7) Important properties of functions: homogeneity, homotheticity, convexity and quasi-convexity

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1 30C00300 Mathematical Methods for Economists (6 cr) 7) Important properties of functions: homogeneity, homotheticity, convexity and quasi-convexity Abolfazl Keshvari Ph.D. Aalto University School of Business Slides originally by: Timo Kuosmanen Updated by: Abolfazl Keshvari 1

2 Outline Concave and convex functions Quasiconcave and quasiconvex functions Definition of homogeneity Homogenizing a function Definition of homotheticity Examples of concave and quasi-concave functions in microeconomics Examples of homogeneous functions in microeconomics 2

3 Concavity and convexity Definition: Let U be a convex subset of R n. A real valued function is concave if f : U R f ( x (1 ) y) f ( x) (1 ) f ( y) [0,1], x, y R n A real valued function g: U R is convex if g( x (1 ) y) g( x) (1 ) g( y) [0,1], x, y R n 3

4 Concavity and convexity Note 1: If function f is concave, then its additive inverse f is convex. Thus, all properties of concave functions carry over to convex functions, and vice versa. Note 2: Convexity of a function and convexity of a set should not be confused. A notion of concave set does not exist. Definition: set S is convex if x, y S x (1 ) y S [0,1] 4

5 Concave function vs convex set Recall that production function f and production possibility set T are equivalent representations of technology: y f ( x) ( x, y) T Proposition: Production function f is a concave function if and only if production possibility set T is a convex set. 5

6 Recall the cost function km C : R R, Concavity of cost function C( w, y) min w x ( x, y) T x Theorem: The cost function is concave in input prices w: C C C ( w (1 ) w, y) ( w, y) (1 ) ( w, y) 1 2 [0,1],, k m w w R, y R Note: concavity of C does not depend on T. 6

7 Concavity of profit function Recall the profit function km : R R, xy, ( w, p) max p y w x ( x, y) T Theorem: The profit function is concave in prices w, p ( w (1 ) w, p (1 ) p ) ( w, p ) (1 ) ( w, p ) [0,1],, k m w w R, p, p R Note: concavity of π does not depend on T. 7

8 Concavity and convexity univariate case Recall that the following inequality holds for all convex functions f : R R f ( x h) f ( x) f ( x) h x, h R where f is the first derivative (or the sub-derivative). Analogously, the following inequality holds for all concave functions f ( x h) f ( x) f ( x) h x, h R 8

9 Derivative test for univariate functions Assume function f is twice continuously differentiable. Then f is convex if and only if f ( x) 0 x R Analogously, f is concave if and only if f ( x) 0 x R 9

10 Concavity and convexity multivariate case The following inequality holds for all convex functions n f : R R f ( x h) f ( x) f ( x) h x, h R n Analogously, the following inequality holds for all concave functions f ( x h) f ( x) f ( x) h x, h R n Note: inequalities apply to the gradient vectors of differentiable functions as well as all subgradients in the subdifferential. 10

11 Derivative test for multivariate functions Assume function f is twice continuously differentiable. The Hessian matrix is defined as f 11( x) f 12( x) f 1n( x) f ( ) f ( ) f ( ) x x x n f ( x) f n1( x) f n2( x) f nn( x) Function f is convex if and only if the Hessian matrix is positive semidefinite for all x in the domain of f. Analogously, f is concave if and only if the Hessian matrix is negative semidefinite for all x in the domain of f. Note: We will examine the specific criteria for positive/negative semidefiniteness later in the context of optimization. 11

12 Quasi-concavity and quasi-convexity Definition: a function f defined on a convex subset U of R n is quasiconcave if the upper level set C x U f ( x) a a is a convex set for every real number a. Similarly, f is quasiconvex if the lower level set C x U f ( x) a a is a convex set for every real number a. Note: Concavity implies quasiconcavity, but the converse does not hold. Similarly, convexity implies quasiconvexity, but the converse does not hold. 12

13 Quasiconcavity of production function Definition: Input correspondence L is the mapping k m R k L : R 2, L( y) x R ( x, y) T The input set for a given output vector y, L(y), is the set of all input vectors x that can produce output y. Theorem: production function f is quasiconcave if and only if the input sets L(y) are convex for all non-negative y. 13

14 Concavity and quasiconcavity of the Cobb-Douglas function Consider the Cobb-Douglas (CD) production function CD 1 2 k f ( x) x x... x 1 2 k Proposition: The CD function is quasiconcave at all nonnegative parameter values α, β 1, β 2,, β k. Proposition: The CD function is concave if all parameters are non-negative and k i1 1 i 14

15 Homogeneity Definition: For any scalar k, a real valued function is homogeneous of degree k if f : R n R f k ( x) f( x) 0, x R n Although k can be any scalar, in economics, we are typically interested in cases where k = -1 k = 0 (zero homogeneity) k = 1 (linear homogeneity) 15

16 Constant returns to scale k Consider production function f : R R. f exhibits constant returns to scale if and only if it is homogeneous of degree 1: f ( x) f( x) 0, x R n Equivalently, the production possibility set T satisfies T T 0 16

17 Cobb-Douglas function Consider the CD production (or utility) function CD 1 2 n f ( x) x x... x 1 2 n Proposition: The f CD function is homogeneous of degree k n i i1 17

18 Euler s homogenous function theorem Theorem Let f be a continuously differentiable homogeneous function of degree k. Then for all x xf ( x) kf ( x) 18

19 Homogeneity of cost function Consider again the cost function km C : R R, C( w, y) min w x ( x, y) T x Theorem: The cost function is homogeneous of degree one in input prices w: k C( w, y) C( w, y) 0, w R, y R m Interpretation: if the input prices are doubled, the cost doubles as well. 19

20 Homogeneity of output distance function Recall Shephard s output distance function O D ( x, y) inf R ( x, y / ) T Theorem: The output distance function is homogenous of degree one in outputs y: D O O ( x, y) D ( x, y) 20

21 Homogenizing a function Homogeneity property allows us to estimate the distance function from empirical data. Starting from choose one of the outputs, say output 1, and set Then D D O O O ( x, y) D ( x, y) y 1 O ( x, ) D ( x, y) y y 1 1 1/ y 1 Taking logs of both sides, we have y ln O O D ( x, ) ln y1 ln D ( x, y) y 1 y ln 1 ln O O y D ( x, ) ln D ( x, y) y 1 21

22 Homogenizing a function Suppose the outputs are subject to random disturbance according to such that y O D y * exp( ) * ( xy, ) 1 O Then ln D ( xy, ) This yields the regression equation O y ln y1 ln D ( x, ) y 1 Note: normalization by output 1 ensures that linear homogeneity is satisfied. 22

23 Homogenizing a function Suppose we want to estimated a CD function from data, k subject to the linear homogeneity constraint (CRS): We can use the homogeneity property to normalize the inputs by dividing all inputs one of the inputs, say input 1: 1 2 x1 x2 x... k y x1 x1 x1 x1 k i1 1 i y x x 2 k ln ln 2 ln... k ln x1 x1 x1 Having estimated the parameters β 2,, β k, we use 1 1 i k i2 23

24 Homothetic functions n Definition: A function v : R R is homothetic if it is a monotone transformation of a homogeneous function, that is, if there exist a monotonic increasing function such that and a homogeneous function g : R R : n u R n v( x) g( u( x)) x R R Note: the level sets of a homothetic function are radial expansions and contractions of each other. A homogeneous function is also homothetic, but the converse does not hold. 24

25 Next week Unconstrained optimization Simon & Blume, Ch