Rice ECO501 Lectures

Rice UniversityECO 501

������������������Lecture Notes: Microeconomic

Theory I������������������

Christian Roessler

Fall 2008

Contents

1 Preference 31.1 Consumption Set . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Rational Preference . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Utility 82.1 Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Quasiconcavity . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Demand I: Utility Maximization Problem 153.1 Budgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 The Utility Maximization Problem (UMP) . . . . . . . . . . . 163.3 Indirect Utility Function . . . . . . . . . . . . . . . . . . . . . 21

4 Demand II: Expenditure Minimization Problem 234.1 EMP and Hicksian Demand . . . . . . . . . . . . . . . . . . . 234.2 Expenditure Function . . . . . . . . . . . . . . . . . . . . . . . 264.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5 Comparative Statics 305.1 Wealth E¤ects . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.2 Price E¤ects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Law of Demand . . . . . . . . . . . . . . . . . . . . . . . . . . 335.4 Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.5 Money Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.6 Welfare Comparisons . . . . . . . . . . . . . . . . . . . . . . . 36

6 Choice-Based Approach 416.1 Choice Structures . . . . . . . . . . . . . . . . . . . . . . . . . 416.2 Weak Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436.3 Relationship with the Law of Demand . . . . . . . . . . . . . 476.4 Strong Axiom . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

7 Integrability 557.1 Slutsky and Hicks Compensation . . . . . . . . . . . . . . . . 557.2 Aside: Dot Product . . . . . . . . . . . . . . . . . . . . . . . . 577.3 Substitution Matrix . . . . . . . . . . . . . . . . . . . . . . . . 58

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7.4 Substitution Matrix with Preference . . . . . . . . . . . . . . . 627.5 Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

8 Aggregation 718.1 Aggregate Demand Function . . . . . . . . . . . . . . . . . . . 718.2 Representative Consumer . . . . . . . . . . . . . . . . . . . . . 768.3 Failure of the Weak Axiom . . . . . . . . . . . . . . . . . . . . 80

9 Expected Utility 849.1 Lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 849.2 Preference over Lotteries . . . . . . . . . . . . . . . . . . . . . 869.3 Expected Utility Theorem . . . . . . . . . . . . . . . . . . . . 889.4 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.5 State-Space Approaches . . . . . . . . . . . . . . . . . . . . . 98

10 Risk 10010.1 Money Lotteries . . . . . . . . . . . . . . . . . . . . . . . . . . 10010.2 Risk Attitude . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.3 Stochastic Dominance . . . . . . . . . . . . . . . . . . . . . . 107

11 Pro�t Maximization Problem 11111.1 Production Set . . . . . . . . . . . . . . . . . . . . . . . . . . 11111.2 Transformation Function . . . . . . . . . . . . . . . . . . . . . 11411.3 Pro�t Maximization . . . . . . . . . . . . . . . . . . . . . . . 11611.4 Law of Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

12 E¢ ciency of Aggregate Supply 12012.1 E¢ cient Production . . . . . . . . . . . . . . . . . . . . . . . . 12012.2 Cost Minimization . . . . . . . . . . . . . . . . . . . . . . . . 12212.3 Aggregate Supply . . . . . . . . . . . . . . . . . . . . . . . . . 125

13 Partial Competitive Equilibrium 12613.1 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . 12613.2 Partial Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 12913.3 The Long Run . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

14 Welfare Analysis 13214.1 Pareto E¢ ciency and Surplus . . . . . . . . . . . . . . . . . . 13214.2 E¢ ciency of Competitive Equilibrium . . . . . . . . . . . . . . 134

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14.3 E¢ cient Allocations through the Market Mechanism . . . . . 135

15 Externalities 13615.1 Externalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13615.2 Ine¢ ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13815.3 Remedies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13915.4 Public Goods . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

16 Monopoly and Product Di¤erentiation 14516.1 Monopoly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14516.2 Bertrand Price Competition . . . . . . . . . . . . . . . . . . . 14816.3 Repetition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15016.4 Product Di¤erentation . . . . . . . . . . . . . . . . . . . . . . 153

17 Capacity Constraints 15617.1 Capacity-Constrained Pricing . . . . . . . . . . . . . . . . . . 15617.2 Cournot Quantity Competition . . . . . . . . . . . . . . . . . 15817.3 Competitive Limit . . . . . . . . . . . . . . . . . . . . . . . . 161

18 Precommitment and Entry 16218.1 Precommitment . . . . . . . . . . . . . . . . . . . . . . . . . . 16218.2 Entry Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 16618.3 Socially Optimal Entry . . . . . . . . . . . . . . . . . . . . . . 168

1 Preference

1.1 Consumption Set

� In the �rst part of these lectures, we consider the consumer decisionproblem in a market economy, where goods are o¤ered at posted prices.Most of our attention will be on the primitive approach that starts with"rational" individual preferences and derives optimal choices for givenprices and endowments. We will also look at connections with the"revealed preference" approach that makes assumptions directly aboutchoices.

� Let there be L di¤erent commodities. A consumption bundle is a list

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x1; x2; : : : ; xL of quantities of each commodity, represented by a vector

x =

264 x1...xL

375in the commodity space RL.

� Note thatRL includes consumption bundles that list negative quantitiesof some goods. These can be interpreted as debts or giving some of one�sendowment to others.

� Any two items that could sell at di¤erent prices should be modeled asdistinct commodities. Strictly speaking, the description of a commoditymay have to include detailed context information, such as "a diet cokecan from a vending machine at Grand Central Station in the summerof 2008."

� The de�nition of the commodity space tells us what kind of object aconsumption bundle is. It may be the case that not all such objects(i.e. not all vectors) can be feasibly consumed, or we may wish toimpose restrictions in a particular model. Physical constraints include:we need time to consume, and time is limited; we cannot consume intwo places simultaneously; we have to consume enough of certain things(food, shelter) to survive.

� A restriction we will impose for modeling purposes is that the set offeasible consumption bundles, or the consumption set, is convex. Thatis, if x and y are feasible, then any mixture z = �x + (1� �) y with� 2 (0; 1) is also feasible. This assumption rules out indivisible com-modities.

Example. Suppose the commodities are time (in minutes) spent watch-ing television, and time (in minutes) spent in a rollercoaster car on "TheBeast" (Kings Island near Cincinnati), a ride which takes approximately �veminutes. It is possible to watch no television and ride "The Beast" once (callthis bundle x), and it is also possible to watch television for one hour andnot ride "The Beast" (call this bundle y). But one cannot watch television

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for half an hour and ride "The Beast" for two-and-a-half minutes (unless itis a one-period model ....). Since this bundle (call it z) is a mixture of x andy, the consumption set for these two commodities is not convex.

� Any economic model is an idealized representation of reality. We willmake many assumptions that are in some sense too strong to ever holdexactly. In some cases, they preserve the spirit of the problem, andpropositions for the idealized world are also informative about reality.In others, the assumptions limit the applicability of the model to a morespeci�c problem. Convexity of the consumption set is a conveniencerestriction that should not alter the thrust of our results, even thoughmany commodities are actually indivisible.

� Most of the time, we will assume that the consumption set X containsall bundles with only non-negative quantities:

X = RL+ ��x 2 RL s.t. x` � 0 for ` = 1; : : : ; L

,

which is convex.

1.2 Rational Preference

� The �rst step in analyzing individual choices from the consumptionset is to de�ne preference over its elements. For the moment, we canthink of X more generally as a choice set, where the elements may inparticular be consumption bundles. Or they may be something elseone can express a preference about, for example sports teams.

� A preference % is a binary relation on X: a subset of X � X. Weinterpret (x; y) 2%, which is commonly denoted x % y, as "x is at leastas good as y."

� The preference % associates with every x 2 X its better-than (or uppercontour) set and worse-than (or lower contour) set:

% (x) � fa 2 X s.t. a % xg- (x) � fb 2 X s.t. x % bg :

The intersection of the upper and lower contour sets is the indi¤erenceset at x:

� (x) � fb 2 X s.t. x � bg :

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� From % derive the strict preference and indi¤erence relations � and �:

x � y i¤ x % y and not y % x

x � y i¤ x % y and also y % x:

� In almost all economic theory, preferences are assumed to be rationalin the following sense.

De�nition. Preference % is rational i¤(i) % is complete: 8x; y 2 X, x % y or y % x (or both).(ii) % is transitive: 8x; y; z 2 X, x % y % z =) x % z.

� Completeness says that any two elements of X can be compared: oneis always preferred, or else they are indi¤erent. But it is never the casethat the agent could not say whether he prefers x or y (or �nds themindi¤erent), for example because he is seeking more information. Thisscenario could be accommodated by state-dependent preference, wherethe state is de�ned by what the agent learns.

� Transitivity is most closely related to the usual notion of rationality: itrequires the agent to rank alternatives consistently and predictably. Inpractice, people often violate transitivity unwittingly, especially whenthe choice objects are complex and unfamiliar. However, most wouldrevise a stated preference when one points out to them that it is in-transitive.

� Moreover, a transitivity violator is exposed to "Dutch book" trades.Suppose the agent initially has x. Since the preference is cyclical, e.g.x � y % z % x, and the agent should be willing to pay some non-zeroamount to exchange y for x, one could o¤er z for x, then y for z, thenx for y (at a price). Which leaves the agent with the initial bundle x,but he has made a payment. If these trades are repeated often enough,they will bankrupt the agent.

Exercise 1 (MWG 1.B.1, 1.B.2). Show: if% is rational, then 8x; y; z 2 X(i) x � x, (ii) relations � and � are transitive, (iii) x � y % z =) x � z.

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1.3 Utility Functions

� To apply tools from calculus in analyzing choices, it is useful to repre-sent the preference relation by a function.

De�nition. The function u : X ! R is a utility function that representspreference relation % if 8x; y 2 X,

x % y () u (x) � u (y) :

� If u represents %, then any strictly increasing function f : R! R canbe composed with u to give a new function v : X ! R (where v (x) =f (u (x)) for all x 2 X), with the property that v (x) � v (x0) ()u (x) � u (x0). Thus v also represents %. Clearly, utility functions arenon-unique.

� On the other hand, a utility function may fail to exist. It never existsfor a preference that is not rational.

Proposition. Only a rational preference relation can be represented by autility function.

Proof. Suppose a utility function u : X ! R represents preference %on X. Then % is complete: for any x; y 2 X, we have u (x) � u (y) oru (y) � u (x), so x % y or y % x. And % is transitive: when x % y % z, wehave u (x) � u (y) � u (z), hence u (x) � u (z) and then x % z. So existenceof a utility function implies that % is rational. The contrapositive, "if % isnot rational, then there does not exist a utility function," follows.�

Exercise 2 (MWG 1.B.5). Show: if X is �nite and % is a rational pref-erence relation on X, then there exists a utility function u : X ! R thatrepresents %.

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2 Utility

2.1 Continuity

� In this lecture, we discuss which properties of preferences ensure theexistence of the type of utility function to which the standard optimiza-tion techniques apply. When is % represented by a utility function uthat is twice di¤erentiable and has a (unique) maximum? Conditionsfor di¤erentiability are fairly di¢ cult to establish, so we focus on therelated, but weaker, property of continuity.

� In addition to continuity, di¤erentiability requires the absence of kinksin the utility function.

De�nition. Preference relation % is continuous if x % y whenever x andy are the limits of sequences fxng1n=1 and fyng

1n=1 such that x

n % yn for alln.

� Informally, a continuous preference ranks x and y the same as it ranksobjects that are very similar to x and y.

� In particular, suppose fyng1n=1 converges to y, and yn % x for all n.Then continuity requires y % x. Hence the limit of every convergentsequence in % (x), the upper contour set, is also in % (x), so that % (x)is closed. By the analogous argument, - (x), the lower contour set, isalso closed.

� The converse is true, too. Thus, a continuous preference is equiva-lently de�ned by the closedness of all lower and upper contour sets.Informally, x is preferred to y if objects that are very similar to x arepreferred to y.

� In the context of functions, continuity preserves closeness, rather thanpreference, under limits: whenever x is very close to y in the domain,f (x) is very close to f (y) in the image.

De�nition. The function u : X ! R is continuous if f (x) is the limit ofsequence ff (xn)g1n=1 whenever x is the limit of sequence fxng

1n=1 � X.

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� The proof of existence of a continuous utility function is simpli�ed by(but actually valid without) imposing a property called monotonicityon preferences. Its de�nition presupposes that the consumption set isX = RL+.

� The notation x� y stands for x` > y` for ` = 1; : : : ; L.

De�nition. Preference relation % is monotone if x� y =) x � y.

� Monotonicity is a fairly strong assumption. It rules out bads, but thisis not a real problem, since bads can be relabeled as goods (e.g. replace"waste" with "waste disposal"). Many things are, however, desirableonly up to a point (you may like some ice cream, perhaps a lot, butnot tons of it delivered to your home). Monotonicity welcomes more ofanything; it is inconsistent with limited wants.

Proposition. Any continuous rational preference relation can be representedby a continuous utility function.

Proof. Assume that preference % is continuous (and monotone) on RL+.We will construct utility values and show that the resulting function repre-sents % and is continuous.Let e = (1; : : : ; 1) 2 RL+ be a vector of 1s. Monotonicity implies: for every

consumption bundle x, there exists � � 0 such that x % �e and � <1 suchthat �e % x (just let � = 0 and � large enough so that all quantities in�e are greater than the corresponding quantities in x). This means that thesets A+ (x) � f� 2 R+ s.t. �e % xg and A� (x) � f� 2 R+ s.t. x % �eg arenonempty. Given x, we have � 2 A� (x) or � 2 A+ (x) for any � 2 R+by completeness, so R+ � A� (x) [ A+ (x). Moreover, A� (x) and A+ (x)are closed, since - (x) and % (x) are closed, so that preference must bepreserved under limits of sequences f�neg1n=1.(i.e. sequences of bundles thathave equal quantities of all commodities). To sum, A� (x) and A+ (x) arenonempty, closed, and together cover R+, which is a connected set. Hence

A� (x) \ A+ (x) =�� 2 R+ s.t. x � �e

is nonempty for all x 2 RL+.

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Monotonicity implies further that there is only one � such that x � �e,since �0e � x for all �0 > � and x � �e for all �00 < �. We may thereforede�ne a function u : X ! R+ by

u (x) = � 2 A� (x) \ A+ (x)

for all x 2 X. It remains to be shown that u represents % and is continuous.Suppose x % y and x = �e and y = �e. Then � � � by monotonicity,

so u (x) � u (y) : Conversely, if u (x) � u (y), then �e � �e, so x % y bymonotonicity.Continuity of u requires: for any sequence fxng1n=1 with limit x, the se-

quence fu (xn)g1n=1 converges to u (x). Note that, because fxng1n=1 converges,

there exists for any " a number n (") such that kxn � xk < " for n � n (").Hence fu (xn)g1n=n(") lies in a compact set: namely, in the interval [�; �]where � = 0 and � is the highest quantity of any commodity in fxng1n=n(").Therefore fu (xn)g1n=1 must have a convergent subsequence.Furthermore, every convergent subsequence has limit u (x), which is there-

fore the limit of the sequence fu (xn)g1n=1. To see this, suppose there is aconvergent subsequence

�u�xm(n)

�1n=1

(m is an increasing function) thatconverges to � 6= u (x). If e.g. � > u (x), then �e � u (x) e by monotonicity,and also �e � u (x) e, where � = 1

2(�+ u (x)) lies between � and u (x).

Because u�xm(n)

�! � > �, we have, for some N , u

�xm(n)

�> � for all

n > N . Then xm(n) � u�xm(n)

�e � �e, which implies x % �e, because % is

continuous and % (x) therefore closed. But x � u (x) e, which con�icts with�e � u (x) e. Analogously, � < u (x) leads to a contradicton.�

Exercise 3 (MWG 3.C.2). Prove the converse: if a continuous utilityfunction represents %, then % is continuous.

Example. Lexicographic preferences are not continuous and do not admita utility representation. With lexicographic preferences, there exists an or-dering of commodities, so that x(1) > y(1) =) x % y (where (1) refers tothe highest-priority commodity), x(1) = y(1) and x(2) > y(2) =) x % y, etc.(As in alphabetic entries in a telephone book or dictionary.)These preferences violate continuity: e.g. every bundle in the sequence

fxng1n=1, where xn(1) = 1nand xn(2) = 0 for all n, is preferred to y such that

y(1) = 0 and y(2) = 1, but the limit x = (0; : : : ; 0) is worse than y.

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Suppose u is a utility function representing lexicographic preference %.Let x(1) = y(1) = � and x(2) = 1; y(2) = 2. Whatever values u (x) andu (y) > u (x) the utility function assigns to x and y, we can �nd a rationalnumber r (�) 2 (u (x) ; u (y)). Lexicographic preference implies � > �0 =)r (�) > r (�0) (since x with x(1) = � is preferred to y0 with y0(1) = �0, i.e.� > u (x) > u (y0) > �0). Hence the function r maps one-to-one from R (anuncountable set) to Q (a countable set), a contradiction.One can understand the argument intuitively by noting that, with lexi-

cographic preferences, the indi¤erence sets are singletons, so that a di¤erentutility value has to be assigned for every bundle, i.e. every point of RL. Butthe utility values come from the (in some sense smaller) set R.

Exercise 4 (MWG 3.C.4). Find a preference relation that is not contin-uous but has a utility function that represents it.

2.2 Quasiconcavity

� Besides continuity (di¤erentiability), which allows us to apply calculusin solving the utility maximization problem, we would like to knowwhether the utility function has a maximum. This property (quasicon-cavity) relates to the convexity of preference.

De�nition. Preference relation.% is convex if 8x 2 X the upper contourset % (x) is convex: y; z 2% (x) =) 8� 2 [0; 1]

�y + (1� �) z 2% (x) :

� I.e. % is convex if y % x and z % x imply 8� 2 [0; 1], �y+(1� �) z %x.

� Preference relation % is strictly convex if y % x and z % x imply8� 2 (0; 1), �y + (1� �) z � x, provided y 6= z.

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� Convex preference can be interpreted as a desire for variety: if I liketwo bundles equally, then I �nd a mixture of the two (a more balancedbundle) more appealing. By the same token, I dislike extremes. If Ihave a lot of one commodity and little of the others, I am willing totrade aggressively (give up a lot) for a more balanced bundle. Thisresults in a "diminishing marginal rate of substitution": a high relativevaluation for commodities of which I have little, which decreases as Iacquire more.

Exercise 5 (MWG 3.C.1). Show that lexicographic preference is rational,monotone and strictly convex.

De�nition. Utility function u is quasiconcave if 8x 2 X the upper contourset % (x) � fa 2 X s.t. u (a) � u (x)g is convex.

� Quasiconcavity is weaker than concavity, which is easily apparent froman equivalent de�nition: u is quasiconcave if 8x; y 2 X and 8� 2 [0; 1]

u (�x+ (1� �) y) � min fu (x) ; u (y)g :

(Why are the de�nitions equivalent? Suppose x % y, then convexity of% (y) implies �x+ (1� �) y 2% (y), hence u (�x+ (1� �) y) � u (y).Conversely, suppose x; y 2% (z), and note that u (�x+ (1� �) y) �min fu (x) ; u (y)g � u (z) implies �x+ (1� �) y 2% (z).)

� Of course, a convex function satis�es: 8x; y 2 X and 8� 2 (0; 1)

u (�x+ (1� �) y) � �u (x) + (1� �)u (y) � min fu (x) ; u (y)g :

� Utility function u is strictly quasiconcave if 8x; y 2 X and 8� 2 (0; 1)

u (�x+ (1� �) y) > min fu (x) ; u (y)g ;

provided y 6= z.

� The graph of a convex function always lies above a line segment betweentwo of its points. The graph of a quasiconcave function only has to lieabove the lower of the two points. Figure 1 illustrates the concave andquasiconcave cases in panels (a) and (b).

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Figure 1: Concave and quasiconcave functions

� Quasiconcavity is su¢ cient to guarantee that a local extremum is aglobal maximum. If there existed a greater maximum or a minimum,then the utility function would have a trough somewhere and increaseon both sides of it, resulting in a "gap" (non-convexity) in the uppercontour set. However, the global maximum may not be unique if it ispart of a plateau.

Proposition. If a utility function represents a (strict) convex preferencerelation, then it is (strictly) quasiconcave.

Proof. Consider a weakly convex preference relation; I demonstrate thatit implies quasiconcavity (the strict case is analogous). To show:

u (�x+ (1� �) y) � u (x)

oru (�x+ (1� �) y) � u (y)

for all x; y 2 X, with � 2 (0; 1). With convex preference, �x+ (1� �) y % xwhenever y % x, and �x + (1� �) y % y whenever x % y. If u represents

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%, it must satisfy u (�x+ (1� �) y) � u (x) (i.e. the �rst inequality) ifu (y) � u (x), and u (�x+ (1� �) y) � u (y) (i.e. the second inequality) ifu (x) � u (y).�

Exercise 6 (MWG 3.B.1). Show: if % is monotone, then % is locallynonsatiated.

Exercise 7 (MWG 3.C.5).(a).Preference relation % is homothetic if 8x; y 2 X, y � x =) y � x,

and x � y =) 8� � 0, �x � �y. Utility function u is homogeneousof degree 1 if 8� > 0 u (�x) = �u (x). Show: a continuous preference ishomothetic i¤ it admits a utility function that is homogeneous of degree 1.(b) Let e1 = (1; 0; : : : ; 0) denote the bundle that contains 1 unit of com-

modity 1 and none of the other commodities. Preference relation.% is qua-silinear with respect to good 1 if 8x; y 2 X, 8� > 0, x + �e1 � x, andx � y =) 8� 2 R, (x+ �e1) � (y + �e1). Show: a continuous preferenceis quasilinear with respect to commodity 1 if it admits a utility function ofthe form u (x) = x1 + � (x2; : : : ; xL). (No need to show a converse.)

Exercise 8 (MWG 3.C.6). Consider preferences in a two-commodityworld represented by the CES (constant elasticity of substitution) utilityfunction

u (x) = (�1x�1 + �2x

�2)1=� :

Show:(a) When � = 1, the graphs of the indi¤erence.sets are linear.(b) As � ! 0, the CES utility function represents in the limit the same

preferences as the Cobb-Douglas utility function u (x) = x�11 x�22 .

(c) As � ! �1, the CES utility function has in the limit the sameindi¤erence sets as the Leontief utility function u (x) = min fx1; x2g.

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3 Demand I: Utility Maximization Problem

3.1 Budgets

� A consumer is constrained in her choices by a limited budget. If shehas wealth w, and commodities sell at market prices

p =

264 p1...pL

375 2 RL++;then her chosen bundle x 2 RL+ must satisfy

p � x = p1x1 + � � �+ pLxL � w:

� The set of bundles that meet the budget contraint, for given prices andwealth, is called the budget set.

De�nition. The Walrasian budget set is the set of all bundles x 2 RL+a¤ordable with wealth w at prices p:

Bp;w =�x 2 RL+ s.t. p � x � w

:

� Notice that a decline in any price enlarges the budget set: more bundlessatisfy the constraint.

� In a two-commodity world, the budget set is the area under the budgetline p1x1 + p2x2 = w.

� More generally, the budget set is bounded by an (L� 1)-dimensionalhyperplane, de�ned by p � x = w.

� The budget set is convex: if p � x � w and p � x0 � w, then 8� 2 [0; 1]we have p � x00 = p � �x + p � (1� �)x0 � w. It is also compact: closedand bounded, since x` � w=p` for ` = 1; : : : ; L (you can at most spendall your wealth on one commodity)

Exercise 9 (MWG 2.D.2). For an individual who consumes amount x ofa commodity priced at p, and h hours of leisure, when the hourly wage is 1,what is the Walrasian budget set?

15

3.2 The Utility Maximization Problem (UMP)

� Let the consumer preferences be rational, continuous, and locally non-satiated - which implies, from the previous lecture, that there is acontinuous and quasiconcave utility function representing it. The con-sumption set is X = RL+.

� The "utility maximization problem" is one of two equivalent ways toframe the consumer choice problem (the other is the "expenditure min-imization problem," which we will get to later on). In the UMP, theconsumer picks the most-preferred bundle in her Walrasian budget set,which gives utility

maxx�0

u (x) s.t. p � x � w:

� By the Weierstrass theorem, a continuous real-valued function on acompact, nonempty set has a maximum. Therefore the UMP has asolution as long as preferences are continuous.

� A solution to the UMP is in principle a set of consumption bundlesin the budget set, all of which give maximal utility. This set dependson the parameters of the budget constraint: commodity prices andindividual wealth. The map from prices and wealth to the set of utility-maximizing consumption bundles in the associated budget set is calledthe Walrasian demand correspondence.

� In general, the solution can be stated in terms of Kuhn-Tucker condi-tions: for ` = 1; : : : ; L,

@u (x)

@x`= �

@g (x)

@x`+

LX`=1

�`@h` (x)

@x`;

where � � 0 and the �` � 0 are Lagrange multipliers (shadow prices),

g (x) � p � x� w � 0;

is the budget constraint, and the

h` (x`) � �x` � 0;

` = 1; : : : ; L are the non-negativity constraints. If the relevant con-straint is non-binding, � = 0, respectively �` = 0 (i.e. the interior�rst-order condition holds).

16

� The conditions reduce to

@u (x)

@x`= �p` � �;

or equivalently@u (x)

@x`� �p`

with equality if x` > 0. More concisely we can write this as5u (x) � �p(with equalities where x` > 0) in terms of the gradient vector

5u (x) �

264@u(x)@x1...

@u(x)@xL

375 :� In a two-commodity world, where the budget constraint binds and bothgoods are consumed, these conditions imply

@u (x) =@x1@u (x) =@x2

=p1p2;

which is incidentally the tangency condition familiar from diagrams inintermediate micro books. The left side is the marginal rate of substi-tution, i.e. the slope of the indi¤erence curve (solve du = MU1dx1 +MU2dx2 = 0 for dx2=dx1). The right side is the slope of the bud-get line (found by totally di¤erentiating the budget line constraint, i.e.p1dx1+p2dx2 = 0, and solving for dx2=dx1.). To be tangent, the slopeshave to be equal. See Figure 2.

� Tangency is necessary for an interior optimum, but there are in�nitelymany points satisfying it for di¤erent levels of wealth. Therefore, thebudget constraint is needed to �x the solution.

Example. Consider preferences represented by the Cobb-Douglas utilityfunction u (x1; x2) = x�1x

1��2 . Note that, if u represents preferences, then so

must~u (x1; x2) = lnu (x1; x2) = � lnx1 + (1� �) ln x2

17

Figure 2: Optimal choice at the point of tangency

represent them, since the logarithm is an increasing transformation. Because~u (x1; x2) is strictly increasing in x1 and x2, the budget constraint

p1x1 + p2x2 � w

must bind.Thus, the constrained choice is the solution to the Lagrangean problem

maxx1;x2

L (x1; x2; �) = � lnx1 + (1� �) ln x2 + � (w � p1x1 � p2x2) ;

which satis�es �rst-order conditions

@L (x1; x2; �)

@x1=

�

x1� �p1 = 0

@L (x1; x2; �)

@x2=

1� �

x2� �p2 = 0

@L (x1; x2; �)

@�= w � p1x1 � p2x2 = 0:

The �rst two reduce to�

1� �

x2x1=p1p2;

18

Thus�p2x2 = (1� �) p1x1:

From the budget constraint (the third �rst-order condition), we have

w � p1x1 = p2x2;

hencex1 = �

w

p1and x2 = (1� �)

w

p2:

� If preference is monotonic, the budget will always be exhausted, sinceall commodities are valuable. But a weaker property, local nonsatia-tion, actually su¢ ces.

De�nition. Preference relation.% is locally nonsatiated if 8x 2 X and8" > 0, 9y 2 X such that ky � xk � " and y � x.

� Local nonsatiation (which also has the e¤ect of ruling out thick indif-ference sets) is a more plausible property than monotonicity, since itallows for limited wants.

De�nition. The Walrasian demand correspondence x (p; w) satis�es Walras�law if 8p� 0,8w > 0 and 8x 2 x (p; w), p � x = w.

� Walras�law is satis�ed if the consumer�s choice exhausts the budget.

� The demand correspondence is homogeneous of degree k if x (�p; �w) =�kx (p; w).

Proposition. If u is a continuous utility function that represents locallynon-satiated preference relation % on X = RL+, then 8p � 0, 8w > 0,8x 2 x (p; w) the Walrasian demand correspondence is homogeneous of degreezero and satis�es Walras�law. If u is in addition quasiconcave, then x (p; w)is convex, and if u is strictly quasiconcave, then x (p; w) is a singleton.

19

Proof. Homogeneity of degree zero follows from the fact that the budgetconstraint is unchanged when prices and wealth are scaled by �:

�p � x � �w () p � x � w:

Walras�law re�ects non-satiation. If p � x < w for x 2 x (p; w), then bynon-satiation there exists in every neighborhood of x an x0 such that x0 � x.If we pick a su¢ cienty small neighborhood, then p � x0 < w, so that x0 is inthe budget set. But then x is not a utility-maximizing choice, a construction.If u is quasiconcave, then its upper contour sets are convex. Since all

elements of x (p; w) are equally preferred, x (p; w) is also the upper contourset of any of its members: if x 2 x (p; w), then

x (p; w) =% (x) = fa 2 X s.t. u (a) � u (x)g :

Hence x (p; w) inherits the convexity of % (x). If u is strictly quasiconcaveand x (p; w) has two distinct elements x and x0, then

u (x00) = u (�x+ (1� �)x0) > min (u (x) ; u (x0)) :

But since u (x) = u (x0) = min (u (x) ; u (x0)), this implies x00 (which is in thebudget set, because it is convex) is strictly preferred to both x and x0, whichcontradicts x; x0 2 x (p; w) :�

Exercise 10 (MWG 3.D.1). Verify that the above proposition holds forthe Walrasian demand function with Cobb-Douglas utility.

� The following is a natural generalization of the continuity concept forpoint-valued functions to set-valued correspondences. Informally it saysthat solutions in one constraint set should still be solutions at a verysimilar constraint set.

De�nition. The Walrasian demand correspondence is upper hemi-continuousif 8 (p; w), x 2 x (p; w) whenever (p; w) and x are limits of sequences f(pn; wn)g1n=1and fxng1n=1 such that xn 2 x (pn; wn) for all n.

Proposition. If u is a continuous utility function that represents locallynon-satiated preference relation % on X = RL+, then 8p � 0, 8w > 0 theWalrasian demand correspondence is upper hemi-continuous.

20

Proof. Suppose sequences f(pn; wn)g1n=1 and fxng1n=1 converge to (p; w)

and x, and we have xn 2 x (pn; wn) for all n, but x =2 x (p; w). Then thereexists x0 in Bp;w such that u (x0) > u (x). By continuity of u, there existsalso y 2 Bp;w such that u (y) > u (x). Since (pn; wn) converges to (p; w), itmust be the case for all su¢ ciently large n that y 2 Bpn;wn. This impliesu (xn) � u (y), since xn is in the choice set for Bpn;wn. But as fxng1n=1converges to x, this argument leads to u (x) � u (y), a contradiction.�

� Of course, the result implies that, if the Walrasian demand correspon-dence is in fact a function, then this function is continuous.

3.3 Indirect Utility Function

� The value of the utility function at a solution to the UMP is the highestit can attain on the budget set, i.e. for a particular set of prices andwealth. The map from prices and wealth to the highest attainableutility value is called the independent utility function.

� The indirect utility function v is quasiconvex if 8�v the lower contourset f(p; w) s.t. v (p; w) � �vg is convex.

Proposition. If u is a continuous utility function that represents locallynonsatiated preference relation % on X = RL+, then 8p � 0, 8w > 0, theindirect utility function is homogeneous of degree zero, i.e. v (�p; �w) =v (p; w), strictly increasing in w and non-increasing in p` for ` = 1; : : : ; L,quasiconvex, and continuous in p and w.

Proof. The indirect utility function is homogeneous of degree zero becausethe Walrasian demand correspondence is: since the scaling of prices andwealth does not a¤ect the set of utility-maximizing choices, it cannot a¤ectthe utility derived from them.Since wealth increases and price decreases enlarge the budget set, the

best available choice can only improve, so that utility associated with itcannot decrease. Because of local nonsatiation, which implies that the utility-maximizing choice lies in the boundary of the budget set, the indirect utilityfunction must strictly increase in wealth.

21

To establish quasiconvexity, suppose v (p; w) � �v and v (p0; w0) � �v andconsiderv (p00; w00), the maximal utility attainable in the budget set de�nedby prices

p00 = �p+ (1� �) p0

and wealthw00 = �w + (1� �)w0:

For any x in this budget set,

�p � x+ (1� �) p0 � x � �w + (1� �)w0;

so p � x � w or p0 � x � w0 must be true. In the �rst case, x 2 Bp;w, sou (x) � v (p; w). In the second case, x 2 Bp0;w0, so u (x) � v (p0; w0). Thusu (x) � �v, i.e. x 2 f(p; w) s.t. v (p; w) � �vg, the lower contour set at �v.Because x was arbitrary, all bundles in Bp00;w00 belong to the lower contourset at �v, and because �v was arbitrary, v is quasiconvex.In the more restrictive case that u is strictly quasiconcave, x (p; w) is

continuous, and u (x (p; w)) is a composition of continuous functions, so thatit is also continuous.�

Exercise 11 (MWG 3.D.2). Verify that the above proposition holds for theinverse utility function with the log transformation of Cobb-Douglas utility.

Exercise 12 (MWG 3.D.6). In a three-commodity setting, let preferencesbe represented by the utility function

u (x) = (x1 � b1)� (x2 � b2)

� (x3 � b3) :

(a) Why is there no loss of generality from imposing � + � + = 1?Assume this for the remaining parts.(b) What are the �rst-order conditions for the UMP? Derive Walrasian

demand and the indirect utility function.(c) Verify that the general properties of Walrasian demand and indirect

utility functions hold in this case.

22

4 Demand II: ExpenditureMinimization Prob-lem

4.1 EMP and Hicksian Demand

� In the UMP ("utility maximization problem"), an optimal choice max-imizes utility on a �xed budget set. Analogously, we can de�ne optimalchoice as minimizing expenditure on a �xed upper contour set (i.e. aset in which all bundles yield at least a certain utility). This approachis called the EMP ("expenditure minimization problem").

� In the EMP, a consumer picks the consumption bundle for which ex-penditure is

minx�0

p � x s.t. u (x) � u

= maxx�0

(�p � x) s.t. u (x) � u:

� We assume p� 0 and u > u (0) throughout (so that at least one com-modity must be consumed, and none in in�nite quantity). Further-more, we assume that u represents a continuous, locally non-satiatedpreference on RL+, and is di¤erentiable.

� The set of consumption bundles that are solutions to the EMP at pricesp and required utility u is denoted as h (p; u) � RL+ and called theHicksian demand correspondence (or function, if single-valued).

Exercise 13 (MWG 3.E.3). Argue that a solution to the EMP exists ifp� 0 and u (x) � u for some x 2 RL+.

� In parallel to the UMP, we apply the Kuhn-Tucker conditions: at asolution x�, we have for ` = 1; : : : ; L,

�@ (p � x�)

@x`= �

@ (u� u (x�))

@x`+

LX`=1

�`@ (�x�`)@x`

;

where � � 0 and the �` � 0 are Lagrange multipliers (shadow prices).If the utility constraint is non-binding at x, i.e. u (x) > u, we have� = 0. If the `th non-negativity constraint is non-binding at x, i.e.x` > 0, then �` = 0.

23

� Resolving the derivatives,

�p` = ��@u (x�)

@x`� �`;

i.e. (after rearranging and relabeling the multiplier ~� � 1=�)

@u (x�)

@x`� ~�p`

with equality if x` > 0 (so that �` = 0).

� This is exactly analogous to the �rst-order conditions in the UMP. In atwo-commodity world, where the utility constraint binds (so that ~� <1) and both goods are consumed, the solution x� is again characterizedby the tangency condition:

@u (x�) =@x1@u (x�) =@x2

=p1p2:

� The only di¤erence is that the remaining constraint that �xes x� is nownot the budget constraint, but the utility constraint u (x�) = u.

Example. Reconsider preferences represented by the Cobb-Douglas utilityfunction u (x1; x2) = x�1x

1��2 . The utility constraint has to bind, since expen-

diture p � x is strictly decreasing in x1 and x2. (If it did not bind, you couldslightly lower consumption of one or both of the goods and attain a lowerexpenditure within the utility constraint.) Moreover, u (x) = u (0) if x1 orx2 is zero, so they must be strictly positive for x to attain u > u (0).The constrained choice therefore satis�es the "tangency" condition, which

specializes to�

1� �

x�2x�1=p1p2:

given the Cobb-Douglas marginal utilities.From the utility constraint, we have

x��1 x�1��2 = u;

hence

x�2 =

�x�2x�1

��u

24

and

x�2 =

�1� �

�

p1p2

��u and x�1 =

��

1� �

p2p1

�1��u:

� Notice that x�1 and x�2 are functions of prices and the required utility(not of wealth) in the EMP. The wealth that is needed to attain u isallowed to vary. In this sense, Hicksian demand is also called com-pensated demand, because if prices increase, expenditure is implicitlyadjusted as needed in order to keep utility constant. But the consump-tion bundle x may change so as to make the increase in expenditure assmall as possible.

� Hicksian demand has properties that correspond to those of Walrasiandemand. The Hicksian demand correspondence is said to have no excessutility if 8x 2 h (p; u), u (x) = u.

Proposition. If u represents continuous, locally non-satiated preferenceson RL+, and p � 0, then the Hicksian demand correspondence h (p; u) ishomogeneous of degree zero in p, has no excess utility, is convex if preferenceis convex, and single-valued if preference is strictly convex.Proof. Homogeneity of degree zero follows from the fact that the upper

contour set at u (the constraint set), and therefore any solution to EMP, isuna¤ected by scaling prices:

x 2 h (�p; u) () x 2 h (p; u) :

(It is only the expenditure that changes, not the bundle that minimizes it -the scaling leaves relative prices unaltered.)No excess utility re�ects continuity of the utility function, which is in-

herited from preferences. If there were a solution x 2 h (�p; u) such thatu (x) > u, then we could construct a scaled-down bundle x0 = �x with� 2 (0; 1) that satis�es p �x0 < p �x (since x0 � x) and u (x0) � u (for � closeenough to 1, by continuity). But this means x0, and not x, can be a solutionto EMP at u, a contradiction.Let x; x0 2 h (p; u), so that p � x = p � x0. If preference is convex (utility

quasiconcave), then upper contour sets are convex, so

x00 = �x+ (1� �)x0

25

attains utility u. Moreover,

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

It follows that x00 also minimizes expenditure and belongs to h (p; u). Ifpreference is strictly convex (utility strictly quasiconcave), then x00 � x (andx00 � x0 since h (p; u) satis�es no excess utility, so that x0 � x). Continuityimplies there exists � 2 (0; 1) such that �x00 � x, i.e. u (�x00) > u (x). Butsince �x00 � x00 and p � x00 = p � x, we have p � �x00 < p � x, which impliesthat x does not minimize expenditure on the constraint set, i.e. x =2 h (p; u),a contradiction. Hence two such elements x; x0 of h (p; u) cannot exist, andh (p; u) must be a singleton.�

4.2 Expenditure Function

� The value of p � x� at a solution x� to the EMP is denoted as e (p; u)and called the (minimum) expenditure.

� The expenditure function is homogeneous of degree one in p if 8� > 0,e (�p; u) = �e (p; u).

Proposition. If u represents a continuous, locally non-satiated preferenceon RL+, and p � 0, then the expenditure function e (p; u) is homogeneousof degree one in p, strictly increasing in u and non-decreasing in p` for` = 1; : : : ; L, concave in p, and continuous in p and u.

Proof. Since scaling up the price does not change the constraint set(i.e. the upper contour set at u), it does not a¤ect the solution x�. Hencee (�p; u) = �p � x� = �e (p; u), so that the expenditure function is homoge-neous of degree one in p.If e (p; u) were not strictly increasing in u, then there would exist solutions

x0 and x00, respectively at u0 and u00 > u0, such that p � x0 � p � x00 > 0. (Wemaintain u > u (0), so x0; x00 6= 0.) Continuity implies there is a bundlex = �x00 for some � 2 (0; 1) that satis�es u (x) > u0 and p � x < p � x00 � p � x0(since x� x00). But then x0 cannot be a solution to EMP.Suppose e (p; u) were strictly decreasing in p` for some `. I.e. if we

compare expenditure at two price vectors p0 and p00 di¤ering only in thatp00` � p0`, then e (p

00; u) < e (p0; u). Since the constraint set is not a¤ected

26

by the price di¤erence, the same x is a solution with both p and p0, soe (p00; u) = p00 � x � p0 � x = e (p0; u), a contradiction.If x00 solves EMP at u with prices p00 = �p+(1� �) p0 for � 2 [0; 1], then

e (p00; u) = p00 � x00

= �p � x00 + (1� �) p0 � x00

� �e (p; u) + (1� �) e (p0; u) ;

since x00 is available at prices p and p0 (so expenditure with prices p and p0

at the minimizing bundles x and x0 cannot be larger then at x00). Hence theexpenditure function is concave in p.We do not prove continuity.�

� Concavity follows from the fact that, if p is increased to p0 and x kept�xed, expenditure (to maintain utility level u) increases linearly toe (p0; u). The consumer can always attain u at an expenditure no greaterthan e (p0; u), but may be able adjust x to x0 to attain u at reducedexpenditure. Since this is true for all p, e (p0; u) increases less thanlinearly in p, thus is concave.

Exercise 14 (MWG 3.E.2). Con�rm that the general properties of theHicksian demand function and the expenditure function hold with Cobb-Douglas preferences.

Exercise 15 (MWG 3.E.6). For the CES (constant elasticity of substitu-tion) utility function

u (x) = (x�1 + x�2)1=� ;

derive the Hicksian demand function and the expenditure function, and verifytheir general properties in this case.

Exercise 16 (MWG 3.E.7). If preferences are quasilinear with respect tothe �rst good, show that the Hicksian demand functions for the remaininggoods are invariant to u, and �nd the form of the expenditure function.

27

4.3 Duality

� The connection between Walrasian and Hicksian demand is that UMPand EMP have the same solutions when wealth w in the UMP is �xedat the level of minimized expenditure p � x� � e (p; u) in the EMP,and (equivalently) when utility u in the EMP is �xed at the level ofmaximized (or indirect) utility u (x�) � v (p; w) in the UMP. I.e.

h (p; u) = x (p; e (p; u)) and x (p; w) = h (p; v (p; w)) :

Example. Compare the Walrasian and Hicksian demands for Cobb-Douglaspreferences:

xw1 = �w

p1and xw2 = (1� �)

w

p2and

xh1 =

��

1� �

p2p1

�1��u and xh2 =

�1� �

�

p1p2

��u:

Equating either the x1s or x2s gives

w =�p1�

��� p21� �

�1��u:

Since

u (xw) =

��w

p1

���(1� �)

w

p2

�1��=

��

p1

���1� �

p2

�1��w � v (p; w)

in the UMP, we see that xw = xh if u = u (xw). On the other hand, since

p � xh = p1

��

1� �

p2p1

�1��u+ p2

�1� �

�

p1p2

��u

=

��

1� �

�1��+

�1� �

�

��!p�1p

1��2 u

=�p1�

��� p21� �

�1��u � e (p; u) ;

in the EMP, xw = xh if w = p � xh.

28

� The UMP and EMP are generally equivalent in the following sense.

Proposition. If u represents continuous, locally non-satiated preferenceson RL+, and p� 0, then:(i) if x� solves the UMP at wealth w > 0, then x� solves the EMP at

u = u (x�), and e (p; u) = w;(ii) if x� solves the EMP at utility u > u (0), then x� solves the UMP at

w = e (p; u), and u (x�) = u.

Proof. (i) Suppose x� is a solution to UMP at w, but not to EMP atu = u (x�). Let instead x0 be a solution to EMP at u = u (x�), so thatp � x0 < p � x� and u (x0) � u. Local nonsatiation implies that there exists x00

such that u (x00) > u (x0) su¢ ciently close to x0 for p � x00 < p � x� to still hold.But then x00 2 Bp;w and u (x00) > u (x�), so that x� was not a solution toUMP. By contradiction, x� solves EMP, and therefore e (p; u) = p � x� = w.(ii) Conversely, suppose x� is a solution in EMP at u > u (0), but not

in UMP at w = p � x�. Let instead x0 be a solution to UMP at w, so thatu (x0) > u (x�) and p � x0 � p � x�. By continuity, we can scale x0 down tox00 = �x0, with � 2 (0; 1) su¢ ciently close to 1, such that u (x00) > u (x�) stillholds. (Note that u > u (0) implies x� 6= 0 and x0 6= 0, so p � x� > 0.) Sincex00 � x�, we have p � x00 < p � x�, and x� cannot be a solution to EMP. Bycontradiction, x� solves UMP, and therefore u (x�) = u.�

Exercise 17 (MWG 3.E.9). Using the equivalence of the UMP and EMP,show that the general properties of the indirect utility function (homogeneousof degree zero, strictly increasing in w, non-increasing in prices, quasiconvex,continuous in p and w) imply the general properties of the expenditure func-tion (homogeneous of degree one in p, strictly increasing in u, non-decreasingin prices, concave in p, continuous in p and u), and vice versa.

Exercise 18 (MWG 3.E.10). Using the equivalence of the UMP andEMP and the properties of indirect utility and expenditure functions, showthat properties of the Walrasian demand function for continuous, locallynon-satiated preferences (homogeneous of degree zero, Walras�law, convex /single-valued if preferences are convex / strictly convex) imply properties ofthe Hicksian demand function (homogeneous of degree zero in p, no excessutility, convex / single-valued if preferences are convex / strictly convex),

29

and vice versa. (Note there is a typo in the book - you are to show thatProposition 3.D2 implies Proposition 3.E3, not 3.E4).

� The relationship between UMP and EMP is just a special case of a farmore general theory of duality. In this connection, duality means thata constrained maximization problem can be expressed as a constrainedminimization problem, swapping objective and constraint.

5 Comparative Statics

5.1 Wealth E¤ects

� In this lecture, we examine the behavior of the demand function asprices and wealth change. We assume for now that demand is single-valued (i.e. preferences are strictly convex).

� The wealth e¤ect for the `th good is @x` (p; w) =@w.

� If the wealth e¤ect is positive, i.e. @x` (p; w) =@w � 0, we say that thegood is normal. If the wealth e¤ect is negative, i.e. @x (p; w) =@w < 0,we say that the good is inferior.

� Goods are inferior when there are higher-quality, costlier substitutesfor the agent to switch to as wealth increases. (E.g. supermarket breadvs. fresh bread from a bakery.)

� The total wealth e¤ect is given by the derivative vector with respect tow:

Dwx (p; w) =

264@x1(p;w)@w...

@xL(p;w)@w

375 2 RL:5.2 Price E¤ects

� The price e¤ect for the `th good is @x` (p; w) =@p`.

� Typically, we expect the price e¤ect to be negative. A good for whichthe price e¤ect is positive, so that the agent consumes more of it afterits price increases, is called a Gi¤en good.

30

� A Gi¤en good is similar to an inferior good - actually, it is just a veryinferior good. As price increases, the agent�s budget set shrinks, and ifthe wealth e¤ect (having more of an inferior good when you are poorer)is su¢ ciently powerful, we have a Gi¤en good.

� The total price e¤ect is given by the derivative matrix with respect top:

Dpx (p; w) =

264@x1(p;w)@p1

� � � @x1(p;w)@pL

.... . .

...@xL(p;w)@p1

� � � @xL(p;w)@pL

375 :� We consider now some relationships between price and wealth e¤ects.Recall that Walrasian demand is homogeneous of degree zero. An im-pliciation is that changes to the consumption bundle in response tosimultaneous (and proportionate) increases in prices and wealth can-cel. This merely re�ects the invariance of the budget constraint.

Proposition. If Walrasian demand x (p; w) is homogeneous of degree zero,then 8p and 8w,

Dpx (p; w) p+Dwx (p; w)w = 0:

Proof. By zero-homogeneity, 8� > 0,

x (�p; �w)� x (p; w) = 0;

i.e. 264 x1 (�p; �w)...

xL (�p; �w)

375�264 x1 (p; w)

...xL (p; w)

375 =24 000

35This is true, in particular, for � = 1. Totally di¤erentiating with respect to�, we have for ` = 1; : : : ; L,

LXk=1

@x` (�p; �w)

@pkpk +

@x` (�p; �w)

@ww = 0;

which corresponds to the claim in matrix notation when � = 1.�

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� Walras� law implies that an increase in prices (while wealth remains�xed) must be accompanied by an o¤setting decrease in consumption,and that an increase in wealth (while prices remain �xed) brings withit a corresponding increase in consumption.

� To understand how the following results relate to this intuition, no-tice that a price increase for all commodities would increase the re-quired wealth in proportion x (p; w), and therefore the decrease in re-quired wealth from reduced consumption, p�Dpx (p; w), has to be of thesame magnitude. Similarly, an increase in available wealth has to bematched by the increase in required wealth from greater consumption,p �Dwx (p; w).

Proposition. If Walrasian demand x (p; w) satis�es Walras� law, then 8pand 8w,

x (p; w)T + pTDpx (p; w) = 0T

andpTDwx (p; w) = 1:

Proof. Di¤erentiating Walras�law,

p � x (p; w) = w;

�rst with respect to prices, we have for ` = 1; : : : ; L

x` (p; w) +LXk=1

pk@xk (p; w)

@p`= 0;

which corresponds to the �rst claim in matrix notation.Di¤erentiating Walras� law with respect to wealth, we have for ` =

1; : : : ; LLX`=1

p`@x (p; w)

@w= 1;

which corresponds to the second claim in matrix notation.�

32

Exercise 19 (MWG 2.E.3). If x (p; w) is homogeneous of degree zero, i.e.8� > 0, x (�p; �w) = x (p; w), and satis�es Walras�law, show that

p �Dpx (p; w) p = �w:

Interpret.

Exercise 20 (MWG 2.E.4). If x (p; w) is homogeneous of degree one withrespect to w, i.e. 8� > 0, x (p; �w) = �x (p; w), and satis�es Walras�law,show that "`w (p; w) = 1 for ` = 1; : : : ; L. Interpret.

Exercise 21 (MWG 2.E.6). In the case of the demand function

x1 (p; w) =p2

p1 + p2 + p3

w

p1

x2 (p; w) =p3

p1 + p2 + p3

w

p2

x3 (p; w) =p1

p1 + p2 + p3

w

p3;

verify the three derivative conditions for a demand function x (p; w) that ishomogeneous of degree zero and satis�es Walras�law.

5.3 Law of Demand

� The "law of demand" refers to the intuitive property that a price in-crease should reduce demand for a commodity. This statement is, how-ever, not generally valid for Walrasian demand.

� Exceptions are the Gi¤en goods. They arise because a price increasee¤ectively makes the agent poorer and may lead to an increase in theconsumption of relatively cheap commodities.

� In the Hicksian de�nition of demand, a price increase is accompaniedby an increase in expenditure, so it does not "impoverish," and theGi¤en e¤ect is absent. Hence the two de�nitions of demand di¤er incomparative statics.

� Hicksian demand is said to obey the "compensated law of demand,"i.e. (single-valued) demand for any commodity diminishes if its priceincreases.

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Proposition. If u represents continuous, locally non-satiated preferenceson RL+, and the Hicksian demand correspondence h (p; u) is single-valued forall p� 0, then 8p0; p00,

(p00 � p0) � (h (p00; u)� h (p0; u)) � 0:

Proof. Simply observe that

p00 � h (p00; u) � p00 � h (p0; u)p0 � h (p0; u) � p0 � h (p00; u) ;

since h (p00; u) minimizes expenditure when prices are p00, and h (p0; u) mini-mizes expenditure when prices are p0. Subtracting p0 � h (p0; u) on the rightand p0 � h (p00; u) on the left of the �rst inequality preserves signs and givesthe result.�

5.4 Elasticity

� The partial derivatives are not unit-free measures. E.g. @x` (p; w) =@p`would depend on the currency in which we quote prices: clearly, a $1increase in the price of petrol would cause a greater response in demandthan a U1 increase (which translates into about one cent). Moreover,@x` (p; w) =@p` depends in this case on whether petrol is sold by gallonor by liter (about a quarter gallon).

� An alternative way to describe price and wealth e¤ects is in terms ofelasticities, which are unit-free.

De�nition. The price elastisticity of demand x` (p; w) with respect to thekth good is:

"`k (p; w) �@x` (p; w)

@pk

pkx` (p; w)

:

If ` = k, "`k (p; w) is called "own-price elasticity." If ` 6= k, "`k (p; w) iscalled "cross-price elasticity." The wealth elastisticity of demand x` (p; w) is:

"`w (p; w) �@x` (p; w)

@w

w

x` (p; w):

34

� Elasticities can be interpreted as the approximate relative percent changein two variables (this relationship holds exactly only in the limit, whenthe change is very small): e.g. the percent change in x` (p; w) in re-sponse to a small percent change in p`.

Exercise 22 (MWG 2.E.8). Demonstrate that the price elasticity of de-mand x` (p; w) with respect to the kth good can be expressed as

"`k (p; w) =@ ln (x` (p; w))

@ ln pk;

and derive a corresponding expression for "`w (p; w).

� In terms of elasticities, we can restate the relationship between theresponses in demand for commodity ` to changes in prices and wealth(as implied by homogeneity) as:

LXk=1

"`k (p; w) + "`w (p; w) = 0:

5.5 Money Metric

� In the remainder, we consider the welfare implications of price changes.First, we derive a quantitative measure of welfare change in terms of theexpenditure function. Since data availability is normally a signi�cantconstraint, we are also interested in conditions under which a price iswelfare-improving that are based on minimal information (such as theinitial price and demand and the new price).

� Let the consumer�s preference relation be rational, continuous, locallynonsatiated and the expenditure and indirect utility functions di¤eren-tiable. Suppose wealth is �xed at w > 0, and the price changes fromp0 to p1.

� If consumer�s preferences are known, then she is better o¤ after theprice change if and only if v (p1; w)� v (p0; w) � 0.

35

� Due to the duality of the UMP and EMP, we can express the indirectutility at p1 and p0 in terms of the expenditure required to attain it.Recall that the expenditure function e (p; u) is strictly increasing in u.Hence, at given prices, the expenditure function is itself an indirectutility function when u is evaluated at u = v (p; w).

� I.e. we have, for any �xed price �p� 0,

e (�p; v (p; w)) � e (�p; v (p0; w0)) () v (p; w) � v (p0; w0) ;

and e (�p; v (p; w))�e (�p; v (p0; w0)) is a meaningful quantitative measureof welfare change. It is the change in wealth needed to buy the optimalbundle when the budget set changes from Bp;w to Bp0;w0.

� The expenditure function evaluated at u = v (p; w) is called the moneymetric (indirect utility function). It is independent of the utility repre-sentation we choose for the agent�s preference, as all utility representa-tions select the same consumption bundle at given prices and wealth.Hence it is unique up to the choice of �p.

� Note that because indirect utility is decreasing in prices, so is the moneymetric. (Expenditure increases in its price arguments, but those pricesare �xed in the money metric.)

� Clearly, the agent is better o¤ at prices p1 than at price p0 if and onlyif e (�p; v (p1; w))� e (�p; v (p0; w)) � 0 (for any indirect utility function),i.e. if a bundle that yields the maximal utility attainable at prices p1 ismore expensive than a bundle that yields the maximal utility attainableat prices p0.

5.6 Welfare Comparisons

� Let u0 � v (p0; w) and u1 � v (p1; w). Then we can de�ne the changein welfare by

e�p0; u1

�� e

�p0; u0

�= e

�p0; u1

�� e

�p1; u1

�;

since e (p0; u0) = e (p0; v (p0; w)) = w = e (p1; v (p1; w)) = e (p1; u1)through the equivalence of UMP and EMP.

36

� This is the additional wealth the agent would have needed at the oldprices p0 in order to attain the level of utility u1 that is available underthe new prices p1.

� The following is also a plausible way to de�ne the change in welfarefrom the money metric:

e�p1; u1

�� e

�p1; u0

�= e

�p0; u0

�� e

�p1; u0

�;

This is what the agent has to spend at the new prices p1 in order tomaintain the original utility u0 that was available at the old prices p0.

� Note, however, that if there were wealth e¤ects, then these measureswould not coincide. Consider a price change in good 1 only. If p01 > p11,so that u1 > u0, then u1 is associated with greater wealth at givenprices. If good 1 is normal, this means more of it is demanded atoptimum. If good 1 is inferior, less of it is demanded. Hence the chosenbundles and their associated expenditures would depend on how u is�xed.

Exercise 23 (MWG 3.I.3). Welfare change as measured by

EV�p0; p1; w

�� e

�p0; u1

�� e

�p1; u1

�is called the "equivalent variation" (EV), and

CV�p0; p1; w

�� e

�p0; u0

�� e

�p1; u0

�is called the "compensating variation" (CV). (Respectively, these tell us thechange in wealth that is required to maintain utility at u1 and u0 after aprice change.) Suppose the price of good ` falls (other price remained �xed),giving the new price vector p1 � p0. Demonstrate that CV (p0; p1; w) >EV (p0; p1; w) if good ` is inferior.

Exercise 24 (MWG 3.I.5). Suppose u (x) is quasilinear with respect tothe �rst good (and p = 1 is �xed). Show that CV (p0; p1; w) = EV (p0; p1; w)for any prices p0 and p1, at all wealth levels w.

Exercise 25 (MWG 3.I.6). Let there be a population of consumers indexedby i = 1; : : : ; I, with utility functions ui (x) and individual wealths wi. For

37

any change in prices from p0 to p1 such thatP

iCVi (p0; p1; w) > 0, show that

it is possible to compensate everyone for lost utility. I.e. there exist wealthlevels fw0ig

Ii=1 such that

Piw

0i �

Piwi and vi (p

1; w0i) � vi (p0; wi) for all i.

� Absent wealth e¤ects, the two de�nitions of welfare change are equiv-alent. We can use the fact that, for ` = 1; : : : ; L,

@e (p; u)

@p`= h` (p; u)

(by the envelope theorem, since e (p; u) = p �h (p; u) is locally invariantto changes in its minimizer h (p; u)) to derive

e�p0; u

�� e

�p1; u

�=

LX`=1

Z p0`

0

h` (p; u) dp` �LX`=1

Z p1`

0

h` (p; u) dp`

=LX`=1

Z p0`

p1`

h` (p; u) dp`:

� This is the change in consumer surplus as a result of the price change.See Figure 3 for illustrations of the consumer surplus from good 1 beforeand after a price cut.

� What if only partial information is available about demand, such asthe initial price p0 and choice x0 � x (p0; w), and we wish to evaluatethe impact of a change in prices to p1?

Proposition. If preference is locally nonsatiated, then the agent is strictlybetter o¤ under (p1; w) than under (p0; w) if (p1 � p0) � x0 < 0.Proof. By Walras�law, p0 � x0 = w, so (p1 � p0) � x0 < 0 implies p1 � x0 < w.But then x0 is in the interior of the budget set at p1, and by local nonsatiationthere exists a bundle x 2 Bp1;w that is strictly preferred to x0.�

Exercise 26 (MWG 3.I.12). Extend this test of welfare improvement tochanges in prices and wealth from (p0; w0) to (p1; w1), where it is now notnecessarily the case that w1 = w0. (No need to do this for equivalent andcompensating variation.)

38

Figure 3: Change in consumer surplus

� The converse is not always true: (p1 � p0) � x0 > 0 does not implythat the agent is worse o¤ after the price change. To appreciate thedi¤erence, look at Figure 4, where the two scenarios are depicted inprice space.

� The set of prices that keep expenditure constant at e (p0; x0) is drawnas a convex curve, because the expenditure function is concave in eachprice. (I.e. keeping p1 �xed, expenditure increases at a diminishingrate as p2 increases, hence smaller reductions in p1 are required too¤set increases of given size in p2.)

� Since p0 is an optimal choice, it attains e (p0; x0), and therefore lies onthe curve. The gradient of the �xed-expenditure curve at this pointis rpe (p

0; x0) = x0, by the envelope theorem: since x0 minimizese (p0; x0) at p0, i.e. rxe (p

0; x0) = 0, we have for ` = 1; 2,

de (p0; x0)

dp`=@e (p0; x0)

@p`+@e (p0; x0)

@x`

@x`@p`

=@e (p0; x0)

@p`= x`:

� The gradient vector x0 of e (p; u0) is orthogonal to the tangent of thelevel set fp s.t. e (p; u0) = e (p0; u0)g at p0. To see this, think of a vector

39

Figure 4: Scenarios (p1 � p0) � x0 < 0 (left) and (p1 � p0) � x0 > 0 (right)

in the tangent space (i.e. a vector in the direction of the tangent). Itmust be a multiple of v � (1;�x01=x02), since a one-unit increase in p1requires p2 to be reduced by (x01=x

02) p1 to keep expenditure constant.

Clearly, x0 � v = 0.

� Since p1�p0 is the vector "from" p0 to p1, it must lie below the tangentto the level curve, which is orthogonal to x0, when (p1 � p0) � x0 < 0,and above when (p1 � p0) � x0 > 0. The concavity of the expenditurefunction therefore implies that e (p1; u0) < e (p0; u0) in the �rst case (sothat there is a welfare improvement), but not necessarily in the second.As drawn in the right panel of Figure 4, welfare decreases, since p1 liesabove the level curve at e (p0; u0), but if the price change (in the samedirection) were su¢ ciently large, p1 could lie beneath the level curve.

Exercise 27 (MWG 3.I.11). Suppose x1 = x (p1; w) is known in additionto p0, p1 and x0. Argue that the agent is worse o¤ at p1 than at p0 if(p1 � p0) � x1 > 0, or equivalently if p0 � (x1 � x0) < 0 (wealth is �xed).Give graphic intuition for these results. (They can be established via �rst-order approximation of the expenditure function at p1, but direct proofs areenough.)

40

6 Choice-Based Approach

6.1 Choice Structures

� So far we have assumed that choice behavior is consistent with an un-derlying rational preference ordering. Since preferences are unobserv-able, their existence and properties are not directly testable. In thislecture, we will build a largely parallel theory of demand that startsfrom the properties of observable choices.

� A choice structure (B; C (�)) consists of a family B of budget sets anda choice rule C (�) that assigns a nonempty subset C (B) � B to everyB 2 B.

� In this context, a budget set B 2 B may be thought of as a speci�cdecision problem, of many possible problems, the agent may face. Theproblem the agent solves is to choose one or more elements from thisset.

� Since existence of a preference is not assumed, it is not meaningful tosay that objects in C (B) are indi¤erent to each other and preferred toeverything else in B. The set C (B) simply describes the objects theagent might be observed to choose when presented with budget set B,whatever the reasons.

� One may de�ne a "revealed preference" relation from a choice rule asfollows.

De�nition. %� is the "revealed preference" derived from choice structure(B; C (�)) if x %� y ("x is revealed preferred to y") if and only if x 2 C (B)for some B 2 B such that x; y 2 B.

� The relation%� need have none of the rationality properties we assumedfor primitive preferences. For example, x and y are only comparable ifx 2 C (B) or y 2 C (B) for some B 2 B such that x; y 2 B. If x and yare never chosen, then we have no information about them. Hence %�is not necessarily complete.

41

� There is also no guarantee of transitivity. For example, if C (fx; yg) =fxg, C (fy; zg) = fyg, and C (fx; zg) = fzg, then x �� y �� z, butz �� x.

Exercise 28 (MWG 2.F.4). Laspeyres and Paasche indices measure thechange in consumption between two points in time at �xed prices. Letp0 and q0 denote prices and quantities at time 0, and let p1 and q1 de-note the new prices and quantities at time 1. The Laspeyres index, LQ �(p0 � x1) = (p0 � x0), is based on initial prices. The Paasche index, PQ �(p1 � x1) = (p1 � x0), uses new prices. Consider also the expenditure changeEQ � (p1 � x1) = (p0 � x0), which allows prices to vary. (If demands refer toaggregate consumption, this is the percent change in GDP.) Argue:(a) If LQ < 1, then x0 is revealed preferred to x1.(b) If PQ > 1, then x1 is revealed preferred to x0.(c) If EQ < 1 or EQ > 1, no revealed preference between x0 and x1 can

be established.

� Given a budget set and a preference relation %, one may derive a choicerule as follows.

De�nition. (B,C� (�;%) ; ) is the choice struture generated by preference %if and only if 8B 2 B,

C� (B;%) = fx 2 B s.t. 8y 2 B, x % yg :

� In principle, the generated choice rule could be empty for some B 2B. (I.e. there is no most-preferred element in the budget set.) Aswe know, rationality and continuity of % ensure that there exists acontinuous utility representation and a solution to the UMP, hence thatC� (B;%) 6= ?. Whenever we refer to a generated choice structure inthis lecture, we will impicitly assume that it satis�es the de�nition of achoice structure, i.e. that the generated choices are always nonempty.

42

6.2 Weak Axiom

� In order to have anything substantive to say about observed choices, weneed to impose some minimal consistency. The weak axiom of revealedpreference says that, if x is ever a choice when y is available, then xmust be a choice whenever x is available and y is a choice. In otherwords, if we ever observe a preference for x over y, then we can neverobserve a strict preference for y over x.

De�nition. Choice structure (B; C (�)) satis�es the weak axiom (WA) if,whenever x 2 C (B) for B 2 B with x; y 2 B, and y 2 C (B0) for B0 2 Bwith x; y 2 B0, we also have x 2 C (B0).

Exercise 29 (MWG 2.F.3). The following is partial information about aconsumer�s purchases:

Year 1 Year 2Quantity Price Quantity Price

Good 1 100 100 120 100Good 1 100 100 ? 80

:

Give the range of quantities of good 2, consumed in year 2, such that(a) the choices violate the weak axiom,(b) the consumption bundle in year 1 is revealed preferred to that in year

2,(c) the consumption bundle in year 2 is revealed preferred to that in year

1,(d) neither (a), (b) nor (c) can be concluded based on the data,(e) given WA holds, good 1 is inferior at some price,(f) given WA holds, good 2 is inferior at some price.

Exercise 30 (MWG 1.C.2). Argue that WA is equivalent to the followingproperty. If B;B0 2 B, with x; y 2 B \ B0, then x 2 C (B) and y 2 C (B0)imply fx; yg � C (B) \ C (B0).

Example. In the non-transitive scenario above, WA is not violated. Sinceeach pair (fx; yg, fy; zg, fx; zg) belongs to only one B 2 B, the axiom isnot tested. It would be a di¤erent matter if we added a fourth budget setfx; y; zg to B. Now any choice rule violates WA. If x 2 C (fx; y; zg), then

43

WA stipulates that x is a choice for any subset of fx; y; zg that contains xand where y or z is a choice. But this was not the case. The same argumentapplies to y 2 C (fx; y; zg) and z 2 C (fx; y; zg), so that C (fx; y; zg) mustbe empty (which is impossible by de�nition).

� Hence, for an arbitrary family of budget sets B, it is not the casethat the revealed preference %� derived from (B; C (�)) which satis�esWA is necessarily rational. The choice rule needs to be de�ned on asu¢ ciently comprehensive family of budget sets. (This makes it morerestrictive, in the sense that it has to "commit" to a choice in moredecision situations.)

Proposition. Let B include all subsets of X that contain one, two or threeelements. Then, if the choice structure (B; C (�)) satis�es WA, the revealedpreference %� derived from it is rational.

Proof. The relation %� derived from (B; C (�)) is complete: 8x; y 2 X,we have fx; yg 2 B. Either x 2 C (fx; yg), i.e. x %� y, or y 2 C (fx; yg),i.e. y %� x, or both. Moreover, %� derived from (B; C (�)) is transitive:suppose x %� y and y %� z. Then there exists B such that x; y 2 Band x 2 C (B), and there exists B0 such that y; z 2 B0 and y 2 C (B0).Consider the budget set fx; y; zg. If z 2 C (fx; y; zg), then WA requiresy 2 C (fx; y; zg). If y 2 C (fx; y; zg), then WA requires x 2 C (fx; y; zg).Hence x 2 C (fx; y; zg), whereby x %� z.�

� Since the revealed preference ordering is fully determined by choicesfrom two-object sets, it is in fact unique. (Provided every possible setof two objects is included in B.)

Exercise 31 (MWG 1.C.3). Let choice structure (B; C (�)) satisfy WA,and de�ne revealed strict preference such that x �� y () 9B 2 B withx; y 2 B, x 2 C (B) and y =2 C (B).(a) Compare �� to ��� de�ned such that x ��� y () x %� y andnot y %� x (where %� is the revealed preference derived from the choice

44

structure). Show that the two de�nitions are equivalent. Does this dependon WA?(b) Give an example where �� is not transitive.(c) Argue that �� is transitive if B includes all three-element subsets of X.

� When the choice rule C (�) in choice structure (B; C (�)) is generated bya rational preference %, i.e. 8B 2 B, C (B) = C� (B;%), we say that% rationalizes C (�).

� One can think of generating a choice structure from a rational pref-erence as simulating empirical choice data with a rational preferencemodel. If the model exactly predicts choices, then the choices are ex-plained (rationalized) by the model (preference).

� If the revealed preference %� derived from choice structure (B; C (�)) isrational, then it rationalizes (B; C (�)). This is not quite obvious; wemust verify that %� generates (B; C (�)). In fact, that statement is trueonly if WA holds for (B; C (�)).

Proposition. The revealed preference %� derived from a choice structure(B; C (�)) that satis�es WA generates (B; C (�)).Proof. Our task is to show that 8B 2 B, C (B) = C� (B;%). Let x 2

C (B). Since x %� y, 8y 2 B, we have x 2 C� (B;%�). Thus C (B) �C� (B;%�). In the other direction, let x 2 C� (B;%�), i.e. x %� y, 8y 2 B.Then there must exist, for every y 2 B, a budget set By such that x; y 2By and x 2 C (By). Now, some such y 2 B must be chosen from B, i.e.y 2 C (B). By WA, this implies x 2 C (B). Thus C� (B;%�) � C (B).Combining the inclusions, C (B) = C� (B;%�).�

� Hence, if WA holds for the choice structure, then the revealed preferencederived from it is rational and generates the choice structure. I.e. WAimplies that the choice structure is rationalizable (provided the choicerule covers all sets of up to three objects).

� It turns out that WA is not only su¢ cient (with restrictions on B), butalso necessary for a choice structure to be rationalizable.

45

Proposition. A choice structure (B; C� (�;%)) that is generated by a rationalpreference % satis�es WA.Proof. Suppose x 2 C� (B;%) for B 2 B and y 2 B. Since the choice

structure (B; C� (�;%)) is generated by %, x % y. Let y 2 C� (B0;%) andx 2 B0. WA requires that x 2 C� (B0;%). Suppose now that % is rational,i.e. in particular transitive. Because y 2 C� (B0;%), we have 8z 2 B0, y % zand by transitivity x % z. Then x 2 C� (B0;%), so WA holds.�

� Put together, these results establish that a choice rule, de�ned (at least)on all sets with up to three elements, re�ects a rational preference ifand only if it satis�es WA.

� One might be tempted to conclude that the preference- and choice-based approaches are basically equivalent, since we could include allpossible budget sets in B. But in the theory of demand, budget setshave a special form (they satisfy p � x � w), which is restrictive (e.g.convex). Arbitrary budget sets may not make sense, nor does a solutionto the UMP necessarily exist for them.

� In a meaningful sense, the preference-based approach (using rational-ity) is less general than the choice-based approach (using the weakaxiom). Rational preference always gives us the weak axiom in a gen-erated choice structure, but choices that satisfy the weak axiom neednot be consistent with rational preference.

Exercise 32 (MWG 1.D.4). If choice structure (B; C (�)) is rational-izable, show that it satis�es path-invariance: 8 fB1; B2g � B such thatB1 [B2 2 B and C (B1) [ C (B2) 2 B, it is the case that C (B1) [ C (B2) =C (C (B1) [ C (B2)).

Exercise 33 (MWG 1.D.5). On the set of objectsX = fx; y; zg, de�ne thefamily of budget sets B = ffx; yg ; fy; zg ; fx; zgg. Think of the choice ruleC (�) as assigning, to each budget set B 2 B, a probability distribution C (B)over objects in B. This stochastic choice rule C (�) is said to be rationalizableby preferences if there exists a probability distribution over (strict) preferencerelations on X (here there are six such relations) that 8B 2 B induces C (B).

46

(a) Show that C (fx; yg) = C (fy; zg) = C (fx; zg) = (1=2; 1=2) is ratio-nalizable in this sense.(b) Show that C (fx; yg) = C (fy; zg) = C (fx; zg) = (1=4; 3=4) is not

rationalizable in this sense.(c) Find �; � 2 [0; 1] such that C (fx; yg) = C (fy; zg) = C (fx; zg) =

(�; 1� �) is rationalizable if and only if � 2 [�; �].

6.3 Relationship with the Law of Demand

� It seems plausible that Walrasian demand could satisfy a version of thelaw of demand if we mimic the compensation for price changes that isimplicit in Hicksian demand by adjusting wealth with prices.

� This intuition is correct. In addition to homogeneity of degree zero andWalras�law, the weak axiom is the minimal property we need to imposeon (single-valued) Walrasian demand to make it satisfy a compensatedlaw of demand.

� In the context of (single-valued) demand, WA has the following speci�cform: 8 (p; w) and 8 (p0; w0),

p � x (p0; w0) � w and x (p; w) 6= x (p0; w0) =) p0 � x (p; w) > w0:

� I.e. if x (p0; w0) was available at prices p and wealth w, but x (p; w)was chosen instead, then x (p; w) must be unavailable at prices p0 andwealth w0, when x (p0; w0) is chosen.

� Notice how the single-valuedness of choices a¤ects WA: since we cannotrequire that x 2 C (B0) when x 2 C (B 3 y) and y 2 C (B0 3 x), westipulate instead that x =2 B0. Single-valuedness implies, when x ischosen over y in B, that x is revealed strictly preferred to y, and strictpreference is antisymmetric.

Exercise 34 (MWG 2.F.12). Verify that a Walrasian demand functionx (p; w) which is generated by a rational preference satis�es WA.

Exercise 35 (MWG 2.F.14). Argue that a Walrasian demand functionx (p; w) that satis�es WA is homogeneous of degree zero.

47

� Figure 5 illustrates how WA restricts choices given budget sets Bp;wand Bp0;w0. The top left panel depicts the budget sets; the other panelsshow all possible locations for x = C (Bp;w) and x0 2 C (Bp0;w0) thatsatisfy WA. In the top right panel, x 2 Bp;w\Bp0;w0. Since x is availablein both budget sets, the (single) choice of x0 in Bp0;w0 reveals it (strictly)preferred to x. Then x0 =2 Bp;w, else not choosing x0 violates WA. Thesame type of argument applies in the other cases.

� Before I give an analytical proof of the equivalence of WA and thecompensated law of demand, a graphic sketch will be helpful. Supposewe start with bundle x, which lies on the boundary of the budgetset Bp;w, in accordance with Walras� law. Consider a compensatedprice change to p0, where wealth is adjusted to w0 such that x remainsa¤ordable. The pivoting of the budget line through x is visible in Figure6.

� In the left panel, the set of bundles that satis�es WA is dotted. Sincex is a¤ordable in both Bp;w and Bp0;w0 (by the construction of the com-pensation), and was revealed preferred to everything in Bp;w, the choicex0 in Bp0;w0 must lie outside Bp;w if it is distinct from x. (Otherwise, x0

would have been chosen prior to the compensated price change.) If weassume that x0 also satis�es Walras�law, then it lies on the bold-stripedportion of the boundary of Bp0;w0.

� In the right panel, the set of bundles that satis�es CLD is dotted. Sincep02 < p2 as drawn, and the price change is compensated, CLD requiresthat x02 > x2. Then x0 must lie in the triangle above x below the newbudget line. By Walras�law, x0 belongs to the bold-striped portion ofthe boundary of Bp0;w0. This is exactly the same set that satis�es WA.

� Are we done? Actually no. We have only shown that WA is equivalentto CLD for compensated price changes. But WA is a property thatpertains to arbitrary (possibly uncompensated) price changes. We muststill demonstrate that such price changes will also satisfy WA.

� This point can be made via the contrapositive: if a price change violatesWA, then we can construct a compensated price change that violatesCLD, contrary to the assumption. (Hence no price change can violateWA.)

48

Figure 5: WA for single-valued demand

49

Figure 6: Equivalence of WA and CLD for compensated price changes

� In Figure 7, we have situation that is at odds with WA: x and x0 bothbelong to Bp;w and Bp0;w0 (other violations are straightforward to dealwith). We can construct a new budget set Bp00;w00 that represents acompensated price change for both x and x0 (graphically, the boundaryof Bp00;w00 pivots through both x and x0).

� One can see that p002 < p2 and p002 > p02. Hence CLD requires, for thesecompensated changes in demand, that x002 > x2 and x002 < x02. I.e. x

00

must lie in both dotted triangles, above x and below x0. But these aredisjoint sets, so CLD is violated. Since failures of WA (with arbitraryprice changes) lead to contradiction, when CLD holds for compensatedprice changes, CLD is in fact su¢ cient for WA.

Proposition. For a Walrasian demand function x (p; w) that is homoge-neous of degree zero and satis�es Walras�law, WA holds if and only if 8 (p; w)and 8 (p0; w0) such that w0 = p0 � x (p; w) and x (p; w) 6= x (p0; w0),

(p0 � p) � (x (p0; w0)� x (p; w)) < 0:

50

Figure 7: Impossibility of WA failures

Proof. Since x (p0; w0) satis�es Walras�law, we have p0 � x (p0; w0) = w0 =p0 � x (p; w), so that the CLD inequality reduces to the equivalent inequality

p � (x (p0; w0)� x (p; w)) > 0:

(If) WA holds vacuously if x (p; w) = x (p0; w0) and is satis�ed if p0 �x (p; w) >w0 or p �x (p0; w0) > w. Consider therefore (p; w) and (p0; w0) where x (p; w) 6=x (p0; w0) and w0 � p0 � x (p; w). If w0 = p0 � x (p; w), then the inequalityapplies, and p � x (p0; w0) � w would imply p � x (p; w) < w, violating Walras�law. Hence p � x (p0; w0) > w, so that WA is not tested. Similarly for the casew = p � x (p0; w0).Thus we can concentrate on p0 � x (p; w) < w0 and p � x (p0; w0) < w.

This appears to be a violation of WA, but we will show that there existsa compensated price change for which the CLD cannot hold. Hence thescenario does not satisfy the assumptions.Given the strict inequalities, there exists � 2 (0; 1) such that

(�p+ (1� �) p0) � x (p; w) = (�p+ (1� �) p0) � x (p0; w0)

(because at � su¢ ciently close to 1, the right side is close to p � x (p; w) =w > p � x (p0; w0), and at � su¢ ciently close to 0, the left side is close to

51

p0 � x (p0; w0) = w0 > p0 � x (p; w) < w0, where the equalities are due to Walras�law).Let p00 � �p+(1� �) p0 andw00 � (�p+ (1� �) p0)�x (p0; w0) = (�p+ (1� �) p0)�

x (p; w). Sincep00 � x (p; w) = w00

andp00 � x (p0; w0) = w00;

we have constructed compensated price changes from p to p00 and from p0 top00.By Walras�law, p00 � x (p00; w00) = w00, so

p00 � (x (p00; w00)� x (p; w)) = 0

andp00 � (x (p00; w00)� x (p0; w0)) = 0:

Because p to p00 and p0 to p00 are compensated price changes, CLD must holdfor both:

p � (x (p00; w00)� x (p; w)) > 0

andp0 � (x (p00; w00)� x (p0; w0)) > 0:

From the de�nition of p00, we have p = (1=�) p00 � ((1� �) =�) p0. There-fore, CLD for p to p00 implies

(p00 � (1� �) p0) � (x (p00; w00)� x (p; w)) > 0;

and, since p00 � (x (p00; w00)� x (p; w)) = 0, this means

p0 � x (p; w) > p0 � x (p00; w00) :

Now, since p0 � x (p0; w0) = w0 > p0 � x (p; w) by assumption, it follows that

p0 � x (p0; w0) > p0 � x (p00; w00) :

But then the CLD for p0 to p00 cannot hold. Hence p0 � x (p; w) < w0 andp � x (p0; w0) < w is not possible, and we have shown that WA holds forall (p; w) and (p0; w0) if CLD holds for all compensated price changes (andWalras�law is in e¤ect).

52

(Only if) Because p0 � x (p; w) = w0, the bundle x (p; w) is available at(p0; w0). But since x (p0; w0) is chosen instead at (p0; w0) (revealing it pre-ferred), WA requires that x (p0; w0) is unavailable at (p; w), i.e. p �x (p0; w0) >w. From Walras�law, p � x (p; w) = w, which gives the strict inequality

p � (x (p0; w0)� x (p; w)) > 0:

�

� Observe that we are not comparing initial demand x (p; w) to the un-compensated demand x (p0; w) after the price change to p0. Instead,wealth is adjusted to w0 = p0 � x (p; w), so that the initial bundle is stillin the budget set at the new prices.

� The remarkable aspect of the result is that nothing more than WA isneeded for the compensated law of (Walrasian) demand (other thanhomogeneity of degree zero and Walras�law, which require only thatpreference is locally nonsatiated). In particular, preference does nothave to be rational or continuous.

� Since it is an "if and only if" proposition, WA can be said to be equiv-alent to the compensated law of demand under local nonsatiation.

Exercise 36 (MWG 2.F.5). Let a di¤erentiable Walrasian demand func-tion x (p; w) satisfy homogeneity of degree zero, Walras�law and the weakaxiom. Suppose x (�; �) is also homogeneous of degree one with respect towealth w, so that consumption of all goods increases proportionately: 8p; wand 8� > 0, x (p; �w) = �x (p; w). Show that the law of demand holds alsofor uncompensated price changes: 8p; p0, (p0 � p) � (x (p0; w)� x (p; w)) � 0.

Exercise 37 (MWG 2.F.13). Consider a multivalued Walrasian demandcorrespondence x (p; w).(a) Give the Walrasian version of WA in this case, generalized to choice

sets.(b) If x (p; w) satis�es WA and Walras�law, show that x (p; w) also has

the following property: 8x 2 x (p; w) and 8x0 2 x (p0; w0),

p � x0 < w =) p � x > w:

53

(c) Moreover, show that if x (p; w) satis�es WA and Walras� law, thenthe following compensated law of demand holds: 8 (p; w) and 8x 2 x (p; w),and 8 (p0; w0) such that w0 = p0 � x,

(p0 � p) � (x0 � x) � 0:

6.4 Strong Axiom

� A choice-based theory that is fully equivalent to the preference-basedapproach with rationality can be based on the strong axiom of revealedpreference.

De�nition. Choice structure (B; C (�)) satis�es the strong axiom (SA) if,for any collection of budget sets

�B1; : : : ; BN

� B, and x1; : : : ; xN such that

xn 2 C (Bn) and xn+1 2 Bn for n = 1; : : : ; N � 1, we have x1 2 C�BN�if

x1 2 BN .

� SA is just WA if we consider only collections of N = 2 budget setsfB;B0g � B: then x 2 C (B) and y 2 C (B0) with y 2 B and x 2 B0

implies x 2 C (B0). I.e. if x1 is chosen when x2 is available, then x1must be chosen whenever it is available and x2 is chosen.

� Beyond WA, SA says, if x1 is chosen when x2 is available, and x2 ischosen when x3 is available (so that x1 is indirectly revealed preferredto x3), then x1 must be chosen whenever it is available and x3 is chosen.(And we could make the chain arbitrarily long.) You will recognize the�avor of transitivity in SA.

� Transitivity is precisely what WA failed to guarantee in a revealedpreference relation derived from it. SA, however, is su¢ cient withoutany requirements on the domain of the choice rule. Thus, If the choicestructure (B; C (�)) satis�es SA, then the revealed preference %� derivedfrom it is rational. (The proof is non-elementary.)

� Since SA implies WA, it inherits all properties of WA. Therefore, achoice structure is generated by a rational preference if it satis�es SAand only if it satis�es WA.

Exercise 38 (MWG 3.J.1). Show for the consumption set R2+ that Wal-rasian demand satis�es WA if and only if it satis�es SA.

54

7 Integrability

7.1 Slutsky and Hicks Compensation

� If choices are generated from rational preferences, then they satisfy theweak axiom, which is equivalent to the compensated law of demand(as long as choices are homogeneous of degree zero and satisfy Walras�law): 8 (p; w) and 8 (p0; w0) such that w0 = p0 � x (p; w),

(p0 � p) � (x (p0; w0)� x (p; w)) � 0:

Hence rational preferences imply the version of the compensated lawof demand we posed in the discussion of the choice-based approach.

� The reverse conclusion is problematic: while the compensated law ofdemand implies the weak axiom, the weak axiom is not su¢ cient forthe existence of a rational preference that generates the choices.

� In the context of the expenditure minimization problem, we also deriveda compensated law of demand (for continuous and locally nonsatiatedpreferences): 8p and 8p0,

(p0 � p) � (h (p0; u)� h (p; u)) � 0:

� Are the two statements of the compensated law of demand equiva-lent? In the choice-based approach, the consumer�s wealth is explic-itly adjusted so that the initial choice x remains a¤ordable at pricesp0. Graphically, a price increase (inward pivot of the budget line) isaccompanied by a wealth increase (outward shift of the budget line,which passes through x), as shown in the left panel of Figure 8. Thistype of compensation is called Slutsky compensation.

� In the preference-based approach, the consumer�s utility is held �xedand expenditure is allowed to vary, so that the initial utility is stillattained at the optimal choice h (p0; u) after the price increase. Graph-ically, a price increase is accompanied by a wealth increase (outwardshift of the budget line, which remains tangent to the indi¤erence curveat u), as shown in the right panel of Figure 8. This is known as Hickscompensation.

55

Figure 8: Compensation in the choice- and preference approaches

� Visually, it is clear that the two versions of the compensated law ofdemand are not the same. However, for in�nitesimal price changes,they do lead to the same adjustments in choices.

� Our interest in this lecture is in whether the preference-based com-pensated law of demand implies stronger restrictions on choices thatguarantee rationalizability. I.e. if we observe that choices satisfy it,can they always be constructed from a rational preference?

� To this end, we will characterize the compensated law of demand in thechoice-based and preference-based approaches in terms of the substi-tution matrix, which contains price e¤ects. Negative semide�nitenessof the substitution matrix is (almost) equivalent to the choice-basedcompensated law of demand and necessary (but not su¢ cient) for thepreference-based compensated law of demand.

� The additional property that the substitution matrix must have if thepreference-based compensated law of demand holds is symmetry. Wethen argue that symmetry is enough to recover a rational preferencefrom choices, so that the preference-based compensated law of demandindeed guarantees rationalizability.

56

7.2 Aside: Dot Product

� Because we will make heavy use of vector notation and transformationsof vector equations in this lecture, I brie�y review the properties of thedot product.

� The dot product of vectors x and y is the sum of the products ofcorresponding elements (that have the same indices):

x � y �LX`=1

x`y`:

� Because multiplication is commutative, so is the dot product: x�y = y�xsince

x � y =LX`=1

x`y` =LX`=1

y`x` = y � x:

� Because multiplication is distributive, so is the dot product: x�(y + z) =x � y + x � y since

x � (y + z) =LX`=1

x` (y` + z`) =LX`=1

x`y` +LX`=1

x`z` = x � y + x � z:

� The dot product is not associative: (x � y) � z 6= x � (y � z) since, ingeneral,

(x � y) � z =KXk=1

LX`=1

x`y`

!zk =

LX`=1

x`y`

!z1 + � � �+

LX`=1

x`y`

!zK

6=

LX`=1

y`z`

!x1 + � � �+

LX`=1

y`z`

!xK :

� The dot product of x and y can be written in matrix notation:

x � y = xTy:

Commutativity, xTy = yTx, and distributivity, xT (y + z) = xTy+xT z,are of course inherited.

57

� The outer product of vectors x and y is a matrix (as opposed to theinner product xTy, which is a scalar):

xyT �

264 x1y1 � � � x1yL...

. . ....

xLy1 � � � xLyL

375 :� In matrix notation, there is a version of associativity, which involves aswitch from outer product to inner product:

�xyT

�z = x

�yT z�since

�xyT

�z =

264 x1y1 � � � x1yL...

. . ....

xLy1 � � � xLyL

375264 z1...zL

375 = LX`=1

y`z`

!x1 + � � �+

LX`=1

y`z`

!xL

=

264 x1...xL

3750B@� y1 � � � yL

� 264 z1...zL

3751CA = x

�yT z�:

Notice that the matrix xyT cannot be expressed in terms of the dotproduct between two vectors, so there is no con�ict with the claim thatthe dot product is not associative.

7.3 Substitution Matrix

� The substitution matrix (alternatively known as the Slutsky matrix)S (p; w) lists, for every two commodities ` and k, how much more (orless) is chosen of commodity ` per di¤erential increase in the price ofcommodity k at (p; w). The entry in row ` and column k is

s`k (p; w) �dx` (p; w)

dpk=

@x` (p; w)

@pk+@x` (p; w)

@w

@ (p � x (p0; w0))@pk

=@x` (p; w)

@pk+@x` (p; w)

@wxk (p; w) ;

where x (p0; w0) is the �xed initial demand.

� The �rst term measures the direct e¤ect of the price change on x` (p; w)and is called the (pure) substitution e¤ect. The second term measuresthe e¤ect of the Slutsky compensation on x` (p; w) and is called thewealth e¤ect.

58

� Even though we denote these choices as Walrasian demands, we donot necessarily assume here that they arise from the utility maximiza-tion problem. These are observed choices from the budget set Bp;wthat may or may not be rationalizable by an underlying preference.(They are Walrasian in the sense that there is no implicit adjustmentin expenditure after a price change, via a �xed utility level. Any suchcompensation is explicit.)

� Hence

S (p; w) =

264 s11 (p; w) � � � s1L (p; w)...

. . ....

sL1 (p; w) � � � sLL (p; w)

375=

264@x1(p;w)@p1

+ @x`(p;w)@w

x1 (p; w) � � � @x1(p;w)@pL

+ @x1(p;w)@w

xL (p; w)...

. . ....

@xL(p;w)@p1

+ @xL(p;w)@w

x1 (p; w) � � � @xL(p;w)@pL

+ @xL(p;w)@w

xL (p; w)

375= Dpx (p; w) +Dwx (p; w)x (p; w)

T :

� The following is a characterization of the choice-based compensatedlaw of demand.

Proposition. If Walrasian demand function x (p; w) is di¤erentiable, ho-mogeneous of degree zero and satis�es Walras� law and the weak axiom,then 8 (p; w) the substitution matrix S (p; w) is negative semide�nite, i.e.8v 2 RL, vTS (p; w) v � 0.Proof. Under the assumptions, the compensated law of demand holds:

8 (p; w) and 8 (p0; w0) such that w0 = p0 � x (p; w),

(p0 � p) � (x (p0; w0)� x (p; w)) � dp � dx (p; w) � 0:

This inequality implies negative semide�niteness of the substitution matrix.From the total derivative of x, and the fact that, for a compensated price

change,dw � w0 � w = (p0 � p) � x (p; w) = dp � x (p; w) ;

59

we have

dx = Dpx (p; w) dp+Dwx (p; w) dw

= Dpx (p; w) dp+Dwx (p; w) (dp � x (p; w))= Dpx (p; w) dp+Dwx (p; w)

�x (p; w)T dp

�=

�Dpx (p; w) +Dwx (p; w)x (p; w)

T�dp:

(using commutativity of the dot product and, after switching to matrix no-tation, associativity and distributivity).Substituting into the �rst inequality,

dp ��Dpx (p; w) +Dwx (p; w)x (p; w)

T�dp � 0:

Because the magnitude of the price change dp 2 RL was unrestricted, thisimplies dx (p; w) =dp = S (p; w) is negative semide�nite.�

� The converse, that a negative semide�nite substitution matrix impliesthe weak axiom, is true provided that S (p; w) is in addition negativede�nite, i.e. vTS (p; w) v < 0, whenever v is not proportional to p(v 6= �p for any �).

� As we will see in a moment, the substitution matrix cannot be negativede�nite for all v 2 RL (i.e. it is in fact never de�nite) since, lettingv = p, we get pTS (p; w) p = 0.

� An implication of negative semide�niteness is that all diagonal entries(the own-price e¤ects) are non-positive:

dx` (p; w)

dp`=@x` (p; w)

@p`+@x` (p; w)

@wx` (p; w) � 0:

Therefore, a Gi¤en good is necessarily inferior, since @x` (p; w) =@p` > 0only if @x` (p; w) =@w < 0.

60

Proposition. If Walrasian demand function x (p; w) is di¤erentiable, homo-geneous of degree zero and satis�es Walras� law, then 8 (p; w), S (p; w) p =pTS (p; w) = 0.

Proof. Previously (in Lecture 5), we established

Dpx (p; w) p+Dwx (p; w)w = 0:

when x (p; w) is homogeneous of degree zero, and

pTDpx (p; w) + x (p; w)T = 0

as well as pTDwx (p; w) = 1 when x (p; w) satis�es Walras�law.Then

S (p; w) p = Dpx (p; w) p+Dwx (p; w)�x (p; w)T p

�= Dpx (p; w) p+Dwx (p; w)w = 0

(where the last equality uses Walras�law), and furthermore

pTS (p; w) = pTDpx (p; w) + pTDwx (p; w)x (p; w)T

= pTDpx (p; w) + x (p; w)T = 0:

�

� Hence negative semide�niteness is exactly what is implied by the choice-based compensated law of demand (equivalently the weak axiom).

� Since S (p; w) p = 0, zero is an eigenvalue of S (p; w), which meansits determinant jS (p; w)j = 0. (Recall that � is an eigenvalue ifS (p; w) p = �p for some p or if jS (p; w)� �Ij = 0. The latter im-plies jS (p; w)j = 0 if � = 0.) This is useful to know when checkingnegative semide�niteness from the signs of the principal minors. Wewill see an example later on.

Exercise 39 (MWG 2.F.17). Let the Walrasian demand function x (p; w)have the form: for k = 1; : : : ; L,

xk (p; w) =wPL`=1 p`

:

61

Determine whether x (p; w)(a) is homogeneous of degree zero;(b) satis�es Walras�law;(b) is consistent with the weak axiom;(c) has a negative semide�nite, symmetric substitution matrix.

Exercise 40 (MWG 2.F.10) Compute the substitution matrix for Wal-rasian demand function x (p; w) where

x1 (p; w) =1

p1 + p2 + p3

p2p1;

x2 (p; w) =1

p1 + p2 + p3

p3p2;

x3 (p; w) =1

p1 + p2 + p3

p1p3:

Demonstrate that the substitution matrix is negative semide�nite, but notsymmetric, at p = (1; 1; 1). Show that it does not satisfy the weak axiom.

Exercise 41 (MWG 2.F.16). Let the Walrasian demand function x (p; w)be as follows:

x1 (p; w) =p2p3;

x2 (p; w) = �p1p3;

x3 (p; w) =w

p3:

(a) Con�rm that x (p; w) is homogeneous of degree zero and that Walras�law holds.(b) Demonstrate that x (p; w) does not satisfy the weak axiom.(c) Show that 8v 2 R3, v � S (p; w) v = 0.

7.4 Substitution Matrix with Preference

� If we allow choices to arise from a standard preference, then the du-ality between utility minimization and expenditure minimization lets

62

us express the substitution matrix in terms of Hicksian demand atu = v (p; w):

@h (p; u)

@p=dh (p; v (p; w))

dp=dx (p; w)

dp� s`k (p; w)

for `; k = 1; : : : ; L.

� In order for a di¤erentiable Hicksian demand function h (p; u) to existas a solution to the expenditure minimization problem, u (�) has torepresent a continuous, locally nonsatiated, strictly convex preferenceon X = RL+. I emphasize that the equivalence with a given Walrasiandemand function is only valid if the conditions for the existence of aHicksian demand function hold.

� Rewriting the substitution matrix accordingly as

S (p; w) =

264@h1(p;u)@p1

� � � @h1(p;u)@pL

.... . .

...@hL(p;u)@p1

� � � @hL(p;u)@pL

375 = Dph (p; u) ;

a further property can be derived, because S (p; w) is now implicitlycon�ned to choices the re�ect an underlying standard preference.

Proposition. If u (�) represents a continuous, locally nonsatiated, strictlyconvex preference on X = RL+, and the Hicksian demand function h (p; u)is continuously di¤erentiable at u, then the matrix Dph (p; u) = S (p; w) isnegative semide�nite and symmetric (i.e. dh` (p; u) =dpk = dhk (p; u) =dp` for`; k = 1; : : : ; L).Proof. Recall �rst that h (p; u) = rpe (p; u) by the envelope theorem, so

thatDph (p; u) = D2

pe (p; u) :

Since e (p; u) is a concave function, we can appeal to the fact that the Hessian(i.e. the second-derivative matrix) of any concave function is negative semi-de�nite.I will prove this general result. The Taylor expansion of e (p; u) around

� = 0 in displacement direction v 2 RL is

e (p+ �v; u) = e (p; u) +rpe (p; u)T (�v) +

�2

2vTD2

pe (p+ "v; u) v

63

for some " 2 [0; �]. Then

vTD2pe (p+ "v; u) v � 0;

because the concavity of the expenditure function implies

e (p+ �v; u) � e (p; u) +rpe (p; u)T (�v) :

This is apparent when, in the de�nition of concavity, i.e. 8p, 8p0 and8� 2 [0; 1],

e (�p+ (1� �) p0; u) � �e (p; u) + (1� �) e (p0; u) ;

we substitute �v � p0 � p for p0, so that (after rearrangement)

e (p+ �v; u) � e (p; u) +e (p+ (1� �) �v; u)� e (p; u)

1� �;

and we apply the limit as � ! 1 (making the last term a derivative withrespect to p, in direction v).Now, if the magnitude � of the displacement in the Taylor expansion is

chosen su¢ ciently small, then " is very close to zero, and we have

vTD2pe (p; u) v � 0;

where the direction v 2 RL was arbitrary. This establishes that D2pe (p; u) =

Dph (p; u) is negative semide�nite.Symmetry comes from Clairaut�s theorem, which states that the Hessian

of any function that has continuous second partial derivatives at all pointsin the domain is symmetric. Since this is true for e (p; u) by virtue of thefact that its second derivatives are the �rst derivatives of h (p; u), which wereassumed to be continuous, D2

pe (p; u) = S (p; w) must be symmetric.�

� The preference-based compensated law of demand, which is implicit inHicksian demand, therefore requires both negative semide�niteness andsymmetry of the substitution matrix. Symmetry was not an implicationof the choice-based compensated law of demand.

64

� Before we examine this di¤erence in examples, we con�rm that thesubstitution matrix will not be negative de�nite.

Proposition. If u (�) represents a continuous, locally nonsatiated, strictlyconvex preference on X = RL+, and h (p; u) is di¤erentiable at u, thenDph (p; u) p = S (p; w) p = 0.Proof. Under these conditions, Hicksian demand is homogeneous of degree

zero in p, so that h (�p; u) = h (p; u). Di¤erentiating both sides with respectto �, we have for ` = 1; : : : ; L,

LXk=1

@h` (�p; u)

@�pk

@�pk@�

=LXk=1

@h` (�p; u)

@pkpk = 0;

which corresponds to the claim.�

Example. Suppose choices have the Cobb-Douglas form:

x1 (p; w) =1

3

w

p1; x2 (p; w) =

1

3

w

p2; x3 (p; w) =

1

3

w

p3:

Observe that the choices satisfy homogeneity of degree zero and Walras�law(they add up to w). The substitution matrix is easily calculated from partialderivatives to be

S (p; w) =1

9w

264 �21p21

1p1p2

1p1p3

1p1p2

�2 1p22

1p2p3

1p1p3

1p2p3

�2 1p23

375 :Clearly, this is symmetric. One can check directly, that it is negative semi-de�nite because 8v 2 RL,

vTS (p; w) v =1

9w

��2v

21

p21+ 2

v1v2p1p2

+ 2v1v3p1p3

� 2v22

p22+ 2

v2v3p2p3

� 2v23

p23

�= �1

9w

�v1p1� v2p2

�2+

�v1p1� v3p3

�2+

�v2p2� v3p3

�2!� 0

(with equality only if v = �p for any �). Hence Cobb-Douglas choices sat-isfy the both the choice-based and the preference-based compensated law ofdemand.

65

Example. Consider now the following modi�cation:

x1 (p; w) =1

3

w

p1+p2p1; x2 (p; w) =

1

3

w

p2� 1; x3 (p; w) =

1

3

w

p3:

Neither homogeneity of degree zero nor Walras�law are violated, e.g.

3X`=1

p`x` (p; w) =1

3w + p2 +

1

3w � p2 +

1

3w = w:

The substitution matrix

S (p; w) =1

9w

264 �1p21

�2 + 6p2

w

�1

p1p2

�1 + 3p2

w

�1

p1p3

�1 + 3p2

w

�1

p1p2

�1 + 6p2

w

�� 1p22

�2 + 3p2

w

�1

p2p3

�1� 3p2

w

�1

p1p31

p2p3�2 1

p23

375is obviously not symmetric.In this case, it is more convenient to check negative semide�niteness in-

directly. One technique is to compute the prinicipal minors, which are de-terminants of n� n matrices derived by deleting n�m corresponding rowsand columns. A matrix is negative semide�nite if all odd principal minors(i.e. with m odd) are non-positive and all even principal minors (i.e. with meven) are non-negative.The �rst-order principal minors are the diagonal entries, which are nega-

tive. The second-order minors have the following form. If row r and columnc are deleted, and i and j are the lowest and highest non-deleted rows, andk and l are the lowest and highest non-deleted columns,

Mrc = sik (p; w) sjl (p; w)�sil (p; w) sjk (p; w) = (�1)I(r;c)w

pipjpkpl

�1

3+p2w

�;

where I (r; c) = 1 if r+ c is odd, and I (r; c) = 0 if r+ c is even. Since Mrc isnegative if and only if r + c is odd, and r = c (i.e. Mrc is a principal minor)implies r + c is even, the second-order principal minors are all positive.The third-order principal minor is the determinant of the matrix, which

we know is always zero for S (p; w). It is worth checking once that, indeed,

jS (p; w)j = s11 (p; w)M11 � s12 (p; w)M12 + s13 (p; w)M13 = 0:

66

Hence the matrix is negative semide�nite, which implies that the weak axiomholds (because all lower-order principal minors have strict signs, the negativede�niteness for v 6= �p obtains). These choices do not satisfy the preference-based compensated law of demand, since the substitution matrix fails to besymmetric.

Exercise 42 (MWG 3.G.14). The following are Walrasian substitutione¤ects for an agent who has rational preferences and faces prices p = (1; 2; 6):

S (p; w) =

24 �10 ? ?? �4 ?3 ? ?

35 :Find the missing entries.

Exercise 43 (MWG 2.F.11). Show that, in a two-commodity world, wherex (p; w) is di¤erentiable, homogeneous of degree zero and sats�es Walras�law,S (p; w) is always symmetric.

7.5 Integrability

� There is a deeper signi�cance to the observation that the substitutionmatrix is symmetric in the derivation from preference-based demand,but not in the choice-based approach using the weak axiom. We knowthat the weak axiom guarantees that the choice structure is rational-izable (i.e. admits an underlying preference), and is also equivalent tothe compensated law of demand.

� The preference-based approach is more special in nature, as is evidentfrom the fact that it implies both the compensated law of demandand symmetry of the substitution matrix. Hence there is hope that ademand function that has both properties is rationalizable.

� From a modeling point of view, when we build a theory from a demandfunction with certain properties, rather than from preferences, it isimportant to know whether the demand function could in fact occurif the agent were optimizing with respect to some rational preference.This is known as the integrability problem: are homogeneity of degreezero, Walras�law, the compensated law of demand and symmetry ofthe substitution matrix su¢ cient for rationalizability?

67

� The answer is yes. I.e. choices are rationalizable (given that they sat-isfy homogeneity of degree zero, Walras�law and the compensated lawof demand) if the substitution matrix is symmetric. (We also presumethat preferences are convex and continuous on in this discussion, sorationalizability refers here to the existence of a generating preferencethat has all the standard properties, not just completeness and transi-tivity. The symmetry of the substitution matrix is also necessary forrationalizability in terms of such preferences.)

� To understand the connection, we must think about how a preferencewould be recovered from choices x (p; w). A preference is fully describedby its upper contour sets

% (x) =�a 2 RL+ s.t. a % x

=

�a 2 RL+ s.t. u (a) � u (x)

for some utility function that represents %.

� Equivalently we can express % (x), using the duality between utilitymaximization and expenditure minimization, as

%u(x)=�a 2 RL+ s.t. 8p� 0, e (p; u (x)) � p � a

:

This is the set of bundles that are more costly than the cheapest bundlerequired to attain u (x), at any given prices. To see why these bundlesform the upper contour set of x, we need insights from abstract dualitytheory.

� Duality theory revolves around the idea that any closed, convex setcan be described in terms of its linear approximation by hyperplanesthat separate the set from points not in the set. The sets of interestto us are the upper contour sets of a continuous and convex preferencerelation.

� Fix a utility function u (�) that represents %, and a level u, and denotethe upper contour set

% (x) =�a 2 RL+ s.t. u (a) � u (x)

by %u. A hyperplane that separates the bundle y =2%u from the set %uis parameterized by some vector p and scalar eu such that

p � y < eu � p � x

68

for all x 2%u.

� Given %u, the separating hyperplane theorem says that such a hyper-plane (i.e. p and eu) exists for every y =2%u. Each hyperplane separatesRL into two halfspaces, one containing the point y and the other con-taining the set %u. The intersection of the latter halfspaces is theoriginal set %u, since it excludes all points not in the set.

� For every p, there is an element x� 2%u that generates the smallestvalue of p � x attainable in %u. The function that maps p to this valueat x� is called the support function of %u:

eu (p) � inf fp � x s.t. x 2%ug :

� This function is necessarily concave in p (by the argument we alreadymade for the concavity of the expenditure function - it increases atmost linearly in p).

� Clearly, we have eu (p) � p � x for all x 2%u, at all p. It follows fromthe separating hyperplane theorem that there exists some p � 0 forevery y =2%u such that eu (p) > p � y. Hence, no y =2%u can satisfyeu (p) � p � y for all p� 0. Then %u can be expressed as

%u=�x 2 RL s.t. 8p� 0; eu (p) � p � x

:

� The upper contour set at u in the expenditure minimization problemis convex and closed if preferences are convex and continuous. Thisset can be approximated by tangent budget hyperplanes that containbundles x� in the upper contour set at which p � x is minimized, givenprices. By �nding such a budget hyperplane for every p, we tracethe boundary ot the upper contour set. Knowledge of the minimumexpenditure p � x� at every price, i.e. the expenditure function (whichis the support function) would allow us to recover the upper contourset. See Figure 9.

� Duality theory lets us infer the upper contour set at every utility levelu and therefore recover the preference relation, once we have an expen-diture function. What remains is to obtain the expenditure functionfrom choices.

69

Figure 9: Upper contour set approximated by e (p; u)

� This is the proper integrability problem, the question whether the sys-tem of partial di¤erential equations rpe (p; v (p; w)) = h (p; v (p; w)) =x (p; w) has a solution. The Frobenius theorem gives necessary andsu¢ cient conditions for integrability that are, in particular, satis�ed ifthe derivative matrix is symmetric.

� As we have seen, the derivative matrix of rpe (p; v (p; w)) is the substi-tution matrix S (p; w) = Dph (p; v (p; w)). Hence symmetry is exactlywhat is needed for recovering preferences from choices, i.e. for choicesto be rationalizable.

Exercise 44 (MWG 3.H.6). Derive expenditure and utility function fromthe Walrasian demand function x` (p; w) = �`w=p` for ` = 1; : : : ; L withPL

`=1 �` = 1.

Exercise 45 (MWG 3.H.5). How can expenditure and utility function berecovered from the indirect utility function?

70

8 Aggregation

8.1 Aggregate Demand Function

� In general, aggregate demand is not a function of aggregate wealth, butrather a correspondence even if individual demands are functions. Ata given level of aggregate wealth, di¤erent wealth distributions lead todi¤erent choices, individually and in the aggregate. In this lecture, weconsider the special circumstances under which aggregate demand is awell-de�ned function of aggregate wealth.

� We also ask whether an aggregate demand function has welfare con-tent. Individual demands are outcomes of utility maximization andtherefore represent the best choices available to an agent. Can we saythat aggregate demand re�ects the best consumption choices availableto society?

� Finally, we consider whether aggregate demand inherits the weak ax-iom from individual demands, which is an important condition for theuniqueness of general equilibria.

� Let there be I consumers i = 1; : : : ; I with rational preference relations%i and individual wealths wi. Their Walrasian demands are denotedby xi (p; wi). Then aggregate demand is:

x (p; w1; : : : ; wI) =IXi=1

xi (p; wi) :

Exercise 46 (MWG 4.C.11). Let two agents have identical wealthsw1 = w2 = w=2, and let their preferences of over bundles of two goods haveutility representations

u1 (x11; x21) = x11 + 4px21

andu2 (x12; x22) = 4

px12 + x22:

(a) Derive the individual demand functions and the aggregate demand func-tion.

71

(b) Find the individual Slutsky matrices Si (p; w=2) for i = 1; 2 and theaggregate Slutsky matrix S (p; w). (With two goods, the entire matrix isdetermined by one element.) Demonstrate that dp � S (p; w) dp < 0 for alldp 6= 0 that are not proportional to p. Does aggregate demand satisfy theweak axiom?

� In general, aggregate demand depends on prices and all individualwealth levels. Can we build a theory where only the aggregate wealthPI

i=1wi � w a¤ects aggregate demand? I.e. when isPI

i=1 xi (p; wi) =PIi=1 xi (p; w

0i) for all (w1; : : : ; wI) and (w

01; : : : ; w

0I) that distribute the

same aggregate wealth w, so that we can write

x (p; w1; : : : ; wI) =IXi=1

xi (p; wi) = x

p;

IXi=1

wi

!?

� Consider a wealth distribution (w1; : : : ; wI) and some di¤erential changein wealth (dw1; : : : ; dwI) such that

PIi=1 dwi = 0. Since dw is a redis-

tribution of wealth that does not a¤ect total wealth, it cannot a¤ectaggregate consumption of any commodity if x is to be a function ofaggregate wealth only. I.e.

IXi=1

@x`i (p; wi)

@widwi = 0

for commodity ` = 1; : : : ; L.

� Because dw might a¤ect only two individuals, and leave everyone�swealth unchanged, it must be that the wealth e¤ect for each commoditymust be the same for any two individuals, i.e. 8i; j 2 I,

@x`i (p; wi)

@wi=@x`j (p; wj)

@wj

for ` = 1; : : : ; L.

� Demand functions have this property at any prices and wealth distrib-ution if and only if preferences admit an indirect utility function of theGorman form:

vi (p; wi) = ai (p) + b (p)wi:

72

� For the "if" part, we need Roy�s identity.

Proposition. If u represents a continuous, locally nonsatiated and strictlyconvex preference on X = RL+, and the indirect utility function v (�; �) isdi¤erentiable at (�p; �w)� 0, then

x (�p; �w) = � rpv (�p; �w)

@v (�p; �w) =@w:

Proof. Suppose v (�p; �w) = �u. By duality of the utility maximization andexpenditure minimization problems, 8p, v (p; e (p; �u)) = �u. Hence

rpv (p; e (p; �u)) +@v (p; e (p; �u))

@e (p; �u)rpe (p; �u) = 0:

Evaluating at p = �p, and using rpe (�p; �u) = h (�p; �u) = x (�p; �w), and replacinge (�p; �u) with �w, we have

rpv (�p; �w) +@v (�p; �w)

@w�x (p; �w) = 0

as claimed.�

� Substituting the Gorman form of the indirect utility function into Roy�sidentity, we have

x (p; w) = � rpv (p; w)

@v (p; w) =@w= �rpai (p)

b (p)� rpb (p)

b (p)wi:

Di¤erentiating with respect to wealth,

rwix (p; w) = �rpb (p)

b (p);

which is independent of wi. Since the wealth e¤ect is the same for allindividuals and all wealth levels, aggregate demand depends only onaggregate wealth.

73

� This argument only provides su¢ ciency, but the Gorman form is infact necessary for the existence of an aggregate demand function.

Example. Preference % is homothetic if x � y implies tx � ty for allx; y and all t > 0. If u is a utility function that represents homotheticpreference %, then u (x) = u (y) () u (tx) = u (ty) for all t > 0. It followsthat the utility function is homogeneous of degree one, i.e. u (tx) = tu (x).Furthermore, the expenditure function is homogeneous of degree one in u:

te (p; u) = t (p � x) = p � (tx) = e (p; tu) :

Denote the expenditure required to reach u = 1, given prices, by a newfunction e (p; 1) � ~b (p). Now one can write the expenditure at an arbitraryutility u as e (p; u) = ue (p; 1) = u~b (p). Since e (p; u) = w and u = v (p; w)in the corresponding utility maximization problem, we have

v (p; w) = b (p)w

where b (p) � 1=~b (p), for a homothetic preference. Hence the indirect utilityfunction has the Gorman form.

We can see directly how the Gorman form relates to linear wealth expan-sion paths in this case. Using Roy�s identity,

x (p; w) = �rpb (p)

b (p)w:

Hence, at �xed prices, demand increases linearly (and proportionately forall commodities) in wealth. Thus rwx (p; w) = x (p; w) =w, and the incomeelasticity of demand is, for commodity ` = 1; : : : ; L,

"`w (p; w) = �@x` (p; w)

@w

w

x (p; w)= 1:

This means a �xed share of the budget is spent on each commodity, a propertythat you know from demand functions associated with Cobb-Douglas utilityfunctions, which belong to the class of homothetic preferences.

Example. Preference is quasilinear in good k if x � y implies x + �ek �y+�ek, where � > 0 and ek is a bundle of one unit of commodity k (and zero

74

of any other commodity). I.e. consuming commodity k does not a¤ect howthe agent values other commodities. As a consequence, xk (p; w) must enteradditively into the utility function, e.g. u (x) = xk (p; w) + f (x�k (p; w)).(The notation x�k (p; w) refers to quantities of all commodities other thank.) The function f is nonlinear - if other commodities enter additively, thenit is a condition of optimality that only one of these is consumed.Recall that utility maximization entails the "tangency"

@u (x) =@x`@u (x) =@xk

=p`pk

for all ` (at an interior solution). Since

@u (x)

@xk= 1;

all marginal utilities are constant with respect to consumption.For all commodities that enter nonlinearly, such a condition �xes the

quantity at a speci�c level. Only commodity k has a constant marginalutility at all levels of xk (p; w), hence only xk (p; w) can adjust to a wealthchange at a solution. It follows that

xk (p; w) =1

pk

w �

LX` 6=k=1

p`�x` (p)

!=w

pk� g (p)

(all remaining wealth is spent on commodity k). Letting a (p) � f (�x�k (p))�g (p) and b (p) = 1=pk, the indirect utility function therefore has the Gormanform:

v (p; w) = a (p)� b (p)w:

The wealth e¤ects of quasilinear demands are constant

@xk (p; w)

@w=1

pk;@x` (p; w)

@w= 0

for all ` 6= k. Thus, the wealth expansion path is in this case parallel to thek-axis. With homothetic preference, it is a straight line through the origin(i.e. the zero bundle).

75

� In some cases, there may be a �xed wealth distribution rule (w1 (p; w) ; : : : ; wI (p; w))that depends only on prices and aggregate wealth w. Then aggregatedemand can be written as a function of aggregate wealth,

IXi=1

xi (p; wi (p; w)) =

IXi=1

~xi (p; w) ;

without imposing uniform wealth e¤ects. (Recall that we normally needthem because aggregate wealth could be distributed in many ways. Ifwe know what the distribution is, then multiplicity is not an issue.)

� If wealth e¤ects are non-uniform, and the distribution of wealth is not�xed, aggregate demand may depend on certain statistics of the wealthdistribution. One statistic is aggregate wealth (i.e. the mean); we mayalso have to observe variance and higher-order moments. Then aggre-gate demand may be a function of prices and distributional statistics(rather than full information linking each preference to a particularindividual wealth).

8.2 Representative Consumer

� Given that we have an aggregate demand function, we ask now whetherit is rationalizable in the sense that there exists a preference based onwhich a hypothetical consumer could make these choices on behalf ofthe entire population.

� This is a precondition for the aggregate demand function to have somewelfare content. Without it, one cannot talk about improving on agiven aggregate bundle, e.g. by redistributing wealth.

De�nition. The aggregate demand function x (p; w) admits a positive rep-resentative consumer if there exists a preference % such that x (p; w) is theWalrasian demand function generated by %. I.e. 8 (p; w) and 8x, x 6= x (p; w)and p � x � w =) x (p; w) � x.

� To actually compare aggregate demands, we need to specify how wewould evaluate a particular list of individual outcomes. The rule could

76

be utilitarian (adding up individual utilities) or egalitarian (preferringless variation between individual utilities), or something else.

De�nition. A Bergson-Samuelson social welfare function W : RI ! Rassigns a value to every utility vector (u1; : : : ; uI) for the I agents.

� The optimal (feasible) distribution of wealth (w1; : : : ; wI), given a socialwelfare function W (�), is that which attains

max(w1;:::;wI)

W (v1 (p; w1) ; : : : ; vI (p; wI)) s.t.IXi=1

wi � w

� v (p; w) :

� The aggregate indirect utility function v (p; w) (given social welfarefunction W (�)) arises as follows. For every price vector p and aggre-gate wealth level w, we record the value of W (�) for every possible dis-tribution of w among the I individuals, who are assumed to maximizetheir utility from consumption within their budget sets Bp;w1 ; : : : ; Bp;wIas determined by the distribution and prices. The highest achievablevalue ofW (�) among all distributions, i.e. the maximum at given pricesand aggregate welath, is the indirect utility at (p; w).

Exercise 47 (MWG 4.D.2). Con�rm that v (p; w), thus constructed hasthe usual properties of an indirect utility function (i.e. homogeneous of degreezero, increasing in w, decreasing in p and quasiconvex).

� SupposeW (�) is increasing, concave and di¤erentiable, and the distrib-ution function that solvesW (�) is such that individual wealth wi (p; w)is di¤erentiable in price and homogeneous of degree one for all i.

� It can be shown that v (p; w) is the maximum of a utility function thatrepresents the preference of a positive representative consumer. I.e. theWalrasian demand function derived from v (p; w) via Roy�s identity isthe aggregate of the individual demands underlying (v1 (p; w1) ; : : : ; vI (p; wI)).

77

� Since the aggregate demand function attains maximal utility v (p; w)at all (p; w), it is chosen by the preference underlying v (p; w) (we couldconstruct a complete utility function from all values of W (�)). Hencethere exists a preference with respect to which the aggregate demandat every (p; w) is the utility-maximizing bundle. This means we havea positive representative consumer.

� In this particular case, the positive representative consumer is moreovera normative representative consumer. The normative representativeconsumer�s preference chooses an aggregate demand function x (p; w)that is 8 (p; w) consistent with individual choices x1 (p; w1) ; : : : ; xI (p; wI)at the wealth distribution (w1 (p; w) ; : : : ; wI (p; w)) that maximizesW (�).

� Note that a positive representative consumer could choose some otheraggregate demand function, which arises from suboptimal distributionsof wealth (according to W (�)). There may well be a preference thatrationalizes it. The normative requirement is much stronger.

De�nition. The aggregate demand function x (p; w) admits a normativerepresentative consumer with respect to welfare function W (�) if there ex-ists a preference % that generates the welfare-maximizing aggregate demandfunction x (p; w) (which re�ects at every (p; w) the wealth distribution thatmaximizes W (�)).

Example. Suppose all agents have homothetic preferences and the socialwelfare function is

W (u1; : : : ; uI) =IXi=1

�i lnui

with �i > 0 for all i, andPI

i=1 �i = 1. What is the aggregate demandfunction of a normative representative consumer? The �rst-order conditionsto maximize W (�), subject to

PIi=1wi = w, are, for i = 1; : : : ; L,

@W (u1; : : : ; uI)

@ui

@vi (p; wi)

@wi= �;

where � is the Lagrange multiplier and derivatives are evaluated at thewelfare-maximizing wealth distribution (w1; : : : ; wI).

78

Since@W (u1; : : : ; uI)

@ui=�iui=

�ivi (p; wi)

and@vi (p; wi)

@wi=vi (p; wi)

wi

(follows from v (p; w) = b (p)w as derived above for homothetic preference),we have �i=wi on the left side of the �rst-order conditions. Summing wi =�i=� over i,

w =

IXi=1

wi =

PIi=1 �i�

=1

�:

So the optimal wealth distribution is wi (p; w) = �iw for all i. Then thewelfare-maximizing aggregate demand function is

x (p; w) =IXi=1

xi (p; �iw)

(remember, the �i are given by the social welfare function).

Exercise 48 (MWG 4.D.8). Suppose for any distribution (w1; : : : ; wI) ofw there is a distribution (w01; : : : ; w

0I) of w

0 such that vi (p0; w0i) > vi (p; wi) forall i. Argue that any normative representative consumer must then prefer(p0; w0) to (p; w).

Exercise 49 (MWG 4.D.1). Show that v (p; w) can alternatively derivedby solving the problem

maxx1;:::;xI

W (u1 (x1) ; : : : ; uI (xI))

s.t. p �IXi=1

xi � w

for (x1 (p; w1 (p; w)) ; : : : ; xI (p; wI (p; w))), the individual demands at the op-timal wealth distribution rule (w1 (p; w) ; : : : ; wI (p; w)). Since this formal-ization, where a planner chooses consumption bundles, is equivalent to theone where the planner chooses wealth levels (and lets consumers make theirconsumption decisions in the market), we have a version of the second welfaretheorem.

79

8.3 Failure of the Weak Axiom

� Suppose aggregate demand is a well-de�ned function x (p; w) of pricesand aggregate wealth.

� It is clear that aggregate demand inherits homogeneity of degree zeroand Walras�law from individual demand functions.

� The de�nition of the weak axiom in the Walrasian demand settingextends directly from individual to aggregate demands: it says if p �x (p0; w0) � w and x (p; w) 6= x (p0; w0), then p0 � x (p; w) > w. (Ifaggregate bundle x (p; w) was chosen over x (p0; w0) in one situation,then x (p0; w0) can only be chosen if x (p; w) is unavailable.)

� In general, the weak axiom does not survive aggregation.

Example. Let wealth w = 10 be distributed equally (w1 = w2 = 5) andconsider demands

x1 (p; w1) =

�0;5

2

�; x2 (p; w1) = (3; 1)

at prices (p1; p2) = (1; 2), as well as

x1 (p0; w2) =

�3

2; 2

�; x2 (p

0; w2) = (2; 1)

at prices (p01; p02) = (2; 1). for agent 2. (Here, the subscripts refer to individ-

uals, not commodities.)These demands satisfy the weak axiom, since bundle x1 (p0; w=2) costs

more than w1 = 5 at prices p, and bundle x2 (p; w=2) is costs more thanw2 = 5 at prices p0. (We have well-de�ned orderings x1 (p; w1) �� x1 (p0; w1)and x2 (p0; w2) �� x2 (p; w=2).)Aggregate demands are

x (p; w) =

�3;7

2

�; x (p0; w) =

�7

2; 3

�:

Since both aggregate bundles cost w = 10 at the prices at which they arechosen and less than 10 when they are not chosen, they violate the weak

80

Figure 10: Failure of the aggregate weak axiom

axiom, since each is available in budget sets Bp;w and Bp0;w, but x (p; w) 6=x (p0; w). (There is no well-de�ned ordering, since x (p; w) �� x (p0; w) onBp;w, and x (p0; w) �� x (p; w) on Bp0;w.) As can be seen in Figure 10, thescaled-down aggregate bundles lie inside both individual budget sets (whichare, in this case, scaled-down versions of the aggregate budget sets).

� Even though the weak axiom fails in general, it does hold when indi-vidual demands are always decreasing in prices (for compensated anduncompensated price changes). We know from individual consumertheory that this is not a compelling property. It requires (pure) sub-stitution e¤ects to be su¢ ciently large to cancel out any income e¤ectsof inferior goods.

De�nition. Individual demand function xi (p; wi) satis�es the uncompen-sated law of demand (ULD) if 8p; p0 and 8wi,

(p0 � p) � (xi (p0; wi)� xi (p; wi)) < 0

when xi (p; wi) 6= xi (p0; wi). ULD for the aggregate demand function xi (p; wi)

is the same property without subscripts.

81

� Unlike the weak axiom, ULD does survive aggregation.

Proposition. If every individual Walrasian demand function xi (p; wi) sat-is�es ULD, then aggregate demand x (p; w) satis�es ULD.

Proof. If x (p; w) 6= x (p0; w), then for some i,

(p0 � p) � (xi (p0; wi)� xi (p; wi)) < 0

(for all other agents, the inequality is non-positive). Summing over I, wehave 8p; p0 and 8w,

(p0 � p) � (x (p0; w)� x (p; w)) < 0:

�

� The aggregate version of ULD, in combination the other standard prop-erties, implies the weak axiom. Thus, we can give conditions for onwhich aggregate demand satis�es the weak axiom, but these conditionsdo not have choice-theoretic foundations.

Proposition. If aggregate demand x (p; wi) satis�es homogeneity of degreezero, Walras�law and ULD, then it satis�es the weak axiom.

Proof. Given two price-wealth pairs (p; w) and (p0; w0) with x (p; w) 6=x (p0; w), let p � x (p0; w0) � w. The weak axiom requires p0 � x (p; w) > w0, sothat x (p; w) is revealed preferred to x (p0; w0). Let p00 � (w=w0) p0 and notethat, because p00=w = p0=w0, homogeneity of degree zero implies x (p00; w) =x (p0; w0). Thus. p00 � x (p00; w) = (w=w0) (p0 � x (p0; w0)) = w (by Walras�law).From ULD,

(p00 � p) � (x (p00; w)� x (p; w)) < 0:

Since p � x (p00; w) = p � x (p0; w0) � w and p � x (p; w) = w, it is necessary that

p00 � (x (p00; w)� x (p; w)) = w � p00 � x (p; w) < 0;

i.e. p00 � x (p; w) > w. Then (substituting for p00), p0 � x (p; w) > w0.�

82

� The compensated law of demand holds if the price derivative matrixof Hicksian demand, Dphi (p; ui), which is equal to the substitutionmatrix Si (p; wi) of compensated price e¤ects, is negative semide�nite(and de�nite when pre- and post-multiplied by vectors that are notproportional to p).

� Similarly, choices satisfy ULD if the price derivative matrix of Wal-rasian demand, Dpxi (p; wi), has the same properties.

Exercise 50 (MWG 4.C.1). Show that xi (p; wi) satis�es ULD only ifDpxi (p; wi) is negative semide�nite, and conversely, if Dpxi (p; wi) is negativede�nite (except vTDpxi (p; wi) v = 0 when v = �p for some �), then xi (p; wi)satis�es ULD.

� Homothetic preferences are a special case where choices respect ULD(and therefore the weak axiom).

Example. With a homothetic preference, we saw that rwixi (p; wi) =xi (p; wi) =wi. Rearranging Si (p; wi) = Dpxi (p; wi) +rwixi (p; wi)xi (p; wi)

T

gives

Dpxi (p; wi) = Si (p; wi)�1

wixi (p; wi)xi (p; wi)

T :

Since vTSi (p; wi) v � 0 (strictly if v is not proportional to p) and vTxi (p; wi)xi (p; wi)T v =�vTxi (p; wi)

�2 � 0, we have 8v 2 RL, vTDpxi (p; wi) v � 0 (strictly if v isnot proportional to p). This makes Dpxi (p; wi) negative de�nite for everyindividual, so that ULD holds in the aggregate and implies the weak axiom.

Exercise 51 (MWG 4.C.6). Verify that the following claim is true inthe case of a homothetic preference %i. If %ican be represented by a twicecontinuously di¤erentiable, concave utility function ui (�).and 8x,

�xi �D2ui (xi)xi

xi � rui (xi)< 4;

then xi (p; wi) satis�es ULD.

83

Exercise 52 (MWG 4.C.7). If every individual has the same consumptionfunction ~x (p; w), and individual wealth w is distributed on [0; �w] with densitynon-increasing in wealth, show that the aggregate demand function

x (p) =

Z �w

0

~x (p; w) dw

satis�es ULD. On the other hand, demonstrate that there are unimodal distri-butions of wealth (where the density function is single-peaked, �rst increasingand then decreasing) for which ~x (p; w) does not satisfy ULD.

9 Expected Utility

9.1 Lotteries

� In this lecture, we introduce risk into the choice framework.

� The set of N possible outcomes will be denoted by C (for consequences,e.g. C = X could be the set of consumption bundles). The decisionmaker�s choice induces a probability distribution on C: which of theconsequences will be realized is not certain at the time the choice ismade.

De�nition. A simple lottery is a probability distribution L = (p1; : : : ; pN) onthe set of consequences C. I.e. pn � 0 for n = 1; : : : ; N , and

PNn=1 pn = 1.

� A simple lottery is an element of the (N � 1)-dimensional simplex, i.e.the set

� =

(p 2 RN+ s.t.

NXn=1

pn = 1

):

(It is not N -dimensional because pN is determined by p1; : : : ; pN�1 andthe requirement that probabilities sum to 1. E.g. if N = 2, the distri-bution can be represented by a point p 2 [0; 1] in the one-dimensionalline space.)

84

Figure 11: Simplex

� Figure 11 depicts such a simplex for three consequences. For eachconsequence, there is a vertex. The simplex is drawn with height 1,and a lottery L = (p1; p2; p3) is mapped to the unique point whoseperpendicular distance from the three edges re�ects the probabilitiesof the consequences.

� Speci�cally, the distance of point L from the edge opposite 1, along theperpendicular line labeled p1, is equal to the probability of consequence1. The distance from the edge opposite 2, which is labeled p2, is theprobability of consequence 2, etc. This technique takes advantage ofthe fact that the lengths of these perpendicular lines from a point to theedges always sum to 1. Hence the points in the simplex can representprobability distributions.

De�nition. A compound lottery is a probability distribution (�1; : : : ; �K) ona set of K simple lotteries L1; : : : ; LK , where �k � 0 for k = 1; : : : ; K, andPK

k=1 �k = 1.

� It is then an easy matter to reduce a compound lottery on L1; : : : ; LK ,where Lk =

�pk1; : : : ; p

kN

�, to a simple lottery L = (p1; : : : ; pN) such that

pn =

KXk=1

�kpkn

85

for n = 1; : : : ; N . I.e. the probability of consequence n in the reducedlottery is the result of adding probabilities p1n; : : : ; p

Kn over simple lot-

teries L1; : : : ; LK , where each pkn is weighted by the probability �k thatLk is realized in the compund lottery.

� We can write the reduced lottery as a vector sum:

L = �1L1 + � � �+ �KLK 2 �:

� Our focus will be entirely on simple lotteries, on the premise that thedecision maker views any compound lottery as equivalent to the simplelottery it reduces to. Denote the set of all simple lotteries by L.

9.2 Preference over Lotteries

� Analogously to the certain setting, we start by endowing the agentwith a rational preference relation that ranks all elements of L, i.e. allsimple lotteries. Then we add further axioms.

� From two lotteries L and L0 we can obtain a new lottery �L+(1� �)L0,where the probability of consequence n in �L+(1� �)L0 is a weightedaverage of n�s probabilities in L and L0, and � 2 [0; 1] is the weight ofL.

De�nition. Preference % on L is mixture-continuous if 8L;L0 2 L and8L00 2 L, the set of mixtures of L and L0 that are preferred to L00,

f� 2 [0; 1] s.t. �L+ (1� �)L0 % L00g ;

is closed, as is the set of mixtures of L and L0 to which L is preferred,

f� 2 [0; 1] s.t. L00 % �L+ (1� �)L00g :

� Mixture continuity is a weaker property than continuity of% (i.e. upperand lower contour sets of L00 are closed). Whereas continuity impliesthat all convergent sequences in % (L00) have limits in % (L00), mixturecontinuity only requires this of a restricted set of sequences (those thatcan be constructed by mixing two lotteries and varying the weight �).The same applies, of course, to - (L00).

86

� The �avor is, however, much the same. Mixture continuity means thatpreference between two lotteries is robust to su¢ ciently small changesin their probabilities (in certain directions): if ~L is very similar to L,then L % L0 only if ~L % L0.

� One can think of some instances where people may violate continuity forextreme �. (Because their preferences over consequences are essentiallylexicographic.) E.g. most of us would never commit a violent crime(lottery L). We might prefer a completely law-abiding life (lotteryL0) to small-scale tax evasion (lottery L00), and prefer small-scale taxevasion to a mostly law-abiding life with a small chance of committinga violent crime (mixing L and L0).

� Such violations sound plausible, but they are often sensitive to framing.We have trouble imagining very small probabilities and treat them asif they are substantial. If you ask, "would you risk your life to watch asports game?," you are likely to get a di¤erent answer than if you ask,"would you drive to your friend�s house to watch the game?"

� Continuity plays the same role here as under certainty: it guaranteesthat there exists a utility representation U : L ! R such that L %L0 () U (L) � U (L0). Note that it is conventional to use the capitalletter U to distinguish a utility representation for a preference overlotteries from a utility representation for a preference over consumptionbundles or consequences.

� There is an obvious relationship between consequence n and the de-generate lottery Ln that assigns probability 1 to consequence n (andzero to all others). The utility value of such a degenerate lottery willbe denoted by un � U (Ln).

De�nition. Preference % on L satis�es independence if 8L;L0,

L % L0 () �L+ (1� �)L00 % �L0 + (1� �)L00

8L00 2 L and 8� 2 (0; 1).

� The independence axiom says that preference between L and L0 shouldnot be a¤ected by rescaling the probabilities in both lotteries in a givendirection.

87

� A more intuitive rationale can be given by thinking of � as a random-ization. Suppose you prefer L to L0 and are o¤ered compound lotteriesthat result in (1) L with probability � and L00 with probability 1 � �and (2) L0 with probability � and L00 with probability 1 � �. If � isthe probability of some state of the world (e.g. outcome of a toin coss),then whichever state is realized, you will be at least as happy with theoutcome of compound lottery (1). Hence you should prefer (1).

� The independence axiom is the fundamental di¤erence between theformalizations of lottery preferences and consumption preferences. Itis primarily responsible for the stronger results we will derive in thepresent context.

� It is important to understand that the independence axiom only makessense in the absence of complementarities, hence it could not be im-posed on preferences over consumption bundles. While the commodi-ties that make up a bundle are consumed together, the consequences oflotteries are mutually exclusive. I.e. mixing lotteries does not changethe nature of the consequences in any way, only their probabilities.

9.3 Expected Utility Theorem

� What the independence axiom gives us, in conjunction with mixturecontinuity, goes beyond the existence of a continuous utility functionthat represents lottery preference. It implies that the utility functionis linear in character. This statement is the expected utility theorem.We will get to the actual theorem in a few steps.

De�nition. Utility function U : L ! R has the expected utility form if thereexists (u1; : : : ; uN) 2 RN such that 8L = (p1; : : : ; pN) 2 L,

U (L) =NXn=1

pnun:

� A utility function with the expected utility form is called a von Neumann-Morgenstern (vNM) utility function.

88

� Notice that the de�nition requires (u1; : : : ; uN) 2 RN to be a utilityfunction for the restriction of the lottery preference to degenerate lot-teries. For if L = Ln (the degenerate lottery that assigns probability 1to consequence n), then U (Ln) = pnun = un if U (�) has the expectedutility form.

Exercise 53 (MWG 6.B.2). If U (�) represents a preference % on simplelotteries L and has the expected utility form, demonstrate that % satis�esthe independence axiom.

Proposition. Utility function U : L ! R has the expected utility form ifand only if it is linear:

U

KXk=1

�kLk

!=

KXk=1

�kU (Lk)

for any K lotteries L1; : : : ; LK 2 L and weights �1; : : : ; �K 2 [0; 1] such thatPKk=1 �k = 1.Proof. (If) Any lottery L = (p1; : : : ; pN) can be written in terms of

degenerate lotteries L1; : : : ; LN as L =PN

n=1 pnLn. If U (�) has the linearity

property, then

U (L) = U

NXn=1

pnLn

!=

NXn=1

pnU (Ln) =

NXn=1

pnun;

so that U (�) has the expected utility form.(Only if) Consider the compound lottery that assigns probabilities (�1; : : : ; �K)

to lotteries L1; : : : ; LK , where Lk =�pk1; : : : ; p

kN

�for k = 1; : : : ; K. If U (�) has

the expected utility property, then it assigns to the corresponding reducedlottery

PKk=1 �kLk =

�PKk=1 �kp

k1; : : : ;

PKk=1 �kp

kN

�the value

U

KXk=1

�kLk

!=

NXn=1

KXk=1

�kpkn

!un =

KXk=1

�k

NXn=1

pknun =KXk=1

�kU (Lk)

(swapping the order of summation is permitted by distributivity). HenceU (�) is linear.�

89

Figure 12: Indi¤erence curves with the expected utility form

� Linearity of the utility function implies linear indi¤erence curves forlotteries in the simplex, as in Figure 12. Suppose L � L0, so thatU (L) = U (L0) = �U (L) + (1� �)U (L0). If U (�) is linear, then�U (L) + (1� �)U (L0) = U (�L+ (1� �)L0), i.e. any convex combi-nation of L and L0 is indi¤erent to L and L0.

� A utility function that has the expected utility form is therefore asso-ciated with linear indi¤erence curves.

� According to the expected utility theorem, the independence axiomessentially implies the expected utility form (the only other propertyneeded is continuity). The connection is easy to see graphically, sincenonlinear indi¤erence curves violate the independence axiom.

� Consider the left panel of Figure 13. The curved indi¤erence set con-tains L and L0, but not L00 = �L+ (1� �)L0. However, since L � L0,the independence axiom says �L + (1� �)L0 � L0 + (1� �)L0, i.e.L00 � L0. Such a contradiction arises whenever an indi¤erence curve isnonlinear.

� Furthermore, the independence axiom requires the indi¤erence curvesto be parallel (as in Figure 11). The right panel of Figure 13 depicts hownonparallel indi¤erence curves cause a contradiction. Lotteries L1 andL01 yield utility U1, and we construct lotteries L2 and L

02 by respectively

90

Figure 13: Independence induces linear and parallel indi¤erence curves

mixing L1 and L01 identically with a lottery L. I.e. L2 = �L1+(1� �)Land L02 = �L01 + (1� �)L for some � 2 (0; 1). Since L1 � L01, theindependence axiom says �L1+(1� �)L � �L01+(1� �)L, i.e. L2 �L02. But if the indi¤erence line through L2 is not parallel to that for L1and L01, then it cannot contain L

02.

� Here is the expected utility theorem.

Proposition. If rational preference % on L satis�es continuity and inde-pendence, then % admits a utility representation that has the expected utilityform.Proof. I start with a few observations about % that are intuitive, but

should and can be proven formally from the axioms. There exist lotteriesL and L that are respectively preferred to all and none of the L 2 L. If�; � 2 [0; 1], then � > � () �L+ (1� �)L � �L+ (1� �)L.For any L 2 L it is possible to �nd a unique �L 2 [0; 1] such that

L � �LL + (1� �L)L. (We veri�ed it in constructing a utility functionfor a continuous rational preference. This is where the continuity axiom isneeded.) Now we will show that U (L) = �L is a utility function for % andhas the expected utility form.By the de�nition of �L, L % L0 if and only if

L � �LL+ (1� �L)L % �L0L+ (1� �L0)L � L0;

91

i.e. if and only if �L � �L0. Therefore, if U (L) = �L for all L 2 L, thenL % L0 () U (L) � U (L0), so that U (�) represents %.The expected utility form is equivalent to linearity, i.e. U

�PKk=1 �kLk

�=PK

k=1 �kU (Lk) for all L1; : : : ; LK and all �1; : : : ; �K � 0 such thatPK

k=1 �k =1. Since Lk � �LkL+(1� �Lk)L, induction on the independence axiom gives

L �KXk=1

�kLk �KXk=1

�k��LkL+ (1� �Lk)L

�=

KXk=1

�k�LkL+

1�

KXk=1

�k�Lk

!L:

Thus �L =PK

k=1 �k�Lk , so by the construction of U (�),

U

KXk=1

�kLk

!=

KXk=1

�k�Lk =KXk=1

�kU (Lk) :

�

Exercise 54 (MWG 6.B.3). If the set of outcomes C is �nite and rationalpreference % on L satis�es independence, demonstrate that there are bestand worst lotteries L and L in L, such that L % L % L for all L 2 L.

� A utility function that represents a preference over consumption bun-dles is unique up to strictly increasing transformation. Hence it re-�ects the ordinal nature of the preference: there is no signi�cance inthe magnitude ju (x)� u (x0)j, only in the fact that u (x) � u (x0) oru (x) � u (x0). If ju (x)� u (x0)j > ju (x)� u (x00)j, we can �nd an-other utility function that represents the same preference, but wherej~u (x)� ~u (x0)j < j~u (x)� ~u (x00)j.

� In contrast, the next result indicates that a vNM expected utilityfunction is cardinal in nature, because it is unique only up to pos-itive linear transformation, which preserves the (relative) magnitudeof di¤erences. (If jU (L)� U (L0)j > jU (L)� U (L00)j, then 8� > 0,j�U (L)� �U (L0)j > j�U (L)� �U (L00)j.)

92

Proposition. If utility functions U : L ! R and ~U : L ! R representpreference % on L, and U (�) has the expected utility form, then ~U : L ! Rhas the expected utility form if and only if it is a positive linear transformationof U (�), i.e. 9� > 0 and 9 2 R such that 8L 2 L, ~U (L) = �U (L) + .Proof. (If) If U (L) has the expected utility form and ~U (L) = �U (L)+ ,

then ~U (L) is linear: since L =PK

k=1 �kLk for some set of lotteries L1; : : : ; LKand weights �1; : : : ; �K ,

~U (L) = ~U

KXk=1

�kLk

!= �U

KXk=1

�kLk

!+

= �

KXk=1

�kU (Lk)

!+ =

KXk=1

�k (�U (Lk) + ) ;

where the last two equalities re�ect linearity of U (�) and thatPK

k=1 �k = 1.Linearity of ~U (�) implies the expected utility form.(Only if) Suppose U (�) and ~U (�) both have the expected utility form

(thus, both are linear). We will construct constants � > 0 and such that8L 2 L, ~U (L) = �U (L) + . Unless all lotteries are indi¤erent (in whichcase U (�) and ~U (�) are constant functions, so that the property holds), thereare best and worst lotteries L and L in L such that L � L. (Since U (�) iscontinuous, it has a maximizer and minimizer on the probability simplex L,which is a compact set.)For every L 2 L, de�ne

�L �U (L)� U (L)

U�L�� U (L)

:

Rearrangement gives

U (L) = �LU�L�+ (1� �L)U (L)

and, because U (�) is linear, U (L) = U��LL+ (1� �L)L

�. Thus L � �LL+

(1� �L)L.Because ~U (�) represents % and is linear,

~U (L) = ~U��LL+ (1� �L)L

�= �L ~U

�L�+ (1� �L) ~U (L) = �U (L) + ;

where

� �~U�L�� ~U (L)

U�L�� U (L)

> 0; � ~U (L)� �U (L) :

93

�

Exercise 55 (MWG 6.B.4). A safety agency is looking for an evacuationcriterion for an area that has a 1% probability of �ooding. Four things canhappen: (A) No evacuation is necessary, and none is performed. (B) An un-necessary evacuation is performed. (C) A necessary evacuation is performed.(D) A necessary evacuation is not performed.The agency is indi¤erent between sure outcome (B) and a scenario where

(A) occurs with probability p 2 (0; 1) and (D) occurs with probability 1� p.The agency is also indi¤erent between sure outcome (C) and a scenario where(A) occurs with probability q 2 (0; 1) and (D) occurs with probability 1� q.Moreover, it prefers (A) to (D). Suppose the expected utility theorem applies.(a) Construct an expected utility function for the agency.(b) Compare the following policy criteria: (1) Evacuate in 90% of �ood-ing instances, and evacuate unnecessarily in 10% of the instances where no�ooding occurs. (2) Evacuate in 95% of �ooding instances, and evacuate un-necessarily in 15% of instances where no �ooding occurs. Derive probabilitydistributions over the four outcomes under these criteria and decide, basedon the expected utility function, which criterion the agency prefers.(Note that this version of the problem corrects two typos in MWG.)

9.4 Paradoxes

� Expected utility (or, rather, the underlying independence axiom) en-tails some speci�c predictions about choices. These predictions fre-quently fail in some famous experiments that have led to various alter-native axiomatizations in order to explain the observed behavior.

� Consider four lotteries that o¤er the following probabilities over threeprizes:

$2:5 million $0:5 million $0 millionL1 0 1 0L01 0:10 0:89 0:01L2 0 0:11 0:89L02 0:10 0 0:90

:

The decision maker is asked to compare L1 to L01 and L2 to L02.

� If is often observed (roughly half the time) that L1 � L01 (giving upthe chance to win a greater prize to avoid a small risk of zero), while

94

L02 � L2 (accepting a slightly increased risk of zero for the prospect ofa greater prize).

� These preferences are not necessarily irrational, but they are incon-sistent with expected utility theory. Notice that in both cases L0 isobtainable from L by taking 0:11 out of the probability of winning $0:5million and distributing 0:01 to the probability of $0 and 0:10 to theprobability of $2:5 million. Only the initial probabilities di¤er, but ac-cording to the independence axiom they should not a¤ect preferencefor the adjustment.

� Explicitly, if L1 � L01, then U (L1) = u0:5 � 0:10u2:5+0:89u0:5+0:01u0,which implies, after adding 0:89 (u0 � u0:5) on both sides, 0:11u0:5 +0:89u0 � 0:10u2:5 + 0:90u0, i.e. L2 � L02. But L

02 � L2 was observed.

This is known as the Allais paradox.

� Perhaps people worry about regrets they might have if a bad outcomeoccurs. (I.e. if L01 results in zero, it is viewed as a loss of $0:5 millionthat the agent would have had with L1, and therefore seems somehowworse than zero. In contrast, if L02 results in zero, there was no way toguarantee a non-zero prize by choosing L2.)

Exercise 56 (MWG 6.B.5). The following property is known as the"betweenness axiom": 8L;L0 2 L and 8� 2 (0; 1), if L � L0, then �L +(1� �)L0 � L. Suppose there are three possible outcomes.(a) Show that a preference relation on lotteries satis�es independence onlyif it satis�es betweenness.(b) Depict in the simplex that, if the continuity and betweenness axiomshold, then indi¤erence curves of a lottery preference must be straight lines.Conversely, depict that straight-line indi¤erence curves imply betweenness.(c) Argue (from a graphic comparison) that betweenness is weaker thanindependence.(d) Draw an indi¤erence map that satis�es betweenness and produces thechoices from the Allais paradox.

� Consider now the following experiment. The alternatives are lotteriesover outcomes: go to Venice, watch a movie about Venice, stay at home.

95

Even though you prefer the outcomes in this order, you may prefer torandomize between going to Venice and staying at home, rather thanbetween going to Venice and watching the movie. This would makesense if you expect that, in case you cannot go to Venice (but werehoping to), you will no longer enjoy the movie.

� But it clearly violates the independence axiom. This is known asMachina�s paradox. It reminds us that preference need not be �xed,but may depend on realizations of events. If you want to think aboutit in terms of regrets, the regret here is not over a choice the decisionmaker failed to make, but over the outcome "nature" chose (which isbeyond the agent�s control).

� Perhaps the most in�uential critique of expected utility theory is basedon the Ellsberg paradox. Suppose there are two urns, R and H, thateach contain 100 balls of white or black color. The proportion of colorsis known for R (there are 49 white and 51 black balls), but not for H.The decision maker will win a $1; 000 prize if he can pick a ball of aspeci�ed color (i.e. either white or black) from one of the urns. All hehas to do is choose the urn from which to take the ball.

� Many people will always take the ball from R in successive experi-ments where �rst a black ball and then a white ball wins. If a blackball wins, then R induces a lottery RB over outcomes $1; 000 and $0with probabilities 0:51 and 0:49, whereas H induces a lottery HB withunknown probabilities � and 1 � �. If a white ball wins, R inducesRW = (0:49; 0:51) and H induces HB = (1� �; �).

� The problem is that there is no way to assign a probability � (thata black ball is chosen from H) that could justify choosing R in bothexperiments. Since the agent wants to maximize expected winnings,he can prefer RB = (0:51; 0:49) to Hb only if � < 0:51. But then1� � > 0:49, so that he must prefer HW to RW .

� This failure of expected utility theory is fundamental: behavior appearsto be at times inconsistent with the notion that individuals choose be-tween known (or even estimated) probability distributions over out-comes.

96

� As we will see, expected utility theory can be extended to replace theobjective probabilities in the de�nition of lotteries with subjective (im-plicitly believed) probabilities of events. But this does not address theEllsberg paradox, which contradicts the existence of unique probabili-ties altogether.

� An important current research area in decision theory is therefore non-expected utility theory, where agents may e.g. have ambiguous beliefs(allowing for multiple probability distributions), and choices could re-�ect optimistic or pessimistic expectations. These situations, whereagents do not use simple probabilistic information or beliefs, are saidto be characterized by (Knightian) uncertainty, rather than risk.

Exercise 57 (MWG 6.F.2). In the setting of the Ellsberg paradox, letu (0) = 0 and u (1000) = 1 represent the decision maker�s preferences oversure amounts of money. A probabilistic belief that the color of the H-ballis white can be expressed as � 2 [0; 1]. Suppose, however, that the decisionmaker has a set P � [0; 1] of such beliefs. The available actions are R and H(respectively, picking the ball from urn R and H). Denote by W the choicesituation where the $1000 prize is won if the ball is white and $0 otherwise.In choice situation B, a black ball wins $1000 dollars, and the decision makergets $0 otherwise.For each choice situation, let the utility function over actions R and

H be as follows. In W , UW : fR;Hg ! R is such that UW (R) = 0:49and UW (H) = min f� s.t. � 2 Pg. In B, UB : fR;Hg ! R is such thatUW (R) = 0:51 and UW (H) = min f1� � s.t. � 2 Pg. I.e. UW (R) is theexpected utility of $1000 given the (objective) probability implied by thenumber of white and black balls in urn R. But UW (H) is the expectedutility of $1000 based on the most pessimistic probability in P . (This is aninstance of Gilboa and Schmeidler�s theory of nonunique prior beliefs.)(a) Show that if P consists of a single belief, then UW and UB are derivedfrom a vNM utility function, and UW (R) > UW (H) () UB (R) < UB (H).(b) Find a set P such that UW (R) > UW (H) and UB (R) > UB (H).

Exercise 58 (MWG 6.B.6). Sometimes, an agent�s preference over lotter-ies depends on a prior action a 2 A (e.g. when you have to bring wine todinner, you would like to know what kind of meat will be served, so you prefer

97

a degenerate lottery). Such a preference has an induced utility representation

U (L) = maxa2A

NXn=1

pnun (a)

for all L = (p1; : : : ; pN) 2 L, where un (a) is the utility assigned to degeneratelottery Ln if action a 2 A is taken. Show that U (�) is convex, but (byexample) not necessarily linear.

9.5 State-Space Approaches

� It is often possible and useful to impose a bit more structure wherewe have so far assumed given probabilities. Let S be a set of states(decision-relevant situations that may materialize), where the proba-bility �s of state s 2 S is objectively known (we will relax this tosubjective knowledge later).

� The choice objects are now taken to be random variables x : S ! C, i.e.functions that determine which consequence will occur in each state.(These functions are also called acts.)

� Such random variables are closely related to lotteries via the probabilitydistribution over states. While a lottery assigns probabilities to conse-quences, a random variable, or act, assigns consequences to events thatare characterized by their probabilities. Both types of choice objectstherefore induce a probability distribution over consequences.

� Preference is de�ned on the space of random variables X and, if itsatis�es continuity and a variant of independence called the sure-thingprinciple, can be represented by a function that has a modi�ed expectedutility form.

De�nition. Utility function U : X ! R has the extended expected util-ity form if there exists 8s 2 S, a function us : C ! R such that 8x =(x1; : : : ; xS) 2 X ,

U (x) =

SXs=1

�sus (xs) :

98

� Comparing this to the expected utility form, U (L) =PN

n=1 pnun, theprobability pn of consequence n (that is determined by the choice oflottery L) is replaced by state probability �s. And utility un of thelottery that is degenerate in consequence n is replaced by utility us (xs)of consequence xs (which the act x associates with state s).

� An extended expected utility function represents preferences on X thatsatisfy continuity and the sure-thing principle (provided there are atleast three states). Informally, the sure-thing principle says that pref-erence between x and x0 should be determined on states in which theydisagree.

� To express the sure-thing principle formally, I introduce some new lan-guage. Denote by xE the restriction of random variable x on S to theevent E � S, and by x�E the restriction of x to the complement of E.(I.e. xE assigns states s 2 E to consequences.)

� Say that x %E x0 if xEx0�E % x0 (i.e. x is preferred to x0 when x�E isreplaced by x0�E, so that x di¤ers from x0 only on E). One can readx %E x0 as "x is preferred to x0 on E."

De�nition. Preference % on X satis�es the sure-thing principle if 8E � S,x %E x0 () �x %E �x0 whenever xE = �xE and x0E = �x0E.

� The axiom applies when consequences are identical for states outsidethe event E, so that the choice between x and x0, respectively �x and�x0, matters only if E occurs. (Hence the name "sure thing" - if whathappens outside E cannot be changed, it should not a¤ect preference.)So if x % x0, and xE = �xE, and x0E = �x0E, then it should be the casethat �x % �x0.

� The sure-thing principle takes the place of the independence axiom inthe present setting. It is analogous in that it requires preference to beindependent of events on which the acts do not di¤er. (Or, in alterna-tive formulations, events that have zero probability of occurring.)

� So far, we have taken for granted that there are objective probabilitiesfor the states. It is, of course, rare to have such information. In fact,

99

nothing forces us to interpret the �1; : : : ; �S in the extended expectedutility function as objective probabilities.

� Suppose preference % on X respects continuity and the sure-thingprinciple, so that it has an extended expected utility representationU (x) =

PSs=1 �sus (xs). This tell us that, for every state s and act x,

the value �sus (xs) is uniquely determined up to positive linear trans-formation. If �s is not given objectively, then it is arbitrary.

� Is there a compelling way to "disentangle" �s from us (xs), i.e. �x �s sothat we can interpret it as the subjective probability the agent assignsimplicitly to state s? If we are willing to require that preference isstate-uniform (depends only on the consequence chosen in state s, butranks consequences the same in every state), then us (xs) � u (xs) forall s 2 S, and �s is determined through �su (xs) up to scaling. If the�1; : : : ; �S are to be interpreted as probabilities, then

PSs=1 �s = 1, so

that they are completely determined.

� The result that a preference on X that satis�es continuity, the sure-thing principle and state uniformity admits a utility representation ofthe form U (x) =

PSs=1 �su (xs), with unique probabilities �1; : : : ; �S,

is known as the subjective expected utility theorem, due to Savage. Itimplies that expected utility theory does not depend on factual knowl-edge of probabilities, but can be built around personal beliefs that areprobabilistic.

10 Risk

10.1 Money Lotteries

� A lottery over continuous amounts of money x 2 R can be describedmost generally in terms of its cumulative distribution function F : R![0; 1]. (The more direct approach would be to use density functionsf (�), but these do not always exists and exclude the case of discreteoutcomes. If f (�) does exist, then F (x) =

R x�1 f (t) dt.)

� We now take the lottery space L to be the set of distribution func-tions on R. The continuous version of the expected utility theorem

100

guarantees that a continuous preference % on L, that satis�es the in-dependence axiom, can be represented by

U (F ) =

Zu (x) dF (x) ;

(Note that f (x) � dF (x) =dx, so that dF (x) = f (x) dx, and thusRu (x) dF (x) =

Ru (x) f (x) dx whenever the density exists.)

� The function u (�), which records the values of degenerate lotteries, i.e.certain amounts of money, is called Bernoulli utility function. Assumethat u (�) is increasing, continuous and bounded. (If it were unbounded,small probability events could make a lottery in�nitely desirable - theSt. Petersburg paradox.)

Exercise (MWG 6.C.2). Suppose an individual�s Bernoulli utility functionu (�) has the quadratic form u (x) = �x2 + x. Show that utility from adistribution is determined by the mean and variance of the distribution, andonly these moments. (No need to do part b.)

10.2 Risk Attitude

� A risk-averse agent is someone who rejects fair gambles (that haveneither an expected gain nor loss).

De�nition. An agent is risk-averse if she prefers the expected money valueof a lottery to the lottery itself: 8F 2 L,

u

�ZxdF (x)

�� U (F ) :

(Strictly risk-averse if this is an equality only if F (�) is degenerate.) Theagent is risk-neutral if always indi¤erent between the expected value of alottery and the lottery itself, i.e. the above is an equality.

� The criterion for risk aversion implies, when U (�) has the expectedutility form,

u

�ZxdF (x)

��Zu (x) dF (x) :

101

This is Jensen�s inequality that characterizes a concave function u (�).(If you think of the integral over the probability distribution F (�) as aweighted sum, the inequality relates to the basic de�nition of a concavefunction u (�x+ (1� �)x0) � �u (x) + (1� �)u (x0).)

� Hence the Bernoulli utility function of a risk-averse agent is concave,and anyone with a concave Bernoulli function is risk-averse. This facthas a straightforward explanation: the risk-averse agent�s Bernoulliutility increases more slowly with gain than it decreases with a loss.Since the agent has, in utility terms, more to lose than gain from alottery that is fair in money terms, she declines the lottery unless theodds are strictly favorable.

� In Figure 14, the expected utility of a fair lottery, i.e. random vari-able X : fx1; x2g ! [0; 1] with equal probabilities, is labeled U (X).The Bernoulli utility of certain amount E (X) = (1=2)x1 + (1=2)x2is labeled u (E (X)). The individual in the left panel is risk-averse,and U (X) � u (E (X)). (Observe how x1 and x2 along the x-axis areequidistant from E (X), while the corresponding utility value u (E (X))is closer to the utility of the better state, u (x2).)

� On the right panel, we have the contrasting case of a risk-seeker, whoseBernoulli utility is convex.

De�nition. The certainty equivalent c (F; u) of lottery F (�) is its moneyvalue to an agent whose preference is represented by Bernoulli utility functionu (�):

u (c (F; u)) =

Zu (x) dF (x) :

� It is intuitive that a risk-averse person has a certainty equivalent lessthan the expected value of the lottery, i.e. she only values the lotterythe same if it produces on average a gain relative to the certainty equiv-alent. Indeed, concavity of the (increasing) Bernoulli utility function

102

Figure 14: Bernoulli utility functions for risk-averter (left) and risk-seeker(right)

implies:Zu (x) dF (x) � u

�ZxdF (x)

�() u (c (F; u)) � u

�ZxdF (x)

�() c (F; u) �

ZxdF (x) :

Figure 15 shows the certainty equivalent for risk-averse and risk-seekingagents.

� Beyond these formal characterizations, risk-averse behavior is evidentin the propensity to buy insurance, even when premia are not "actuar-ially fair" (i.e. the expected payout is less than the premium).

Example. A strictly risk averse agent with initial wealth w faces a possibledamage of $D with probability �. The agent is o¤ered insurance at a fairpremium � per dollar-payout in the event of a loss. If � dollars of insurance

103

Figure 15: Certainty equivalent for risk-averter (left) and risk-seeker (right)

(i.e. conditional payout) are purchased at this premium, the agent�s wealthwill be either w � �� (if no damage occurs) or w � �� � D + � = w +(1� �)� � D (if there is damage). Expected utility from a choice of �,which induces a lottery over w+(1� �)��D and w��� with probabilities(�; 1� �), is then

U (�) = �u (w + (1� �)��D) + (1� �)u (w � ��) :

The �rst-order condition with respect to �,

u0 (w + (1� �)�� �D) = u0 (w � ���) ;

can be solved for w + (1� �)�� �D = w � ���, i.e. �� = D because u0 (�)is strictly decreasing from strict concavity. Thus, a risk-averse agent insuresfully if the premium is actuarially fair.

Example. Suppose an amount of wealth can be invested in a safe asset,which yields $1, and a risky asset with earnings distribution F (z) such thatexpected return

RzdF (z) is greater than 1. The portfolio choice problem is

to determine the optimal shares � and � to invest in these assets, such that� + � = 1. Since the random return, given � and �, is � + �z, utility is a

104

random variable u (�+ �z) = u (1� � + �z). An expected utility maximizersolves:

max�2[0;1]

Zu (1 + � (z � 1)) dF (z) ;

which has �rst-order condition, with respect to �,Z(z � 1)u0 (1 + �� (z � 1)) dF (z) = 0

(at an interior solution). Since the left side is greater than zero at � = 0,given

RzdF (z) > 1 and that u (�) is increasing everywhere, we must have

�� > 0, whether or not the individual is risk-averse. The general principleis that an agent will always invest some share of wealth in an actuariallyfavorable asset.

Exercise (MWG 6.C.19).

� Risk aversion can be quanti�ed and compared by means of the absoluteand relative coe¢ cients of risk aversion.

De�nition. The Arrow-Pratt coe¢ cient of absolute risk aversion is, for atwice di¤erentiable Bernoulli utility function u (�) at x,

rA (x; u) = �u00 (x)

u0 (x):

� This is essentially a measure of the concave curvature of the Bernoulliutility function (for an increasing concave function, u0 (x) > 0 andu00 (x) < 0). Note that we cannot compare the degree of concavity basedon �u00 (x) alone, since this derivative can be scaled by a positive lineartransformation (which yields another utility function that representsthe preference). But such a transformation would also scale u0 (x), soit cannot a¤ect rA (x; u).

� For a risk-neutral agent, u0 (x) is constant and u00 (x) = 0, so rA (x; u) =0.

105

� The more risk-averse of two agents has a lower certainty equivalentfor any given lottery F (�). Moreover, 2 is more risk-averse than 1 inthe sense that rA (x; u2) � rA (x; u1) if and only if u2 (�) is a concavetransformation of u1 (�). I.e. there exists an increasing concave function (�) such that 8x, u2 (x) = (u1 (x)).

Exercise (MWG 6.C.6, C.7).

Example. It is possible to recover the preference from the Arrow-Pratt coef-�cient. Suppose rA (x; u) = a for all x. Integrating u00 (x) = �au0 (x) on bothsides, we have u0 (x) =u (x) = @ lnu (x) =@x = �a, and integrating once moreon both sides, u (x) = e�ax, i.e. the utility function is exponential when theArrow-Pratt coe¢ cient is constant. I have constructed one particular utilityfunction, assuming the integration constants are zero, but others are stillexponential (solve the di¤erential equation to see this). Exponential utilityfunctions therefore constitute the CARA (constant absolute risk aversion)class.

� The DARA class of Bernoulli utility functions has the plausible prop-erty that wealthier people tend to be less risk-averse.

De�nition. Bernoulli utility function u (�) exhibits decreasing absolute riskaversion (DARA) if rA (x; u) is a decreasing function of x, i.e. 8x, rA (x; u) >rA (x

0; u) whenever x < x0.

Exercise (MWG 6.C.8).

De�nition. The coe¢ cient of relative risk aversion is, for a twice di¤eren-tiable Bernoulli utility function u (�) at x,

rR (x; u) = �xu00 (x)

u0 (x):

106

� If rR (x; u) = xrA (x; u) is decreasing in x, then clearly rA (x; u) mustbe decreasing in x. The converse is not true. Therefore, non-increasingrelative risk reversion is a stronger property than decreasing absoluterisk aversion.

Example. Consider the Bernoulli utility function u (x) = x1��= (1� �),where � 2 (0; 1). Since rA (x; u) = �=x is decreasing in x, this function is inthe DARA class. But rR (x; u) = xrR (x; u) = � is constant, so DARA doesnot imply DRRA.

Exercise (MWG 6.C.18).

Exercise (MWG 6.C.12).

10.3 Stochastic Dominance

� Up to now, we have compared agents in terms of the risk aversionexhibited by their utility functions. Now we are interested in compar-ing lotteries and �nding criteria by which they can be ranked, givenproperties of preference, such as risk attitude.

� If distribution F yields a higher expected utility than lottery G, re-gardless of risk attitude (i.e. the speci�c form of the Bernoulli utilityfunction), F is said to �rst-order stochastically dominate G.

De�nition. Distribution F (�) �rst-order stochastically dominates G (�), writ-ten F %FOSD G, if Z

u (x) dF (x) �Zu (x) dG (x)

for any non-decreasing function u : R! R.

� If distribution F yields a higher expected utility than lottery G for arisk-averse agent (i.e. a concave Bernoulli utility function), then F issaid to second-order stochastically dominate G.

107

De�nition. Distribution F (�) second-order stochastically dominates G (�),written F %SOSD G, if F (�) and G (�) have the same expectation of x, i.e.RxdF (x) =

RxdG (x), andZ

u (x) dF (x) �Zu (x) dG (x)

for any non-decreasing concave function u : R+ ! R.

� F %FOSD G is equivalent to the property: for any x, getting more thanx is more likely under F than under G.

� A related statement can be made for second-order dominance. F %SOSDG is equivalent to the property: for any x, probability mass accumulatesfaster toward x under G than under F . (I.e. G spreads the probabilitymass more evenly and gives more weight to the extremes. I will give aprecise characterization in a moment.)

� The next results make use of some integral relationships that may befound through "integration by parts." The technique is based on theproduct rule applied to u (x)F (x):

d

dxu (x)F (x) = u0 (x)F (x) + u (x) f (x)

implies

u (x) f (x) =d

dxu (x)F (x)� u0 (x)F (x)

and, integrating both sides,Zu (x) f (x) dx = ��

Zu0 (x)F (x) dx;

where � is a constant (since du (x)F (x) =dx integrated on (�1;1) isu (1)F (1)� u (�1)F (�1), and any distribution function satis�esF (�1) = 0 and F (1) = 1).

� Applied to u0 (x)R x�1 F (t) dt, the product rule gives

d

dxu0 (x)

Z x

�1F (t) dt = u00 (x)

Z x

�1F (t) dt+ 2u0 (x)F (x)

108

(since the derivative ofR x�1 F (t) dt with respect to x is

R x�1 f (t) dt +

F (x) = 2F (x) by Leibniz�rule). Thus

u0 (x)F (x) =1

2

d

dxu0 (x)

Z x

�1F (t) dx� 1

2u00 (x)

Z x

�1F (t) dt

and Zu0 (x)F (x) dx = � � 1

2

Z �u00 (x)

Z x

�1F (t) dt

�dx:

(where � is a constant, since 12u0 (1)

R1�1 F (t) dt =

12u0 (1)).

� HenceZu (x) dF (x) = ��

Zu0 (x)F (x) dx

= �� � +1

2

Z �u00 (x)

Z x

�1F (t) dt

�dx:

Proposition. The payo¤ distribution F (�) �rst-order stochastically domi-nates G (�) if and only if 8x, F (x) � G (x).

Proof. (If) Let F (x) � G (x) for all x. From integration by parts,we have

Ru (x) dF (x) = � �

Ru0 (x)F (x) dx and

Ru (x) dG (x) = � �R

u0 (x)G (x) dx, soZu0 (x)F (x) dx �

Zu0 (x)G (x) dx ()

Zu (x) dF (x) �

Zu (x) dG (x) :

If u (�) is an increasing function, the �rst inequality holds, given that F (x) �G (x) for all x. Hence F �rst-order dominates G.(Only if) Suppose F (�x) > G (�x) for some �x. To show that F fails to

�rst-order dominate G, we need to �nd a non-decreasing function u (�) suchthat

Ru (x) dF (x) <

Ru (x) dG (x) at some x. Consider u (x) = 1 for x > �x

and u (x) = 0 for x � �x, which is non-decreasing. ThenZu (x) dF (x) =

Z 1

�x

dF (x) = 1� F (�x)

< 1�G (�x) =

Z 1

�x

dG (x) =

Zu (x) dF (x) :

109

�

Exercise (MWG 6.D.1).

Exercise (MWG 6.D.2).

Proposition. The payo¤ distribution F (�) second-order stochastically dom-inates G (�) if and only if 8x,Z x

�1G (t) dt �

Z x

�1F (t) dt:

Proof. (If) LetR x�1 F (t) dt �

R x�1G (t) dt for all x. From integrating

by parts, we haveRu (x) dF (x) = � � � + 1

2

R �u00 (x)

R x�1 F (t) dt

�dx andR

u (x) dG (x) = ���+ 12

R �u00 (x)

R x�1G (t) dt

�dx. If u (�) is concave, then

u00 (x) < 0, soZ x

�1F (t) dt �

Z x

�1G (t) dt ()

Zu (x) dF (x) �

Zu (x) dG (x) :

Hence F second-order dominates G.(Only if) Suppose

R x�1 F (t) dt >

R x�1G (t) dt for some x. To show that

F fails to second-order dominate G, we need to �nd a concave functionu (�) such that

Ru (t) dF (t) <

Ru (t) dG (t) at x. Let u (t) = t, except

in an interval [x; x] containing x whereR x0�1 F (t) dt >

R x0�1G (t) dt for all

x0 2 [x; x]. On this interval, let u (t) be strictly concave.From integration by parts,Zu (t) dF (t) >

Zu (t) dG (t) ()

Z �u00 (x)

Z x

�1F (t) dt

�dx >

Z �u00 (x)

Z x

�1G (t) dt

�dx:

Since u00 (x) = 0 for all x =2 [x; x] and u00 (x) < 0 for all x 2 [x; x], whereR xxF (t) dt >

R xxG (t) dt, the left inequality cannot hold, so

Ru (x) dF (x) <R

u (x) dG (x). This means G �SOSD F , a contradiction.�

110

� F %SOSD G is also equivalent to the property that G is a "mean-preserving spread" of F . I.e. if G is obtainable from F by replac-ing every certain outcome x with a lottery that yields x + z, where zis a zero-mean random variable, distributed according to Hx (z) withRzHx (z) = 0.

Example. Consider two lotteries that reward outcomes of rolling a fair die.F gives $1 if a number up to 3 is rolled and $2 if the number is greaterthan 3. G pays nothing for a 1 and $5 for a 6, and $1 otherwise. Theselotteries have the same mean, 3=2, and probabilities (1=2; 1=2) over ($1; $2),respectively (1=6; 2=3; 1=6) over ($0; $1; $5). To obtain G from F , replacethe $1 and $2 wins in F with lotteries that give ($0; $1; $5) respectively withprobabilities (1=3; 7=12; 1=12) and (0; 3=4; 1=4). Observe that the expectedvalues of these lotteries are $1 and $2. The compound lottery over ($0; $1; $5)that plays (1=3; 7=12; 1=12) and (0; 3=4; 1=4) with equal probability reducesto (1=6; 2=3; 1=6) = G. So we have constructed G as a mean-preservingspread of F , i.e. F %SOSD G.

Example. Continuing in the previous scenario, distribution H is called an"elementary increase in risk" from G if it redistributes all probability massto the extreme points in G�s domain (while preserving the mean). I.e. His the lottery (7=10; 0; 3=10). This is a mean-preserving spread via lotteries(1; 0; 0), (4=5; 0; 1=5) and (0; 0; 1) in place of the $0, $1 and $5 wins.

Exercise (MWG 6.D.4).

Exercise (MWG 6.D.3).

11 Pro�t Maximization Problem

11.1 Production Set

� Firms exist in order to transform some goods (inputs) into other goods(outputs). Which goods are inputs and which are outputs is a matterof choice for every �rm within the technological constraints. Feasibleproduction plans are described by the production set Y , which lists allpossible combinations of input and output quantities. These are simplybundles, or vectors, in the commodity space.

111

� However, unlike consumption bundles, production vectors necessarilycontain negative entries (every technology requires inputs). The pro-duction set can therefore not be restricted to R+. A typical elementis (y1; : : : ; yL) 2 RL, where y` < 0 identi�es the `th commodity as aninput for the particular �rm, and y` > 0 means ` is an output. The setof all such feasible vectors is the production set Y � RL.

� It is commonly assumed that Y is nonempty and closed (i.e. there aree¢ cient production plans), and that there can be "no free lunch" (nooutput without input, y 2 Y and y � 0 =) y = 0).

� The following are typical properties that may fail to apply in specialcircumstances. If the �rm is able to shut down its operations without"sunk costs," it has the option of inaction (0 2 Y ). If the �rm canalways use more inputs without reducing output, free disposal applies(y 2 Y and y0 � y =) y0 2 Y ). If it is impossible to fully recoverinputs from outputs, we have irreversibility (y 2 Y and y 6= 0 =)�y =2 Y ).

Exercise (MWG 5.B.5).

� Finally, one often assumes some form of convexity, which can be brokendown into additivity and returns-to-scale properties. Additivity saysthat two feasible production plans can be combined (y; y0 2 Y =)y + y0 2 Y ). (Free entry would imply additivity.) With returns toscale, we have 8y 2 Y , �y 2 Y for � 2 [0; 1] (non-increasing), � � 1(non-decreasing) or � � 0 (constant). (Note these are all versions ofwhat is conventionally called "constant returns to scale;" they restrictthe scale parameter to di¤erent intervals.)

Proposition. The production set Y is additive and exhibits non-increasingreturns to scale if and only if it is a convex cone, i.e. 8y; y0 2 Y and 8�; � �0, �y + �y0 2 Y .Proof. (If) Take any y; y0 2 Y and �; � > 0. Suppose Y has the additivity

and non-increasing returns properties. By k times adding y, we have ky 2 Y .This can be done for any integer k, so let k > max f�; �g. Because �=k <

112

Figure 16: Typical convex cone production set (shaded)

1, nonincreasing returns implies (�=k) (ky) = �y 2 Y . By the analogousconstruction, �y0 2 Y . Now additivity gives �y + �y0 2 Y , so that Y is aconvex cone.(Only if) With � = 1 and � = 1, the convex cone is seen to satisfy

additivity. Similarly, with � = 0, it satis�es nonincreasing returns. HenceY is a convex cone only if Y has the additivity and non-increasing returnsproperties.�

� If Y is a convex cone, then Y is convex (y; y0 2 Y and � 2 [0; 1] =)�y + (1� �) y0 2 Y ). The converse is true only if 0 2 Y (inactivity ispossible), so that non-increasing returns hold (let y0 = 0), but this isnot su¢ cient (for additivity).

� We will typically assume that Y is convex, which is a weaker prop-erty than convex cone, or that Y is strictly convex (y 6= y0 and � 2

113

(0; 1) =) �y + (1� �) y0 is in the interior of Y ). (A convex cone isnot strictly convex.)

Exercise (MWG 5.B.2).

Exercise (MWG 5.B.3).

Exercise (MWG 5.C.8).

11.2 Transformation Function

� The production set can be expressed in terms of a "transformationfunction" F : RL ! R that assigns values F (y) � 0 to y 2 Y andF (y) > 0 to y =2 Y , with F (y) = 0 if and only if y is a boundarypoint of Y . (Such a function could be de�ned for any set, including theconsumption set, which is however easy enough to express as X = R+.)

Example. Suppose each of goods q1 and q2 can be made from K andL. Technologies are Cobb-Douglas: q1 = k�1 l

1��1 and q2 = k�2 l

1��2 , where

k1 + k2 = K or l1 + l2 = L. Given k1 and l1, the maximal output of q2 is�q2 = (K � k1)

� (L� l1)1��. The (production possibility) frontier is therefore

described by

K =

��q1

l1��1

�1=�+

�q2

(L� l1)1��

!1=�(rearrange the production functions for k1 and K � k1, add up). A transfor-mation function is:

F (q1; q2;�k;�l) = k1 + k2 �K =

�q1

l1��1

�1=�+

�q2

l1��2

�1=�� k1 � k2:

Notice that F (q1; q2;�k;�l) = 0 if and only if k = k1 + k2 = K and l =l1 + l2 = L, and else F (q1; q2;�k;�l) < 0 unless k1 + k2 > K or l1 + l2 > L(which are infeasible).The transformation function contains all relevant information about pro-

duction possibilities. You can recognize outputs and inputs by the fact thatan increase in q1 and q2 increases F (�), and an increase in a k or l decreases

114

F (�). (When inputs are �xed, an increase in output indicates greater e¢ -ciency. When outputs are �xed, an increase in input means lower e¢ ciency.)The production plan (�q1; �q2;�K;�L), where F (�q1; �q2;�K;�L) = 0, is fullye¢ cient.

� Suppose F (�) is di¤erentiable at a boundary point �y. Then, holding y`�xed for ` = 3; : : : ; L,

dF (�y) =@F (�y)

@y1dy1 +

@F (�y)

@y2dy2 = 0;

and the slope of the transformation function is

dy2dy1

= �@F (�y) =@y1@F (�y) =@y2

:

� If y1 and y2 are outputs, this ratio is called the marginal rate of trans-formation (MRT). It measures the amount by which, at �xed inputlevels, one output has to be reduced in order to produce more of theother. If y1 and y2 are intputs, the ratio is the marginal rate of tech-nical substitution (MRTS). Then it measures the amount by which, at�xed output levels, one input can be reduced when using more of theother.

Example. In the Cobb-Douglas case above, where the transformation func-tion was

F (�q1; �q2;�K;�L) =��q1

l1��1

�1=�+

�q2

(L� l1)1��

!1=��K;

we �nd

MRT21 �dq2dq1

= ���

(�q1=l1)(1��)=�

(�q2= (L� l1))(1��)=� = �

�

�

(k1=l1)1��

((K � k1) = (L� l1))1��

and

MRTSLK �dL

dK= � �

1� �

�L� l1�q2

�1=�= � �

1� �

L� l1K � k1

:

115

11.3 Pro�t Maximization

� The standard behavioral premise about �rms is that they maximizepro�ts. This standpoint is not as immediately compelling as maxi-mization with respect to consumer preferences. However, if �rms areowned by consumers, and �xed shares of pro�t accrue to the owners,higher pro�ts enlarge the owners�budget sets and therefore increaseindirect utilities.

� However, this argument relies on the "price-taking" assumption thateach �rm regards p = (p1; : : : ; pL)� 0 as independent of its productionplan. Else, an owner may �nd it optimal to manipulate prices byincreasing the production of the goods she likes. We do assume price-taking behavior throughout.

� Pro�t maximization also requires that the technology is certain: else, arisk-averse owner may favor less risky (but potentially less pro�table)production plans.

� Finally, if �rms are operated by agents instead of owners, they maypursue other objectives than pro�t maximization, since the pro�t doesnot accrue to them.

Exercise (MWG 5.G.1).

� Pro�t, in the conventional sense, is p �y =PL

`=1 p`y` (recall that inputsenter negatively, as costs). From a nonempty and closed productionset, the �rm chooses production plan y at p� 0 that attains

maxy2Y

p � y = maxy2RL

p � y

s.t. F (y) � 0:

� Clearly, the �rm must choose a production vector in the boundary of Y ,else it is possible to reduce inputs without reducing outputs, which atstrictly positive prices increases pro�t. Hence the constraint specializesto F (y) = 0.

116

� A solution to the �rm�s problem is the supply correspondence

y (p) = fy 2 Y s.t. p � y = � (p)g :

� We de�ne the pro�t function as � (p) � maxy2Y p � y, i.e. it is the valuefunction associated with the �rm�s maximization problem (analogousto the indirect utility function and the expenditure function in demandtheory). It depends only on the parameter p; the optimal choice y (p)is implicit.

� Suppose the transformation function is di¤erentiable. Then the La-grangean �rst-order conditions for pro�t-maximization are, for ` =1; : : : ; L,

�@F (y�)

@y`= p`;

or in matrix notation, �rF (y�) = p. (Since the gradient of the trans-formation function is proportional to p at a solution y�, the pro�t-maximizing direction to adjust y, if the technological constraint is re-laxed, is in the direction of prices, such that expensive goods are pro-duced using cheap goods.)

� For any pair of commodities k; l, the condition implies

@F (y�) =@yk@F (y�) =@yl

=pkpl:

Example. The �rm from the previous examples uses only inputs K and L,at prices r and w. Thus, a condition for pro�t-maximization is

�MRTSLK =�

1� �

L� l1K � k1

=r

w;

and another is

�MRT21 =�

�

(k1=l1)1��

((K � k1) = (L� l1))1�� =

p1p2:

(Additional �rst-order conditions place implicit restrictions on prices.)

117

Solving jointly, we obtain the optimal input ratios in the production ofeach output:

�1 � k1l1=

��

�

p1p2

�1=(1��)��

1� �

w

r

�(1��)=(1��);

�2 � K � k1L� l1

=�

1� �

w

r:

The scale of production is not determined, since the technology exhibitsconstant returns. We can express the outputs in terms of the levels of oneinput

�q1 = k�1 l1��1 = �1l1;

�q2 = (K � k1)� (L� l1)

1�� = �2 (L� l1) ;

or use the identity l1 = (�2L�K) = (�2 � �1) to express outputs in terms ofthe total use of both inputs:

�q1 =�1

�2 � �1(�2L�K) ;

�q2 =�2

�2 � �1(K � �1L) :

Since �1 increases in p1, it makes sense that �q1 increases, and �q2 decreases,in �1.

Exercise (MWG 5.C.9).

Exercise (MWG 5.C.12).

Proposition. The supply correspondence y (�) for a production set Y that isclosed and satis�es free disposal is homogeneous of degree zero, convex if Yis convex, and single-valued if Y is strictly convex. The pro�t function � (�)is homogeneous of degree one and convex.Proof. Since a price increase from p to kp by k > 0 does not a¤ect the pro-duction set and can be factored out of p�y, it does not change the maximizers,i.e. the supply correspondence y (�). Then it is clear that � (p) = p � y� withy� 2 y (p) is homogeneous of degree one.Suppose Y is convex. If y 2 y (p) and y0 2 y (p), then p � y = p � y0 (since

all elements of the supply correspondence maximize pro�t at p), so

p � y = �p � y + (1� �) p � y0 = p � (�y + (1� �) y0) ;

118

i.e. �y+(1� �) y0 2 y (p). Moreover, if Y is strictly convex, then 8y; y0 2 Y ,where y 6= y0, and 8� 2 (0; 1), the convex combination �y + (1� �) y0 is inthe interior of Y . Then it is possible to reduce inputs and increase outputsin Y , which must lead to strictly higher pro�t than p � (�y + (1� �) y0).Therefore, if y 2 y (p) and y0 2 y (p), then �y + (1� �) y0 =2 y (p), whichcontradicts convexity of y (p). Hence y (p) cannot contain distinct y and y0,and must be single-valued.To see that � (�) is convex, let � 2 [0; 1] and p; p0 � 0. For any y 2

y (�p+ (1� �) p0), by de�nition of � (�) as a maximum, p � y � � (p) andp0 � y � � (p0). Therefore,

� (�p+ (1� �) p0) = (�p+ (1� �) p0) � y � �� (p) + (1� �)� (p0) ;

which means � (�) is convex.�

11.4 Law of Supply

� The law of supply says that more is produced of an output, less usedof an input, whose price increases. It arises from the convexity of thepro�t function, which is an intuitive property: a price increase causesa linear increase in pro�t at �xed supply, so the �rm�s ability to adjustsupply can only further enhance pro�t.

Proposition. If the supply function y (�) for a production set Y that isclosed and satis�es free disposal is di¤erentiable at �p, then y (�p) = r� (�p)and Dy (�p) = D2� (�p) is a symmetric and positive semide�nite matrix, i.e.8v 2 RL, vTDy (�p) v � 0 (equality if v = �p).Proof. That y (�p) = r� (�p) follows immediately from the envelope theorem(y (�p) is a maximizer at � (�p) = �p � y (�p), hence locally constant). Positivesemide�niteness of D2� (�p) is also immediate from the fact that � (�) is con-vex. Since y (�) is homogeneous of degree zero, y (��p) = y (�p) for � > 0.Di¤erentiating on the left and on the right with respect to �p, we have for` = 1; : : : ; L,

LXk=1

@y` (��p)

@�pk

@�pk@�

=

LXk=1

@y` (��p)

@pkpk = 0:

This is the `th entry in the vector Dy (�p) �p. Therefore, �pTDy (�p) �p = 0.

119

�

� The statement y (�p) = r� (�p), that the supply function can be recov-ered from the (maximal) proft function, is known as Hotelling�s lemma.

� Positive semide�niteness of Dy (�p) is an expression of the law of sup-ply and implies, in particular, that own-price e¤ects @y` (p) =@p` arepositive. Its non-di¤erential equivalent can be understood directly:

(p� p0) � (y � y0) = p � (y � y0) + p0 � (y0 � y) � 0

for all p, p0 and y 2 y (p), y0 2 y (p0). This is true because p � y � p � y0and p0 � y0 � p0 � y (by the optimality of the supply correspondence).

12 E¢ ciency of Aggregate Supply

12.1 E¢ cient Production

� The focus of this lecture are the e¢ ciency properties of supply andaggregate supply. First, we look at a basic notion of e¢ ciency that isde�ned from the properties of the production plan alone. Second, wediscuss the stronger notion of cost minimization, which incorporatesprices. Both are essentially implied by pro�t maximization and surviveaggregation.

� E¢ cient production is non-wasteful.

De�nition. Production plan y 2 Y is e¢ cient if there exists no y0 2 Y suchthat y 6= y and y0 � y.

� While all e¢ cient production plans are boundary points of the produc-tion set Y , not all boundary points are e¢ cient. (Consider a naturalresource that cannot be produced. Any feasible production plan thatdoes not use the resource is a boundary point, but it may be possibleto reduce waste in other commodities.)

120

Proposition. Production plan y 2 Y maximizes pro�t at p � 0 only if itis e¢ cient.

Proof. If y is not e¢ cient, i.e. there exists y0 � y in Y and y0 6= y,then p � y0 > p � y (given strictly positive prices), so that y cannot be pro�t-maximizing.�

� This result becomes the �rst welfare theorem under aggregation. Thepartial converse corresponds to the second welfare theorem.

Proposition. If the production set Y is convex, then there exists for everye¢ cient production plan y 2 Y a nonzero price vector p � 0, given which yis pro�t-maximizing on Y .

Proof. If y 2 Y is e¢ cient, then Y must be disjoint from the set

Py ��y0 2 RL+ s.t. y0 � y

of production plans that contain more of each output and less of each input.Since Py is convex (if y0 � y and y00 � y, then any convex combinationstrictly exceeds y), there exists a hyperplane that separates Y and Py, i.e.there is some p 6= 0 such that p � y0 � p � y00 for any y0 2 Py and any y00 2 Y .We need to show that p � 0 and that y is pro�t-maximizing at prices p.

Suppose p` < 0 for some commodity `. Then one can �nd y0 � y with y0`�y`su¢ ciently large that p �y0 < p �y. But this was ruled out by the constructionof p.Suppose there is a feasible production plan y00 2 Y such that p �y00 > p �y.

Then p � y00 > p � y0 for every y0 in some (su¢ ciently close) neighborhood ofy. Because such a neighborhood contains y0 � y, the construction of p suchthat p � y0 � p � y00 is again violated. It follows that p � y � p � y00 for anyy00 2 Y .�

Example. Consider a production plan (q;�z) that is e¢ cient on the single-output production set Y given by concave production function f (�), i.e.f (z) = q. From the �rst-order conditions, we can construct prices (q; w)at which z is pro�t-maximizing. Represent Y by the transformation functionF (q;�z) = q � f (z). Fix the output price p, say at p = 1. Then for every

121

`, divide the �rst-order condition �@F (q;�z�) =@z` = �@f (z�) =@z` = w` bythe �rst-order condition for the optimal output q, �@F (q;�z�) =@q = � =p = 1: for ` = 1; : : : ; L� 1,

w` =@f (z�)

@z`:

This determines the desired price vector at which the e¢ cient productionplan (q;�z) is pro�t-maximizing.

� The de�nition of production e¢ ciency makes no reference to prices. Astronger criterion that selects among e¢ cient production plans is costminimization.

12.2 Cost Minimization

� I focus on the single-output case, where a quantity q is produced usinginput vector z 2 RL�1, and the technological contraint is expressed bya production function f (z). The production function can be viewed asa transformation function F (q;�z) = q � f (z), since 8 (q;�z) =2 Y ,q > f (z) and 8 (q;�z) 2 Y , q � f (z) with equality if and only if(q;�z) is in the boundary of Y .

� Denote by w � 0 the vector of input prices. The cost-minimizingchoice of inputs, at a given output, de�nes the cost function:

c (w; q) � minz�0

w � z

s.t. f (z) � q:

� Notice that the optimal choice of inputs is implicit in the cost function;its value c (w; q) is the lowest cost required to produce q at prices w.

� A solution to the cost minimization problem at di¤erent (w; q) is the(conditional) factor demand correspondence z (w; q). (The quali�er"conditional" refers to the fact that the factor demand depends onoutput.)

122

� The factor demands at any solution (q�;�z�) to the pro�t maximizationproblem must be cost-minimizing given q�, since pro�t at q�, i.e. p�q��w � z, is strictly decreasing in the cost w � z. Hence pro�t maximizationimplies not just e¢ ciency, but also that factor demand z solves the costminimization problem.

� If the production set is convex (i.e. the production function is concave),the following �rst-order conditions (together with the production func-tion) characterize z� 2 z (w; q). Letting the `th commodity be theoutput, for all ` = 1; : : : ; L� 1,

�@f (z�)

@z`� w`

with equality if z�` > 0. In matrix notation, �rf (z�) � w.

Exercise (MWG 5.C.10).

� Since q must be the pro�t-maximizing output, given the cost func-tion, i.e. solve maxq (pq � c (w; q)), the �rst-order condition for pro�tmaximization, p = @c (w; q) =@q, must hold. The multiplier � in thecost-minimization problem is the marginal value of relaxing the tech-nological constraint, i.e. � = @c (q; w) =@q. Therefore, � = p whenevercost is minimized at the pro�t-maximizing output level.

� The direction of the inequality in the �rst-order condition for cost-minimization is then intuitive: whenever � (@f (z�) =@z`) > w`, theuse of input ` should be increased, because ` adds value pMP` thatis greater than its price w`. Hence z�` > 0 at an optimum (and� (@f (z�) =@z`) = w`) unless � (@f (z�) =@z`) < w` at z�` = 0.

� If z�1 ; z�2 6= 0, the �rst-order conditions imply MP1=MP2 = w1=w2 and,since �MP1=MP2 is the slope of f (z) = q, and �w1=w2 is the slope ofc (w; q) = �c, tangency of the "isoquant" (graph of f (z) = q) and the"isocost" (graph of c (w; q) = �c). See Figure 17.

� This picture is reminiscent of the expenditure minimization problem ifyou think of q (or the implied pro�t pq) as the �rm owner�s "utility." Infact, at given prices and costs, the owner�s wealth and indirect utilityv (p; w) are increasing in q, so there is a direct connection.

123

Figure 17: Tangency solution to cost minimization

� This parallel implies that c (w; q) inherits properties of the expenditurefunction (homogeneous of degree one and concave in input prices w,and non-decreasing in q). Moreover, the factor demand correspondencehas properties of the Hicksian demand correspondence for commoditybundles (homogeneous of degree zero in w, z (w; q) = rwc (w; q) wheredi¤erentiable, Dwz (w; q) is symmetric and negative semide�nite, i.e.satis�es the law of demand). (The identity z (w; q) = rwc (w; q) iscalled Shepard�s lemma.)

Exercise (MWG 5.C.3). Consider a single-output technology Y , with freedisposal, that is is closed and characterized by production function f (�).Show: if f (�) is homogeneous of degree one (constant returns to scale), thenc (�) and z (�) are homogeneous of degree one in q. If f (�) is concave (non-increasing returns to scale), then c (�) is a convex function of q, so thatmarginal cost is non-decreasing in q.

� The (minimum) cost function is better behaved than the pro�t functionunder non-decreasing returns to scale (when it is optimal to producein�nite or zero output). Moreover, the cost function always exists,whereas the pro�t function only exists if �rms are price-takers.

124

Exercise (MWG 5.D.4).

Exercise (MWG 5.D.5).

12.3 Aggregate Supply

� Suppose now there are J �rms with production sets Y1; : : : ; YJ (eachnonempty, closed and with free disposal). Denote �rm j�s pro�t func-tion and supply correspondence by �j (p) and yj (p).

� The aggregate supply correspondence is

y (p) �(y 2 RL s.t. 9y1; : : : ; yJ with yj 2 yj (p) and

JXj=1

yj = y

):

It includes every sum of production plans y1; : : : ; yJ that are individu-ally optimal for the �rms.

� The aggregated supply correspondence admits a representative �rm inthe sense that such a �rm, faced with the aggregated production setY1+ � � �+ YJ , would choose a vector in y (p), i.e. an aggregate of plansthat �rms j = 1; : : : ; J would choose in the individual production setsY1; : : : ; YJ .

� Let y� (p) denote the supply correspondence on Y1 + � � �+ YJ , and callthe associated pro�t function �� (p).

Proposition. If p� 0, then any y 2 y� (p) satis�es

y =JXj=1

yj

for some y such that yj 2 yj (p) for j = 1; : : : ; J . Moreover, �� (p) =PJj=1 �j (p).

Proof. It is helpful to establish the second part �rst: �� (p) =PJ

j=1 �j (p).

Let yj 2 Yj for j = 1; : : : ; J . ThenPJ

j=1 yj 2 Y � Y1 + � � � + YJ implies

�� (p) � p �PJ

j=1 yj =PJ

j=1 (p � yj), i.e. �� (p) �PJ

j=1 �j (p). On the other

125

hand, if y 2 Y , there are yj 2 Yj for j = 1; : : : ; J such thatPJ

j=1 yj = y. Then

p � y = p �PJ

j=1 yj =PJ

j=1 (p � yj) �PJ

j=1 �j (p), i.e. �� (p) �

PJj=1 �j (p).

For the �rst part, we need y� (p) � A �nPJ

j=1 yj s.t. yj 2 yj (p) for j = 1; : : : ; Jo

and also A � y� (p). Let y 2 y� (p) and y =PJ

j=1 yj for some yj 2 Yj, j =

1; : : : ; J . Because y is pro�t-maximizing, p �PJ

j=1 yj = �� (p) =PJ

j=1 �j (p).I.e. p � yj < �j (p) for some j is only possible if p � yk > �k (p) for somek, which con�icts with the de�nition of �k (p) as a maximum value func-tion. It follows that p � yj = �j (p), hence yj 2 yj (p), for all j. I.e.y� (p) � A. Now let yj 2 Yj, j = 1; : : : ; J . Since p � yj = �j (p) for all j,p �PJ

j=1 yj =PJ

j=1 (p � yj) = �� (p). ThusPJ

j=1 yj 2 y� (p), i.e. A � y� (p).�

� A key implication of this result is that e¢ ciency aggregates: since every�rm produces a given output at minimal cost, and a hypothetical plan-ner, who has the economy�s combined production possibilities available,can do not better than to replicate individual choices, aggregate supplymust be cost-minimizing.

� Recall that the "law of supply" (optimal production plans increase incommodities whose prices increase) applies under mild conditions (Yis closed and has free disposal).

� If every yj (p) satis�es the law of supply, then y (p) also does. This isso because symmetry and positive semide�niteness are properties thatare preserved under matrix addition.

� In fact, the law of supply inequality for �rm j, (p� p0)�(yj (p)� yj (p0)) �

0, sums over j = 1; : : : ; J to (p� p0) � (y (p)� y (p0)) � 0, its aggregateversion.

13 Partial Competitive Equilibrium

13.1 Competitive Equilibrium

� A private ownership economy is populated by consumers with prefer-ences, endowments and ownership stakes in �rms that have production

126

possibilities. We have discussed the consumer problem of choosing abundle x 2 X from the consumption set X, and the producer problemof choosing a plan y 2 Y from the production set Y . Aggregate de-mand and supply are functions of prices that we took to be exogenous.Now we consider how prices are determined through the interaction ofdemand and supply.

� Let there by I consumers i = 1; : : : ; I, J �rms j = 1: : : : ; J , and Lgoods ` = 1; : : : ; L. Consumer i�s preference over consumption bundlesxi 2 Xi � RL is represented by the utility function ui (�). Initially, iis endowed with a bundle !i = (!`i; : : : ; !Li) 2 Xi. The total endowedquantity of commodity ` in the economy is !` �

PIi=1 !`i.

� Firm j implements the production plan yj 2 Yj � RL that maximizespro�t. Consumer i holds a share �ij in �rm j, which is a proportionalright to the �rm�s net output (i.e. the consumer provides a fraction�ij of the inputs used by �rm j and receives a fraction �ij of the out-puts). Given the production decisions, the total available quantity ofcommodity ` in the economy is !` +

PJj=1 y`j.

� An allocation describes, in this context, the consumption bundle eachconsumer ends up with and the production activity of each �rm.

De�nition. An allocation (x�1; : : : ; x�I ; y

�1; : : : ; y

�J) is a list of consumption

vectors x�i 2 Xi � RL for all consumers i 2 I and production vectors y�j 2Yj � RL for all �rms j 2 J .

� An allocation is said to be feasible, given the economy�s endowments!1; : : : ; !L of commodities ` = 1; : : : ; L, if for each `,

IXi=1

x�`i � !` +

JXj=1

y�`j

(i.e. no more than the available quantity of each commodity is allocatedfor consumption).

127

De�nition. A competitive equilibrium is an allocation (x�1; : : : ; x�I ; y

�1; : : : ; y

�J)

and price vector p� 2 RL such that consumers and �rms optimize, and mar-kets clear:(i) 8i 2 I, u�i (x�) = maxxi2Xi ui (xi) s.t. p� � xi � p� � !i +

PJj=1 �ij

�p� � y�j

�,

(ii) 8j 2 J , p� � y�j = maxyj2Yj fp� � yjg,(iii) for ` = 1; : : : ; L,

PIi=1 x

�`i = !` +

PJj=1 y

�`j.

� It is implicit in the de�nition of competitive equilibrium that consumersand �rms treat prices as independent of their choices. However, in theaggregate these choices determine prices through the market clearingconditions.

� Notice that a proportional change in all prices, from p to �p with � > 0,has no e¤ect on any of the three aspects of competitive equilibrium,since demand and supply are homogeneous of degree zero. Therefore,we can arbitrarily assign a price 1 to some commodity (which is thencalled the numeraire) without changing the equilibrium allocation.

� Suppose markets clear under an allocation for all commodities exceptk. Then, if consumers exhaust their budgets, the market for k mustclear, too. This is apparent from adding up the budget constraints,

IXi=1

(p � xi)�IXi=1

(p � !i)�IXi=1

JXj=1

(�ij (p � yj))

=IXi=1

p � xi � p � !i � p �

JXj=1

�ijyj

!

= p �IXi=1

xi � !i �

JXj=1

�ijyj

!= 0;

(factoring out the price vector is possible by the distributivity of thedot product).

� Now if markets for commodities ` 6= k clear, i.e.

IXi=1

x`i � !`i �

JXj=1

�ijy`j

!= 0;

128

then

pk

IXi=1

xki � !ki �

JXj=1

�ijykj

!= 0;

which (provided pk > 0) implies that the market for commodity kclears.

13.2 Partial Equilibrium

� Often we are interested in the market for a particular good ` and wouldlike to treat the remaining markets as "everything else," absorbingwhatever wealth the consumer does not spend on `. Strictly speaking,that is a quasilinear scenario, but one may argue that it applies ap-proximately whenever only a small portion of the consumer�s wealth isspent on `.

� The main prerequisites for such a "partial equilibrium" analysis arethat changes in the price of ` do not create wealth e¤ects and thatchanges in the quantity demanded of ` do not cause price adjustmentsin other markets. If they did, then the marginal value of residual wealthwould not be �xed, and it is then not clear what it means to measurethe price of a unit of ` in terms of what is given up of other goods. Ifmoney existed, the value of a dollar spent on ` would not be constant;it would depend on how much of ` is consumed.

� Wewill assume a quasilinear environment, but note that it may general-ize a bit beyond to situations where demand for ` is just not signi�cantenough to cause non-negligible wealth e¤ects and price changes else-where. Clearly, this argument is violated if ` has close substitutes orcomplements.

� Suppose then that all goods other than `, bundled together, enter con-sumer i�s utility function linearly, and denote the quantity of this com-posite commodity by mi ("money" left over). It can be treated as thenumeraire, i.e. we normalize its price to 1.

� Let the quantity of good ` be xi, and i�s quasilinear preferences arerepresented by

ui (mi; xi) = mi + �i (xi) ;

129

where �i (�) is concave, i.e. �0i (xi) > 0, �00i (xi) < 0 at all xi � 0, and�i (0) = 0 (a normalization). The price of ` is p.

� Firms require cj (qj) of numeraire commodity to produce qj � 0 units ofgood `. Let cj (qj) be convex and twice di¤erentiable, i.e. c0i (xi) > 0,c00i (xi) � 0 at all qj � 0. (Note that c (0) = 0 only makes sense in thelong run, when there are no �xed costs.)

� Suppose there is no endowment of `, and an endowment !mi > 0 of thenumeraire.

� Firm j sets qj so that p�q�j attains

maxqj�0

fp�qj � cj (qj)g ;

which has �rst-order conditions p� � c0j�q�j�, with equality if q�j > 0.

� Consumer i solves

maxmi2R;xi2R+

fmi + �i (xi)g

s.t. mi + p�xi � !mi +JXj=1

�ij�p�q�j � cj

�q�j��

= maxmi2R;xi2R+

(�i (xi)� p�xi + !mi +

JXj=1

�ij�p�q�j � cj

�q�j��)

;

since the budget constraint holds with equality at a solution (so thatwe can substitute for mi).

� The �rst-order conditions, �0i (xi) � p� (equality if x�i > 0), determinethe equilibrium p� together with market clearing,

PIi=1 x

�i =

PJj=1 q

�j .

Note that none of the equilibrium conditions depend on endowmentsor ownership stakes. This is a property of quasilinear preferences.

Exercise (MWG 10.C.2).

Exercise (MWG 10.G.2).

Exercise (MWG 10.C.6).

Exercise (MWG 10.G.5). (Only part a.)

130

13.3 The Long Run

� Let there be in�nitely many �rms that could potentially produce com-modity ` with the same cost function. Firms are identical, so we maydenote any individual �rm�s output as q (since it is unique if costs arestrictly convex).

� Given enough time, a �rm can attain zero pro�t by shutting downcompletely (c (0) = 0). Assuming free entry and exit, �rms will thenenter as long as positive pro�t can be made, and exit if only negativepro�t can be made. Therefore, all individual �rms must have zeropro�t in long-run equilibrium, else (if positive) more would enter or (ifnegative) more would exit, so some �rms could not be optimizing.

� This leads to the following special version of competitive equilibrium.

De�nition. A long-run competitive equilibrium, given aggregate demandfunction x (p) and cost function c (q) for potentially active �rms (wherec (0) = 0), is a price p�, a per-�rm production level q�, and a number of�rms J� such that �rms optimize, pro�table entry is not possible, and mar-kets clear:(i) p�q� � c (q�) = maxq�0 fp�q � c (q)g,(ii) p�q� � c (q�) = 0,(iii) x (p�) = J�q�.

� Assume strictly positive demand at a price equal to marginal cost,x (c0 (0)) > 0, and constant returns to scale: c (q) = cq for some c > 0.Then we must have p� � c, else every �rm would want to producein�nitely by (i). Hence maxq�0 (p� � c) q = 0, so that p� = c. Sincex (c0 (0)) = x (c) > 0, market clearing (iii) requires q� > 0, but q� isindeterminate by (ii), and therefore J� is indeterminate.

� There is no long-run equilibrium with increasing and strictly convexcosts. In this case, if p > c0 (0), �rms make a positive pro�t at somelevel of production, so total supply is in�nite. Therefore p � c0 (0), butthen supply is zero and demand is positive by assumption, x (c0 (0)) > 0,i.e. market clearing fails.

131

� To get a determinate long-run number of �rms, there must exists ane¢ cient scale �q that minimizes long-run average cost �c = c (�q) =�q. Ifp > �c, output is in�nite. If p < �c, then no nonzero output level ispro�table (since �c is the minimized average cost). So we must havep = �c, and J = x (�c) =�q.

Exercise (MWG 10.F.2). Long-run and short-run competitive equilibriumprice

Exercise (MWG 10.F.3). Tax impact in short run and long run

Exercise (MWG 10.F.6). Short-run and long-run supply function with�xed factor

14 Welfare Analysis

14.1 Pareto E¢ ciency and Surplus

� We now consider the welfare properties of competitive equilibrium.

De�nition. A feasible allocation (x1; : : : ; xI ; q1; : : : ; qJ) is Pareto e¢ cientwith respect to preferences represented by utility functions u1 (�) ; : : : ; uI (�)if there exists no other feasible allocation (x01; : : : ; x

0I ; q

01; : : : ; q

0J) such that

ui (x0i) � ui (xi) for all i = 1; : : : ; I (and the inequality is strict for someone).

� This is a minimal welfare criterion: there is no distributional fairnessimplied (allocating everything to one person is Pareto e¢ cient). How-ever, it makes sense to at least require that not everyone can be madebetter o¤.

� Consider a partial equilibrium setting where individual utility functionsare quasiconcave, i.e.

ui (m;x) = mi + �i (xi)

for i = 1; : : : ; I. As in the previous lecture, mi is i�s consumption of anumeraire commodity, which is a composite of all goods other than `.Commodity ` is consumed by individual i in quantity xi and producedby �rm j in quantity qj at cost cj (qj).

132

� In the quasilinear setting, there is a convenient welfare test. An al-location is Pareto e¢ cient if and only if it maximizes the aggregatesurplus

S (x1; : : : ; xI ; q1; : : : ; qJ) =

IXi=1

�i (xi)�JXj=1

cj (qj) :

� To understand the connection, consider the utility possibility set:

U =

�(u1; : : : ; uI) 2 RI s.t. 9 feasible allocation (x1; : : : ; xI ; y1; : : : ; yJ)

with ui � ui (xi) for i = 1; : : : ; I

�;

i.e. the set of all attainable "utility allocations" u1; : : : ; uI .

� If individual consumption of good ` and production plans are �xed at(�x1; : : : ; �xI ; �q1; : : : ; �qJ), then !m �

PJj=1 cj (�qj) is available to be spent

on the numeraire good. Since the total utility from consuming ` isPIi=1 �i (�xi), the feasible utility constraint is

IXi=1

ui � !m +IXi=1

�i (�xi)�JXj=1

cj (�qj)

= !m + S (�x1; : : : ; �xI ; �q1; : : : ; �qJ) :

� This means, the boundary of the utility possibility set is linear in theconsumption of the numeraire (with slope �1). Altering production orconsumption of good 1 leads to parallel shifts

� Optimal consumption and production levels for good ` are those wherethe boundary of the utility possibility set is as far out as possible. ThenPareto-optimal allocations can only di¤er in the distribution of thenumeraire among consumers (given the �i (�) are strictly concave andcj (�) strictly convex, their levels are uniquely determined at optimalconsumption and production choices)

� An increase in aggregate surplus (in the quasilinear setting) expandsthe utility possibility set and is therefore welfare-improving with respectto any other reasonable measure (that respects the Pareto principle).Hence, in the quasilinear environment, Pareto-optimal and surplus-maximizing is equivalent.

133

14.2 E¢ ciency of Competitive Equilibrium

� Recall that, in the competitive equilibrium problem, p = c0j (qj) for allj, and p = �0i (xi) for all i at a solution. De�ning the inverse demandfunction as x�1 (�) such that x�1 (x (p)) = p, and the inverse supplyfunction as q�1 (�) such that q�1 (q (p)) = p, we have q�1 (q (p)) = p =c0j (qj).

� Note that marginal cost and marginal utility, at a solution, are thesame across all consumers and �rms (since they are both equal to theconstant p). Hence we can replace c0j (�) with the "industry marginalcost" C 0 (�), and x�1 (x (p)) with the price P (�) at which a particularquantity is demanded. Then �0i (�xi) = P (x) for every i, and c0j (�qj) =C 0 (q) for every j.

� Consider a di¤erential change in aggregate surplus:

dS =IXi=1

�0i (xi) dxi �JXj=1

c0j (qj) dqj

= P (x)IXi=1

dxi � C 0 (q)JXj=1

dqj = (P (x)� C 0 (x)) dx

(Because P (x) and C 0 (x) are constants at a competitive equilibrium,they do not vary with individual or �rm. Apply feasibility q = x andPI

i=1 dxi = dx = dq =PJ

j=1 dqj in the last equality.)

� Integrating over quantity consumed of `, we get

S (x) = S0 +

Z x

s=0

(P (s)� C 0 (s)) ds;

which is graphically the area between demand and supply curve belowx, the standard depiction of aggregate surplus. (S0 equals aggregatesurplus when ` is not consumed, i.e. in this case the endowment of thenumeraire.)

� It is clear from the expression that aggregate surplus is maximized at xsuch that P (x) = C 0 (x), i.e. the competitive equilibrium consumptionlevel x where price equals marginal cost of `.

134

� Note that aggregate surplus is the area between the inverse demandfunction P (x) = x�1 (x (p)) = p = �0i (xi) and the inverse supply func-tion C 0 (q) = q�1 (q (p)) = p = c0j (qj). I.e. the inverse demand functionindicates marginal utility of consumption, and the inverse supply func-tion indicates marginal cost of production. (In a well-de�ned sense,since these are equalized across individuals in competitive equilibrium.)

� At the competitive equilibrium solution, we therefore have P (x) =C 0 (x) () �0i (xi) = c0j (qj). I.e. the marginal utility of another unitof ` equals its opportunity cost in terms of the numeraire. (All thispresumes consumers and �rms are price takers.)

� Thus, competitive equilibrium allocation is Pareto-optimal. This is aninstance of the �rst welfare theorem, which says that a competitiveequilibrium allocation is e¢ cient.

Exercise (MWG 10.D.3).

Exercise (MWG 10.D.4).

14.3 E¢ cient Allocations through the Market Mecha-nism

� Since the competitive equilibrium is surplus-maximizing in the quasi-linear case, it can be thought of as the solution to the problem

max(x1;:::;xI)�0(q1;:::;qJ )�0

(!m +

IXi=1

�i (xi)�JXj=1

cj (qj)

)

s.t.IXi=1

xi =IXi=1

qj:

� First-order conditions are: for i = 1; : : : ; I,�0i (xi) � �

and for j = 1; : : : ; J ,�c0j (qj) � �:

(With equality if xi > 0, respectively qj > 0.) And the feasibilityconstraint.

135

� To equate these to �rst-order conditions of the competitive equilibriumproblem, must have � = p. I.e. the shadow price � of the resourceconstraint for good ` is its price p, i.e. price re�ects the marginalsocial value of the good (also implies, through �rst-order conditions,�0i (xi) = � = c0j (qj), marginal utility from consumption equals mar-ginal cost). Then the problem jointly solves for a competitive equilib-rium and Pareto e¢ cient allocation.

� Because the amount of the numeraire commodity !m can be allocatedin any conceivable way among individuals without a¤ecting the solution(but it does, of course, a¤ect individual utility), every possible utilitydistribution can be implemented and remains Pareto optimal. This thesecond welfare theorem.

� The second welfare theorem says that any Pareto-e¢ cient allocation isa competitive equilibrium allocation for some speci�cation of zero-sumtransfers, i.e. T1; : : : ; TI such that

PIi=1 Ti = 0, between individual

endowments of the numeraire commodity.

Exercise (MWG 10.C.3).

Exercise (MWG 10.C.4).

15 Externalities

15.1 Externalities

� Up until now, we have assumed that individuals care only about theirown consumption - they are indi¤erent to the consumption choices ofothers or production choices of �rms, except insofar as these a¤ect one�sown budget set. One can, however, think of many examples whereother people�s actions directly impact one�s personal welfare (tra¢ ccongestion, noise, pollution, to name a few). These e¤ects are calledexternalities.

De�nition. An externality is a welfare e¤ect one agent�s actions have onanother.

136

� To get a sense of the issue, consider initially a bilateral externalitythat is imposed by one person on another (as opposed to a multilateralexternality that a¤ects many others). The two concerned individualsi = 1; 2 are a small part of the economy, i.e. their choices do not a¤ectprices p 2 RL. Their wealths, given these prices, are w1 and w2.

� Besides having preferences over their consumption of goods xi = (x1i; : : : ; xLi),both individuals also care about an action h 2 R+ taken by 1. Thus, 1imposes a (positive or negative) externality on 2.

� Since we wish to focus on the role of the externality, it is convenient tode�ne a "derived" preference over levels of h, given that consumptionof the L goods is optimized. The associated utility function is

vi (p; wi; h) = maxxi�0

ui (xi; h)

s.t. p � xi � wi:

� Let preferences be quasilinear with respect to a numeraire (good 1), sothat

ui (xi; h) = gi (x�1i; h) + x1i

(where x�1i = (x22; : : : ; xL2; h) denotes consumption of goods 2; : : : ; L).Evaluating at the optimal demands x�1i (p; h), which are independentof wealth, we have

vi (p; wi; h) = gi (x�1i (p; h) ; h) + wi � p�1 � x�1i (p; h)� �i (p; h) + wi

where �i (p; h) = gi (x�1i (p; h) ; h)� p�1 � x�1i (p; h).

� Since commodity prices p are �xed throughout the discussion, we cantreat �i (�; �) as a function of h only. Let �i (�) be strictly concave (i.e.�00i (�) < 0), and normalize to �i (0) = 0.

� We should, however, note that concavity may well be violated in prac-tice. For example, if a �rm is able to shut down operations when asu¢ ciently high negative externality is imposed on it, then its pro�tfunction cannot be forever decreasing in h, making it convex some-where. Or if a �rm�s constant-returns-to-scale production function isa¤ected by a positive externality (which acts like another input), thenthe �rm e¤ectively operates an increasing returns technology and there-fore has a convex pro�t function.

137

15.2 Ine¢ ciency

� Denote by h� the equilibrium level of action h, which must be optimalfor 1, and by h�� the (possibly di¤erent) socially optimal level. Re-call that, in the quasilinear setting, the socially optimal allocation isunambiguously the one that maximizes aggregate surplus

S (h) = �1 (h) + �2 (h) + w1 + w2;

(Pareto e¢ ciency holds if and only if surplus is maximized).

� The �rst-order condition that the socially optimal externality h�� mustsatisfy is therefore

�01 (h��) + �02 (h

��) � 0;with equality if h�� > 0.

� However, 1�s choice of h maximizes 1�s derived utility function, whichhas the Kuhn-Tucker �rst-order condition

�01 (h�) � 0;

with equality if h� > 0.

� Suppose h� > 0, so that 1 in fact imposes the externality on 2, and saythat h�� > 0 (otherwise, the externality is obviously ine¢ cient). If h isa negative externality, i.e. �02 (�) < 0 everywhere, then h� = h�� only if

�01 (h��) = ��02 (h��) > 0:

Since �01 (h) was assumed to be decreasing in h (by concavity), and�01 (h

��) > 0 = �01 (h�), we must have h�� < h� (too much of the exter-

nality is imposed).

� If h is a positive externality, i.e. �02 (�) > 0 everywhere, then h� = h��

only if�01 (h

��) = ��02 (h��) < 0:Again, since �01 (h) is decreasing in h and �

01 (h

��) < �01 (h�), it follows

that h�� > h� (too little of the externality is imposed).

Exercise (MWG 11.D.1).

138

15.3 Remedies

� One way to eliminate the ine¢ ciency is by quota, i.e. by mandatingagent 1 to provide no more or no less than the socially e¢ cient levelh��, depending on whether the externality is negative or positive.

� Another option is to tax or subsidize the externality. The optimal tax(subsidy) has to achieve h� = h��, i.e. �01 (h

��) = ��02 (h��). Sincethe tax subtracts from what agent 1 is able to spend on the numerairecommodity, it leaves derived utility

v1 (p; w1; h) = �1 (h)� th+ w1;

which is maximized by h such that

�01 (h�) = t

(given that h�� > 0, so that the government wants to induce h� > 0).

� One can see that h� = h�� if

t = ��02 (p; h��) :

The interpretation is that 1 is made to bear the cost (receives thebene�t) bestowed on 2. Such corrective taxes are called Pigouviantaxes.

Exercise (MWG 11.D.2).

Exercise (MWG 11.B.4).

� Bargaining can also resolve the externality problem. Say, 1 has theright to choose any level of h, but can sign a contract with 2 to set h toa speci�c level in return for compensation. If 2 o¤ers a payment of Tfor the implementation of �h, that is su¢ cient for 1, then 2 maximizesderived utility

v2�p; w2; �h

�= �2

��h�� T + w2

s.t. �1��h�+ T + w1 = �1 (h

�) + w1

= �2��h����1 (h

�)� �1��h��+ w2;

after substituting T = �1 (h�)� �1

��h�from the constraint.

139

� Because �01 (h�) = 0 (if h� > 0), the �rst-order condition is

�02��h�= ��01

��h�;

hence �h = h��.

� Alternatively, if 2 has the right to choose h, then 2 can let 1 purchasethe right to implement �h at a price T . The optimal o¤er maximizes

v2�p; w2; �h

�= �2

��h�+ T + w2

s.t. �1��h�� T + w1 = �1 (0) + w1

= �2��h����1 (0)� �1

��h��+ w2;

which is identical to the case where 1 owns the right to the externality(since �1 (0) = 0) and has the same solution.

� While it does not matter who has the right to externality, it is crucialthat the right is allocated to someone in advance and tradable. Thisarguments is known as the "Coase theorem." In the case of �rms, thetransfer of rights may occur through a merger or acquisition, whichcreates value by internalizing the externality.

Exercise (MWG 11.B.1).

� Related is the "missing markets" view of externalities. Instead of bar-gaining over the amount of the externality, a market for units of theexternality would also �x the problem. An externality is then simplya commodity, and ine¢ ciency arises when it is not traded (the marketis missing).

� We can generalize to a model where J �rms produce amounts h1; : : : ; hJof the externality and I individuals experience amounts h1; : : : ; hI of it.(To connect with the previous two-person environment, you can thinkof 1 as a �rm and 2 as a person.)

� If �rms must buy the right to impose a unit of the externality onsomeone at a price ph, then �rm j demands hj units that maximize itspro�t

�j (p; hj) = fj (hj)� phhj;

140

where fj (hj) is j�s (concave) production function for the numeraire,which requires hj as an input. (This set-up re�ects the interpretationthat the externality is negative. If it is positive, think of fj (hj) < 0as a convex cost function, and of ph < 0, i.e. �rms must be paid toproduce the externality. Conclusions are the same.)

� Consumer i supplies hi units of the externality, maximizing

vi (p; wi; hi) = �i (hi) + phhi + wi:

First-order conditions imply

f 0j (hj) = ph = ��0i (hi)

for �rms that produce, and individuals who consume, a positive amountof the externality. hj > 0, respectively hi > 0. The market price of theexternality turns out to be the Pigouvian tax.

� Socially optimal provision of the externality maximizes aggregate sur-plus

S (h) =IXi=1

(�i (hi) + wi) +JXj=1

fj (hj) ;

(Note that transfersPI

i=1 phhi =PJ

j=1 phhj drop out of the socialobjective due to market clearing.)

� First-order conditions for

maxh1;:::;hI�0h1;:::;hJ�0

S (h) s.t.IXi=1

hi =

JXj=1

hj

givef 0j�p; h��j

�= � = ��0i (p; h��i )

for every i and j such that h��i > 0 and h��j > 0. This implies h��j = h�j ,i.e. the market provision of the externality is Pareto-e¢ cient.

Exercise (MWG 11.D.5). (Assume f (0) = 0.)

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15.4 Public Goods

� The market solution relied on the depletable nature of the externality:a �rm could directly sell a unit to a consumer without a¤ecting others.This is a reasonable assumption for some types of externalities (say,construction noise next to a single house), but not for others (for ex-ample, air pollution). The latter type of externality is called a publicgood (or public bad, as the case may be).

De�nition. A public good is a non-rivalrous (i.e. non-depletable) commod-ity: it can be consumed simultaneously by all agents.

� Suppose �rms j = 1; : : : ; J produce quantities h1; : : : ; hJ of a publicgood at convex cost cj (hj) in terms of the numeraire. They sell units ofit to individuals at price ph. Individuals i = 1; : : : ; I demand quantitiesh1; : : : ; hI , but in fact consume the entire amount of the public goodthat �rms supply, i.e.

PJj=1 hj =

PIi=1 hi � h.

� Thenvi (p; wi; hi) = �i (h)� phhi + wi;

and�j (hj) = phhj � cj (hj) :

� The aggregate surplus is

S (h) =IXi=1

(�i (h) + wi)� c (h) ;

where c (h) =PJ

j=1 cj (hj) is the total cost of providing amount h of

the public good (paymentsPI

i=1 phhi =PJ

j=1 phhj canceled out).

� Surplus is maximized (Pareto e¢ ciency attained) only ifIXi=1

�0i (h��) = c0 (h��)

(provided h�� > 0). This is the Samuelson condition for the optimalprovision of a public good. (It says that the sum of marginal ratesof substitution between a public and private good, in this case thenumeraire, should equal the marginal rate of transformation.)

142

� If the public good is excludable, then government can achieve e¢ ciencyby imposing personal prices (p1; : : : ; pI) on individuals for a unit of thepublic good: so-called Lindahl prices pi = �0i (p; h

�) per unit of thepublic good consumed, for i = 1; : : : ; I.

� These prices are �rst-order conditions for consumers, hence all individ-uals optimize by demanding h� units. A single �rm (or a consortium

of J �rms) maximizes pro�t � (h) =�PI

i=1 pi

�h � c (h) by settingPI

i=1 pi = c0 (h�). HencePI

i=1 �0i (h

�) = c0 (h�), which implies h� = h��.

� The classic (or pure) public good is, however, non-excludable in ad-dition to being non-rivalrous (i.e. non-depletable). Then individuali can choose to pay for amount hi of the public good, but consumeh =

PIi=1 hi. If the good is sold in the market at price ph, the �rst-

order conditions for individuals and �rms are

�0i (h�)� ph � 0; ph � c0j

�h�j�� 0

with equality if h�i > 0, respectively h�j > 0. Note that, in a competitive

equilibrium, marginal cost of production is equalized across �rms, sothat c0j

�h�j�= c0 (h�) when h�j > 0..

� If at least one unit of the public good is provided, then �0i (h�i ) = ph =c0 (h�) for some individual i and �rm j. Given that �0i (h) > 0 for all i,and c0j (hj) > 0 for all j, this implies

IXi=1

�0i (h�) > c0 (h�)

For example, if every �rm produced the public good, and every individ-ual consumed it, then

PIi=1 �

0i (h

�) =PJ

j=1 c0j

�h�j�= Jc0 (h�) > c0 (h�).

� Because �0i (�) is a decreasing function of h, and c0 (�) is an increasingfunction of h�, it follows that h� < h��, i.e. the public good is under-provided. This is an instance of the free-rider problem: individuals donot pay for the bene�ts they receive from purchases by others, so theprice �rms receive for producing a unit of the public good understatesthe social value.

143

� In fact, if there is someone who has a higher marginal utility for thepublic good than everyone else, then �01 (h) = ph for this individualentails �0i (h) < ph for all others, i.e. no one else pays for a unit of thegood.

� The government can either provide a public good directly or use (mini-mum) quotas or subsidies to induce the e¢ cient amount. For example,a per-unit subsidy of the form

si =Xk 6=i

�0k (p; h�)

would cause consumers to take the total bene�t of their public goodpurchases into account.

Exercise (MWG 11.B.5).

Exercise (MWG 11.D.3).

� In general, whenever government must intervene, it faces the funda-mental problem that it has no �rst-hand information about the e¤ectof the public good on individuals. Thus, it may over- or underprovidethe costly public good. A central issue is how to design a �nancingmechanism that elicits the correct information from individuals.

� A solution is a Groves-Clarke (or pivotal) mechanism, which speci�esthat every agent pays the "costs" he in�icts on others. Speci�cally,the government could ask consumers how their well-being is a¤ectedby di¤erent levels of pollution and ask �rms how bene�cial it is forthem to be able to pollute. Based on the responses, the governmentimplements the level of pollution that is socially optimal, given thereports. In addition, it pays consumers the bene�ts reported by �rms,and collects damages from �rms equal to the loss in well-being reportedby consumers.

� No one has an incentive to misrepresent their true costs or bene�tsbecause one�s own report only a¤ects payments the other side makesor receives (not one�s own liabilities). It also determines the level of

144

pollution the government sets, but neither �rms nor individuals havean interest to manipulate it. If �rms report excessive costs, then morepollution will be allowed, but they will also have to pay higher damages.If consumers exaggerate their loss in well-being, pollution will be morerestricted, but they they receive less compensation.

� Because a Groves-Clarke mechanism completely internalizes externali-ties - everyone bears the social cost of their actions (or reaps the socialbene�ts) - it achieves e¢ ciency.

16 Monopoly and Product Di¤erentiation

16.1 Monopoly

� For many industries, the price-taking assumption that is fundamentalto competitive equilibrium is unrealistic. The notion that many small�rms produce a particular good can be relaxed to varying extents. Mostdramatically, to �rms whose products have no close substitutes (mo-nopolists). Then, to unique products that are, however, imperfectlysubstitutable (di¤erentiation). Finally, to perfectly substitutable prod-ucts provided by a limited number of �rms.

� We begin with a monopolist that faces di¤erentiable demand x (�) andcost c (�) for its product (and we also assume that pro�t is quasiconcave,i.e. �rst-order conditions identify a unique maximum).

� The monopolist sets its price to maximize pro�t

� (p) = px (p)� c (x (p)) :

The �rst-order condition

x (p�) + p�@x (p�)

@p=@c (x (p�))

@x

@x (p�)

@p

can be restated in terms of the inverse demand function p (x (p�)) as

@p (x (p�))

@xx (p�) + p� =

@c (x (p�))

@x;

145

i.e. marginal revenue equals marginal cost. This is a general op-timization principle for the �rm, even in competitive environments,where its marginal impact on price is zero: @p (x) =@x = 0, and thusp = @c (x) =@x.

� Note on inverting the derivative @x (p�) =@p: it is generally the casethat

@x (p)

@p

@x�1 (x (p))

@x= 1:

This can be seen by di¤erentiating the left and right side of x�1 (x (p)) =p with respect to p (using the chain rule on the left). Then, if we denotethe inverse demand function x�1 (x (p)) by p (x),

@p (x)

@x=

1

@x (p) =@p:

Alternatively, one can come to the same conclusion by writing pro�tas p (x)x� c (x) and di¤erentiating with respect to x.

Exercise (MWG 12.B.1).

� Rearranging the �rst-order condition to

p� =@c (x (p�))

@x� @p (x (p�))

@xx (p�) ;

we see that the monopoly price strictly exceeds marginal cost, provided@p (x (p)) =@x < 0. The reason is that the monopolist reduces sales fromthe socially optimal level where p = @c (x) =@x, in order to increase theprice consumers are willing to pay per unit. (A competitive �rm, onthe other hand, is too small relative to the market to a¤ect the price.)

Exercise (MWG 12.B.6).

146

Figure 18: Surplus sharing under perfect competition (left) and monopoly(right)

� The restraint in sales causes a "deadweight loss" to society, since theforegone units cost less to produce than people are willing to pay forthem. From society�s point of view, only the allocation matters, andthe price is irrelevant. However, the price determines how surplus fromtrade is divided between the �rm and consumers, so it is not irrelevantto the monopolist. Figure 18 illustrates.

Example. Let inverse demand be linear, p (x) = a� bx, and marginal costconstant at c. The optimal price and sales quantity satisfy marginal revenueequals marginal cost:

@

@xp (x�)x� = a� 2bx� = c;

i.e.x� =

a� c

2b; p� = p (x�) =

a+ c

2:

You can see from the inverse demand function that the �rm would notproduce unless a > c, so p� > c = p�� in the relevant circumstances. Since

147

x (p) = (a� p) =b, the competitive output is x�� = x (c) = (a� c) =b > x�.The deadweight loss is the area of the triange with height p� � p�� = p� � cand base length x�� � x�:

L =1

2(x�� � x�) (p� � p��) =

(a� c)2

4b> 0:

Exercise (MWG 12.B.9).

Exercise (MWG 12.B.10).

� If the monopolist were able to (perfectly) price-discriminate, i.e. chargeevery consumer her valuation of each unit sold, then it would be pro�t-maximizing to sell the socially optimal quantity, and no deadweightloss would occur. This is immediately apparent from the fact that themonopolist then captures the entire surplus and directly maximizes it.

� There are, however, important practical obstacles to price discrimina-tion, from limited information to the possibility of resale by consumerswho obtain the product at lower prices.

Exercise (MWG 12.B.5).

16.2 Bertrand Price Competition

� The two workhorse oligopoly models are Bertrand and Cournot duopoly.Both extend straightforwardly to more players. In this lecture, we de-velop Bertrand competition from its pure case, where �rms interactonce with identical products, to repetition and to di¤erentiated prod-ucts.

� Two �rms operate with constant marginal cost c and face total demandx (p) for their joint output. Each �rm i = 1; 2 faces demand

xi (pi; pj) =

8<:0 if pi > pj

12x (pi) if pi = pjx (pi) if pi < pj

individually.

148

� The �rms simultaneously set prices. The solution of interest is a Nashequilibrium. I will not give a formal de�nition of strategy here: inwords, it speci�es an action for every possible information state theplayer may �nd himself in. For our purposes, think of a player�s strategyas a direct action or, if there is a sequence of moves, as an actionconditional on the prior moves. Let Si be the set of such actions orconditional actions available to player i.

De�nition. A Nash equilibrium is a strategy pro�le s = (s1; s2) such that,for i = 1; 2, si 2 Si and

�i (si; sj) � �i (s0i; sj)

for all s0i 2 Si.

� A Nash equilibrium speci�es the strategy chosen by each player. It isstable in the sense that, given j�s strategy, i cannot gain by chang-ing hers, and vice versa. Since the payo¤s are in this context pro�ts�i (pi) = (pi � c)xi (pi; pj), a Nash equilibrium is a price for each �rmthat is pro�t-maximizing, given the other �rm�s price.

� Pure Bertrand duopoly has a unique Nash equilibrium (p�1; p�2) in whichp�1 = p�2 = c. Given p�j = c, p�i < 0 gains all sales, but makes a negativepro�t per sale, whereas p�i > 0 loses all sales. Hence neither improvesi�s payo¤. This means p�i = c for i = 1; 2 is a Nash equilibrium.

� Suppose there existed a distinct other Nash equilibrium (p0�1 ; p0�2 ). If

p0�i < p0�j , then i can strictly increase pro�t by raising the price (regard-less of whether p0�i � c or p0�i < c). Hence none of the prices can belarger than the other; we must have p0�i = p0�j . Now if p

0�i < c, then

increasing the price would strictly increase i�s pro�t. On the otherhand if p0�i > c, decreasing the price would gain all sales and strictlyincrease i�s pro�t. It follows that only p0�i = p0�j = c quali�es as a Nashequilibrium. (Hence the Nash equilibrium is unique.)

� The �nding that interaction of even just two �rms reverts to marginalcost pricing is striking, but typically not observed in reality. Next, weconsider two departures from pure Bertrand competition that restorehigher prices and positive pro�ts.

149

Exercise (MWG 12.C.1).

Exercise (MWG 12.C.4). (Only part a.)

16.3 Repetition

� One factor the pure Bertrand model ignores is that most �rms knowthat they will face the joint price-setting problem repeatedly over time.This circumstance greatly expands the �rms�options. They can nowcondition their pricing on the rival�s past actions and react aggressivelyto price cuts.

� Above marginal cost pricing may become viable because �rms maxi-mize intertemporal, rather than immediate pro�t and will consider thee¤ect of setting a low price today on the rival�s behavior tomorrow.Intertemporal pro�ts are

1Xt=1

�t�1�it;

where �it is i�s pro�t in period t and � 2 (0; 1) is a discount factor. Thediscount factor may be interpreted in various ways, e.g. as the constantprobability (at each point in time) that the �rms will compete again inthe following period.

Exercise (MWG 12.D.1).

� Note that, if �it is a constant value �i, then the in�nite sum can bereduced to

1Xt=1

�t�1�i =1

1� ��i;

since

(1� �)

1Xt=1

�t�1�i =

1Xt=1

�t�1�i �1Xt=1

�t�i

=

1Xt=1

�t�1�i �1Xt=2

�t�1�i = �i:

150

� Cooperation on high prices (for example, the monopoly price pm) mightbe sustainable in Nash equilibrium if both �rms play the trigger strat-egy

pi (t) =

�pm if t = 1 or pj (�) = pm for � = 1; : : : ; t� 1c otherwise

for i = 1; 2. I.e. at any given time, �rm i sets a high price pm only if jhas set pm at all previous times. Otherwise, i "punishes" j by pricingat marginal cost, so that j can make no more than zero pro�t.

� Such a strategy is called Nash reversion, because it switches from thebest to the worst Nash equilibrium in the punishment phase. It is alsoa "grim" strategy in that it never forgives the rival for setting a lowprice.

� In theory, if monopoly prices can be maintained forever in equilibrium,punishment will never actually occur - all that matters is the threat.This is why an unforgiving trigger strategy such as the one above canbe optimal: the threat is never tested.

� More pragmatic trigger strategies may maintain low prices for a su¢ -ciently long time to deter the rival in the future. In case the punish-ment is ever triggered, say because the rival makes a mistake, marginalcost pricing hurts the �rm in the present, but can be understood asan investment in reputation that permits pro�table cooperation in thefuture.

� If � � 1=2 (i.e. there is not too much discounting of future income),then pi (t) for i = 1; 2 is a Nash equilibrium of an in�nitely repeatedBertrand game. The reasoning is inductive. In the �rst period, thestrategies prescribe (p1 (1) ; p2 (1)) = (pm; pm). Now suppose (p1 (t� 1) ; p2 (t� 1)) =(pm; pm) and p1 (t) = pm. The best alternative for 2 to p2 (t) = pm

is to slightly undercut and earn a pro�t of (pm � c) qm, instead of(pm � c) (qm=2).

� Undercutting triggers p1 (�) = c for � = t + 1; : : : ;1 (so that themaximal pro�t 2 can receive in future periods is zero), so it is optimalfor 2 in period t if and only if

(pm � c) qm >1

1� �(pm � c)

qm

2;

151

i.e. if the pro�t from capturing all sales today exceeds the discountedvalue of future pro�t streams when sharing the sales.

� Because the inequality holds only when � < 1=2, it is optimal for 2to set p2 (t) = pm if � � 1=2. Since the �rms share the market at� = 1, and continue to do so whenever they have shared it previously,induction implies that they share the market in all periods. Hence(p1 (1) ; p2 (1)) = (pm; pm) is a Nash equilibrium of the in�nitely re-peated game.

� Since � < 1=2 implies that the optimal coordination on (p1 (t) ; p2 (t)) =(pm; pm) cannot be sustained, it is intuitively clear that no other kindof coordination can work either. I.e. either p1 (t) = c or p2 (t) = c atall times t.

� However, � � 1=2 admits many other cooperative equilibria besidesthe pro�t-maximizing one. This is an instance of the Folk Theoremfrom game theory, which says that equilibria leading to any payo¤s thatexceed the best each player can achieve independently (the minmax, i.e.the best attainable in the worst-case scenario) exist for some su¢ cientlylarge �.

Exercise (MWG 12.D.4).

� Here, the minmax pro�ts are zero, since every player can always guar-antee zero by setting price to marginal cost, but no more than that.Therefore, any combination of pro�ts exceeding zero is possible (to seethis, just replace pm above with any other prices, possibly asymmetric,and consider � very close to 1, so that threats are arbitrarily damagingin the long run). Which equilibrium occurs often depends on historyand focal points.

� Collusion is often not maintained as smoothly as suggested here. Inpractice, demands and costs �uctuate, and it may not be obvious toother parties whether a �rm is ceasing to cooperate, or simply reactingto environmental changes. To prevent �rms from taking advantage ofthis ambiguity, it becomes necessary to punish deviations, even if it is

152

not clear why they happened, and this leads to intermittent price wars,between periods of high prices.

Exercise (MWG 12.D.5).

16.4 Product Di¤erentation

� When products are di¤erentiated, �rm i�s demand is not perfectly price-elastic. It may instead be a continuously decreasing function xi (pi; pj)of �rm i�s price, given the price pj of the competitor.

� If �rms face constant marginal cost c > 0, they choose their prices tomaximize

�i (pi) = (pi � c)xj (pi; pj) :

Typically, �rms retain some positive demand when pricing above mar-ginal cost, from consumers who value their unique products. Hencepricing above marginal cost will be the pro�t-maximizing strategy, evenif it means getting undercut.

� There are two main approaches to product di¤erentation: in one, theHotelling model, a consumer demands only one of the products; inthe other, the Dixit-Stiglitz model, products are used together, thereis an explicit preference for variety. Monopolistic competition is thespecial case of the Dixit-Stiglitz model where variety is in�nite (thereis a continuum of di¤erentiated products with small market shares, butsome market power). I focus on the Hotelling model.

� Consider duopolists that serve a continuum of consumers whose de-mands arise in the following manner. The �rms are associated withpoints 0 and 1 in a spectrum [0; 1] of possible products. Every indi-vidual occupies at a point z 2 [0; 1] that re�ects her most preferredproduct. In purchasing either 0 or 1, the individual incurs a "travelcost" zt or (1� z) t, which indicates lost satisfaction from consuminga non-ideal product.

� Someone who prefers z is therefore willing to pay at most

P0 (z) = v � zt

153

for product 0, and at most

P1 (z) = v � (1� z) t

for product 1 (v is the good�s undiscounted value).

� Individual demand is for either one or zero units, hence no one consumesboth products, and some might consume neither. If the �rms chargep0 and p1 for a unit of their products, then the person who prefers zwill consume product 0 if P0 (z) � p0, i.e.

z � v � p0t

and P0 (z)� p0 � P1 (z)� p1, i.e.

z � 1

2+p1 � p02t

� z:

On the other hand, this person will consumer product 1 if

z � 1� v � p1t

and the second inequality is reversed.

� The point z, where P0 (z) � p0 = P1 (z) � p, belongs to the "marginalconsumer," who is just indi¤erent between the products.

� Suppose z is uniformly distributed on [0; 1], so that every possible prod-uct is preferred by the same mass of consumers. Then the total demandfor product 0 is the length of the interval [0; z0], where

z0 � min�v � p0t

; z

�;

and total demand for product 1 is the length of the interval [z1; 1], i.e.1� z1, where

z1 � max�1� v � p1

t; z

�:

(Assuming, of course, that prices are such that z0; z1 2 (0; 1). When-ever this is violated, one �rm makes no sales, which cannot be pro�t-maximizing unless the other �rm prices below marginal cost.)

154

� For simplicity, let v be su¢ ciently large that the market is "covered"in equilibrium: every consumer purchases one of the products. Thenthe marginal consumer determines the �rm�s demand functions

x0 (p0; p1) = z; x1 (p0; p1) = 1� z:

� Pro�ts�0 (p0; p1) = (p0 � c) z; �1 (p0; p1) = (p1 � c) (1� z)

are maximized when the �rst-order conditions with respect to p0 andp1,

p�0 (p1) =p1 + c+ t

2;

p�1 (p0) =p0 + c+ t

2;

are met.

� These are the best responses to the other �rm�s price. A joint solution,where prices are mutual best responses, is a Nash equilibrium:

p�0 (p�1) = c+ t = p�1 (p

�0) :

I.e. product di¤erentiation allows the �rms to price above marginalcost in Bertrand equilibrium, and increasingly so the more consumersdiscount for di¤erences from their most preferred products. In thelimiting case, where consumers do not care about such di¤erences andthus t = 0, prices equal marginal cost, and pro�ts are zero.

� This simple case can be extended to multiple �rms, arbitrary consumertaste distributions, individual demands other than zero-one, asymmet-ric costs, as well as endogenous �rm locations (strategic product posi-tioning). Above marginal cost pricing is not merely owed to the factthat few �rms compete in the market. In fact, entry may not lowerprices because it tends to attract consumers on the margin (who havethe lowest willingness to pay for the product). Once the incumbent�rm loses these price-sensitive consumers, it may actually increase itsprice further.

Exercise (MWG 12.C.17). Linear city with di¤erent costs

Exercise (MWG 12.C.16). Circular city with quadratic cost

155

17 Capacity Constraints

17.1 Capacity-Constrained Pricing

� An assumption that was implicit in Bertrand oligopoly, and turns outto be crucial for its marginal cost pricing equilibrium, is that �rmscan serve arbitrarily large demands (i.e. capacity is free or can beincreased instantly). Then, no �rm can a¤ord to be undercut, since allsales would be lost to competitors. These circumstances lead to veryaggressive pricing.

� With limited production capacities, the lower-priced of two identicalproducts is sold to early buyers, and the higher-priced alternative maystill be demanded by latecomers. This tends to soften pricing, becausea �rm can make a pro�t despite being undercut. Capacity constraintslead to the Cournot model of oligopoly, which can be viewed as quantity,rather than price, competition in the sense that the �rm�s fundamentalchoices are their production capacities.

� For simplicity, consider again a duopoly in which �rms i = 1; 2 simul-taneously choose capacities q1 and q2 (at a positive per-unit cost), andsubsequently engage in price competition. At the latter point, each�rm is able to sell up to its capacity at a constant marginal cost c � 0.Both capacities are known to both �rms at the time when prices areset.

� We assume that the products are identical, and demand for the totaloutput is a continuous, strictly decreasing and concave function x (p).Denote the inverse demand by p (�). Which of the two prices e¤ectivelyconstrains demand depends on the capacities. If pi < pj and qi � x (pi),then �rm i serves everyone, so pi determines how much is demanded. Ifqi < x (pi), then �rm j serves those who cannot buy from �rm i, whichsells at capacity, so pj determines how much is demanded.

� We must be speci�c about who buys from i at the lower price whendemand exceeds i�s capacity. This is determined through a rationingrule. We will impose that consumers with the highest willingness to

156

pay are at the head of the queue. Then demand is, for i = 1; 2,

xi (p1; p2) =

8<:min fqi; x (pi)g if pi < pjmin

�qi;max

�x (pi)� qj;

12x (pi)

if pi = pj

min fqi;max fx (pi)� qj; 0gg if pi > pj

:

� This rationing rule where "highest valuations served �rst" is known inindustrial organization as the "e¢ cient rationing rule." (An alternativeis, for example, the proportional rationing rule, where anyone is equallylikely to be at the head of the queue, so that �rm j�s customers havethe same expected willingness to pay as �rm i�s. This would increasej�s demand, and decrease i�s demand, at any given prices.)

� We begin by analyzing the pricing game for some given capacities q1 > 0and q2 > 0. (If one �rm does not invest in capacity, we simply have amonopoly.) Denoting by b (qj) the optimal quantity �rm i would sellif it were not capacity-constrained (and �rm j sold qj), let q1 � b (q2)and q2 � b (q1). (Hence, the capacity constraints bind.)

� Since capacity is costly to build, both �rms must sell a nonzero quantityin equilibrium: else they could avoid losses by not investing in capacity.

� This means prices must be equal. Otherwise, if p�i < p�j , then �rmi would sell at capacity (given that consumers �rst go to the lower-priced seller, and �rm j still manages to sell something). But theni could slightly increase its price and still sell at capacity, giving it astrictly higher pro�t.

� Similarly, it is not possible that p�1 = p�2 < p (q1 + q2), in which casemarket demand would exceed both capacities, so that each �rm couldpro�tably increase its price. On the other hand, if p�1 = p�2 > p (q1 + q2),then market demand does not cover both capacities. At least one �rmhas spare capacity and will want to slightly lower its price (i.e. undercutlike a pure Bertrand competitor) as long as p�1 = p�2 > c (otherwise, ifp�1 = p�2 � c, it is not optimal to invest in capacity).

� The only alternative is p�1 = p�2 = p (q1 + q2). It remains to be shownthat this is a Nash equilibrium given q1 and q2. Suppose thereforethat pj = p (q1 + q2). Neither of i�s deviations from pi = p (q1 + q2) ispro�table. Namely, pi < p (q1 + q2) simply lowers the price, but cannot

157

increase sales, since �rm i is already at capacity. And pi > p (q1 + q2)is undesirable by the assumption b (qj) � qi, which implies that �rm i�spro�t increases in sales when j is at, and i is below, capacity. Hence ishould lower price while pi � p (q1 + q2).

� We conclude that p�1 = p�2 = p (q1 + q2) is the unique Nash equilibriumof the pricing game with capacities qi 2 (0; b (qj)]. (Incidentally, thereis no equilibrium if qi > b (qj) and p (q1 + q2) > c, since i then sellsbelow capacity, but at symmetric prices it would always be pro�tableto slightly undercut.)

� This insight motivates the Cournot model of oligopoly, which takes asgiven that each �rm will produce up to capacity and set the market-clearing price p (q1 + q2). The Cournot model focuses on the capacity-or quantity-setting game preceding price formation.

17.2 Cournot Quantity Competition

� Let the inverse demand p (�) be a decreasing, di¤erentiable function ofqi + qj with p (0) > c. Duopolist i = 1; 2 solves

maxqi�0

(p (qi + qj)� c) qi;

which has �rst-order condition

p0 (q�i + qj) q�i + p (q�i + qj)� c � 0

(equality if q�i > 0).

� Suppose q�i = 0. Then p (q�i + qj) = p (qj) � c, and q�j = 0 contradictsp (0) > c. On the other hand, q�j > 0 would imply p

0 �q�j � q�i +p �q�j � = c

at qi = q�i , i.e. p�q�j�< c (since p0 (�) < 0 everywhere). Again, this

contradicts p (0) > c. Under the assumptions, we must have q�i > 0 fori = 1; 2, so that the �rst-order conditions apply with equality.

� Then they are functions

q�i = �p (q�i + qj)� c

p0 (q�i + qj)

that implicitly determine the best quantity choice for �rm i, given �rmj�s quantity choice.

158

� ANash equilibrium�q�i ; q

�j

�jointly solves these best-response functions,

thus:

p�q�i + q�j

�= c� p0

�q�i + q�j

� q�i + q�j2

= c� p0�q�i + q�j

�q�i :

Proposition. Given identical, constant marginal costs c > 0, and in-verse demand for the industry�s combined output such that p (0) > 0 andp0 (q1 + q2) < 0 whenever q1+ q2 � 0, the Cournot equilibrium price satis�es

c < p (q�1 + q�2) < p (qm) ;

where qm is the optimal monopoly output.

Proof. From the Nash equilibrium condition above, it is immediate thatp (q�1 + q�2) > c. So we show p (q�1 + q�2) < p (qm). Since p0 (�) < 0, this willbe the case if q�1 + q�2 > qm. If q�1 + q�2 < qm, then �rm i can increase itsproduction to qi = qm�q�j , in which case the industry supplies the monopolyquantity, at the monopoly price and monopoly pro�t. Because the increasein q�i lowers the price, while q

�j remains �xed, this must strictly reduce j�s

pro�t. But combined pro�t cannot decrease, since the monopoly pro�t isan upper bound on industry pro�t. Hence i�s pro�t strictly increases. Thisviolates Nash equilibrium, hence we must have q�1 + q�2 � qm.If q�1 + q�2 = qm, then the industry acts like a monopolist, and the �rst-

order condition for Cournot equilibrium should be identical to the �rst-ordercondition for a monopoly equilibrium at q�1 + q�2 = qm, namely p (q) = c �p0 (q) q, where q = q1+q2. You can check that it is not so. Thus q�1+q

�2 > qm.

�

� Note that q�1 = b (q�2) and q�2 = b (q�1) in Cournot equilibrium: thequantity choices can be interpreted as binding capacity choices that leadto p (q�1 + q�2) as a price equilibrium in a Bertrand game with capacityconstraints.

� Unlike pure Bertrand oligopoly, Cournot oligopoly does not lead tocompetitive (marginal cost) pricing. The reason is that, given capacity

159

constraints, neither �rm can capture the entire market by undercutting.Since each �rm�s demand is less than perfectly elastic, it can make apositive pro�t by pricing above marginal cost.

� Yet the �rms are unable to maximize industry pro�t, i.e. set themonopoly price. In determining its capacity, each �rm only takes intoaccount how an increase in sales, and therefore a lower market-clearingprice, a¤ects its own revenue. It ignores the e¤ect on the rival�s revenue.Hence the �rms do not respond to the full bene�t (to the industry) ofkeeping sales low and price high as a monopolist would; they sell "toomuch."

� This is easy to see from the Nash equilibrium condition: Cournotduopolists mark-up on marginal cost by �p0

�q�i + q�j

�q�i , i.e. the ef-

fect of the price reduction on own revenue. By contrast, a monopolistmarks up by �p0 (q�) q�, with q� = q�i + q�j , i.e. the e¤ect on industryrevenue.

Example. If inverse demand is linear, p (q1 + q2) = a � b (q1 + q2) (wherea > c and b > 0), then pro�ts are (a� bqi � bqj � c) qi, resulting in �rst-orderconditions (best-response functions)

q�1 (q2) =a� bq2 � c

2b

and

q�2 (q1) =a� bq1 � c

2b:

(In this context, best-response functions are often referred to as reactionfunctions.) Solving jointly, we obtain the Nash equilibrium

q�1 (q�2) =

a� c

3b= q�2 (q

�1) ;

with market-clearing price

p (q�1 + q�2) =1

3a+

2

3c:

Since a > c, this means p (q�1 + q�2) > c. The monopoly quantity is �rm i�sbest response to qj = 0. Hence qm = q�i (0) = (a� c) = (2b) and p (qm) =a=2 + c=2 > p (q�1 + q�2).

160

Exercise (MWG 12.C.9).

Exercise (MWG 12.C.20).

Exercise (MWG 12.D.3).

17.3 Competitive Limit

� In the generalization to J �rms producing nonzero quantities with iden-tical marginal costs c, �rst-order conditions imply that the outputs areequal, and adding up over j = 1; : : : ; J , i.e.

Jq� = �J p (Jq�)� c

p0 (Jq�);

can be arranged for the equilibrium price

p (Jq�) = c� p0 (Jq�) q�:

� Intuitively, as J !1, each �rm�s production level becomes very small,so that p (Jq�) ! c. (I.e. competitive pricing is approached in thelimit.) If the number of �rms in the industry is determined by free entryin response to pro�t opportunities, the competitive limit is obtained asdemand increases.

Exercise (MWG 12.C.7).

� Consider parameterized demand functions x� (p) = �x (p), where x (p)is some aggregate demand function, and � > 0 is the "market size."Let p� be the equilibrium price, and Q� the equilibrium total output(after all pro�table entry has occurred), when the market size is �.

� Assume that each �rm has the same minimum average cost �c at somelevel of output �q > 0 (we do not impose constant marginal cost here).Then Q� + �q � �x (�c) because otherwise, if demand exceeded supply(plus an additional quantity �q) at p = �c, the market-clearing price aftera further entry (at the minimum average cost level �q) would be greaterthan �c, and entry would be pro�table.

161

� This imposes a lower bound on the equilibrium supply Q�: it cannotfall short by more than �q of the quantity that would be supplied ifp = �c. It follows that the equilibrium price p� is bounded above: itcannot exceed �c by so much that the post entry price, after another�rm enters and supplies �q, is still greater than �c.

� Let p� (�) be the inverse supply function at market size �, and de�nethe di¤erence between this upper bound on p� and �c (which is a lowerbound such that no �rm exits) as

4p� � p� (�x (�c)� �q)� p� (�x (�c)) :

� In order to see what happens to the equilibrium price interval as marketsize increases from � = 1 toward in�nity, rewrite 4p� in terms of p (�),the inverse supply function at � = 1. Since q� = x� = �x = �q, wehave p (q) = p� (q�=�), i.e.

4p� = p

��x (�c)� �q

�

�� p (x (�c)) ;

which goes to zero as �!1.

� This means the equilibrium price converges to p� (x� (�c)) = �c, the long-run competitive price, as market size gets large. The industry outputis then also competitive (and �rms are small relative to the market,since they produce at the �xed minimum average cost level �q). Withconstant marginal cost c, �c = c.

Exercise (MWG 12.F.3).

Exercise (MWG 12.F.2).

Exercise (MWG 12.F.4).

18 Precommitment and Entry

18.1 Precommitment

� In some industries, �rms act sequentially, rather than simultaneouslyas assumed so far. In fact, it may super�cially appear that sequential

162

moves are always more realistic, but this view misses the point of thedistinction between the game forms. It is a matter of information,rather than timing.

� In a sequential game, strategy sets di¤er in that some players are ableto condition their actions on what other players are observed to do.This is not necessarily a bene�cial power to have: �rst movers canpotentially exploit it by manipulating the incentives for late movers.

� The e¤ect of an observed action by �rm i on the behavior of another�rm j is captured by j�s best response function. If j best-respondsto a change in i�s strategy with a change in the same direction, i.e.dbj (si) =dsi > 0, then si and sj = bj (si) are strategic complements. Ifj�s best response is in the opposite direction, i.e. dbj (si) =dsi < 0, thenwe have strategic substitutes.

� For instance, in pure Bertrand competition, a price cut by one �rm(above marginal cost) is answered by a price cut by the other �rm, sothese are strategic complements. In a Cournot setting, a sales increaseby one �rm often creates incentives for the other �rm to reduce itssales, so these are strategic substitutes.

Example. Suppose costs depend on an investment k that �rm i can make(for example, in process innovation), i.e. c0 (k) < 0. Let the investment stagebe followed by Cournot competition with linear inverse demand p (qi + qj) =a � b (qi + qj) (where a > c � 0 and b > 0) and constant marginal costs(namely, c (k) for i, and c for j). With �rm-speci�c marginal costs, thebest-response functions are

q�i (qj; k) =a� c (k)

2b+1

2qj;

q�j (qi (k)) =a� c

2b+1

2qi (k) ;

leading to equilibrium

q�i�q�j ; k

�=

a� 2c (k) + c

3b;

q�j (q�i (k)) =

a� 2c+ c (k)

3b:

163

The best-response level of output for i is increasing in the cost-reduction: forany qj, q�0i (k) > 0. Di¤erentiating j�s best response with respect to k gives

@q�j (qi (k))

@k= �1

2q0i (k) < 0,

so output levels are strategic substitutes, and i�s investment in cost reduc-tion leads to less aggressive behavior from j, which reinforces the inherentadvantage of lower cost.

Exercise (MWG 12.G.1).

� A Cournot duopolist may take advantage of the strategic substitutesrelationship by precommitting to high sales, so that the competitor willconcede a large share of the market.

� The sequential version of Cournot duopoly is called Stackelberg duopoly.One �rm (the "leader") sets its quantity �rst, and the choice is observedby the other �rm (the "follower"). The follower�s optimal reaction isanticipated by the leader. Given the leader�s choice q1, the followermaximizes

�2 (q1; q2) = (p (q1 + q2)� c) q2:

The resulting best-response function for the follower is

q�2 (q1) = �p (q1 + q�2)� c

p0 (q1 + q�2):

This is identical to a Cournot �rm�s best response function.

� The leader, however, maximizes against the follower�s best responsefunction (rather than a �xed value):

�1 (q1; q�2 (q1)) = (p (q1 + q�2 (q1))� c) q1:

This leads to the leader�s best response

q�1 (q�2) = �

p (q�1 + q�2 (q�1))� c

p0 (q�1 + q�2 (q�1))

1

1 + q�02 (q�1)=

q�2 (q�1)

1 + q�02 (q�1):

If q1 and q2 are strategic substitutes, i.e. q�02 (q�1) < 0, then q�1 (q

�2) >

q�2 (q�1). If q1 and q2 are strategic complements, i.e. q

�02 (q

�1) > 0, then

q�1 (q�2) < q�2 (q

�1).

164

� Since q�2 (q1) is the Cournot best response function, it is clear that totaloutput increases in the case of strategic substitutes, and total outputdecreases in the case of strategic complements. If inverse demand p (�)is decreasing in combined sales, then the industry price decreases withstrategic substitutes and increases with strategic complements relativeto Cournot competition.

Example. With linear inverse demand, p (q1 + q2) = a � b (q1 + q2), thefollower�s best response in the Stackelberg model is

q�2 (q1) =a� c

2b� q�12:

Since 1 + q�02 (q�1) = 1=2,

q�1 (q�2) = 2q

�2 (q1) =

a� c

b� q�1 =

a� c

2b;

whereasq�2 (q

�1) =

a� c

4b:

This is the strategic substitutes case (q�02 (q�1) = �1=2); the leader sells a

larger quantity, knowing that the follower will practice restraint in orderavoid too much price deterioration. Compared to the simultaneous Cournotoutcome q�1 (q

�2) = q�2 (q1) = (a� c) = (3b), the overall quantity sold increases

from q�1+q�2 = (2=3) (a� c) =b to q�1+q

�2 = (3=4) (a� c) =b under Stackelberg

competition. As a result, price decreases from p (q�1 + q�2) = a=3 + (2=3) c top (q�1 + q�2) = a=4+ (3=4) c (keeping in mind that a > c). You can check thatthe leader�s pro�t is larger, and the follower�s pro�t smaller, than in Cournotcompetition.

� In the Stackelberg model, both �rms produce a nonzero quantity, aslong as price exceeds marginal cost. (There is always at least a smallstrictly positive pro�t to be made.) When production requires a �xedinitial outlay (e.g. in product development or a plant), the leader mightdeter the follower from incurring the start-up cost by committing to anaggressive response (high sales at a low price). This intention can besignaled credibly (in the sense that it becomes the best response) byinvesting in high capacity and low marginal cost.

165

� The leader might then be able to operate as a monopolist, possiblyunder constraints to keep the price low enough to continually discourageentry.

18.2 Entry Equilibrium

� In the remainder of the lecture, we return to a symmetric environmentand consider how much entry will occur in equilibrium with a �xedentry cost K > 0. There is an in�nite number of potential entrantswho decide, at stage 1 of the game, whether to invest K or not, andthe entrants compete at stage 2 as oligopolists.

� Assume that, for any number J of entrants, there is in stage 2 a unique,symmetric Nash equilibrium, yielding pro�t �J for each entrant (thisexcludes the entry cost K). The equilibrium number of entrants J�

is such that no �rm wants to either enter or exit, given the prevailingpro�t �J�:

�J� � K and �J�+1 < K:

(We assume �rms enter if indi¤erent, i.e. if �J�+1 = K.)

� If �J is decreasing in J , and �J ! 0 as J ! 1, then the equilibriumJ� is unique.

Example. Suppose stage 2 is a pure Bertrand game, where �rms have con-stant marginal costs c, and inverse demand is linear, p (q) = a � bq witha > c � 0 and b > 0. We know that �J = 0 when J > 1. If the monopolypro�t �m exceeds entry cost K, a single �rm will enter (i.e. J� = 1)and set the monopoly price. Since �m (q) = (p (q)� c) q is maximized byq� = (a� c) = (2b), the optimal pro�t is �m (q�) = (a� c)2 = (4b). The entrycriterion �m (q�) � K is therefore equivalent to K � (a� c)2 = (4b).

Exercise (MWG 12.E.1).

� In an alternative set-up, entry cost may be incurred only if the �rmmakes non-zero sales. Thus, a �rm can observe how many other �rmsenter, and bear the �xed cost only if it can make a non-negative pro�t.This approach restores something close to marginal cost pricing in the

166

Bertrand entry game, since �rms enter if and only if there is positivepro�t to be made.by undercutting the prevailing price.

� Amonopolist could in this case not set a price above p� = (K + cx (p)) =x (p),which approaches c when industry demand x (p) is large. Else, another�rm could enter without pre-paying the entry cost K. In other words,the entrant would not have to worry that it may be undercut in theensuing Bertrand competition and be left with a loss of K. It can sim-ply respond to the incumbent�s current price, since it can exit freelyif the incumbent�s behavior changes. Industries where such "hit andrun" entry is possible are called contestable markets.

Example. Under the same cost and demand conditions, consider a Cournotgame in stage 2. Each entrant maximizes

�J = (p (JqJ)� c) qJ = (a� (J � 1) b�qJ � bqJ � c) qJ ;

so that

qJ� =a� c

2b� J � 1

2�qJ

=a� c

b

1

J� + 1;

since all �rms symmetrically set �qJ = q�J . Thus

�J� =1

b

�a� c

J� + 1

�2is strictly decreasing in J�, and �J� ! 0 as J� ! 0.Since � ~J = K at

~J =a� cpbK

� 1;

the equilibrium number of entrants J� is the greatest integer below ~J . AsK ! 0 or b! 0, J� !1 and the industry price

p (J�qJ�) = a� J�bqJ� = a� J�

J� + 1(a� c)

=1

J� + 1a+

J�

J� + 1c

approaches marginal cost.

167

18.3 Socially Optimal Entry

� Entry in an industry provides, on the one hand, valued goods to con-sumers and, on the other, duplicates entry cost for �rms. Given that�rms choose their production levels to maximize pro�t at the post-entrystage, given the market structure, there exists an e¢ cient number ofentrants from a social perspective that optimally resolves this tradeo¤.

� This number J�� maximizes consumption bene�ts (as measured by thewillingness to pay for each unit) less production cost (which includeseach �rm�s entry cost K):

S (J) =

Z JqJ

0

p (s) ds� J (c (qJ) +K) :

Example. Returning to the pure Bertrand example, the socially optimalnumber of entrants can be no more than two, since price equals marginalcost if two �rms compete (hence surplus is maximized, and it cannot besocially desirable to incur further entry costs). Since a single �rm enters inequilibrium, the number is either socially optimal, or one fewer than sociallyoptimal.

Example. In the Cournot example that was already described,

S (J) = J

�(a� c) qJ �

1

2Jbq2J �K

�=

J

J + 1� 12

�J

J + 1

�2!(a� c)2

b� JK:

The welfare-maximizing number of �rms J�� satis�es

S 0 (J��) =1

(J�� + 1)3(a� c)2

b�K = 0;

so

(J�� + 1)3 =

�a� cpbK

�2:

168

Recall that ~J = (a� c) =pbK � 1 was the equilibrium number of entrants.

It exceeds the socially optimal number, since J�� + 1 =�~J + 1

�2=3< ~J + 1.

Exercise (MWG 12.E.3).

� Entry may be ine¢ cient with Bertrand as well as with Cournot com-petition: in the �rst case, we have (possibly) too little, in the secondgenerally too much, entry. There are two kinds of failures.

� No �rm may �nd it pro�table to enter an industry as a monopolistbecause the maximal pro�t the �rm can make with per-unit pricingdoes not recover the cost of entry. However, from a social perspectiveit may be desirable that the monopolist operate, since the consumersurplus may o¤set losses in pro�t. Of course, the potential monopolistignores these bene�ts to consumers, unless it can extract them throughprice discrimination or is o¤ered a subsidy.

� The same logic could apply to an additional potential entrant in anindustry with some number of incumbents - it may be socially valu-able, but not privately optimal, for the entry to occur. This reasoningsuggests that underentry by one �rm (relative to J��) is possible.

� More typically, there is overentry because �rms consider only whethertheir pro�ts can cover the cost of entry, but ignore the reduction inincumbents�pro�ts. Because the incumbents sell fewer units after anadditional entry, their contribution to surplus declines, while the com-bined entry cost they incurred remains �xed. From a social point ofview, the net increase in total output (and consumption bene�ts) maynot justify the duplication of entry costs.

� But for the �rm, entry may still be pro�table because its sales exceedthe increase in total output (the di¤erence being output reductionsby incumbents) and could therefore cover the entry cost. In short,overentry is possible due to the "business-stealing" e¤ect.

� These insights can be stated as a general result: under standard condi-tions, equilibrium entry may well exceed the socially optimal level, butmay fall short by at most one �rm.

169

Proposition. If marginal �rm pro�ts are positive for any number of entrantsJ (i.e. p (JqJ) � c0 (qJ) � 0), and more entry increases industry output(J > J 0 =) JqJ � J 0qJ) but decreases �rm output and pro�t (J > J 0 =)qJ � qJ 0), and if furthermore p0 (�) < 0 and c00 (�) � 0, then the equilibriumnumber of entrants J� � J�� � 1, where J�� is the socially optimal numberof entrants.Proof. Since the claim is obviously true when J�� = 1, let J�� > 1. By

de�nition, W (J��) � W (J�� � 1), hence:Z J��qJ��

(J���1)qJ���1p (s) ds� J�� (c (qJ��)� c (qJ���1))� c (qJ���1) � K:

Under the assumptions, J��qJ�� > (J�� � 1) qJ���1, and price is a decreasingfunction of industry supply, so p ((J�� � 1) qJ���1) is the maximum in theprice interval from p (J��qJ��) to p ((J�� � 1) qJ���1). Therefore,Z J��qJ��

(J���1)qJ���1p (s) ds � p ((J�� � 1) qJ���1) (J��qJ�� � (J�� � 1) qJ���1) :

(Think of the fact that the area under the curve p (q) between q0 � (J�� � 1) qJ���1 andq1 � J��qJ�� is contained in the rectangle formed by q1� q0 and the maximalheight of the function p (q).)Then, by rearranging the �rst inequality,

J�� (p ((J�� � 1) qJ���1) (qJ���1 � qJ��) + c (qJ��)� c (qJ���1))

� p ((J�� � 1) qJ���1) qJ���1 � c (qJ���1)�K:

Now, the convexity of the cost function (c00 (�) � 0) implies

c (qJ��)� c (qJ���1) =

Z qJ��

qJ���1

c0 (s) ds � c0 (qJ���1) (qJ�� � qJ���1)

since c0 (qJ���1) is the minimum in the cost interval between c0 (qJ���1) andc0 (qJ��). Thus,

J�� (p ((J�� � 1) qJ���1)� c0 (qJ���1)) (qJ���1 � qJ��)

� p ((J�� � 1) qJ���1) qJ���1 � c (qJ���1)�K:

Since qJ���1 > qJ�� and p ((J�� � 1) qJ���1) � c0 (qJ���1) � 0 by assump-tion, the right side is positive, i.e.

p ((J�� � 1) qJ���1) qJ���1 � c (qJ���1) � �J���1 � K:

170

Then �rms must enter at least until their number is J�� � 1, because �J isdecreasing in J :

�J � �J�1 = p (JqJ) qJ � p ((J � 1) qJ�1) qJ�1 � (c (qJ)� c (qJ�1))

� p (JqJ) qJ � p ((J � 1) qJ�1) qJ�1 � c0 (qJ�1) (qJ � qJ�1)

= (p (JqJ)� c0 (qJ�1)) qJ � (p ((J � 1) qJ�1)� c0 (qJ�1)) qJ�1 � 0;

since qJ � qJ�1 and p (JqJ) � p ((J � 1) qJ�1) from JqJ � (J � 1) qJ�1 andp0 (�) < 0.�

Exercise (MWG 12.E.2).

Exercise (MWG 12.E.4).

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