CONSUMER DEMAND KENNETH R. DRIESSEL Consumer Demand The most basic unit in microeconomics is the consumer. In this section we discuss the consumer optimization problem: The consumer has limited wealth and decides what to buy. The decision is determined by the consumer s utility function and wealth constraint. In this section, we use the following notation: R + := [0, ) := {α R : 0 α} denotes the set of nonnegative real numbers. R n + := {x R n : ( i = 1,..., n) 0 x i } denotes the nonnegative orthant in real n-space. For X R n, u : X R and a, b in X, [a, + ) := {x X : u(a) u(x)}, [a, b] := {x X : u(a) u(x) u(b)}, etc. Let X be a subset of R n +, let u : X R, let p be a nonzero vector in R n + and let w be a positive real number. We use the following phrases to describe these items; these phrases indicate the intended application to consumer economics. The elements x of the set X are possible consumption bundles for the consumer. The elements x i of the vector x represent the quantity of good i in the bundle. The set X is the consumption set of the consumer. The function u is the utility function of the consumer; it represents the consumer s preferences; in particular, if u(x) u(y) then the Date: November 8, 2012. 1
2 KENNETH R. DRIESSEL consumer (weakly) prefers the bundle y to the bundle x. The vector p represents the prices of the goods; in particular, p i represents the price of a unit of good i. The positive real number w represents the wealth possessed by the consumer. The consumer faces the following optimization problem: Let B(p, w) := {x X : p, x w}. This set represents the possible consumption bundles for the consumer with wealth w at prices p. This set is called the budget set for the consumer with wealth w at prices p. In particular, note that the quanity p, x represents the cost of the bundle x. The consumer wants to maximize utility u(x) on this budget set. This is a precise statement of the consumer s optimization problem. The following notions will be used below. Definition: Let X be a subset of R n. Then X is convex if, for every pair of points x, y in X, and every real number t in the interval [0, 1], the point tx + (1 t)y is also in X. The set X is strictly convex is for all t in the open interval (0, 1) the point tx + (1 t)y is in the interior of X. Definition: Let X be a convex subset of R n. Then a function u : X R is concave if, for all x, y in X, and all t [0, 1], u(tx + (1 t)yu) tu(x) + (1 t)u(y) and is strictly concave if, for all t (0, 1), u(tx + (1 t)yu) > tu(x) + (1 t)u(y). Proposition 1. Let X R n be a convex set and let u : R. If u is a concave function then, for all a X, the upper level set [a, + ) determined by u is convex. If u is strictly concave then, for all z X, the upper level set [a, + ) is strictly convex.
CONSUMER DEMAND 3 Proof. Consider a pair of points x, y in [a, +infty) and a real number t in the closed interval [0, 1]. We have u(tx + (1 t)y) tu(x) + (1 t)u(y) tu(a) + (1 t)u(y) u(a). This completes the proof of the first part of the result. A similar argument proves the second part. Proposition 2. Properties of the budget set. If X is closed then B(p, w) is closed. If X is convex the B(p, w) is convex. If X is bounded then B(p, w) is bounded. If X R n + and ( i = 1,..., n) 0 < p i then B(p, w) is bounded. Let 0 < λ R. Then B(λp, λw) = B(p, w). Remark: The last conclusion is called the homogeneity property of the budget set. Proof. Let C(p, w) := {x R n : p, x w}. Claim: The set C(p, w) is closed. Note that the map φ : R n R : x p, x is continuous. Note C(p, w) is the inverse image of the closed set (, w] under φ; that is, C(p, w) = φ 1 {α R : α w}. Hence, C(p, w) is closed. Claim: If X is closed then the set B(p, w) is closed. Note B(p, w) = C(p, w) X. Thus B(p, w) is the intersection of two closed sets. Claim: The set C(p, w) is convex. Consider any x, y in C(p, w) and 0 t 1. We have p, tx + (1 t)y = t p, x + (1 t) p, y tw + (1 t)w = w. Claim: If X is convex then the set B(p, w) is convex.
4 KENNETH R. DRIESSEL Note again that B(p, w) = C(p, w) X. Thus B(p, w) is the intersection of two convex sets. Claim: If X is bounded then B(p, w) is bounded. Note B(p, w) is a subset of X. Claim: If X R n + and ( i = 1,..., n) 0 < p i then B(p, w) is bounded. We have, for any x B(p, w) and any index j, 0 p j x j n p i x i = p, x w. i Hence x j w/p j. Claim:. Let 0 < λ R. Then B(λp, λw) = B(p, w). We have the following equivalent conditions: x B(λp, λw), λ p, x λw, p, x w, x B(p, w). Proposition 3. Existence of a solution of the consumer optimization problem. Let (p, w) be a price-wealth pair. Assume the set B ( p, w) is nonempty and compact. Assume that u is continuous. Then u has a maximum on B(p, w). Proof. Recall that a continuous function on a compact set has a maximum. Let (p, w) be a price-wealth pair for a consumer with utility function u : X R. Assume that the budget set B(p, w) for the consumer is nonempty. Let M(p, w) := {x X : u(x) is maximal on B(p, w)}. This set represents the set of solutions of the consumer s optimization problem. We call this set the demand set for the consumer with wealth w at prices p. The consumer maximizes utility u(x) on the consumption bundles
CONSUMER DEMAND 5 in this set. Note that if x M(p, w) iff, for all y X, p, y w u(y) u(x). We shall repeatedly use this implication below. The following notions are used in the next result. Definition: Let u : X R be a utility function. A point x X is satiated ( or a satiated consumption bundle) if no other point is strictly preferred; in symbols, if not ( y X) u(x) < u(y). The utility function u is locally non-satiated at x if the following condition is satisfied: For every open neighborhood U of x there is a y U such that u(x) < u(y). Example: Let x be an element of X. Assume that the utility function u is continuously differentiable in a neighborhood of x. Then, if the derivative Du(x) (which can be identified with the gradient u(x)) is nonzero then u is locally non-satiated at x. In particular, the gradient provides a direction in which u is strictly increasing. Proposition 4. Properties of the demand set. Assume that the consumption set X is closed and convex. Assume that the utility function u : X R is continuous. Let (p, w) be a price-wealth pair. Then the demand set M(p, w) is closed and bounded; let 0 < λ R. Then M(λp, λw) = M(p, w); if u satisfies ( x X) [x, + ) is convex then M(p, w) is convex; if u satisfies ( x X) [x, + ) is strictly convex then M(p, w) consists of a single point; let x be an element of M(p, w). If u is locally non-satiated at x then p, x = w. Remark: The second conclusion is called the homogeneity property of the demand set. It loosely says the following: Prices themselves don t matter; it is the ratios of prices that matter. The last conclusion is called Walras Law. Let H(p, w) := {x X : p, x = w}. This set is called the
6 KENNETH R. DRIESSEL wealth hyperplane determined by the price-wealth pair (p, w). Walras law says that if x is optimal then x is on the wealth hyperplane. An indifference class [x, x] is thick if its interior in X is not empty. Note that if x is optimal and [x, x] is thick then x might not be on the wealth hyperplane. Proof. Claim: The demand set is closed. Consider any sequence x k of elements of the demand set that tends to an element x. Since X is closed, the budget set is closed. Hence x is in the budget set. Since u is continuous, we also have that the limit of the sequence u(x k ) is u(x). Let r denote the maximum value of u on the budget set. If each x k is in the demand set then the sequence k u(x k ) is a constant sequence with value r. Hence u(x) = r and x is in the demand set. Claim: The demand set is bounded. Note that the budget set is bounded. Claim: The demand set is homogeneous. This conclusion follows from the homogeneity of the budget set and the fact that the utility function does not depend on the price-wealth pair. Claim: Assume ( x X) [x, + ) is convex. Then the demand set is convex. Consider any pair of points x, y in the demand set. Note that u(x) u(y) and u(y) u(x) since x and y are optimal points in the budget set. Hence u(x) = u(y). Consider any real number t in the interval [0, 1]. Let z := tx + (1 t)y. Note that z in in the budget set since it is convex. Hence u(z) u(x) because x is optimal. By the convexity assumption on u, the interval [x, ) is convex. Hence z is in that interval and u(x) u(z). Thus u(x) = u(z). Claim: Assume ( x X) [x, + ) is strictly convex. Then the demand set consists of a single point. Consider any two points x and y in M(p, w). Suppose that x y. Consider any t in the open interval (0, 1). Let z := tx + (1 t)y. Note z is an
CONSUMER DEMAND 7 element of M(p, w) since this set is convex. But, by the strict convexity of [x, + ), we have u(x) < u(z). This contradicts the optimality of x. Claim: (Walras Law) Let x be an element of M(p, w). If u is locally non-satiated at x then p, x = w. Suppose p, x < w. Note that the set {y R n : p, y < w} is an open set since the map R n R : y p, y is continuous. Hence, since u is locally non-satiated at x, there exists y B(p, w) such that p, y < w and u(x) < u(y). This contradicts the optimality of x.