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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.
Figure 1.1. Supply equals demand
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) ⊆
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) ≠
; 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) ⊆ A.
For instance, we can imagine a world of k commodities, where a commodity bundle is a vector x = (x1, …, xk) ∈ Rk+, the positive orthant in k-dimensional Euclidean space. (In this book, the positive orthant means all components nonnegative, or Rk+ = {x ∈ Rk : x ≥ 0}. The strict positive orthant, denoted by Rk++, means elements of Rk all ...
