Convexity

1. Convexity MICROECONOMICS Principles and Analysis Frank Cowell June 2006

2. Convex sets • Ideas of convexity used throughout microeconomics • Restrict attention to real space Rn • I.e. sets of vectors (x1, x2, ..., xn) • Use the concept of convexity to define • Convex functions • Concave functions • Quasiconcave functions

3. Overview... Convexity Sets Basic definitions Functions Separation

4. Convexity in R2 • A set A in R2 • Draw a line between any two points in A x2 Any point on this line also belongs to A... ...so A is convex x1

5. Strict Convexity in R2 • A set A in R2 • Draw a line between any two boundary points of A x2 Any intermediate point on this line is in interior of A... ...so A is strictly convex Examples of convex sets in R3 x1

6. The simplex x1 x1+ x2 + x3= const x2 • The simplex is convex, but not strictly convex 0 x3

7. The ball x1 • A ball centred on the point (a1,a2,a3) > 0 • It is strictly convex Si [xi– ai]2= const x2 0 x3

8. Overview... Convexity Sets For scalars and vectors Functions Separation

9. Convex functions • A function f: RR • Draw A, the set "above" the function f y A := {(x,y): y  f(x)} • If A is convex, f is a convex function • If A is strictly convex, f is a strictly convex function y = f(x) x

10. Concave functions (1) • A function f: RR y • Draw the function –f • Draw A, the set "above" the function –f y = f(x) • If –f is a convex function, f is a concave function • Equivalently, if the set "below" f is convex, f is a concave function • If –f is a strictly convex function, f is a strictly concave function x

11. Concave functions (2) y • A function f: R2R y = f(x) • Draw the set "below" the function f • Set "below" f is strictly convex, so f is a strictly concave function x2 0 x1

12. Convex and concave function y = f(x) • An affine function f: RR y • Draw the set "above" the function f • Draw the set "below" the function f • The graph in R2 is a straight line. • Both "above" and “below" sets are convex. • So f is both concave and convex. x • Corresponding graph in R3 would be a plane. • The graph in Rn would be a hyperplane.

13. Quasiconcavity x2 • Draw contours of a function f: R2R • Pick the contour for some specific y-value y0 . • Draw the "better-than" set for y0 . • If the "better-than" set B(y0) is convex, f is a concave-contoured function • An equivalent term is a "quasiconcave" function • If B(y0) is strictly convex, f is a strictly quasiconcave" function B(y0) y0= f(x) x1

14. Overview... Convexity Sets Fundamental relations Functions Separation

15. Convexity and separation convex • Two convex sets in R2 • Convex and nonconvex sets • Convex sets can be separated by a hyperplane... • ...but nonconvex sets sometimes can't be separated non-convex convex convex

16. A hyperplane in R2 x2 • Hyperplane in R2 is a straight line {x: Sipixi c} • Parameters p and c determine the slope and position • Draw in points "above" H • Draw in points "below" H H(p,c):={x: Sipixi=c} {x: Sipixi c} x1

17. A hyperplane separating A and y • A convex set A H • A point y "outside" A x2 • The point x* in A that is closest to y y • The separating hyperplane A x* • y  A. • y lies in the "above-H" set • x* lies in the "below-H" set x1

18. A hyperplane separating two sets • Convex sets A and B. • A and B only have no points in common. H • The separating hyperplane. A • All points of A lie strictly above H • All points of B lie strictly below H B

19. Supporting hyperplane • Convex sets A and B. • A and B only have boundary points in common. H • The supporting hyperplane. A • Interior points of A lie strictly above H • Interior points of B lie strictly below H B