CONSUMER DEMAND. Consumer Demand

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1 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,

2 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.

3 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 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

5 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 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

7 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.

Microeconomics. Joana Pais. Fall Joana Pais

Microeconomics. Joana Pais. Fall Joana Pais Microeconomics Fall 2016 Primitive notions There are four building blocks in any model of consumer choice. They are the consumption set, the feasible set, the preference relation, and the behavioural assumption.