Winter Lecture 10. Convexity and Concavity

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1 Andrew McLennan February 9, 1999 Economics 5113 Introduction to Mathematical Economics Winter 1999 Lecture 10 Convexity and Concavity I. Introduction A. We now consider convexity, concavity, and the general consequences of these concepts in economic modelling. B. We then describe some specic economic applications. II. Convex Sets A. A set S IR n is convex if it contains the line segment between any pair of points in S, which means that for any x 0 ;x 1 2 S, f(1, )x 0 + x 1 :01gS: B. Examples of Convex Sets. 1. Both open and closed half spaces are convex. Consider the halfspace H = f x 2 IR n : p x 0 g dened by some nonzero p 2 IR n. If x 0 ;x 1 2 H and 0 1, then p ((1, )x 0 + x 1 )=(1,)px 0 +p x 1 (1, )0 + 0 =0; so that (1, )x 0 + x 1 2 H. 2. Both the open and the closed unit balls are convex. If kx 0 k; kx 1 k < 1 1

2 and 0 1, then (applying the Cauchy-Schwartz inequality!) k(1, )x 0 + x 1 k 2 = ((1, )x 0 + x 1 ) ((1, )x 0 + x 1 ) =(1,) 2 kx 0 k 2 +2(1, )x 0 x kx 1 k 2 (1, ) 2 kx 0 k 2 +2(1, )kx 0 kkx 1 k+ 2 kx 1 k 2 = < so that k(1, )x 0 + x 1 k < 1. B. Properties of Convex Sets. (1, )kx 0 k + kx 1 k 2 (1, )+ 2=1; 1. If S and T are convex subsets of IR n, then S + T = f x+y : x 2 S; y 2 T g is convex. To show this we observe that for x 0 + y 0 ;x 1 +y 1 2 S+T and 0 1, (1, )(x 0 + y 0 )+(x 1 +y 1 ) = ((1, )x 0 + x 1 ) + ((1, )y 0 + y 1 ): T 2. If fs g 2B is a collection of convex sets, then the intersection 2B S is convex. (It is easy to draw a picture how the union of two convex sets need not be convex.) a. For any set S IR n the convex hull of S is the intersection of all convex sets that contain S. 3. We now see that any intersection of halfspaces is convex, so we are able to construct a lot of convex sets. Visually, it seems reasonable to guess that all convex sets are intersections of half spaces, and a little thought shows that this is the same as the assertion of the following, which isa twentieth century result, due to Minkowski. The generalization to innite dimensional spaces, which is known as the Hahn-Banach theorem, is one of the most important theorems of functional analysis. Theorem: If A IR n is closed convex and x =2 A, there is p 2 IR n, p 6= 0, such that p x<pyfor all y 2 A. 2

3 Proof: If A is empty, any nonzero p will do, so we may suppose that there is a point y 0 2 A. Then the set A 0 := f y 2 A : ky, xk ky 0,xkg is closed and bounded, hence compact, and since distance d(x; y) = ky,xkis a continuous function, there must be a point y 2 A 0 with ky,xk ky,xkfor all y 2 A 0 and thus for all y 2 A. Let p = y,x. Picking y 2 A arbitrarily, it suces to show that p y p y, since then p y p x + p (y, x) =px+ky,xk 2 >px: We this goal in mind, we observe that (1, )y + y 2 A for any 01, so that 0 < k(1, )y + y, xk 2,ky,xk 2 =k(1, )(y, y )+(y,x)k 2,ky,xk 2 =k(1, )(y, y )k 2 + 2(1, )(y, y )(y, x) =(1,) 2 k(y,y )k 2 + 2(1, )p (y, y ): Since this must hold even when 1, is very small, it must be the case that p(y, y ) 0, as desired. a. In economic applications, the normal p of the separating hyperplane can often be interpreted as a vector of prices. II. Convex and Concave Functions A. Fix a function f : U! IR where U IR n is both open and convex. 1. The subgraph of f and the epigraph of f are, respectively subf = f (x; y) :x2u; y f(x) g and epif = f (x; y) :x2u; y f(x) g: 2. We say that f is convex if the epigraph of f is convex. 3

4 a. Equivalently, for any x 0 ;x 1 and any between 0 and 1, f((1, )x 0 + x 1 ) (1, )f(x 0 )+f(x 1 ): b. We say that f is strictly convex if, for any distinct x 0 ;x 1 2 U and any with 0 <<1, f((1, )x 0 + x 1 ) < (1, )f(x 0 )+f(x 1 ): 3. We say that f is concave if the subgraph of f is convex. a. Equivalently, for any x 0 ;x 1 2 U and any between 0 and 1, f((1, )x 0 + x 1 ) (1, )f(x 0 )+f(x 1 ): b. We say that f is strictly concave if, for any distinct x 0 ;x 1 2 U and any with 0 <<1, f((1, )x 0 + x 1 ) < (1, )f(x 0 )+f(x 1 ): 4. Example: a linear function f(x) =mx + b is both concave and convex, but neither strictly concave nor strictly convex. B. Characterizations of Concavity 1. In the following discussion we only consider concavity, since the corresponding analysis for convexity is exactly the same, except with the obvious signs and inequalities reversed. Proposition 1: If f is concave and dierentiable at x 0, then, for all x 2 U, f(x 0 )+Df(x 0 )(x, x 0 ) f(x): Proof: Suppose, for some x 2 U, that f(x) >f(x 0 )+Df(x 0 )(x, x 0 ). For any 0<<1 4

5 we have f(x 0 +(x,x 0 )), f(x 0 ), Df(x 0 )((x, x 0 )) k(x, x 0 )k = f((1, )x 0 + x), f(x 0 ), Df(x 0 )(x, x 0 ) kx, x 0 k (1, )f(x 0)+f(x), f(x 0 ), Df(x 0 )(x, x 0 ) kx, x 0 k = f(x), f(x 0), Df(x 0 )(x, x 0 ) : kx, x 0 k Since the last expression is positive and independent of, this contradicts the denition of Df(x 0 ). c. When f is strictly concave, the inequality above is strict. Proposition 2: If f is strictly concave and dierentiable at x 0, then f(x 0 )+f 0 (x 0 )(x, x 0 ) >f(x) for all x 2 U, x 6= x 0. Proof: Suppose there is x 2 IR, x 6= x 0 with f(x 0 )+f 0 (x 0 )(x, x 0 ) f(x). Then f( 1 2 x x 0) > 1 2 f(x) +1 2 f(x 0)f(x 0 )+ 1 2 Df(x 0)(x, x 0 ) so that f(x 0 )+Df(x 0 )(x 0, x 0 ) <f(x 0 ) where x 0 = x (x, x 0). Since f is concave, this is a contradiction of Proposition 1. D. The consequences of these results for optimization are extremely important. Theorem: If f is concave and dierentiable at x, then x is a global maximizer for f if and only if Df(x ) = 0. If f is strictly concave and Df(x ) = 0, then x is the unique global maximizer of f. Proof: We know that Df(x )=0ifx is a global maximizer for f, and if Df(x )=0, then Proposition 1 implies that x is a global maximizer for f, and Proposition 2 implies that it is the unique maximizer. 5

6 III. Concavity and the Second Derivative A. For C 2 functions, concavity is equivalent to negative semideniteness of the second derivative: Theorem: If f is C 2 and concave, then, for every x 2 U, D 2 f(x) is negative semidenite. Proof: If, for some x 2 U, D 2 f(x) is not negative semidenite, there is some v 2 IR n such that v 0 D 2 f(x)v >0. Let = v 0 D 2 f(x)v=2. For suciently small >0wehave f(x+v) f(x) +Df(x)(v) +(v) 0 D 2 f(x)(v), kvk 2 = f(x) +Df(x)(v) + 2 (v 0 D 2 f(x)v,kvk 2 ) >f(x)+df(x)(v); which contradicts Proposition 1. Theorem: If f is C 2 and for every x 2 U, D 2 f(x) is negative semidenite, then f is concave. Remarks on the Proof: Fix x 0 ;x 1 2 U. Dene g :[0;1]! IR by g(t) =f((1, t)x 0 + tx 1 ) : It suces to show that g is concave. Briey, the key idea is to show that g 00 (t) 0 for all t 2 (0; 1), at which point the concavity ofgis a matter of elementary calculus. The formal development of these ideas involves certain details (developing the chain rule for multivariate functions, in a way that can be applied to the analysis of g 00 ) that we are not well prepared for, so we will not go into detail. IV. Economic Applications A. For a production function y = f(l), concavity off is expressed economically by saying that f exhibits diminishing returns. When f is convex we say that it exhibits increasing returns. 6

7 1. Exercise: Prove that the prot function (L) = pf(l),wl is concave when f is. (Implicitly we are assuming that the price p of output and the wage w of labor are beyond the rm's control.) B. Consider a consumer whose preferences are characterized by a continuous function u : IR n +! IR. The associated expenditure function describes the minimum amount of money required to purchase a bundle yielding some utility level. Specically, for a price vector p =(p 1 ;:::;p n )2 IR n ++ and a utility level u 2 IR let e(p; u) = minf px = p 1 x 1 + :::+p n x n :u(x) ug: Theorem: For given u the expenditure function is concave as a function of prices. Proof: Consider price vectors p 0 ;p 1 2 IR n ++, and let x 0 and x 1 be consumption bundles that minimize expnditure at these price vectors, respectively, while attaining utility level u. For 0 <<1 let x be a consumption bundle that minimizes expnditure at the price vector p =(1,)p 0 +p 1 = ((1,)p p 1 1 ;:::;(1,)p0 n + p 1 n) while attaining utility level u. Then e(p ;u)=p x =, (1, )p 0 + p 1 x =(1,)p 0 x +p 1 x (1, )p 0 x 0 + p 1 x 1 =(1,)e(p 0 ;u)+e(p 0 ;u): The inequality here is derived from the fact that, at price vector p 0, x cannot be less expensive than x 0, while x cannot be less expensive than x 1 at price vector p 1. IV. Valuation of Risky Prospects A. For each of the following two problems, decide which alternative you would choose: 7

8 1. Problem 1: Choose between the following two probability distributions over monetary prizes: a. Lottery 1A: $2,000,000 with certainty. b. Lottery 1B: $2,000,000 with probability 0.89, $10,000,000 with probability 0.10, and nothing with probability Problem 2: Choose between the following two probability distributions over monetary prizes: a. Lottery 2A: $2,000,000 with probability 0.11 and nothing with probability b. Lottery 2B: $10,000,000 with probability 0.10 and nothing with probability B. The expected utility theory of decision making under uncertainty asserts that, inachoice between \lotteries" over certain outcomes, lotteries are valued by taking the average (according to the probabilities) of numerical valuations of the certain outcomes. 1. The most important applications have agents valuing dierent levels of wealth. Let U: IR +! IR be a continuous function, where U(w) might be thought of as the \lifetime psychic pleasure" of having wealth level w. 2. Thus the expected utility of a lottery that pays w 1 with probability p 1, w 2 with probability p 2, :::, w K with probability p K,is KX k=1 p k U(w k ): (Here we require that p 1 + +p K = 1.) 3. The theory has the property that your choice between two lotteries does not depend on whether they are choices in and of themselves, or are presented as possible continuations in a larger \tree" of possible events. 4. The statistician Leonard \Jimmy" Savage is, perhaps, the person most 8

9 responsible for showing that the expected utility theory is essentially the unique theory with this property. 5. If you choose 1A and 2B in the choice problems above, you violated this principle. a. Don't feel bad: L. J. Savage made the same choices. C. As above, consider an agent whose preferences over lotteries are governed by expected utility, with underlying utility function U: IR +! IR. 1. We will always assume that U is strictly increasing. 2. The agent is said to be risk averse if U is concave. 3. For any lottery `w 1 with probability p 1 ;:::;w K with probability p K,' there is a expectation: p 1 w 1 + +p K w K : Theorem: If U is concave, then any lottery is (weakly) less desirable than receiving the lottery's expectation with certainty: p 1 U(w 1 )++p K U(w K )U(p 1 w 1 ++p K w K ): Proof: In the case K = 2 the claim amounts to the denition of concavity. For larger K we argue by induction. =p 1 U(w 1 )+(1,p 1 ) p2 (1, p 1 ) U(w 2)++ p K U(w K ) 1,p 1 p 1 U(w 1 )++p K U(w K ) p 1 U(w 1 )+(1,p 1 )U p2 w 2 ++ p K w K 1,p 1 1,p 1 U p 1 w 1 +(1,p 1 ) p2 1,p 1 w 2 ++ p K = U(p 1 w 1 + p 2 w 2 + +p K w K ): 1,p K w K 9