Lecture 8: Basic convex analysis

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1 Lecture 8: Basic convex analysis 1 Convex sets Both convex sets and functions have general importance in economic theory, not only in optimization. Given two points x; y 2 R n and 2 [0; 1]; the weighted average x + (1 )y is called the convex combination of x and y. (As a simpli cation, in this lecture we use normal fonts instead of bold to denote vectors.) De nition 1.1. A subset S R n is convex if for any 0 1 and x; x 0 2 S, x + (1 )x 0 2 S. It s straightforward to see that if S is convex, then any nite convex combination 1 x k x k of points in S such that P k l=1 l = 1 also belongs to S, and the intersection of any number of convex sets is convex. De nition 1.2. The convex hull of a set S R n ; denoted by Conv(S); is the smallest convex set containing S. Geometrically, Conv(S) consists of all points that can be written as nite convex combinations of points in S. That is, it is the convex span of S. De nition 1.3. For a set S; x 2 S is an extreme point of S if x cannot be written as the convex combination of other points in S. Theorem 1.1 (Krein-Milman). A compact and convex set S is the convex hull of its extreme points. The example below is useful in giving you a more general sense of the concepts de ned above. Example 1. The Minkowski sum of two sets A; B S is de ned as A + B = fx + y : x 2 A; y 2 Bg: Be careful that although A + f0g = A; unless A is a singleton, A A 6= 1

2 f0g. The Hausdor distance (metric) between two sets A and B is de ned as d H (A; B) = maxfd(a; B); d(b; A)g, where d(a; B) = supfd(x; B); x 2 Ag and d(x; B) = inffd(x; y); y 2 Bg. Let S = f(x 1 ; x 2 ; x 3 ) 2 R 3 : x 1 + x 2 + x 3 = 1g: Consider the set of all closed convex subsets of S; and with a little abuse of noation, denoted it as 2 S. From the above, for any A; B 2 2 S ; their convex combination A + (1 )B S is well-de ned and is also convex. The set 2 S is a large set and its structure is not so transparent. Nevertheless, since it is convex and compact (under the Hausdor metric), due to the Krein-Milman theorem, we can picture 2 S as the convex hull of its extreme points. Therefore, the work reduces to nd out which subsets of S are extreme points of 2 S ; that is, which subsets of S cannot be written as convex combinations of other subsets of S. 2 Hyperplane separation theorem De nition 2.1. Let p 2 R n and p 6= 0; and let a 2 R: The set H(p; a) = fx 2 R n : p x = ag is called a hyperplane in R n. The sets fx 2 R n : p x ag and fx 2 R n : p x ag are called the half-space above and below the hyperplane H(p; a); respectively. Recall that p x = jjpjj jjxjj cos, where is the angle between the vectors p and x. The vector p is the normal of the hyperplane H(p; a) and is thus orthogonal (perpendicular) to it, and a half-space is above H(p; a) if it is by the side of the hyperplane as the direction of p points to. Each hyperplane in R n is a subset of R n with dimension n 1. A hyperplane in R 2 is a straight line and a hyperplane in R 3 is just a two-dimensional plane. Example 2. If p = (p 1 ; p 2 ) 2 R 2 and I 2 R; then the hyperplane H(p; I) = f(x 1 ; x 2 ) 2 R 2 : p 1 x 1 + p 2 x 2 = Ig can be understood as the budget line, given income I and prices p 1 ; p 2. De nition 2.2. Let X; Y be subsets of R n. We say that the hyperplance H(p; a) separates X from Y if p x < a; 8x 2 X and p y > a; 8y 2 Y: 2

3 Theorem 2.1 (Hyperplane separation). Let C be a closed and convex subset of R n, and x 2 R n nc. Then there exists p 2 R n ; p 6= 0; and a 2 R such that the hyperplance H(p; a) separates C and x. Proof. Let y be the point in C that the distance between x and y; jjx yjj; obtains its minimum. (Such y exists due to the Weierstrass theorem). If we let p = x y and a 0 = p y, then p(x y) > 0; hence px > a 0. Also, for any z 2 C, denote the angle between the vectors z y and p by, then cannot be an acute angle, hence p (z y) = jjpjjjjz yjj cos 0: That is, p z p y = a 0 ; 8z 2 C. Now we can see that there must be " > 0; " small enough such that p z < a 0 + "; 8z 2 C and p x > a 0 + ". Let a = a 0 + "; then H(p; a) separates C and x. There are many other versions of hyperplane separation theorems on how hyperplanes separate points or (open or closed) convex subsets from (open or closed) convex subsets; these theorems all share the same geometric intuition as that of the theorem above. When one of the two sets is not closed, the separation theorem may ensure only weak separation. Example 3. Let u 1 (x) and u 2 (x) be utility functions of two agents de ned on a convex set X. Suppose fv 2 R 2 : (v 1 ; v 2 ) (u 1 (x); u 2 (x)) for some x 2 Xg is convex (e.g., this holds true when u 1 ; u 2 are concave functions, which we will formally de ne very soon). The following result is on the Pareto frontier of allocations between two agents; it can be generalized to more agents. Claim 2.2. Every Pareto e cient allocation x 2 X maximizes u 1 (x) + (1 some 2 [0; 1]. )u 2 (x) for We now sketch the proof of this claim using a version of the Hyperplane separation theorem. Suppose x is Pareto e cient. Then the sets A = fv : v u(x) for some x 2 Xg and B = fv : v > u(x )g are disjoint. Since A is convex by assumption and B is also convex, there exist a vector p 2 R 2 and a scalar a such that H(p; a) separates the two sets: that is, p v a for all v 2 A and p v a for all v 2 B (since B is not closed, we don t get strict separation). Since the half-space above H(p; a) contains B, it must be that p 0. Geometrically, since u(x ) 2 A and is on the boundary of B; we have u(x ) 2 H(p; a): That is, p u(x ) = 3

4 a p v; 8v 2 A. Hence p u(x ) p u(x); 8x 2 X: Let = p 1 =(p 1 + p 2 ); then we have a proof. The hyperplane separation theorems are also applied in proving for example, the second theorem of welfare economics, and in game theory, in showing that a mixed strategy is never a best-response if and only if it is strictly dominated. In general, the theorem is potentially useful when we need to show the existence of a vector of weights. De nition 2.3. The support function h X : R n! R of a closed and convex subset X R n is de ned as h X (p) = supfp x : x 2 Xg; where h ; (p) 1 and h X (p) +1 if the sup is in nite. Example 4. Consider X R 2. If X = fxg; then h X (p) = p x, and if X = f(x 1 ; x 2 ) : x x 2 2 = 1g; then h X (p) = jjpjj: The support function h X () is uniquely determined by X, and by de nition, X fx 2 R n : p x h X (p)g: That is, X is included in the half-space below the hyperplane H(p; h X (p)). Furthermore, due to the notation sup in the de nition, h X (p) is chosen to be the smallest number a such that the half-space below H(p; a) includes X. Claim 2.3. h X (p) = inffa 2 R : p x a; 8x 2 Xg: Geometrically, for any closed and convex set X and a vector p; h X (p) is chosen in a way that the hyperplane H(p; h X (p)) is tangent to X and X is included in the half-space below the hyperplane. That is, in direction p; H(p; h X (p)) supports X from the above. Theorem 2.4 (Support-function theorem). For a closed and convex subset X R n ; X = \ p2r nfx 2 R n : p x h X (p)g: Since this result is geometrically straightforward, we don t give its proof. Intuitively, it says that if we know that h X : R n! R is the support function of some closed and convex subset X of R n ; then we can uniquely recover X by taking intersection of the half-spaces below the respective hyperplane H(p; h X (p)); p 2 R n. And in practice, we don t need to consider all p 2 R n when taking the intersection: it is su cient to consider all directions p in the unit ball fx 2 R n : jjxjj = 1g: 4

5 3 Concave functions De nition 3.1. Let f : S! R; where S R n is convex. The function f is (a) concave if f(x + (1 )x 0 ) f(x) + (1 )f(x 0 ), for any x; x 0 2 S and 2 (0; 1), and is (b) convex if f(x + (1 )x 0 ) f(x) + (1 )f(x 0 ), for any x; x 0 2 S and 2 (0; 1) That is, a function is concave if its value at the average of two points is always great than or equal to the average of its values at the two points. When the inequality is strict for all x; x 0 ;, then f is said to be strictly concave (or respectively, strictly convex). The functions which are both concave and convex are of the form f(x) = a x + b and are called a ne functions. Example 5. The function f : R +! R de ned as f(x) = x a is strictly concave if 0 < a < 1 and is strictly convex if a > 1. The function f : R 2 +! R de ned as f(x 1 ; x 2 ) = x a 1x b 2 is called the Cobb-Douglas function; it is concave if a + b 1 and is neither concave nor convex if a + b > 1. Example 6. The pro t function (p) = max x2x p x (similarly, the support function h X (p) in the previous section) is convex. For any concave function f; f is convex and vice versa. Hence it is without loss of generality to concentrate on concave functions. The result below can be viewed as an alternative way of de ning concave functions. Theorem 3.1. Let A f(x; y) : x 2 S; f(x) yg be the set of points "on and below" the graph of f. Then f is concave if and only if A is a convex set. Proof. ()) Suppose f is concave. Let (x 1 ; y 1 ); (x 2 ; y 2 ) be two points in A. Then by de nition, f(x 1 ) y 1 and f(x 2 ) y 2. Since f is concave, for any 2 [0; 1]; f(x 1 + (1 )x 2 ) f(x 1 ) + (1 )f(x 2 ) y 1 + (1 )y 2 : 5

6 This means that (x 1 + (1 )x 2 ; y 1 + (1 )y 2 ); which is (x 1 ; y 1 ) + (1 )(x 2 ; y 2 ); also belongs to A. Hence A is a convex set. (() Suppose A is a convex set. Let x 1 ; x 2 be two points in S. Since f(x 1 ) f(x 1 ) and f(x 2 ) f(x 2 ); we have (x 1 ; f(x 1 )); (x 2 ; f(x 2 )) 2 A: As A is convex, for any 2 [0; 1]; (x 1 ; f(x 1 )) + (1 )(x 2 ; f(x 2 )) 2 A: That is, (x 1 + (1 )x 2 ; f(x 1 ) + (1 )f(x 2 )) 2 A. By the de nition of A; f(x 1 + (1 )x 2 ) f(x 1 ) + (1 )f(x 2 ): Hence f is concave. Theorem 3.2. Let f : S! R be a concave function. Then, if S is open, f is continuous on S. Proof. Pick x 2 S. Since S is open, there exists r > 0 such that N r (x) S and its boundary A = fy : jjx yjj = rg S. Pick any fx n g S such that x n! x. Then there exists N such that for all n N; jjx n xjj < r: Then for all n N; there exist n 2 (0; 1) and z n 2 A such that x n = n x + (1 n )z n : Since f is concave, f(x n ) n f(x) + (1 n )f(z n ): Taking limits on both sides, lim inf n f(x n) f(x): Similarly, for all n N; there exist n 2 (0; 1) and w n 2 A such that x = n x n + (1 n )w n : Since f is concave, f(x) n f(x n ) + (1 n )f(w n ): 6

7 Taking limits on both sides, f(x) lim sup f(x n ): n That is, f(x n ) converges and lim n f(x n ) = f(x). Since fx n g is picked arbitrarily, we know that f(x) is continuous on S. Nevertheless, a function f is concave does not imply that it is always di erentiable on the interior of its domain. The positive result on this is that if f : (a; b)! R is concave, then it is di erentiable at all but countably many points in (a; b). That is, it is di erentiable almost everywhere (cf. Rockafeller, Convex Analysis, 1970). Theorem 3.3. If f : (a; b)! R is twice continuously di erentiable, then f is concave if and only if f 00 (x) 0 for all x 2 (a; b). Proof. (() Suppose f 0 (x) 0; 8x 2 (a; b). Then f 0 (x) is non-increasing. For x; y such that a < x < y < b, pick 2 (0; 1) and let z = x + (1 )y. Then z x = (1 )(y x); y z = (y x): Note that since f 0 (x) is non-increasing, f(z) f(x) = f(y) f(z) = Z z x Z y z f 0 (t)dt f 0 (t)dt Z z x Z y z f 0 (z)dt = f 0 (z)(z f 0 (z)dt = f 0 (z)(y x); z): Hence f(z) f(x) + f 0 (z)(1 )(z x); f(z) f(y) f 0 (z)(y x): 7

8 We have f(z) = f(z) + (1 )f(z) [f(x) + f 0 (z)(1 )(z x)] + (1 )[f(y) f 0 (z)(y x)] = f(x) + (1 )f(y): ()) Suppose f is concave and instead there exists x 0 2 (a; b) such that f 00 (x 0 ) > 0. Since f 00 (x) is continuous, there exists " > 0 such that f 00 (x) > 0 for all x 2 (x 0 "; x 0 +") (a; b). Reverse the proof in the (() part, we can nd x; y 2 (x 0 "; x 0 + ") and 2 (0; 1) such that f(ax + (1 )y) < f(x) + (1 )f(y); of f. which means f is strictly convex on (x 0 "; x 0 + "); this contradicts with the concavity De nition 3.2. An n n matrix A = (a ij ) i;jn is symmetric if a ij = a ji ; 8i; j n. A symmetric A is negative semi-de nite if for all vectors z 2 R n ; z T Az 0. It is negative de nite if for all z 6= 0; z T Az < 0. Theorem 3.4. If C is an open convex subset of R n and f : C! R is twice continuously di erentiable, then f is concave if and only if D 2 f(x) is negative semi-de nite for all x 2 C. Proof. Part I. Pick arbitrary y; z 2 R n such that f : y + z 2 Cg is nonempty. Let g y;z () = f(y + z). We will show that f is concave i each g y;z () is concave in. First, Suppose f is concave. Pick ; 0 such that x := y + z 2 C; x 0 := y + 0 z 2 C. Then f(x) = g y;z () and f(x 0 ) = g y;z ( 0 ). Furthermore, f(x + (1 )x 0 ) = f(y + ( + (1 ) 0 )z) = g y;z ( + (1 ) 0 ): We see straightforwardly that the concavity of f implies that of each g y;z (). Second, suppose g y;z () is concave for each y; z. Pick x; y 2 C and let z = x y: Then y + z = y + (x y) = x + (1 )y: 8

9 Therefore, for each 2 (0; 1); f(x + (1 )y) = g y;x y () = g y;x y ( 1 + (1 ) 0) g y;x y (1) + (1 )g y;x y (0) = f(y + (x y)) + (1 )f(y + 0 z) = f(x) + (1 )f(y): Part II. Now we know that f is concave i each g y;z () is concave, that is, i [g y;z ()] 00 0 for each pair of y; z 2 R n. Let x 0 = y + z, then g 00 y;z() f(y + 2 = z T D 2 f(x 0 )z: Hence g 00 y;z() 0 for each pair y; z is equivalent to D 2 f(x 0 ) being negative semi-de nite for all x 0 2 C. 4 Quasi-concave functions De nition 4.1. Let f : S! R; where S R n is convex. The function f is (a) quasiconcave if f(x + (1 )x 0 ) minff(x); f(x 0 )g; for any x; x 0 2 S and 2 (0; 1); and is (b) quasiconvex if f(x + (1 )x 0 ) maxff(x); f(x 0 )g; for any x; x 0 2 S and 2 (0; 1): The function f is strictly quasiconcave, or quasiconvex, respectively, if the inequality holds strict for all x; x 0 ;. If f is quasiconcave, then its value at the average of two points is greater than or equal to the minimum of its values at the two points. Quasiconcave functions are strictly more general than concave functions. If a function is monotonic, then it is both quasiconcave and quasiconvex. Example 7. f(x) = e x is quasiconcave, although it is strictly convex. The Cobb-Douglas function f(x 1 ; x 2 ) = x a 1x b 2 is quasiconcave if a; b > 0. 9

10 The main reason that we care about quasiconcave functions in economics is because of the following result, which intuitively states that a utility function is quasiconcave if and only if its indi erence curves are boundaries of some convex set, and this captures diminishing marginal rate of substitution. Theorem 4.1. Let U(c) = fx 2 S : f(x) cg be the upper contour set of f for level c. Then f is quasiconcave i U(c) is a convex set for all c 2 R. Proof. ()) Suppose f is quasiconcave. If x; x 0 2 U(c); then f(x) c and f(x 0 ) c. Thus for all 2 (0; 1); f(ax + (1 )x 0 ) minff(x); f(x 0 )g c; which implies x + (1 )x 0 2 U(c): (() Suppose U(c) is a convex set for each c 2 R. For any x; x 0 2 S; obviously, f(x); f(x 0 ) minff(x); f(x 0 )g. Therefore, x; x 0 2 U(minff(x); f(x 0 )g): Since U(minff(x); f(x 0 )g) is a convex set, 8 2 [0; 1]; x+(1 )x 0 2 U(minff(x); f(x 0 )g). That is, 8 2 [0; 1]; f(x+ (1 )x 0 ) minff(x); f(x 0 )g. Example 8. Let u : R 2 +! R denote a consumer s utility function. Then the boundary of the upper contour set, u(x; y) = c; is exactly the indi erence curve of the consumer at utility level c. Thus u(x; y) is quasiconcave guarantees that the indi erence curve is of convex shape. That is, the function y(x) that solves u(x; y) = c is convex in x. This implies diminishing marginal rate of substitution, that is, it guarantees that MRS xy = dy dx = u x(x; y) u y (x; y) is decreasing in x. Intuition: the more of good x that you are consuming right now, the less of good y is needed to compensate you for your loss of 1 unit of good x. 5 A digression on ordinal and cardinal utility De nition 5.1. For any nondecreasing function g on R and u : R n! R; the composition g u that maps x 2 R n to g(u(x)) is called a monotonic (nondecreasing) transformation of u. 10

11 Example 9. The utility functions 3x 1 x 2 + 2; (x 1 x 2 ) 2 ; (x 1 x 2 ) 3 + x 1 x 2 ; e x 1x 2 and ln x 1 x 2 are all monotonic transformations of u(x 1 ; x 1 ) = x 1 x 2 : Intuitively, a preference relation is ordinal if it de nes the relative order of the objects (e.g., prefer bundle (x 1 ; x 2 ) over bundle (y 1 ; y 2 )), and it is cardinal if it also de nes the intensity of comparison (e.g., bundle (x 1 ; x 2 ) generates 5 units more utility than bundle (y 1 ; y 2 )). Formally, De nition 5.2. A property of functions is called ordinal if whenever a function u has this property, for any monotonic transformation g, g u also has this property. Otherwise, it is called cardinal. Claim 5.1. Concavity is a cardinal property. This can easily be seen from the following example. Example 10. u(x) = p x is concave on [0; 1); but the monotonic transformation of it, g(u(x)) = ( p x) 4 = x 2, is strictly convex. Claim 5.2. Quasiconcavity is an ordinal property. Proof. This is immediate. Since g is nondecreasing and f is quasiconcave, 8 2 (0; 1) and x; y 2 S; g(f(x + (1 )y)) g(minff(x; f(y))g) = minfg(f(x)); g(f(y)): 11