Microeconomic Foundations I
Choice and Competitive MarketsBy David M. KrepsPRINCETON UNIVERSITY PRESS
Copyright © 2013 Princeton University Press
All right reserved.
ISBN: 978-0-691-15583-8
Contents
Preface.....................................................................................xiiiChapter One. Choice, Preference, and Utility................................................1Chapter Two. Structural Properties of Preferences and Utility Functions.....................30Chapter Three. Basics of Consumer Demand....................................................50Chapter Four. Revealed Preference and Afriat's Theorem......................................67Chapter Five. Choice under Uncertainty......................................................79Chapter Six. Utility for Money..............................................................123Chapter Seven. Dynamic Choice...............................................................148Chapter Eight. Social Choice and Efficiency.................................................166Chapter Nine. Competitive and Profit-Maximizing Firms.......................................197Chapter Ten. The Expenditure-Minimization Problem...........................................233Chapter Eleven. Classic Demand Theory.......................................................258Chapter Twelve. Producer and Consumer Surplus...............................................289Chapter Thirteen. Aggregating Firms and Consumers...........................................306Chapter Fourteen. General Equilibrium.......................................................329Chapter Fifteen. General Equilibrium, Efficiency, and the Core..............................358Chapter Sixteen. General Equilibrium, Time, and Uncertainty.................................386About the Appendices........................................................................437Appendix One: Mathematical Induction........................................................439Appendix Two: Some Simple Real Analysis.....................................................441Appendix Three: Convexity...................................................................452Appendix Four: Correspondences..............................................................469Appendix Five: Constrained Optimization.....................................................479Appendix Six: Dynamic Programming...........................................................485Appendix Seven: The Implicit Function Theorem...............................................534Appendix Eight: Fixed-Point Theory..........................................................535References..................................................................................543Index.......................................................................................551
Chapter One
Choice, Preference, and Utility Most people, when they think about microeconomics, think first about the slogan supply equals demand and its picture, shown here in Figure 1.1, with a rising supply function intersecting a falling demand function, determining an equilibrium price and quantity.
But before getting to this picture and the concept of an equilibrium, the picture's constituent pieces, the demand and supply functions, are needed. Those functions arise from choices, choices by firms and by individual consumers. Hence, microeconomic theory begins with choices. Indeed, the theory not only begins with choices; it remains focused on them for a very long time. Most of this volume concerns modeling the choices of consumers, with some attention paid to the choices of profit-maximizing firms; only toward the end do we seriously worry about equilibrium.
1.1. Consumer Choice: The Basics
The basic story of consumer choice is easily told: Begin with a set X of possible objects that might be chosen and an individual, the consumer, who does the choosing. The consumer faces limits on what she might choose, and so we imagine some collection A of nonempty subsets of X from which the consumer might choose. We let A denote a typical element of A; that is, A is a subset of X. Then the choices of our consumer are denoted by c(A).
The story is that the consumer chooses one element of A. Nonetheless, we think of c(A) as a subset of A, not a member or element of A. This allows for the possibility that the consumer is happy with any one of several elements of A, in which case c(A) lists all those elements. When she makes a definite choice of a single element, say x, out of A—when she says, in effect, "I want x and nothing else"—we write c(A) = {x}, or the singleton set consisting of the single element x. But if she says, "I would be happy with either x or y," then c(A) = {x, y}.
So far, no restrictions have been put on c(A). But some restrictions are natural. For instance, c(A) [subset or equal to] A seems obvious; we do not want to give the consumer a choice out of A and have her choosing something that is not in A. You might think that we would insist on c(A) [not equal to]; that is, the consumer makes some choice. But we do not insist on this, at least, not yet. Therefore ...
A model of consumer choice consists of some set X of possible objects of choice, a collection A of nonempty subsets of X, and a choice function c whose domain is A and whose range is the set of subsets of X, with the sole restriction that c(A) [subset or equal to] A.
For instance, we can imagine a world of k commodities, where a commodity bundle is a vector x = (x1, ..., xk) [member of] Rk+, the positive orthant in k-dimensional Euclidean space. (In this book, the positive orthant means all components nonnegative, or Rk+ = {x [member of] Rk: x ≥ 0}. The strict positive orthant, denoted by Rk++, means elements of Rk all of whose components are strictly positive.) If, say, k = 3 and the commodities are (in order) bread, cheese, and salami, the bundle (3, 0, 0.5) means 3 units of bread, no cheese, and 0.5 units of salami, in whatever units we are using. We can also imagine prices pi for the commodities, so that p = (p1, ..., pk) is the price vector; for convenience, we assume that all prices are strictly positive, or p [member of] Rk++. And we can imagine that the consumer has some amount of income y ≥ 0 to spend. Then the consumer's choice problem is to choose some affordable bundle given these prices and her income; that is, a typical set A is a budget set
{x [member of] Rk+ : p · x ≤ y}.
A model of consumer choice in this context is then a choice function that says which bundles the consumer would be willing to accept, as a function of the prices of the goods p and her level of income y.
This is not much of a model, yet. Economic modeling begins with an assumption that the choices made by the consumer in different situations are somewhat coherent. Imagine, for instance, a customer at a café asking for a cup of coffee and a piece of pie. When told that they have apple and cherry pie, she opts for apple. Then the waiter tells her that they also have peach pie. "If you also have peach," she responds, "I would like cherry pie, please." We want to (and will) assume that choice in different situations is coherent enough to preclude this sort of behavior; we'll formalize this next page, in Definition 1.1b.
This is one sort of coherence. A second is that the consumer's choices are in accord with utility maximization, for some utility function defined on X. That is, there is a function u : X -> R, such that for every A,
c(A) = {x [member of] A : u(x) ≥ u(y) for all y [member of] A}. (1.1)
A third sort of coherence involves a preference relation over X. A preference relation expresses the consumer's feelings between pairs of objects in X. We denote the preference relation by ≥ and imagine that for every pair x and y from X, the consumer is willing to say that either ≥ ≥ y, meaning x is at least as good as y, or not. For any pair x and y, then, one of four mutually exclusive possibilities holds: (1) the consumer says that x ≥ y and that y ≥ x; (2) x ≥ y but not y ≥ x; (3) y ≥ x but not x ≥ y; or (4) neither x ≥ y nor y ≥ x. Then, with these preferences in hand, a consumer chooses from a set A precisely those elements of A that are at least as good as everything in A, or
c(A) = {x [member of] A : x ≥ y for all y ≥ A}. (1.2)
When you look at (most)models in microeconomics that have consumers, consumers make choices, and the choice behavior of the consumer is modeled by either (1) a utility function and the (implicit) assumption that choice from any set A is governed by the rule (1.1) or (2) a preference relation and the (implicit) assumption that choice from any set A is governed by the rule (1.2). (Discrete choice models in econometrics have so-called random utility models, in which choices are stochastic. And in some parts of behavioral economics, you will find models of choice behavior that don't quite fit either of these frameworks. But most models have either utility-maximizing or preference-maximizing consumers.)
The questions before us in this chapter are: How do these different ways of modeling consumer choice compare? If we restrict attention to coherent choice, does one imply the other(s)? Can they be made consistent?
The basic answer is that under certain coherence assumptions, the three ways of modeling consumer choice are equivalent. We begin with the case of finite X. (We worry a lot about infinite X later.) To keepmatters simple, we make the following assumption for the remainder of this chapter (but see Problems 1.15 and 1.16).
Assumption. A is the set of all nonempty subsets of X.
Two properties of choice functions and two properties of a preference relation must be defined:
Definition 1.1.
a. A choice function c satisfies finite nonemptiness if c(A) is nonempty for every finite A 2 A.
b. A choice function c satisfies choice coherence if, for every pair x and y from X and A and B from A, if x, y [member of] A [intersection] B, x [member of] c(A), and y [member of or equal to] c(A), then y [member of or equal to] c(B).
c. A preference relation on X is complete if for every pair x and y from X, either x ≥ y or y ≥ x (or both).
d. A preference relation on X is transitive if x ≥ y and y ≥ z implies that x ≥ z.
Some comments about these definitions may be helpful: Concerning a, if X is finite, finite nonemptiness of c means that c(A) is nonempty for all subsets of X. Later in the chapter, the restriction to finite A will have a role to play. Choice coherence is the formalization intended to preclude the apple, cherry, and peach pie vignette: If apple is the (sole) choice out of {apple, cherry}, then cherry cannot be chosen from {apple, cherry, peach}. An equivalent (contrapositive) form for b is: For every pair x and y from X and A and B from A, if x, y [member o] A [intersection] B, x [member of] c(A), and y [member of] c(B), then y [member of] c(A) and x [member of] c(B).
Proposition 1.2. Suppose that X is finite.
a. If a choice function c satisfies finite nonemptiness and choice coherence, then there exist both a utility function u : X -> R and a complete and transitive preference relation ≥ that produce choices according to c via the formulas (1.1) and (1.2), respectively.
b. If a preference relation ≥ on X is complete and transitive, then the choice function it produces via formula (1.2) satisfies finite nonemptiness and choice coherence, and there exists a utility function u : X -> R such that
x ≥ y if and only if u(x) ≥ u(y). (1.3)
c. Given any utility function u : X -> R, the choice function it produces via formula (1.1) satisfies finite nonemptiness and choice coherence, the preference relation it produces via (1.3) is complete and transitive, and the choice function produced by that preference relation via (1.2) is precisely the choice function produced directly from u via (1.1).
In words, choice behavior (for a finite X) that satisfies finite nonemptiness and choice coherence is equivalent to preference maximization (that is, formula (1.2)) for complete and transitive preferences, both of which are equivalent to utility maximization (via formulas (1.1) and (1.3)). However expressed, whether in terms of choice, preference, or utility, this conglomerate (with the two pairs of assumptions) is the standard model of consumer choice in microeconomics.
A much-used piece of terminology concerns display (1.3), which connects a utility function u and a preference relation ≥. When (1.3) holds, we say that the utility function u represents the preference relation ≥.
In terms of economics, Proposition 1.2 is the story of this chapter. Several tasks remain:
1. We prove the proposition.
2. We consider how (and whether) this proposition extends to infinite X. After all, in the one example we've given, where X = Rk+, we have an infinite X. Most economic applications will have an infinite X.
3. We have so far discussed the binary relation ≥, known as weak preference, which is meant to be an expression of "at least as good as." In economic applications, two associated binary relations, strict preference ("strictly better than") and indifference ("precisely as good as") are used; we explore them and their connection to weak preference.
4. We comment briefly on aspects of the standard model: What if A does not contain all nonempty subsets of X? What is the empirical evidence for or against the standard model? What alternatives are there to the standard model?
1.2. Proving Most of Proposition 1.2, and More
Parts of Proposition 1.2 are true for all X, finite or not.
Proposition 1.3. Regardless of the size of X, if u : X -> R, then
a. the preference relation ≥u defined by x ≥u y if u(x) ≥ u(y) is complete and transitive, and
b. the choice function cu defined by cu(A) = {x [member of] A : u(x) ≥ u(y) for all y [member of] A} satisfies finite nonemptiness and choice coherence.
Proof. (a) Given any two x and y from X, either u(x) ≥ u(y) or u(y) ≥ u(x) (since u(x) and u(y) are two real numbers); hence either x ≥u y or y ≥u x. That is, ≥u is complete.
If x ≥u y and y ≥u z, then (by definition) u(x) ≥ u(y) and u(y) ≥ u(z); hence u(x) ≥ u(z) (because ≥ is transitive for real numbers), and therefore x ≥u z. That is, ≥u is transitive.
(b) Suppose x, y [member of] A [intersection] B and x [member of] cu(A). Then u(x) ≥ u(y). If, moreover, y [not member of] cu(A), then u(z) > u(y) for some z [member of] A. But u(x) ≥ u(z) since x [member of] cu(A) implies u(x) ≥ u(z) for all z [member of] A; therefore u(x) > u(y). Since x [member of] B, this immediately implies that y [member of or equal to] cu(B), since there is something in B, namely x, for which u(y) [??] u(x). This is choice coherence.
If A is a finite subset of X, then {r [member of] R : r = u(x) for some x [member of] A} is a finite set of real numbers. Every finite set of real numbers contains a largest element; that is, some r* = u(x*) in the set satisfies r* ≥ r for all the elements of the set. But this says that u(x*) ≥ u(x) for all x [member of] A, which implies that x* [member of] cu(A), and cu(A) is not empty.
Proposition 1.4. Regardless of the size of X, if ≥ is a complete and transitive binary relation on X, the choice function c≥ defined on the set of all nonempty subsets of X by
c≥(A) := {x [member of] A : x ≥ y for all y 2 A}
satisfies finite nonemptiness and choice coherence.
Proof. Suppose x, y [member of] A [intersection] B, x [member of] c≥(A), and y [not member of] c≥(A). Since x [member of] c≥(A), x ≥ y. Since y [member of] c≥(A), y [??]; z for some z [member of] A. By completeness, z ≥ y. Since x [member of] c≥(A), x ≥ z. I claim that y [??] x: Assume to the contrary that y ≥ x, then x ≥ z and transitivity of ≥ would imply that y ≥ z, contrary to what was assumed. But if y [??] x, then since x [member of] B, y [not member of] c≥(B). That is, c≥ satisfies choice coherence.
I assert that if A is a finite (and nonempty) set, some x [member of] A satisfies x ≥ y for all y [member of] A (hence c≥(A) is not empty). The proof is by induction on the size of A: if A contains a single element, say, A = {x}, then x ≥ x because ≥ is complete. Therefore, the statement is true for all sets of size 1. Assume inductively that the statement is true for all sets of size n - 1 and let A be a set of size n. Take any single element x0 from A, and let A' = A\{x0}. A' is a set of size n - 1, so there is some x' [member of] A' such that x' ≥ y for all y [member of] A'. By completeness of ≥, either x' ≥ x0 or x0 ≥ x'. In the first case, x0 ≥ y for all y [member of] A, and we are done. In the second case, x0 ≥ x0 by completeness, and x0 ≥ y for all y [member of] A' , since x' ≥ y, and therefore transitivity of ≥ tells us that x0 ≥ y. Hence, for this arbitrary set of size n, we have produced an element at least as good as every other element. This completes the induction step, proving the result.
(Continues...)
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