1. Frank Cowell: Microeconomics Convexity Microeconomia III (Lecture 0) Tratto da Cowell F. (2004), Principles of Microeoconomics November 2004

2. Frank Cowell: Microeconomics Convex sets Ideas of convexity used throughout microeconomics Ideas of convexity used throughout microeconomics Restrict attention to real space R n Restrict attention to real space R n I.e. sets of vectors (,,..., ) I.e. sets of vectors (x 1, x 2,..., x n ) Use the concept of convexity to define Use the concept of convexity to define  Convex functions  Concave functions  Quasiconcave functions

3. Frank Cowell: Microeconomics Overview... Sets Functions Separation Convexity Basic definitions

4. Frank Cowell: Microeconomics x 1 Convexity in R 2 l Any point on this line also belongs to A......so A is convex  A set A in R 2  Draw a line between any two points in A x 2 l

5. Frank Cowell: Microeconomics Strict Convexity in R 2 x 1 Any intermediate point on this line is in interior of A......so A is strictly convex  A set A in R 2  Draw a line between any two boundary points of A x 2 l l Examples of convex sets in R 3

6. Frank Cowell: Microeconomics The simplex 0 x1x1 x3x3 x2x2 x 1 + x 2 + x 3 = const  The simplex is convex, but not strictly convex

7. Frank Cowell: Microeconomics The ball 0 x1x1 x3x3 x2x2  i [x i – a i ] 2 = const  A ball centred on the point (a 1,a 2,a 3 ) > 0  It is strictly convex

8. Frank Cowell: Microeconomics Overview... Sets Functions Separation Convexity For scalars and vectors

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

10. Frank Cowell: Microeconomics Concave functions (1) x y y = f(x)  A function f: R  R  Draw A, the set "above" the function –f  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  Draw the function –f

11. Frank Cowell: Microeconomics Concave functions (2) y x1x1 x2x2 0 y = f(x)  A function f: R 2  R  Draw the set "below" the function f  Set "below" f is strictly convex, so f is a strictly concave function

12. Frank Cowell: Microeconomics Convex and concave function x y y = f(x)  An affine function f: R  R  Draw the set "above" the function f  The graph in R 2 is a straight line.  Draw the set "below" the function f  Corresponding graph in R 3 would be a plane.  The graph in R n would be a hyperplane.  Both "above" and “below" sets are convex.  So f is both concave and convex.

13. Frank Cowell: Microeconomics Quasiconcavity x1x1 x2x2 B(y0)B(y0) y 0 = f(x)  Draw contours of a function f: R 2  R  Pick the contour for some specific y-value y 0.  Draw the "better-than" set for y 0.  If the "better-than" set B(y 0 ) is convex, f is a concave-contoured function  An equivalent term is a "quasiconcave" function  If B(y 0 ) is strictly convex, f is a strictly quasiconcave" function

14. Frank Cowell: Microeconomics Overview... Sets Functions Separation Convexity Fundamental relations

15. Frank Cowell: Microeconomics Convexity and separation convex non-convex  Two convex sets in R 2  Convex and nonconvex sets  Convex sets can be separated by a hyperplane... ...but nonconvex sets sometimes can't be separated

16. Frank Cowell: Microeconomics A hyperplane in R 2 x 1 x 2 H(p,c):={x:  i p i x i =c}  Hyperplane in R 2 is a straight line  Parameters p and c determine the slope and position {x:  i p i x i  c}  Draw in points "above" H  Draw in points "below" H {x:  i p i x i  c}

17. Frank Cowell: Microeconomics A hyperplane separating A and y x 1 x 2 l y A x* l  y lies in the "above-H" set  x* lies in the "below-H" set  A convex set A  The point x* in A that is closest to y  y  A.  The separating hyperplane  A point y "outside" A H

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

19. Frank Cowell: Microeconomics Supporting hyperplane B A  Convex sets A and B.  A and B only have boundary points in common.  The supporting hyperplane. l  Interior points of A lie strictly above H  Interior points of B lie strictly below H H