cost at a fixed bundleexaggerates the impact on cost of living.

The Paasche index is the ratio of the costs of purchasing the current periodbundle qA

P (pA, pB) =pA′

qA

pB ′qA

.

By similar arguments the consumer must be worse off if P (pA, pB) > 1 andtotal expenditure is unchanged. Likewise the Paasche index can be shown tobe less than the true cost of living index at current utility υA,

P (pA, pB) ≤ T (υA, pA, pB).

Buying and selling

Suppose an individual has endowments ω = (ω1, ω2, ...) of goods. Theconsumer problem becomes

maxq

u(q) s.t. p′q ≤ y + p′ω

Demands are now

qi = fi(y + p′ω, p) i = 1, 2, . . .

where fi(.) is the standard uncompensated demand function.Changes in endowments have effects like income effects. Changes in prices

have the usual effects plus an effect due to the change in the value of the indi-vidual’s endowment - the endowment income effect. Specifically

∂qi

∂pi=

∂fi

∂pi+

∂fi

∂yωi

=∂gi

∂pi− (qi − ωi)

∂fi

∂y

where gi(υ, p) is the usual compensated demand function. This extends theSlutsky equation to the case of demand with endowments. Notice that thesign of the income effect depends upon whether the individual is a net buyer(qi > ωi) as in the usual case or a net seller (qi < ωi). An increase in the price



of a normal good can now increase demand if the individual is a net seller andthe endowment income effect is therefore strong enough.

Note that there is an important revealed preference argument establishingthat a seller will never become a buyer if the price rises and a buyer will neverbecome a seller if the price fails. In each case, such a change is not possiblesince it would involve consuming a bundle available before the change when thebundle then chosen remains affordable.

Labour supply

The prime example of the importance of considering demand with endow-ments is the analysis of labour supply. Suppose an individual has preferencesover hours not working (“leisure”) h and consumption c. He has unearnedincome of m and endowment of time T. The price of consumption is p and thenominal wage is w. The individual’s budget constraint is

pc + wh = m + wT

which may appear more familiar if written in terms of hours worked l = T − h:

pc = m + wl.

The value of endowments in this context m + wT is referred to as full income.Demand for leisure can be written as an uncompensated demand function,

dependent on full income, wage and output price

h = f(m + wT, w, p)

or as a compensated demand function

h = g(υ, w, p).

The Slutsky equation for leisure is

∂h

∂w

∣∣∣∣m

=∂g

∂w− (h− T )

∂h

∂m.

Since the individual sells time (h < T ) the income effect of a wage changeis opposed to the compensated effect if leisure is normal.

Rephrasing in the more familiar terms of labour supply l

∂l

∂w

∣∣∣∣m

=∂l

∂w

∣∣∣∣υ

+ l∂l

∂m.

Intertemporal choice

Another example of demand with endowments is analysis of intertemporalchoice. Suppose an individual has preferences over consumption when young



c0and consumption when old c1. He has endowed income of y0 and y1in thetwo periods. (If necessary, bequests received can be treated as part of y0 andbequests given as part of c1). Assume no uncertainty about the future. If thereal interest rate on bonds linking the two periods is equal to r for both lendingand borrowing then the budget constraint is

c0 +c1

1 + r= y0 +

y1

1 + r.

which implies that the present value of consumption must equal the presentvalue of income.

Demand for current consumption is

c0 = f0(y0 +y1

1 + r, r)

The effect of interest rate changes clearly depend upon whether the individ-ual is a saver or a borrower since this determines the sign of the income effect.Note that an interest rate rise will never induce a saver to become a borrowerand an interest rate fall will never induce a borrower to become a saver.

Often it is assumed that the utility function can be written as the sum ofutility contributions from the different periods with similar within-period utilityfunctions but with future utility discounted. Thus

u(c0, c1) = ν(c0) +1

1 + δν(c1)

where ν(.) is the within-period utility function and δ is a subjective discount rate.Convexity of preferences, which amounts here to a desire to smooth consumptionover the life-cycle, requires ν(.) to be concave ie ν′′(.) < 0.

Maximising such a utility function subject to the lifetime budget constraint

maxc0

ν(c0) +1

1 + δν(y1 + (y0 − c0)(1 + r))

gives first order conditionν′(c0)ν′(c1)

=1 + r

1 + δ.

Given concavity of ν(.), if r = δ, so that subjective discounting matches themarket interest rate and impatience cancels out the market incentive to save,then c0 = c1 and the consumption stream is flat. If r > δ then c0 < c1 and ifr < δ then c0 > c1.

Asset choice

Suppose that as well as investing in bonds with fixed return of r the indi-vidual can also invest in another asset - say a family enterprise. If X is placedin the family enterprise in the first period then suppose F (X) is returned in thesecond period.



The optimisation problem now has two dimensions

maxc0,X

ν(c0) +1

1 + δν(y1 + F (X) + (y0 − c0 −X)(1 + r))

and first order conditions (assuming an interior solution)

ν′(c0)ν′(c1)

=1 + r

1 + δ

F ′(X) = (1 + r)

Note that the solution to the financial decision is independent of intertemporalpreferences. The individual invests in the family enterprise until the marginalrate of return falls to the market interest rate. This maximises the presentvalue of the individual’s asset portfolio and the first order condition for optimumconsumption choice given that present value is as in the simpler problem above.

The simplicity of the investment decision is a consequence of assuming awayissues concerning risk, liquidity and so on.

Uncertainty

Extending the standard analysis to the case of uncertainty involves regardingquantities consumed in different uncertain states of the world as different goods.Preferences will depend on perceived probabilities of states of the world occur-ring. Budget constraints depend on the mechanisms available for managingrisk.

Some examples of budget constraints in circumstances involving risk are:

• An individual with an asset worth A faces a probability π of losing it. Hecan purchase insurance of K at a cost of γK. Consumption in case of lossis c1 = (1 − γ)K and in the case of no loss is c0 = A − γK. The budgetconstraint for the individual is c0 = A− γ

1−γ c1.

• An individual with income of m has a true tax liability of T but triesto evade an amount d by underdeclaration. There is probability π ofbeing audited in which case he pays the full liability T plus a fine fD.Consumption in case of audit is c1 = m − T − fD and in the case of noaudit is c0 = m − T + D. The budget constraint for the individual isc0 = (1 + f)(m− T )− fc1.

In both of these cases the budget constraint is linear and downward sloping.

Preferences are defined over quantities consumed in the different states(c0, c1...) and depend on perceived probabilities of the states occurring (π0, π1, ...).Under certain assumptions it may be reasonable to regard the consumer as max-imising expected utility

u(c0, c1..., π0, π1, ...) =∑

πiν(ci)



for some state-specific utility function ν(.). We refer to u(·) as a von-Neumann-Morgenstern expected utility function.

The most controversial assumption required to justify an expected utilityformulation is the strong independence axiom or sure thing principle. Considerthe following two choices:

Choice 1:π 1− π

Option A1 α γOption B2 β γ

Choice 2:π 1− π

Option A2 α δOption B2 β δ

In each case the two options deliver the same outcome as each other withprobability 1 − π (though these outcomes differ between the two choices). Itmight therefore be argued that the choice should be driven only by the differentoutcomes occurring with probability π. However these are the same in thetwo choices. Therefore if A1 is preferred to B1 it is argued that A2 shouldbe preferred to B2. This is the sure thing principle. Combined with otherless controversial axioms extending choice to uncertain situations with multipleoutcomes it implies that the MRS between consumption in any two states isindependent of outcomes in any other state.

The sure thing principle is violated by many people’s choices in the followingexample (known as the Allais paradox ):

Choice 1:0.10 0.89 0.01

Option A1 £1m £1m £1mOption B1 £5m £1m 0

Choice 2:0.10 0.89 0.01

Option A2 £1m 0 £1mOption B2 £5m 0 0

where it is common to prefer A1 to B1 and B2 to A2.Note that the function ν(.) is not ordinal. Preferences are changed by arbi-

trary increasing transformations of ν(.). However u(.) is still ordinal.

Risk aversion

To capture risk aversion we need to capture the fact that risk averse in-dividuals prefer to receive the expected value of any gamble with certainty toundertaking the gamble. Thus

ν((1− π)c0 + πc1) > (1− π)ν(c0) + πν(c1).

For this always to be true requires that ν(.) be a concave function. The degreeof concavity is an indicator of the strength of aversion to risk.



Consider the insurance case again. A utility maximising consumer choosesK to maximise

(1− π)ν(A− γK) + πν((1− γ)K).

The first order condition requires

(1− π)γν′(A− γK) = (1− γ)πν′((1− γ)K).

If insurance is actuarially fair then π = γ and therefore ν′(A − γK) =ν′((1− γ)K). If the individual is risk averse then ν′(.) is a decreasing functionand therefore A = K i.e. there is full insurance. This is a typical illustration ofbehaviour under risk. The fairness of insurance means that risk can be elimi-nated without compromising expected consumption and a risk averse individualchooses therefore to eliminate risk.

If insurance is less than fair π < γ then there is underinsurance.

Equilibrium in pure exchange economies

Suppose the hth household has endowment ωh and consumes a bundle qh.An allocation of goods is said to be feasible if the aggregate amount consumedof each good equals the aggregate endowment∑

h

qhi =

∑h

ωhi i = 1, 2, ...

The initial endowments obviously constitute one feasible allocation.Demands if prices are p are qh

i = fhi (p′ωh, p), i = 1, 2, ... where p′ωh is the

value of the individual’s endowment. Market demand is found by adding thedemands across individuals

Qi(p′ω1, p′ω2, ..., p) =∑

h

fhi (p′ωh, p) i = 1, 2, ...

Note the dependence on the complete distribution of endowments.Let zh

i denote the excess demand from the ith household. The aggregateexcess demand zi is given by the excess of market demand over the sum ofendowments

zi(p′ω1, p′ω2, ..., p) =∑

h

zhi =

∑h

[fhi (p′ωh, p)− ωh

i ] i = 1, 2, ...

General equilibrium - referred to as market equilibrium, competitive equilib-rium or Walrasian equilibrium - is a set of prices such that aggregate excessdemand is zero on all markets.∑

h

zhi = 0 i = 1, 2, ...

The competitive allocation is another example of a feasible allocation.



If there are k goods then this seems to define k equations in k unknownprices. However this is misleading. The fact that each household must be onits budget constraint implies that the value of that household’s excess demandis zero ∑

i

piqhi =

∑i

piωhi ⇒

∑i

pizhi = 0.

Adding this equation over households establishes that the value of aggregateexcess demand is also zero ∑

i

pi

∑h

zhi = 0.

This is Walras’ law and is true for any prices (not only the equilibrium prices).It implies that the k excess demands are not independent - in fact there areonly k − 1 independent excess demands to set to zero.

However since demands are homogeneous multiplying all prices by any pos-itive number will give the same excess demands. If any prices constitute a Wal-rasian equilibrium, then so therefore do any positive multiple of those prices. Itis therefore only relative prices which are determined by the equilibrium condi-tions.

There are therefore actually k − 1 independent equations determining k − 1relative prices.

Nothing said so far ensures existence of a Walrasian equilibrium but if ag-gregate demands vary continuously as a function of prices then we may expectthis to be so.

Efficiency

Walrasian equilibrium has the general property of Pareto efficency. Thismeans that there is no feasible allocation such that all consumers are better off(or some are better off without any being any worse off). To prove this supposeit were not the case. Then there would exist an allocation r1, r2,... such that r1

was preferred to q1, r2 was preferred to q2 and so on. But then these bundlescould not be affordable at the equilibrium prices p or the consumers would havepurchased them. Thus∑

i

pirhi >

∑i

piqhi =

∑i

piωhi h = 1, 2, ...

Adding across consumers gives∑i

pi

∑h

rhi >

∑i

pi

∑h

ωhi .

But this conflicts with feasibility which requires∑

h rhi =

∑h ωh

i , i = 1, 2, ...Hence there can be no such alternative allocation.

This is the First Fundamental Theorem of Welfare Economics. Walrasianequilibrium is always Pareto efficient. Note that this says nothing about desir-ability in other respects such as distributional equity - this, for instance, willinevitably depend upon the equity in the allocation of initial endowments.



The Second Fundamental Theorem of Welfare Economics tells us converselythat, under certain further assumptions, any Pareto efficient allocation can besustained as a Walrasian equilibrium given the right allocation of initial endow-ments. In particular, this is true if we assume all agents have convex preferences.

Solving general equilibrium problems for pureexchange economies

Suppose the hth household has endowment ωh and consumes a bundle qh.Demands if prices are p are qh

i = fhi (yh, p), i = 1, 2, ..., k where yh = p′ωh is the

value of the individual’s endowment. We need to find a price vector p solvingthe market clearing equations∑

h

fhi (p′ωh, p) =

∑h

ωhi i = 1, 2, ..., k

From Walras’ law we need only solve for k − 1 relative prices achieving marketclearing on k − 1 of the k markets.

As an example, suppose there are two goods. We can solve only for therelative price so we normalise the price of good 2 to be 1 and let p be the priceof the first. Let the two consumers have Cobb Douglas demands over the twogoods. Individual h therefore has demands

qh1 = αhyh/p qh

2 = (1− αh)yh

where αh is an individual-specific taste parameter.Individual endowments are ωh = (ωh

1 , ωh2 ). Therefore, by substituting the

value of the endowments, demands are

qh1 = αh(pωh

1 + ωh2 )/p qh

2 = (1− αh)(pωh1 + ωh

2 )

To find equilibrium, we know from Walras’ law that we need only find theprice to clear one market. Take the first. Market clearing requires

ω11 + ω2

1 = q11 + q2

1 = α1(pω11 + ω1

2)/p + α2(pω21 + ω2

2)/p

Solving for p gives

p =α1ω1

2 + α2ω22

(1− α1)ω11 + (1− α2)ω2

1

.

Note that this is increasing in endowments of the second good and decreasingin endowments of the first. Note also that it is increasing in the demandparameters α1 and α2. These are readily intelligible demand and supply effectsfor this example.


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