Generalized Convexity in Economics

J.E. Martínez-Legaz Universitat Autònoma de Barcelona, Spain

2nd Summer School: Generalized Convexity Analysis Introduction to Theory and Applications JULY 19, 2008 KAOHSIUNG - TAIWAN References:

John, Reinhard: Uses of generalized convexity and generalized monotonicity in economics. Handbook of generalized convexity and generalized monotonicity, 619–666, Nonconvex Optim. Appl., 76, Springer, New York, 2005.

Martínez-Legaz, Juan Enrique: Generalized convex duality and its economic applications. Handbook of generalized convexity and generalized monotonicity, 237–292, Nonconvex Optim. Appl., 76, Springer, New York, 2005. 1 Preference, Utility and Demand n di¤erent types of commodities

n R+ the set of commodity bundles

n % total preorder on R+ : x; y Rn = x y or y x 2 + ) % % x y; y z = x z % % ) %

% is continuous if graph( ) := (x; y) Rn Rn : x y is closed. % 2 + + % n o n PROPOSITION. A binary relation % on R+ is a continuous total preorder if and only if there is a continuous function u : Rn R such that for all x; y Rn + ! 2 + x y u (x) u (y) : % () n % total preorder on R+

% is convex if x y = (1 ) x + y y for all [0; 1] : % ) % 2

x y x y; y x () % 6%

% is strictly convex if x y; x = y = (1 ) x+y y for all (0; 1) : % 6 ) 2 u : Rn R utility re presentation of + ! % is convex u is quasiconcave % () is strictly convex u is strictly quasiconcave % ()

n n The demand correspondence: X = Xu : R++ R+

X(p) = x Rn : x; p 1; u (x) u (y) f 2 + h i for all y Rn such that y; p 1 : 2 + h i g % is said to be locally nonsatiated if n no relatively open subset of R+ has a %-maximal element.

PROPOSITION. (1) If % is locally nonsatiated then, for every p Rn and x X(p); x; p = 1: 2 ++ 2 h i

(2) If % is convex then X is convex-valued and satis…es GARP: if pi Rn ; xi X(pi)(i = 1; :::; k) then 2 ++ 2 min xi xi+1; pi 0 = xk x1; pk 0. i=1;:::;k 1 ) D E D E

(3) If u is u.s.c. and strictly quasiconcave then X is single-valued and satis…es SARP: if pi Rn ; xi X(pi)(i = 1; :::; k) then 2 ++ 2 min xi xi+1; pi 0 = xk x1; pk < 0. i=1;:::;k 1 ) D E D E 2 Duality in Consumer Theory u : Rn R utility function + ! p Rn price vector 2 +

M > 0 income

( ) maximize u(x) P subject to x; p M h i

The indirect utility function v : Rn R : + ! v(p) = sup u(x): x; p 1 f h i g n S(v) = p R+ : x; p > 1 ( R) 2 h i 2 x:u(\x)> n o THEOREM. Let v : Rn R: There exists a utility + ! function u : Rn R having v as its associated indirect + ! utility function if and only if v is nonincreasing, evenly quasiconvex and satis…es

n v(p) lim v( p)(p bd R+); 1 2 ! v denoting the l.s.c. hull of v:

In this case one can take u nondecreasing, evenly quasiconcave and satisfying

n u(x) lim u( x)(x bd R+): (1) 1 2 !

Under these conditions u is unique, namely, u is the pointwise largest utility function inducing v; furthermore, it satis…es

n u(x) = inf v(p): x; p 1 (x R+): (2) f h i g 2 n COROLLARY. For every v : R+ R; the function n ! v0 : R R de…ned by + ! v (p) = sup u(x): x; p 1 ; 0 f h i g with u given by (2), is the pointwise largest nonincreasing evenly quasiconvex minorant of v that satis…es

n v0(p) lim v0( p)(p bd R+): (3) 1 2 !

THEOREM. Let u : Rn R: There exists a function + ! v : Rn R such that (2) holds if and only if u is + ! nondecreasing, evenly quasiconcave and satis…es (1). In this case one can take v nonincreasing, evenly quasiconvex and satisfying (3). Under these conditions v is unique, namely, v is the pointwise smallest function such that (2) holds; furthermore, it is the indirect utility function associated with u. COROLLARY. For every u : Rn R; the function + ! u0 : Rn R de…ned by + ! u0(x) = inf v(p): x; p 1 ; f h i g v being the indirect utility function associated with u; is the pointwise smallest nondecreasing evenly quasiconcave majorant of u that satis…es

0 0 n u (x) lim u ( x)(x bd R+): 1 2 !

THEOREM. A nondecreasing evenly quasiconcave function u : Rn R satisfying (1) is …nite-valued if and only + ! if its associated indirect utility function v : Rn R is + ! bounded from below and …nite-valued on the interior of n R+:

THEOREM. The indirect utility function v : Rn R + ! induced by a nondecreasing evenly quasiconcave function u : Rn R satisfying (1) is …nite-valued if and only if + ! u is bounded from above and …nite-valued on the interior n of R+: COROLLARY. Let u : Rn R be a nondecreasing + ! evenly quasiconcave function satisfying (1) and let v : Rn R be its associated indirect utility function. The + ! following statements are equivalent: (i) u and v are …nite-valued. (ii) u is bounded. (iii) v is bounded. u : Rn R utility function + !

n The expenditure function: eu : R R R+ + + ! [ f 1g n eu(p; ) = inf x; p : u(x) (p; ) R+ R : fh i g 2 n THEOREM. A function e : R R R+ + + ! [ f 1g is the expenditure function eu for some utility function u : Rn R if and only if the following conditions hold: + !

(i) For each R; either e( ; ) is …nite-valued, con- 2 cave, linearly homogeneous and u.s.c., or it is identically equal to + : 1

(ii) For each p Rn ; e(p; ) is nondecreasing. 2 +

n (iii) @e( ; )(0) = R ; with @e( ; )(0) denoting R + the superdi¤erential2 of the concave function e( ; ) at S the origin, i.e.

n n @e( ; )(0) = x R+ : x; p e(p; ) p R+ : 2 h i 8 2 n o THEOREM. The mapping u eu is a bijection from 7! the set of u.s.c. nondecreasing quasiconcave functions n u : R+ R onto the set of functions n ! e : R R R+ + that satisfy (i)-(iii), + ! [ f 1g

(iv) R @e( ; )(0) = 2 ; T and

(v) < @e( ; )(0) = @e( ; )(0) ( R) : 2 T Furthermore, the inverse mapping is e ue; with ue : 7! Rn R given by + ! ue(x) = sup R : x @e( ; )(0) : f 2 2 g 3 Monotonicity of Demand Func- tions

n n The demand correspondence: X : R++ R+ n X(p) = x R+ : x; p 1; u (x) = v (p) 2 h i n o

THEOREM. If the utility function u : Rn R has no + ! local maximum then the demand function n n X : R++ R+ is cyclically quasimonotone (in the decreasing sense), i.e. if pi Rn ; xi X(pi)(i = 1; :::; k) 2 ++ 2 then

min xi xi+1; pi 0; with xk+1 = x1. i=1;:::;k D E

X is quasimonotone (in the decreasing sense) if n p; q R++; x X(p); y Y (q) 2 2 2 = min x y; p ; y x; q 0: ) fh i h ig X is monotone (in the decreasing sense) if

n p; q R++; x X(p); y Y (q) 2 2 2 = x y; p + y x; q 0: (4) ) h i h i u : R2 R + ! x + x 1 if x + x < 1 u(x ; x ) = 1 2 1 2 1 2 x if x + x 1 ( 1 1 2

maximize u(x1; x2) subject to p x + p x 1 1 1 2 2

For p 1 and 1 p > 0; the optimal solution is 1 2 1 p p 1 x = 2 ; x = 1 : 1 p p 2 p p 1 2 1 2 Let p Rn and x X(p): 2 ++ 2 p is an optimal solution to minimize v(p) subject to x; p 1: h i

There exists 0 such that v(p) + x = 0 and ( x; p 1) = 0: r h i v(p); p + x; p = 0 x; p = hr i h i h i

= v(p); p hr i v(p) v(p); p x = 0: r hr i THEOREM. (Roy’s Identity) If the utility function u is u.s.c. and its associated indirect utility function v is con- tinuously di¤erentiable at p Rn ; with v(p) = 0; 2 ++ r 6 then 1 X(p) = v(p) : ( v(p); p r ) hr i

THEOREM. If the utility function u : Rn R is con- + ! cave, C2; has a componentwise strictly positive gradient n on R++, induces a demand function ' : Rn Rn (i.e. a single-valued demand correspon- ++ ! + dence p Rn X(p) = '(p) ); with ' of class 2 ++ f g C1; and satis…es

x; 2u(x)x n r < 4 x R++ D x; u(x) E 2 h r i then ' is strictly monotone, i.e. it satis…es (4) as a strict inequality whenever p = q: 6 n n X (p) = x R+ : u (x) = v (p) p R++ 2 2 n o 1 n n X (x) = p R++ : v (p) = u (x) x R+ 2 2 n o n u (x) = sup v(p): x; p 1 (x R+) f h i g 2

PROPOSITION. For a C2 nondecreasing quasiconcave n utility function u : R+ R with no stationary points, the following statements! are equivalent:

2 x; u(x)x n (i) h r i 4 x R : x; u(x) 2 ++ h r i 1 1 n 3 3 (ii) The function (x1; :::; xn) R++ u(x1 ; :::; xn ) is convex-along-rays. 2 7!

n (iii) The restriction v : R++ R + of the indirect utility function to the positive! orthant[f 1g has a representation of the type 3 n v(p) = max c ( y; p ) (p R++); (y;c) U h i 2 2 n o with U Rn 0 R: ++ [ f g THEOREM. Let ' : Rn Rn be a C1 demand func- ++ ! + tion induced by a strictly quasiconcave utility function u : Rn R; which is C2 on Rn and has a compo- + ! ++ nentwise strictly positive gradient on Rn ' Rn : ++ [ ++ Then ' is monotone if and only if

2 x; u(x)x x; u(x) r 4 h r i D x; u(x) E 2 1 h r i u(x); u(x) u(x) r r r n 2 x R++ such that u(x) is nonsingular 8 2 r and x; 2u(x)x n 2 r 4 x R++ s. t. u(x) is singular. D x; u(x) E 8 2 r h r i PROPOSITION. Let u : Rn R be a utility function + ! and let v : Rn R + be its associated indirect + ! [ f 1g utility function. If the set

n n (p; x) R++ R+ : u(x) v(p) 0 2 is convexn (in particular, if the function o

n n : R++ R+ R + ! [ f 1g de…ned by (p; x) = u(x) v(p) is quasiconcave) and u has no maximum then the demand correspondence X is monotone.

COROLLARY. Let u : Rn R be a utility function and + ! let v : Rn R + be its associated indirect utility + ! [ f 1g function. If u is concave and has no maximum and v is convex then the demand correspondence X is monotone. THEOREM. If u : Rn R is nondecreasing and the + ! function x Rn u(x 1) is convex-along-rays 2 ++ 7! then the restriction v : Rn R + of the indirect ++ ! [f 1g utility function to the positive orthant is convex. Con- versely, if v : Rn R is bounded, nonincreasing and ++ ! convex then there is a nondecreasing quasiconcave utility function u : Rn R such that x Rn u(x 1) + ! 2 ++ 7! is convex-along-rays and whose associated indirect utility function extends v.

COROLLARY. Let u : Rn R be a C2 nondecreasing + ! quasiconcave utility function. The restriction v : Rn R of its associated indirect utility function ++ ! to the positive orthant is convex if and only if

2 n < x; u(x)x > +2 < u(x); x > 0 (x R++): r r 2 4 Consumer Theory without Util- ity

n Let % be a total preorder on R+:

n For x1; x2 R ; 2 + x1 s x2 x1 x2; x2 x1 () % % x x x x ; x x 1 2 () 1 % 2 2 6% 1

n The indirect preorder induced by : For p;; p2 R ; % 2 + i n p1 p2 x2 B(p2) = x R : x; p2 1 % () 8 2 2 + h i x B(p ) s.n t. x x : o 9 1 2 1 1 % 2

i n % is a total preorder on R+:

i v the associated indi¤erent relation

i the associated strict preorder n Let % be a total preorder on R+:

n The direct preorder induced by : For p1; p2 R ; % 2 + d 1 n x1 x2 p1 B (x1) = p R : x1; p 1 % () 8 2 2 + h i p B 1(x ) s. t.n p p : o 9 2 2 2 1 % 2

d n % is a total preorder on R+: n THEOREM. Let % be a total preorder on R+: The following statements are equivalent:

id (i) % coincides with % :

(ii) % has the following properties:

(a) % is nondecreasing.

n (b) For every x1 R+; n 2 the set x R : x x1 is evenly convex. 2 + % n o n (c) For every x1 R+; n2 if x2 cl x R : x x1 and > 1 then x1 2 2 + % % x2: n o

n (d) For every x1; x2 R ; 2 + if x1 s x2 and x1 is a %-maximal element of B(p1) for n some p1 R+ then x2 is a %-maximal element of B(p2) 2 n for some p2 R : 2 + i Moreover, if conditions (a)-(d) hold, then % has the following properties:

i (a’) % is nonincreasing.

n (b’) For every p1 R+; n 2 i the set p R : p1 p is evenly convex. 2 + % n o n (c’) For every p1 R+; n2 i i if p2 cl p R : p1 p and > 1 then p1 2 2 + % % p2: n o

n (d’) For every p1; p2 R+; i 2i 1 if p1 v p2 and p1 is a % minimal element of B (x1) n i for some x1 R+ then p2 is a % minimal element 1 2 n of B (x2) for some x2 R : 2 +

The duality mapping i is a bijection, with inverse % 7! % d; from the set of all total preorders on Rn 7! + with properties (a)-(d) onto the set of all total preorders i n % on R+ with properties (a’)-(d’). u : R2 R + ! 1 if z 1 u(y; z) = y+1 ( 1 if z > 1

v : R2 R + ! 1 if r < 1 v(q; r) = 0 if r 1 and q = 0 : 8 q > if r 1 and q > 0 < q+1 If v(q; r) = 0 then:> no maximal element exists in B(q; r):

If v(q; r) = 0 then B(q; r) has at least a maximal ele- 6 ment.

i v is a utility representation for % : i (0; 1) v (0; 2)

(0; 1) is a i minimal element of B 1(1; 1) %

2 i No (y; z) R+ exists such that (0; 2) is a % minimal 2 1 element of B (y; z):

If (y; z) R2 ; with y = 0, is such that (0; 2) 2 + 6 2 B 1(y; z) then (0; 2) i 2 3z; 3 B 1(y; z); 2y 2 2 if (0; z) R2 is such that (0; 2) B 1(0; z) then 2 + 2 (0; 2) i 3; 3 B 1(0; z): 2 2 2 i= idi % 6 % u0 : R2 R + ! 1 if z 1 and y = 0 0 z 1 u (y; z) = 8 y z+1 if z 1 and y > 0 > <> 1 if z > 1 > :> 0 id u is a utility representation of % :

For every (y; z) B(0; 2), u0(y; z) < 0 = u0(1; 1) 2

(1; 1) B(0; 1) 2

(0; 1) idi (0; 2)

i (0; 1) v (0; 2) n PROPOSITION. Let % be a total preorder on R+ sat- id isfying properties (a)-(c). Then % is the total preorder whose strict preorder id is de…ned as follows: x id x if and only if 1 2 either x x 1 2 or x1 v x2; x1 is not a % maximal element in B(p) n for any p R and x2 is a -maximal element of B(p) 2 + % for some p Rn : 2 +

id is an extension of :

id % is an extension of % .

n idid THEOREM. For every total preorder % on R+; % id coincides with % : n THEOREM. Let % be a total preorder on R+ such that n for every p R++ the set B(p) has a % maximal 2 i element. Then % has the following properties:

i (a) % is nonincreasing.

n n i (b) For every p1 R ; the set p R : p1 p is 2 + 2 + evenly convex. n o

n n i (c) For every p1 R ; if p2 cl p R : p1 p 2 + 2 2 + and > 1 then p1 % p2: n o

n Conversely, if % is a total preorder on R+ such that (a)- (c) hold with i replaced by then, for every p Rn ; % % 2 ++ the set B(p) has a d maximal element and di % % coincides with % : n THEOREM. Let % be a total preorder on R+ that coin- id cides with % : The following statements are equivalent: n (i) For every p1 R the set B(p1) has a maximal 2 ++ % element. n n i (ii) For every p1 R ; the set p R : p1 p is 2 + 2 + evenly convex. n o n n i (iii) For every p1 R ; the set p R : p1 p is 2 ++ 2 + evenly convex. n o

n % partial preorder on R+

n n The expenditure function: e : R R R+ % + + ! n n e (p; x) = inf x0; p : x0 x p R+; x R+ : % % 2 2 nD E o n (HCP) For all x1; x2 R ; 2 + co Sx1 + n co Sx2 + n = Sx1 Sx2 % R+ % R+ % % ) THEOREM. The mapping % e is a bijection from 7!n the set of all preorders % on R+ whose upper contour x n sets S = x0 R+ : x0 % x satisfy (HCP) onto the % 2 n n set of functionsn e : R R o R+ that satisfy the + + ! following properties:

(a) For every x Rn ; the mapping e( ; x) is concave, 2 + positively homogeneous and upper semicontinuous.

(b) For every x Rn ; the closed convex hull of the 2 + set x Rn : e( ; x ) e( ; x) + Rn coincides with 0 2 + 0 + @e(n; x)(0); the superdi¤erentialo of the concave function e( ; x) at the origin.

The inverse mapping is e %e; with %e denoting n 7! the preorder on R+ de…ned by x1 %e x2 if and only if e ( ; x ) e ( ; x ) (pointwise). 1 2 THEOREM. The mapping e is a bijection % % 7! n from the set of all nondecreasing preorders % on R+ whose upper contour sets are closed and convex onto the n n set of functions e : R R R+ that satisfy the + + ! following properties: (a) For every x Rn ; the mapping e( ; x) is concave, 2 + positively homogeneous and upper semicontinuous. (b’)For every x Rn ; 2 + n x0 R+ : e( ; x0) e( ; x) = @e( ; x)(0): 2 n o The inverse mapping e e is given by: x e x if 7! % 1 % 2 and only if x @e( ; x )(0). 1 2 2 n % total preorder on R+

n n The demand correspondence: X : R++ R+ X(p) = x B(p): x y y B(p) f 2 % 8 2 g

n % on R+ is said to be locally nonsatiated if no relatively n open subset of R+ has a %-maximal element. PROPOSITION. Let % be a locally nonsatiated total pre- n order in R+ and let X be its associated demand correspondence. If the set

n n (x; p) R+ R++ : x y y B(p) 2 % 8 2 is convexn then X is monotone. o

PROPOSITION. Let % be a nondecreasing total preorder n in R+, X be its associated demand correspondence, and assume that % satis…es the following condition: x y 1 % 1 x y = (x + x ) 2 y: 2 % 9 ) 2 1 2 % + > 0; > 0 =>

Then X is monotone.;>

If graph( ) = (x; y) Rn Rn : x y is convex % 2 + + % set then one hasn o

x1 % y x y = 1 (x + x ) +y 2 y: 2 % 9 ) 2 1 2 % 2 + > 0; > 0 =>