x 2 you(x)e~xby~l(y)e台l(x)eu(y)el(x)≥u(y) Here the last equivalence follows from strong monotonicity We now prove the continuity of u(x). Suppose that (x' is a sequence with x)x. We want to prove that u(r)->tx). Suppose this is not true. Then there exists some e>0 and an infinite set of is such that u(x)>u(x)+sor an infinite set of i's such that u(x)<u(x)-E Let us assume the first case.This implies that ux)e uxe+sex+ x+ se On the other hand, since x'x, it follows that for very large value of i in this infinite set, we must ave x se>x which implies that x+ se>x' by strong monotonicity. This leads to a contradiction.■ Properties of Utility Functions Proposition: Let a be represented by u: R"+->RThen (a)u(x)is strictly increasing e 2 is strictly monotonic (b)u(x)is quasiconcave e2isconvex ( c) u(x) is strictly quasiconcave e 2 is strictly convex roof We just need to remind the following definition of (strict)quasiconcavity u(x)is(strictly) quasiconcave o the set (x: u(x)2c) is(strictly)convex for all c 2.3 Indirect Utility Functions and Expenditure Functions 2. 3. IThe Consumer's Utility- Maximizing problen In the basic problem of preference maximization, the set of affordable consumption plans for the consumer is just the set of all bundles that satisfy the budget constraint. Let y be the fixed amount of money available to a consumer and let p=(P,., Pm) be the price vectors of goods, 1,...,m. Then the consumer's problem is to solve the following optimization problem max u(x) st.pX≤y The objective function is continuous. It is clear that the constraint set is compact (closed and bounded ). Then by Weierstrass Theorem(Existence of Extreme Values, in Lecture 1), the above optimization problem does have a global maximum Lemma: If the preference relation satisfies local nonsatiation, then the budget constraint must be binding at the optimal choice of the consumption bundle Proof: Suppose that x *is an optimal solution to the consumer's problem such that Since px is a continuous function of x, it follows that there exists some E>0 such that llx∈X|-x On the other hand, according to local nonsatiation, for the given x* and the above E, there exist ith ly the assumption that x * is an optimal solution Therefore the budget constraint be must binding at5 x y  u(x)e ~ x y ~ u(y)e  u(x)e u(y)e  u(x)  u(y). Here the last equivalence follows from strong monotonicity. We now prove the continuity of u(x). Suppose that {x i}is a sequence with x i → x. We want to prove that u(x i ) → u(x). Suppose this is not true. Then there exists some  > 0 and an infinite set of i's such that u(x i ) > u(x) +  or an infinite set of i's such that u(x i ) < u(x) - . Let us assume the first case. This implies that x i ~ u(x i )e u(x)e + e ~ x + e  x i x + e On the other hand, since x i → x, it follows that for very large value of i in this infinite set, we must have x + e > x i which implies that x + e  x i by strong monotonicity. This leads to a contradiction.  Properties of Utility Functions Proposition: Let be represented by u: Rn + → R. Then (a) u(x) is strictly increasing  is strictly monotonic. (b) u(x) is quasiconcave  is convex. (c) u(x) is strictly quasiconcave  is strictly convex. Proof: We just need to remind the following definition of (strict) quasiconcavity. u(x) is (strictly) quasiconcave  the set {x: u(x)  c} is (strictly) convex for all c. 2.3 Indirect Utility Functions and Expenditure Functions 2.3.1The Consumer's Utility-Maximizing Problem In the basic problem of preference maximization, the set of affordable consumption plans for the consumer is just the set of all bundles that satisfy the budget constraint. Let y be the fixed amount of money available to a consumer and let p = (p1, …, pm) be the price vectors of goods, 1, …, m. Then the consumer's problem is to solve the following optimization problem: max u(x) s.t. px  y x  X Notes: • The objective function is continuous. It is clear that the constraint set is compact (closed and bounded). Then by Weierstrass Theorem (Existence of Extreme Values, in Lecture 1), the above optimization problem does have a global maximum. Lemma: If the preference relation satisfies local nonsatiation, then the budget constraint must be binding at the optimal choice of the consumption bundle. Proof: Suppose that x* is an optimal solution to the consumer's problem such that px* < y Since px is a continuous function of x, it follows that there exists some  > 0 such that px < y for all x  X: ||x - x*|| <  On the other hand, according to local nonsatiation, for the given x* and the above , there exist some y in X with ||y - x*|| <  such that y  x*, which implies that u(y) > u(x*). This contradicts to the assumption that x* is an optimal solution. Therefore the budget constraint be must binding at x*. 