Advanced microeconomics Topic 2: Individual Preferences Primary Readings: DL-Chapter 4; JR-Chapter 3; Varian-Chapter 7 In most economic models, we start with an agent's utility function. The utility function basically maps from bundles that the agent might choose, to the real line. The utility function is quite convenient: it can be maximized and manipulated using mathematical tools. But the question is are we imposing some hidden or desirable assumptions when we take this approach ?prefeent's Is it valid to reduce ple, real- value function, something as complicated preferences over a wide variety of bundles? What does it really mean about the agent's preferences? In this lecture, we will try to answer these questions by analyzing the relationships between axioms about an agent's preferences, and then establishing the existence of a utility function that represents the agent's preferences 2.1 The Consumer's Preferences We let the consumption set, X, represent the set of all alternatives, or complete consumption lans, that the consumer can conceive- whether some of them will be achievable in practice or not Every element of X is called a consumption bundle or a consumption plan X captures the universe of all possible choices a consumer may have. For this reason, the consumption set is also known as the choice set Normally, XcR +-the entire nonnegative orthant of the real space R We will always assume that X is a closed and convex set 2. 1. 2Basic Properties& Axioms of Preferences For x, y e X, when we write x& y, we mean that"the consumer thinks that the bundle x is at least as good as bundle y "We call a a preference relation on X We say, "x is(weakly) preferred to y It is clear that 2 is a binary relation defined on X As the final purpose of introducing a preference relation is to order the set of consumption bundles we need to assume a number of axioms. These axioms of consumer choice are intended to give formal mathematical expression to fundamental aspects of consumer behavior and altitudes toward the objects of choice AXIOM 1:( Completeness)Vx, yE X, (x y)v(y 2 x).(Note:v="or") To satisfy the completeness axiom, the preference 2 cannot be defined so that 12 yox2y, Vj.Reason: it is only a partial ordering. (Note: While this axiom appears innocuous, in combination with the usual confinement of the consumption set to the consumption of the individual only, it rules out externalities in consumption.)
1 Advanced Microeconomics Topic 2: Individual Preferences Primary Readings: DL – Chapter 4; JR - Chapter 3; Varian – Chapter 7 In most economic models, we start with an agent's utility function. The utility function basically maps from bundles that the agent might choose, to the real line. The utility function is quite convenient: it can be maximized and manipulated using mathematical tools. But the question is: Is it valid to reduce a simple, real-value function, something as complicated as an agent's preferences over a wide variety of bundles? What does it really mean about the agent's preferences? Are we imposing some hidden or desirable assumptions when we take this approach? In this lecture, we will try to answer these questions by analyzing the relationships between axioms about an agent's preferences, and then establishing the existence of a utility function that represents the agent's preferences. 2.1 The Consumer's Preferences 2.1.1Consumption Set We let the consumption set, X, represent the set of all alternatives, or complete consumption plans, that the consumer can conceive - whether some of them will be achievable in practice or not. Every element of X is called a consumption bundle or a consumption plan. • X captures the universe of all possible choices a consumer may have. For this reason, the consumption set is also known as the choice set. • Normally, X R m + - the entire nonnegative orthant of the real space Rm. • We will always assume that X is a closed and convex set. 2.1.2Basic Properties & Axioms of Preferences • For x, y X, when we write x y, we mean that "the consumer thinks that the bundle x is at least as good as bundle y." We call a preference relation on X. • We say, "x is (weakly) preferred to y". • It is clear that is a binary relation defined on X. As the final purpose of introducing a preference relation is to order the set of consumption bundles, we need to assume a number of axioms. These axioms of consumer choice are intended to give formal mathematical expression to fundamental aspects of consumer behavior and altitudes toward the objects of choice. AXIOM 1: (Completeness) x, y X, (x y) (y x). (Note: = "or") • To satisfy the completeness axiom, the preference cannot be defined so that x y xj yj, j. (Reason: it is only a partial ordering.) (Note: While this axiom appears innocuous, in combination with the usual confinement of the consumption set to the consumption of the individual only, it rules out externalities in consumption.)
AXIOM 2: (Reflexivity)VxEX, x2x AXIOM 3: (Transitivity)(x y)&(y z) The first assumption says that any two bundles can be compared, the second is trivial, and the third is necessary for any discussions of preference maximization: for if preferences were not transitive, there might be sets of bundles which had no best elements It is useful to extend our notation: We write x >y and say that x is strictly preferred to y. We sometime also write not y 2 x, meaning y is not preferred to x, which is the same as x >y, given completeness We writex-yif(x& y)&(x2 y) and say that x is indifferent to y Examples (a)Finite Set If X is a finite set, then a preference relation on X will partition X into a finite number of subsets such that elements within a subset are all indifferent There will be a strict preference for elements from different subsets (b) Summation Ordering Let X=r Define x a y to mean that∑x≥∑y It is easy to show that this summation ordering is complete, reflective and transitive (c) Lexicographic Ordering Let x=r+ x y if and only if either, there exists some such that xi=yi for iyi; orx= yi for 1≤i≤m Essentially, the lexicographic ordering compares the components one at a time beginning with the first, and determines the ordering based one the first a difference is found This implies that the vector with greatest component is raked the highest The above three axioms are the basic properties of a preference relation. Any relation satisfyin these 3 axioms is called an ordering In order to have a functional representation, we may need a few more axioms(assumptions).(If X is countable, no additional axiom is needed. AXIOM 4:( Continuity) For all y in X, the sets (x: Ia y) and (x: y2 x) are closed sets.It follows that the sets ( x: x>y) and x: y>x) are open sets This assumption is necessary to rule out certain discontinuous behavior It says that, if( x')is a sequence of consumption bundles that are all at least as good as y and if this sequence converges to some bundle x', then x is at least as good as y The key consequence of continuity is as follows: if y is strictly preferred to z and if x is bundle that is close enough to y, then x must be strictly preferred to z E
2 AXIOM 2: (Reflexivity) x X, x x. AXIOM 3: (Transitivity) (x y) & (y z) x z. (Note: & = "and") Note: • The first assumption says that any two bundles can be compared, the second is trivial, and the third is necessary for any discussions of preference maximization: for if preferences were not transitive, there might be sets of bundles which had no best elements. It is useful to extend our notation: • We write x y and say that x is strictly preferred to y. We sometime also write not y x, meaning y is not preferred to x, which is the same as x y, given completeness. • We write x ~ y if (x y) & (x y) and say that x is indifferent to y. Examples: (a) Finite Set • If X is a finite set, then a preference relation on X will partition X into a finite number of subsets such that • elements within a subset are all indifferent; • There will be a strict preference for elements from different subsets. (b) Summation Ordering: • Let X = R m . • Define x y to mean that . =1 =1 m i i m i i x y • It is easy to show that this summation ordering is complete, reflective and transitive. (c) Lexicographic Ordering • Let X = R m +. • x y if and only if • either, there exists some j such that xi = yi for i yj; • or, xi = yi for 1 i m. • Essentially, the lexicographic ordering compares the components one at a time beginning with the first, and determines the ordering based one the first a difference is found. • This implies that the vector with greatest component is raked the highest. The above three axioms are the basic properties of a preference relation. Any relation satisfying these 3 axioms is called an ordering. In order to have a functional representation, we may need a few more axioms (assumptions). (If X is countable, no additional axiom is needed.) AXIOM 4: (Continuity) For all y in X, the sets {x: x y} and {x: y x} are closed sets. It follows that the sets {x: x y} and {x: y x} are open sets. • This assumption is necessary to rule out certain discontinuous behavior. • It says that, if ( x i ) is a sequence of consumption bundles that are all at least as good as y and if this sequence converges to some bundle x * , then x * is at least as good as y. • The key consequence of continuity is as follows: if y is strictly preferred to z and if x is bundle that is close enough to y, then x must be strictly preferred to z. Examples
Summation ordering is continuous Lexicographic order is discontinuous(see the following diagram on R2+) (x1,x2)>(1,1) AXIOM 4A: (Strong Monotonicity)If x 2 y and x+y, then x>y AXIOM 4B: Weak Monotonicity)If xi 2y for all i, then x 2 Weak monotonicity says that "at least as much of everything is at least as good. If the consumer can costlessly dispose of unwanted goods, this assumption is trivial Strong monotonicity says that at least as much of every good, and strictly more of some good is strictly better. This is simply says assuming that goods are good If one of the goods is a"bad", such as garbage or pollution, then strong monotonicity will not be satisfied. But we can easily get around this problem by respecifying the good to be absence of garbage, or absence of pollution, which will normally lead to strong monotonicity AXIOM 5:(Local? Nonsatiation) Given any x in X and E >0, then there is some bundle y in X with lx-y x (An alternative definition: requiring this to hold over some set that contain the set defined by the relevant budget constraint. Local nonsatiation says that one can always do a little bit better, even if one restricted to only small change in the consumption bundle It can be shown that strong monotonicity implies local nonsatiation but not vice versa Key consequence of local nonsatiation rules out"thick"indifference curves The following two assumptions are often used to guarantee nice behavior of consumer demand AXIOM 6A:( Convexity)Given x, y, z E X such that x z and a z, then Ix+(1-Dy 2 zfor ll0≤t≤1. AXIOM 6B: ( Strict Convexity)Given x*y, z e X such that x2 z and y2 z, then x+(1-d)y>z for all 0<(< 1 Convexity implies that an agent prefers average to extremes Convexity is a generalization of the neoclassical assumption of"diminishing marginal rates of substitution Before we move on the functional representation of the preference relation, we must emphasize that the a preference relation is an ordinal, rather than cardinal, concept even though we have attempted to incorporate additional structures by imposing some of the above assumptions
3 • Summation ordering is continuous. • Lexicographic order is discontinuous (see the following diagram on R2 +) AXIOM 4A: (Strong Monotonicity) If x y and x y, then x y. AXIOM 4B: (Weak Monotonicity) If xi yi for all i, then x y. • Weak monotonicity says that "at least as much of everything is at least as good." If the consumer can costlessly dispose of unwanted goods, this assumption is trivial. • Strong monotonicity says that at least as much of every good, and strictly more of some good, is strictly better. This is simply says assuming that goods are good. • If one of the goods is a "bad", such as garbage or pollution, then strong monotonicity will not be satisfied. But we can easily get around this problem by respecifying the good to be absence of garbage, or absence of pollution, which will normally lead to strong monotonicity. AXIOM 5: (Local? Nonsatiation) Given any x in X and > 0, then there is some bundle y in X with || x - y || < such that y x. (An alternative definition: requiring this to hold over some set that contain the set defined by the relevant budget constraint.) • Local nonsatiation says that one can always do a little bit better, even if one is restricted to only small change in the consumption bundle. • It can be shown that strong monotonicity implies local nonsatiation but not vice versa. • Key consequence of local nonsatiation rules out "thick" indifference curves. The following two assumptions are often used to guarantee nice behavior of consumer demand functions. AXIOM 6A: (Convexity) Given x, y, z X such that x z and y z, then tx + (1-t)y z for all 0 t 1. AXIOM 6B: (Strict Convexity) Given x y, z X such that x z and y z, then tx + (1-t)y z for all 0 < t < 1. • Convexity implies that an agent prefers average to extremes. • Convexity is a generalization of the neoclassical assumption of "diminishing marginal rates of substitution." Before we move on the functional representation of the preference relation, we must emphasize that the a preference relation is an ordinal, rather than cardinal, concept even though we have attempted to incorporate additional structures by imposing some of the above assumptions. x2 {(x1, x2) (1, 1)} 1 1 x1
2.2 Utility Functions A utility function is a real-valued function u defined on the consumption set X such that preference rankings are preserved by the magnitude of u. That is, a utility function u has the property that given any two elements x, y in X, u(x)>u(y)if and only if x2y. But not all preference relations can be represented by utility functions. A rather general result is that any continuous preference ordering can be represented by a continuous utility function. This is a very difficult result to prove(Debreu(1959).(Moreover, while any continuous ordering is al ways representable, continuity is not necessary. The necessary and sufficient conditions fo representation is rather technical; see Ng 1979/83, App. IB )We will focus on a somewhat simpler result-the one that can be proved constructively. The main ideas are e select arbitrary fixed line that cuts all of the indifference curves(or surfaces) Once utility is defined along this line, the utility of any other point is found by tracing the appropriate indifference curve to the line and using the utility value there The assumption of strong monotonicity guarantees that the indifference curves exit and that any line of the form ae, a>0 and e>0, cuts them all Existence of Utility functions Suppose that a reference relation on X=R+ is complete, reflexive, transitive, continuous, and strongly monotone. Then there exists a continuous utility function u:R"+->R which represents the preference relation all ones. Then given any vector r, let g of Let e be the vector in r+ consist u (x)= a such that xae We now need to show that u(x)is well-defined, 1. e, it exists and unique Define the following two sets A={a:a≥0,ae2x} B={a:a≥0,xae} Then by strong monotonicity, A is nonempty. B is certainly nonempty since 0 E B. Both A and B are closed by the continuity assumption. On the other hand, by the completeness assumption, we know that every a(e 0)must belong to AUB, that is, AUB=R Note that if a E AnB, then ae -x so that we can let u(x)=a. Therefore, we need to prove that AnB is nonempty By monotonicity, it follows that a E A implies that a'E A for all a'>a. Since A is closed subset of R+, it must be in a form of closed interval [ a, +oo ) which implies that B=[0, a]since B is a nonempty closed set such that AUB=R+ We now have to prove that the value a must be unique. Let aie- x and aze -x. Then it is clear that aie- aie(transitivity property of " -") By strong monotonicity, it must be the case that al Let us prove that the above-defined utility function actually represents the preference relation Consider two bundles x and y, and their associated utility levels u(x)and dy), which by definition satisfy u(x)e-x and u(y)e-y. Now
4 2.2 Utility Functions A utility function is a real-valued function u defined on the consumption set X such that preference rankings are preserved by the magnitude of u. That is, a utility function u has the property that given any two elements x, y in X, u(x) u(y) if and only if x y. But not all preference relations can be represented by utility functions. A rather general result is that any continuous preference ordering can be represented by a continuous utility function. This is a very difficult result to prove (Debreu (1959)). (Moreover, while any continuous ordering is always representable, continuity is not necessary. The necessary and sufficient conditions for representation is rather technical; see Ng 1979/83, App. 1B.) We will focus on a somewhat simpler result - the one that can be proved constructively. The main ideas are: • We select arbitrary fixed line that cuts all of the indifference curves (or surfaces). • Once utility is defined along this line, the utility of any other point is found by tracing the appropriate indifference curve to the line and using the utility value there. • The assumption of strong monotonicity guarantees that the indifference curves exit and that any line of the form e, > 0 and e > 0, cuts them all. Existence of Utility Functions • Suppose that a reference relation on X = R m + is complete, reflexive, transitive, continuous, and strongly monotone. Then there exists a continuous utility function u: R m + → R which represents the preference relation. Proof Let e be the vector in R m + consisting of all ones. Then given any vector x, let u(x) = such that x ~ e. We now need to show that u(x) is well-defined, i.e., it exists and unique. Define the following two sets: A = {: 0, e x} B = {: 0, x e} Then by strong monotonicity, A is nonempty. B is certainly nonempty since 0 B. Both A and B are closed by the continuity assumption. On the other hand, by the completeness assumption, we know that every ( 0) must belong to AB, that is, AB = R+. Note that if * AB, then * e ~ x so that we can let u(x) = * . Therefore, we need to prove that AB is nonempty. By monotonicity, it follows that A implies that ' A for all ' . Since A is closed subset of R+, it must be in a form of closed interval [ * , +), which implies that B = [0, * ] since B is a nonempty closed set such that AB = R+. We now have to prove that the value * must be unique. Let 1e ~ x and 2e ~ x. Then it is clear that 1e ~ 1e (transitivity property of "~"). By strong monotonicity, it must be the case that 1 = 2. Let us prove that the above-defined utility function actually represents the preference relation. Consider two bundles x and y, and their associated utility levels u(x) and u(y), which by definition satisfy u(x)e ~ x and u(y)e ~ y. Now, x2 e x1
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)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 at
5 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 ) 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. px 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 px* 0 such that px u(x*). This contradicts to the assumption that x* is an optimal solution. Therefore the budget constraint be must binding at x*.
2.3. 2Indirect Utility Function Therefore, as a result of the above lemma, under the local nonsatiation assumption, utility-maximizing problem can be restated as w(p, y)=max u(x) s.t. px=y The function v(p, y) that gives the answer to the consumer's problem is called indirect utility function. The value of x that solves this problem is called the consumer's demandable bundle: it gives how much of each good the consumer desires at a given level of prices and income Proposition: The indirect utility function has the following properties (a)Ip, y) is homogeneous of degree 0 in(p, y) (b)wp, y) is nonincreasing in p and increasing in y (c)Ip, y)is quasi-convex with respect to p, that is, the set (p: v(p, y)s) is convex for very y>0 and c. Proof: Parts(a)and (b) are straightforward. Let us prove(c). For any real number c, suppose v(pl, y)scand v(p, y)sc. Then for any t: 0 y and p2x>y →px=|px+(l-D)p2x>y→x∈C Now, 1(p, y)=max(u(x):xE C). Let x* be an optimal solution, then x∈C1∪C2→(x*)≤(p1,y)or以(x*)≤v(p2,y) Hence p=tpI +(l-Op2E (p: v(p, y)sc. max((pl, y),v(pl,y))sc →v(p,y)=l(x*)≤ Example 1( Cobb-Douglas Utility Function): a1>0,i=1, Then the corresponding indirect utility function is v(p, y)=max subject to px=y Then using the first-order conditions(n= Lagrange multiplier) ax∏x 2p;=0, Multiplying xleadsto ∏x-2x1=0,i Summing these equations over i and letting a= a.. we get 0→A which leads to
6 2.3.2Indirect Utility Function Therefore, as a result of the above lemma, under the local nonsatiation assumption, a utility-maximizing problem can be restated as: v(p, y) = max u(x) s.t. px = y The function v(p, y) that gives the answer to the consumer's problem is called indirect utility function. The value of x that solves this problem is called the consumer's demandable bundle: it gives how much of each good the consumer desires at a given level of prices and income. Proposition: The indirect utility function has the following properties: (a) v(p, y) is homogeneous of degree 0 in (p, y); (b) v(p, y) is nonincreasing in p and increasing in y. (c) v(p, y) is quasi-convex with respect to p, that is, the set {p: v(p, y) c} is convex for very y > 0 and c. Proof: Parts (a) and (b) are straightforward. Let us prove (c). For any real number c, suppose v(p1, y) c and v(p2, y) c. Then for any t: 0 y and p2x > y px = tp1x + (1-t)p2x > y x C c Now, v(p, y) = max{u(x): x C}. Let x* be an optimal solution, then x* C1 C2 u(x*) v(p1, y) or u(x*) v(p2, y) v(p, y) = u(x*) max(v(p1, y), v(p1, y)) c. Hence p tp1 + (1-t)p2 {p: v(p, y) c}. Example 1 (Cobb-Douglas Utility Function): ( ) , 0, 1,..., . 1 u x x i i m m i i i = = = Then the corresponding indirect utility function is y v y x m i i i = = = p x p subject to ( , ) max 1 Then using the first-order conditions ( = Lagrange multiplier): 0, 1,..., . 1 x x pi i m j i i i j j i − = = − Multiplying xi leads to 0, 1,..., . 1 x pi xi i m m j i j j − = = = Summing these equations over i and letting = = m i i 1 , we get = = − = = m j j m j j j j x y x 1 1 0 p x which leads to
ai y 夏P the indirect utility function is v(p, y) It is interesting to see that this indirect utility function has the Cobb-Douglas form(although with negative coefficients for the prices) Example 2(Leontief Utility Function) u(x) X x2 a, a, a indifference curves P ←px=y( budget line) Let a =(al, a2,...,am)>0. It is clear that(a) this is a case of perfect complements, which is based on an ideal"composite commodity"a The indirect utility function I(p, y) is normally derived from solving the first-order conditions of the consumer's utility maximization problem. But the Leontief utility function is non-differentiable so we have to use direct argument in finding the maximum. From the above figure. it is evident that the optimal solution is such that Then it follows from the budget constraint that pa→py)==y For a given utility function, the expenditure function is defined as e(p, u)=min px st.l(x)≥l X Proposition: The expenditure function e(p, u) has the following properties
7 . i i i p y x = So the indirect utility function is = = = = m i i i m i i i i i p y p y v p y 1 1 ( , ) . It is interesting to see that this indirect utility function has the Cobb-Douglas form (although with negative coefficients for the prices). Example 2 (Leontief Utility Function) = m m a x a x a x u( ) min , , , 2 2 1 x 1 . Let a = (a1, a2, …, am) > 0. It is clear that (a) this is a case of perfect complements, which is based on an ideal "composite commodity" a. The indirect utility function v(p, y) is normally derived from solving the first-order conditions of the consumer's utility maximization problem. But the Leontief utility function is non-differentiable; so we have to use direct argument in finding the maximum. From the above figure, it is evident that the optimal solution is such that x a * * 2 2 1 1 = = = = m m a x a x a x Then it follows from the budget constraint that ( , ) . * * p a a p p a x p a = = = = y v y y y 2.3.3Expenditure Function For a given utility function, the expenditure function is defined as: e(p, u) = min px s.t. u(x) u x X. Proposition: The expenditure function e(p, u) has the following properties: x2 indifference curves a2 a p px = y (budget line) a2 x1
(a) it is homogeneous of degree I in p: (b) it is nondecreasing in p and increasing in u (c)it is concave in p Proof: Straightforward Proposition: u is continuous, X=R"+. Consider Jv(p, y)=max u(x) s.pX≤y yelp, u)=min p-x t.u(x)≥u (i)Assume that the preference relation satisfies local nonsatiation. If x solves(A), then x solves B)with u=v(, y) (ii) Assume that px*>0. Ifx* solves(B), then x*also solves(A)with y =e(p, u) (i) Suppose x*solves(A), but not(B) →彐 x such that pxM(x)≥u =x' is feasible to(A)and u(x)>v(p, y), contradicting to the optimality of x (ii) Suppose x*solves(B). Let y=px*>0. We want to show: if pxs y, then ux)su(x*) Consider such a x and let x'=ax,00 I(p, 1)=max u(x) t.px≤1 X (b)For x>0 u(x)=min v(p, 1) ≤1 p
8 (a) it is homogeneous of degree 1 in p; (b) it is nondecreasing in p and increasing in u; (c) it is concave in p. Proof: Straightforward. Proposition: u is continuous, X = Rm +. Consider = st y v y u p x p x x . . ( , ) max ( ) (A) = st u u e u . . ( ) ( , ) min x p p x x (B) Then, (i) Assume that the preference relation satisfies local nonsatiation. If x * solves (A), then x * solves (B) with u = v(p, y). (ii) Assume that px* > 0. If x* solves (B), then x* also solves (A) with y = e(p, u). Proof: (i) Suppose x* solves (A), but not (B): x such that p x u(x) u. x 1 is feasible to (A) and u(x 1 ) > v(p, y), contradicting to the optimality of x * . (ii) Suppose x* solves (B). Let y = p x* > 0. We want to show: if p x y, then u(x) u(x*). Consider such a x and let x' = x, 0 0, v(p, 1) = max u(x) s.t. px 1 (C) x X. (b) For x > 0, u(x) = min v(p, 1) s.t. px 1 (D) p 0
Proof. As part(a) is definitional, it needs no proof. Let us prove part (b). For ease of presentation, let U(x)be the optimal value of the following optimization problem St First, from part(a), it is clear that for any x>0 and p >0 such that px= 1, it must be true that i(x) 0, there exists some p>0 such that px=l and u(x)=v(p, 1) Define C=xEX: u(x)>u(r)). Since the utility function u is continuous and quasiconcave, it follows that C is closed and convex and x is a boundary point of C. Supporting Hyperplane Theorem(DL, p 453& p.455)Let C be a convex set and let x be a boundary point of C. Then there is a hyperplane H=(x: ax=a-r)(a* 0)containing xand containing C in one of its closed half spaces, i.e., either z∈H≡{x:ax≥a·x"}, for all z∈C( positive closed half- space z∈H≡{x:ax≤a·x}, for all z∈C( negative closed half- space) Furthermore if C is monotonic. i.e. C+R+=C. then a is a nonnegative vector and C must be in the positive closed half-space, i.e., a z> ax for all ZE C. From the definition of C and from the strong monotonicity of u, we know that C is indeed monotonic Now applying the above theorem to your problem, we know that there exists some nonzero p≥0 such that Since x>0. we can make pro p-x≥px?, for all x∈C I by properly scaling p. Th herefore, r solves the expenditure problem(B)with u=ux). By the previous Proposition, x' must solve the indirect utility problem (A)with v(p, 1)=u(x ). This proves the result. D
9 Proof. As part (a) is definitional, it needs no proof. Let us prove part (b). For ease of presentation, let U(x) be the optimal value of the following optimization problem: min v(p, 1) s.t. px 1 p 0. First, from part (a), it is clear that for any x > 0 and p > 0 such that px = 1, it must be true that u(x) v(p, 1), which implies that u(x) U(x). We now need to prove that u(x) U(x). To prove this, it suffices to show that for any given x 0 > 0, there exists some p > 0 such that px 0 = 1 and u(x 0 ) = v(p, 1). Define C = {xX: u(x) u(x 0 )}. Since the utility function u is continuous and quasiconcave, it follows that C is closed and convex and x 0 is a boundary point of C. Supporting Hyperplane Theorem (DL, p.453 & p.455) Let C be a convex set and let x 0 be a boundary point of C. Then there is a hyperplane H ={x: ax = ax 0} (a 0) containing x 0 and containing C in one of its closed half spaces, i.e., either z H+ {x: ax a x 0}, for all z C (positive closed half-space) or, z H- {x: ax a x 0}, for all z C (negative closed half-space). Furthermore, if C is monotonic, i.e., C + R m + = C, then a is a nonnegative vector and C must be in the positive closed half-space, i.e., az ax 0 for all z C. From the definition of C and from the strong monotonicity of u, we know that C is indeed monotonic. Now applying the above theorem to your problem, we know that there exists some nonzero p 0 such that px px 0 , for all x C. Since x 0 > 0, we can make px 0 = 1 by properly scaling p. Therefore, x 0 solves the expenditure problem (B) with u = u(x 0 ). By the previous Proposition, x 0 must solve the indirect utility problem (A) with v(p, 1) = u(x 0 ). This proves the result.
