Frank Cowell: Convexity CONVEXITY MICROECONOMICS Principles and Analysis Frank Cowell 1 March 2012

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

Frank Cowell: Convexity Overview... 3 Sets Functions Separation Convexity Basic definitions March 2012

Frank Cowell: Convexity x 1 Convexity in R 2 4 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 March 2012

Frank Cowell: Convexity Strict Convexity in R 2 5 x 1 Any intermediate point on this line is in interior of A...so A is strictly convex  A set A in R 2  A line between any two boundary points of A x 2 Examples of convex sets in R 3 March 2012

Frank Cowell: Convexity The simplex 6 0 x1x1 x3x3 x2x2 x 1 + x 2 + x 3 = const  The simplex is convex, but not strictly convex March 2012

Frank Cowell: Convexity The ball 7 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 March 2012

Frank Cowell: Convexity Overview... 8 Sets Functions Separation Convexity For scalars and vectors March 2012

Frank Cowell: Convexity Convex functions 9 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 March 2012

Frank Cowell: Convexity Concave functions (1) 10 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 March 2012

Frank Cowell: Convexity Concave functions (2) 11 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 March 2012

Frank Cowell: Convexity Convex and concave function 12 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  Graph in R 3 is a plane  The graph in R n is a hyperplane  Both "above" and “below" sets are convex  So f is both concave and convex March 2012

Frank Cowell: Convexity Quasiconcavity 13 x1x1 x2x2 B(y0)B(y0) y 0 = f(x)  Draw contours of function f: R 2  R  Pick contour for some 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 March 2012

Frank Cowell: Convexity Overview Sets Functions Separation Convexity Fundamental relations March 2012

Frank Cowell: Convexity Convexity and separation 15 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 March 2012

Frank Cowell: Convexity A hyperplane in R 2 16 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} March 2012

Frank Cowell: Convexity A hyperplane separating A and y 17 x 1 x 2 y A x*  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 y  A  The separating hyperplane  A point y "outside" A H March 2012

Frank Cowell: Convexity A hyperplane separating two sets 18 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 March 2012

Frank Cowell: Convexity Supporting hyperplane 19 B A  Convex sets A and B  A and B only have boundary points in common  The supporting hyperplane  Interior points of A lie strictly above H  Interior points of B lie strictly below H H March 2012