Mathematical Foundations -1- Convexity and quasi-convexity Convex set Convex function Concave function Quasi-concave function Supporting hyperplane
Mathematical Foundations -2- Convexity and quasi-convexity Convex combination x is a convex combination of n x, x, if for some (,1) x (1 ) x x Convex Set n S is convex if, for each x, x S every convex combination x (1 ) x x S Strictly convex set The function f is strictly convex if, for each of S ( x int S) x, x S every convex combination x lies in the interior
Mathematical Foundations -3- Convexity and quasi-convexity We consider functions f defined over a convex set S Strictly convex function n SC1: f is a strictly convex function if, for any x, x S and convex combination x, 1, ( ) (1 ) ( ) ( ) In Fig. 1-2, for any convex combination x, the point D on the chord connecting A and B lies above the point C on the graph C D B of f. A x Fig. 1-2: Graph of the function lies below the chord C1: f is a convex function if, for any x, x S and convex combination x, 1, ( ) (1 ) ( ) ( )
Mathematical Foundations -4- Convexity and quasi-convexity What is the relationship between a convex set and a convex function? n Definition: Graph of a function f : S y G x y xs y n1 {(, ), ( )} A C2: The set A {( x, y) y f ( x), x S} is a convex set Points on or above the graph of f In the figure, the set A is the set of points on or above the graph of the function. C1 C2 1 1 Suppose ( x, y ) and ( x, y ) are both in A. Then y f ( x ) and y f ( x ). We need to show that each convex combination ( x, y ) A. Since f is convex, f ( x ) (1 ) f ( x ) f ( x ) (1 ) y y y. Then ( x, y ) A. QED
Mathematical Foundations -5- Convexity and quasi-convexity C2 C1 Exercise: Prove this Proposition HINT: Consider ( x, y ) ( x, f ( x )) and 1 1 1 1 ( x, y ) ( x, f ( x )). By C2 the convex combination (, ) (,(1 ) ( ) ( )) x y x A C3: For 1 f and for any y, z f S, f ( y) f ( z) ( z) ( y z) x Exercise: Prove that C1 C3 Remark: We showed already that 1 f is concave on the convex set S only if for any x, y S f f ( y) f ( z) ( z) ( y z). The proof that C1 C3 is almost identical x
Mathematical Foundations -6- Convexity and quasi-convexity C3 C1 f For any y, z S f ( y) f ( z) ( z) ( y z). x Consider any x x S. Since S is convex, the convex combination x S. Set z x. Then 1 f 1 f ( x ) f ( x ) ( x ) ( x x ) x f f ( x ) f ( x ) ( x ) ( x x ) x (1 ) Multiply these inequalities as indicated and then add. f x x x x x x f 1 ( x ) [ x (1 ) x x ]. x 1 1 ( ) (1 ) ( ) ( ) ( ) [ ( ) (1 )( )]
Mathematical Foundations -7- Convexity and quasi-convexity C4: For 1 f and for any x, x S, ( )[ g( ) g( )] where g( ) f ( x ) and so 2 1 2 1 f g x x x x 1 ( ) ( ) ( ) Graphically, C4 is the statement that the slope of g( ) is increasing. C4 C3 g g 1 ( ) ( ) (1) () 1 1 1 g( ) d g() d g() d g() But f g x x x x Therefore 1 ( ) ( ).( ). f g x x x x 1 ( ) ( ) () ( ).( ) Graph of the function lies below the chord
Mathematical Foundations -8- Convexity and quasi-convexity C3 C4 Consider x( ), x( ). x, x and convex combinations 1 2 x( ) x ( x x ) and 2 2 x( ) x ( x x ). 1 1 Therefore x x x x 1 ( 2 ) ( 1 ) ( 2 1 )( ) By C3, for any y, z S,where S is convex, f f ( y) f ( z) ( z) ( y z). x Graph of the function lies below the chord Therefore 1 ( ( 2)) ( ( 1)) f ( ( 1)) ( ( 2) ( 1)) f x x x ( x( 1 )) ( 2 1 )( x x ) ( 2 1 ) f ( x( 1 )) ( x x ) x x x Also f 1 f 1 f ( x( 1 )) f ( x( 2)) ( 1 2) ( x( 2)) ( x x ) ( 2 1 ) ( x( 2)) ( x x ) x x Add the inequalities 1 ( f f 2 1)[ ( x( 1)) ( x x ) ( x( 2)) ( x x ) x x But f g x x x x 1 ( ) ( ( )) ( ) Therefore ( 2 1 )[ g( 1 ) g( 2)] and so ( 2 1 )[ g( 2 ) g( 1 )].
Mathematical Foundations -9- Convexity and quasi-convexity An equivalent way of writing this is that for any 1 and 2 1 g( 2) g( 1) 2 1 (*) That is, the second derivative of g( ) must be positive. Also if the second derivative is positive then (*) holds. Therefore if 2 f C4 and C5 are also equivalent. C5: For 2 f and any x, x S g() where g x ( ) ( ( )) ((1 ) ))
Mathematical Foundations -1- Convexity and quasi-convexity Restatement of C5 Suppose that f is twice differentiable on n Define g( ) f ( x ) f ( x ( x x )). 2 n n 2 2 d g 1 f 1 f () ( ) ( ) ( ) ( ) ( ) 2 xi xi x j x j x x x x x d i1 i1 xix j xix j Then we can rewrite C5 as follows: C5: For 2 f defined on S and any x, x S 2 1 f ( x x ) H( x )( x x ) where H( x ) [ ( x )] x x i j If n S then C5 is the statement that the Hessian Matrix H( x ) must be positive semi-definite for all n x
Mathematical Foundations -11- Convexity and quasi-convexity Strictly Concave Function f is a strictly concave function if, for any x, x S and convex combination x, 1, ( ) (1 ) ( ) ( ) Concave Function f is a concave function if, for any x, x S and convex combination x, 1, ( ) (1 ) ( ) ( ) Concave function Reverse all the inequalities in C1 C5 to obtain equivalent definitions of a concave function. Exercise: Show that a function f is concave on S if and only if f is convex on S.
Mathematical Foundations -12- Convexity and quasi-convexity Class exercises (in groups) 1. Prove that a concave function of a concave function is not necessarily concave. 2. Prove that a strictly concave function of a strictly concave function maybe strictly convex. 3. Show that the sum of concave functions is concave.
Mathematical Foundations -13- Convexity and quasi-convexity Shortcuts for checking whether a function is convex or concave With one variable, the easiest way to tell if a function is strictly convex is to see if it has a positive second derivative. For functions of more than one variable, the following propositions are often helpful. Proposition: The sum of convex functions is convex. Proposition: Convex function of a function h( x) g( f ( x)) is convex if (i) f is convex & g is increasing and convex, or (ii) f is linear and g is convex. Proof of (i) Since f is convex, Since g is increasing Define then Since g is convex Appealing to (*) f ( x ) (1 ) f ( x ) f ( x ). g g ( ( )) ((1 ) ( ) ( )) 1 y f ( x ) y f ( x ) g( f ( x )) g((1 ) y y ) (*) g((1 ) y y ) (1 ) g( y ) g( y ) (1 ) g( f ( x )) g( f ( x )). g g g ( ( )) (1 ) ( ( )) ( ( ))
Mathematical Foundations -14- Convexity and quasi-convexity Quasi-concave Function A function f is quasi-concave on the convex set n S if, for any x, x in S, such that 1 ( ) f ( x ) and any convex combination x x x (1 ), 1, ( ) f ( x ). Strictly Quasi-concave Function A function f is quasi-concave on the convex set n S if, for any x, x in S, such that 1 ( ) f ( x ) and any convex combination x x x (1 ), 1, ( ) f ( x ). Example: U( x) x1x 2is quasi-concave on 2 and is strictly quasi-concave on 2 2 { x x, x } But why! Read on.
Mathematical Foundations -15- Convexity and quasi-convexity Proposition: A function is quasi-concave if and only if the upper contour sets of the function are convex Proof of equivalence: Without loss of generality we may assume that 1 ( ) f ( x ). Suppose f is quasi-concave and for any ˆx, consider vectors x and 1 x which lie in the upper contour set U C ( x ˆ) of this function. That is ˆ and ( ) f ( x) ˆ 1 ( ) f ( x). From the definition of quasiconcavity, for any convex combination, x x x (1 ), ( ) f ( x ). Since ˆ, it follows that for all convex combinations f ( x ) f ( xˆ ). Hence all convex ( ) f ( x) combinations lie in the upper contour set. Thus C ( ˆ ) U x is convex. Conversely, if the upper contour sets are convex, then C ( ) U x is convex. Since 1 ( ) f ( x ). Then x and 1 x are both in C ( ) U x therefore, by the convexity of C ( ) U x, all convex combination lie in this set. That is ( ) f ( x ) for all, 1. Q.E.D.
Mathematical Foundations -16- Convexity and quasi-convexity Proposition: If there is some strictly increasing function g such that h( x) g( f ( x)) is concave then f(x) is quasi-concave. Proof: We need to establish that if 1 ( ) f ( x ) then ( ) f ( x ), 1 By concavity h( x ) (1 ) h( x ) h( x ), that is g( f ( x )) (1 ) g( f ( x )) g( f ( x )). (*) Since g is increasing and g ( ( )) g( f ( x )). 1 ( ) f ( x ), g 1 ( ( )) g( f ( x )). Then substituting into (*), By hypothesis, g is a strictly increasing function. Then ( ) f ( x ). Q.E.D. Remark: It is in this last line of the proof that we require that g is strictly increasing. For otherwise g could be constant over some neighborhood g 1 ( ( )) g( f ( x )). Nx (, ) and so we could have 1 ( ) f ( x ) and
Mathematical Foundations -17- Convexity and quasi-convexity Proposition B.2-9: If : n f f ( z) f ( z) ) and quasi-concave then f is concave. is positive, homogeneous of degree 1 (i.e. for all Proof: Because f is homogeneous of degree 1 it follows that for any x, x and (,1) (1 ) f ( x ) f ( x ) f ((1 ) x ) f ( x ), Since f, there exists such that f ((1 ) x ) f ( x ). (1) Then 1 (1 ) f ( x ) f ( x ) (1 ) f ( x ). (2) Because f is homogeneous of degree 1 it follows from (1) that f ((1 ) x ) f ( x ). Note that 1 ((1 ) x x ) (1 ) x x. 1 1 1 Therefore 1 ((1 ) x x ) is a convex combination of (1 ) and x x.
Mathematical Foundations -18- Convexity and quasi-convexity We showed that 1 ((1 ) x x ) Therefore, by the quasi-concavity of f is a convex combination of (1 ) and x x. x 1 ( ((1 ) )) ( ) Because f is homogeneous of degree 1 it follows that. (3) x 1 and hence that ((1 ) ) ( ) f ((1 ) x x ) (1 ) f ( x ). Appealing to (2), f ((1 ) x x ) (1 ) f ( x ) f ( x ). QED
Mathematical Foundations -19- Convexity and quasi-convexity Class examples: In each case is f (i) quasi-concave (ii) strictly quasi-concave (iii) concave (a) (b) (c) (d) (e) (f) (g) 2 f ( x) (1 x1)(2 x2), x 2 f ( x) x1x 2, x 2 f ( x) x1x 2, x 3 f ( x) x1x 2 x1x 3, x f ( x) x x x 1/2 1/2 1/2 1 2 3 f ( x) ( x x x ) 1/2 1/2 1/2 2 1 2 3 f ( x) ( x x x ) 1/2 1/2 1/2 3/2 1 2 3
Mathematical Foundations -2- Convexity and quasi-convexity Class Exercise: Are the following statements true or false? 1. For any convex set then f takes on its global maximum at 2. For any set n S if the quasi-concave function f : S has a local maximum at * x. n S if the strictly quasi-concave function f : S has a local maximum at then f takes on its global maximum at * x. * x, * x,
Mathematical Foundations -21- Convexity and quasi-convexity Supporting hyperplane for a convex set Suppose that for some x, the upper contour set S x { ( ) ( )} is convex. (This will be true if f is quasi-concave.) The linear approximation of f at x is f l x x x x x ( ) ( ) ( ) ( ) Consider the upper contour set of l. This is the set f T x l x l x x x x x x { ( ) ( )} { ( ) ( ) } We will argue that S T..
Mathematical Foundations -22- Convexity and quasi-convexity Proposition: Supporting hyperplane If the set S x { ( ) ( )} is convex, then f S T x x x x x { ( ) ( ) }. The hyperplane hyperplane. f x ( x ) ( x x ) touching the boundary of S at x is called a supporting Proof: For 1 x S define g( ) f ( x( )) f ( x ) f ( x ( x x )). dg Since S is convex, x( ) is in S. Therefore g( ). Since g() it follows that (). d But as we have previously shown, dg d f x () ( x ) ( x x ). Then for all 1 x S, f x ( x ) ( x x ). Q.E.D.
Mathematical Foundations -23- Convexity and quasi-convexity Example 1: Consider the contour set of f ( x) ( x 2 x ) 1/2 2 1 2 through (1,1). The upper contour set is S { x ( x 2 x ) 9} 1/2 2 1 2 Confirm that f ( x) ( x 2 x ) is quasi-concave. 1/2 2 1 2 The linear approximation of f at x is f l x x x x x x x ( ) ( ) ( ) ( ) 9 6( 11) 6( 2 1) Consider the upper contour set of l. This is the set T { x l( x) l( x )} { x 6( x x ) 6( x x ) } 1 1 2 2. Example 2: Production set of a firm Y {( z, q) 3z q } where (,1]. Consider the tangent line through (1,3)