Advances in MATHEMATICAL ECONOMICS ppsx

We characterize insatiate demand functions that are rationalized, in different meanings, by concave utility functions with some additional properties such as upper semi-continuity, conti

Abstract A method in demand analysis based on the Monge-Kantorovich

duality is developed We characterize (insatiate) demand functions that are rationalized, in different meanings, by concave utility functions with some additional properties such as upper semi-continuity, continuity, non-decrease, strict concavity, positive homogeneity and so on The characterizations are some kinds of abstract cyclic monotonicity strengthening revealed preference axioms, and also they may be considered as an extension of the Afriat-Varian theory to an arbitrary (infinite) set of 'observed data' Particular attention is paid to the case of smooth functions

K e y words: demand function, budget set, insatiate demand, utility function,

indirect utility function, rationalizing, strict rationalizing, inducing, strict ducing, Monge-Kantorovich problem (MKP) with a fixed marginal difference, cost function, constraint set of a dual MKP, concave function, strictly concave function, positive homogeneous function, superdifferential

in-Introduction

This article is devoted t o concave-utility-rational d e m a n d functions T h e problem of d e m a n d rationalizing is studied in m a t h e m a t i c a l economics Supported in part by Russian Foundation for Humanitarian Sciences (project 03-02-00027) A part of the material of this paper was presented

at the international conference "Kantorovich memorial Mathematics and economics: old problems and new approaches", St.-Petersburg, January, 8-13, 2004

since 1886 (Antonelli); for history and references see [4], [7] A tial role here is played by theory of revealed preference; see papers by Samuelson [31], [32], Houthakker [6], Uzawa [33], Richter [29] and others The revealed preference axioms, along with some regularity assumptions, give conditions for a demand function to be utility-rational In general,

substan-an utility function rationalizing the demsubstan-and function need not be cave It will be automatically quasiconcave when the revealed preference relation is convex If f/ is a quasiconcave utility function that rationalizes

con-a given demcon-and con-and if con-a function / i : R ^ ' R is (strictly) increcon-asing, then

the composition hoU rationalizes the demand and is quasiconcave as

well In some cases, such a composition proves to be not merely concave but concave [8] (see also [5]) Thus, revealed preference theory together with the concavifiability criteria enables to obtain conditions for demand rationalizing by a concave utility provided that the corre-sponding revealed preference ordering is convex A different approach to concave rationalizability is the nonparametric method of rationalizing a trade statistic; see papers by Afriat [1], [2] and Varian [34], [35]

quasi-In [20], [22] we proposed a new method in demand analysis (and in

an abstract rational choice theory) The method is based on duality sults relating to the Monge - Kantorovich mass transportation problem (MKP), a relaxation of an old 'excavation and embankments' problem due to Caspar Monge [26] As is shown in [20], [22], a demand function

re-is rationalized by a given concave utility function if and only if the responding indirect utility function belongs to the constraint set of an infinite linear program, which is dual to the MKP with a fixed marginal diflFerence and a special cost function The definition of that constraint

cor-set implies that a function u{p) belongs to it if and only if u solves a

system of inequalities extending the Afriat system to an infinite set of 'observed data' In [16], [21] we gave a criterion for such a constraint set

to be nonempty In general, this criterion means some kind of abstract cyclic monotonicity, and for a special cost function determined by a given demand, it is closely connected with axioms of revealed preference In [27], [3] the same approach is applied to problem of rationalizing reverse demand functions by positive homogeneous concave utility functions Furthermore, in case of smooth cost function, applying earlier results

by the author [13],[20] yields conditions in diff^erential form (separately, necessary ones and sufficient ones) for nonemptiness of the corresponding constraint set hence for concave utility rationalizing the demand

In the present paper we give further development of that method

We expose its main features in a more detailed and systematic way than before and generalize considerably our previous results

In what follows, P stands for a nonempty subset of

intR!^ = {p= {pu' ,Pn) : pi > 0 , ,pn > 0},

and P'q:= YliPiQi for any p = {pi, ,Pn), q = (gi, •., 9n) € R""

Consider a consumer buying n commodities ^ = (^i, • •, gn) ^ ^ + at prices p = ( p i , • ,Pn) ^ ^ and having income I{p) > 0 , and denote by B{p) her/his budget set,

B{p):={qeRl:p-q<I{p)}

The function I : P ^ (0, -|-oo) is assumed to be given Important ticular cases are I{p) = I (constant income; in such a case one can take / = 1 by passing to a new unit of money) and I{p) = p-uj (a consumer

par-with endowment a; G M!J: \ {0})

Given a set of data M = {{p^q)} C intE!J: x intR!J: is associated with a multifunction (demand map) D : intWl -^ 2'''^^+,D{p) := {q : (p^q) € M} Let P = domD := {p : D{p) ^ 0} and assume M to be compatible with consumer's budget, so that q e B{p) whenever (p, q) G

M or, equivalently, D{p) C B{p) for all p £ P Also we suppose that all D{p),p E P, are closed hence compact

The economic interpretation of D is as follows Being faced a price

vector p G P, the consumer prefers each bundle of commodities from D{p) to any bundle from B{p) \ D{p) When such consumer's choice is

concave-utility-rational? In other words, when there is a concave utility

function U on W^ that has nice properties and rationalizes D? The latter means that maximum of U on B{p) is attained at bundles q G D{p), i.e.,

D{p) C Arg max U\B{p) Vp G P

Of course, any D is rationalized by a constant function, and this is why

we say about a utility function with nice properties Another way to exclude this degenerate example is to consider the notion of strict ra-

tionalizing We say U strictly rationalizes D, ii D{p) = ArgmaxC/|5(p) whenever p e P

Rationalizing a trade statistic M = {(p*,^*) : t = 1 , ,m} where p* ^p^ iort ^ s (see [1], [2], [34], [35]) is an important particular case of

the general demand rationalizing problem (this variant of the problem

corresponds to a finite P , which is the p-projection oi M)

Our approach to the problem is as follows Given a demand map D (or a single-valued demand function f : P -^ intR!J:) and a Lagrange- Kuhn-Tucker multiplier A : P ^ R+, we take as a cost function (p one

of five specific functions as follows:

c\{p,p') : = KP') ( n^in y • q - I{p') ) ,

\qeD{p) J

C(p,p'):=p'-(/(p)-/(y)),

cHp,p') =

{p'-p)-m-Properties of the corresponding constraint set

Qo{^) := {ueR^: u{p) - u{p') < ^{p,p') \/p,p' € P}

and conditions for this set to be nonempty are the key points in our study

The structure of the paper is as follows Section 1 contains inary information on the Monge-Kantorovich problem including condi-

prelim-tions for Qo{(p) to be nonempty Section 2 is devoted to

concave-utility-rational demand maps We give criteria for concave-utility-rationalizing (strict concave-utility-

rational-izing) a demand map D : P -^ 2'"*"*+ by a concave utility function U with dorall D D{P) in terms related to nonemptiness of Qo{c\) (Theo-

rem 1) Also connections with revealed preference axioms are discussed and some information on the corresponding Lagrange-Kuhn-Tucker mul-tiplier is obtained Section 3 deals with single-valued demand functions

In Theorem 2 we give a criterion for such a function to be utility-rational (strict rational), and in Theorem 3 we describe all utiUty functions (within a broad class of concave functions) that rationalize a given insatiate demand function In Theorem 4 we characterize demand functions that are (strictly) rationalized by non-decreasing strictly con-cave utility functions Also we study demand functions that can be ratio-nalized by continuous (Theorem 5) and by smooth (Theorem 6) utility functions In that Section, all the results are based on nonemptiness conditions for the set QO(CA) and its subset

concave-QI(CA) := {u e QO(CA) : fip) ^ f{p') ^ u(jp) - u{p') < ^x{p,p')}

Section 4 is devoted to demand functions that are rationalized by itive homogeneous utility functions Here the cost function ^ is used

pos-In Theorem 7, that generalizes the corresponding variant of the Varian theory, necessary and sufficient conditions are given for an insa-tiate demand function / to be rationalized by a positive homogeneous (continuous) concave utility function, which is strictly positive on / ( P ) These conditions are equivalent to nonemptiness of Qo(0- ^^ Theorem 8

Afriat-conditions for Q o ( 0 ^^ t)e nonempty are established for a smooth / , and

in Theorem 9 we describe all positive homogeneous use concave utility

functions (within some natural class) that rationalize a given demand

function Finally, Section 5 is devoted to a stronger variant of demand

rationalizing In that variant, the budget constraint is rejected and the

gain to be maximized by a consumer equals utility minus expences We

say that / is induced (resp strictly induced) by a utility function U

if f{p) e ArgmaxC/P \/p e P (resp if f{p) = argmaxC/^ Vp G P ) ,

where the gain U^{q) := U{q) — p - q Theorem 10 characterizes

func-tions f : P -^ intM!J: that are induced by upper semi-continuous (use)

concave utility functions U with domU 2 f{P)- Here cost functions

C and C"^ are used Among other characterizations Theorem 10 asserts

that / has the stated property iff Qo{0 is nonempty or, equivalently,

iff Qo(C*^) is nonempty In Theorem 11 conditions for / to be strictly

induced by U are studied A necessary condition is nonemptiness of the

set Qi(C) := {u G Oo(C) : f{p) ^ f{p') =^ u{p) - u{p') < C(p,p')}, and

if f{P) is open or convex and closed, that condition is also sufficient In

Theorems 12, 13, and 14 we deal with the case where P is a convex

do-main and / is smooth (C^) Theorem 12 says that / is induced by a use

concave utility function C/with domU D f{P) (or, equivalently, Qo(C)

is nonempty) if and only if for every p G P the matrix {dfi{p)/dpj)ij is

symmetric negative semidefinite In Theorem 13 we show that if these

matrices are symmetric and negative definite, then / is strictly induced

by an utility function with the stated properties and Qi {Q is nonempty

Theorem 14 says that in case where f{p) = ( / i ( p i ) , , fnipn)), Qi(C)

is nonempty if and only if every fi is non-increasing

1 Preliminary information on the

Monge-Kantorovich problem

Let X and Y be closed domains in spaces R"^ and R"^, cri and G2 positive

Borel measures on them, aiX = a2Y, and c : X x y ^ R a bounded

con-tinuous cost function The Monge-Kantorovich problem MKP{c\ cri, cr2)

is to minimize a linear functional

{c,ß):= I c{x,y) ß{d{x,y)) (1.1)

JxxY

over the set r(cri,cr2) of positive Borel measures /i on X x y satisfying

TTi/x = <Ji, 7r2/i = (T2' Here, 7ri,7r2 are the natural projecting maps of

XxY onto X, y , and TTI//, 7r2// are the corresponding marginal measures:

for any Borel sets Bx C X and By C y ,

{iT2ß)BY : = ^T^2^{BY) = fi{X X BY)

The optimal value of MKP{c; 0-1,^2) is denoted as C(c; ai,a2) so that

C(c;ai,a2) := inf{(c,//) :/x G r((7i,(72)} (L2) This is the Monge-Kantorovich problem with given marginals a i , (72 and

a cost function c It is a relaxation of the Monge problem MP{c; CTI, (72)

that consists in minimizing the functional

: r ( / ) : = [ c{xj{x))ai{dx) (L3)

Jx over the set ^(cri,(j2) of measure-preserving Borel maps f : X —^ Y

A map / is called measure-preserving if f{(Ti) = (72, that is (72By =

crif~^{BY) for every Borel set By C F The optimal value of the Monge

problem is thus

V(c;(7i,(72) := inf{jr(/) : / e ^((71,(72)} (1.4)

Each measure-preserving map / € $((7i, (72) is associated with a measure

/jLf = ßficFi) € r((7i,(72), where /x/ = (idx x /)(cri) That is, for every

Borel set B CX xY,

fXfB = ai(idx X f)-\B) = ai{x: (x, /(x)) G B}, (1.5)

It is clear that (c, /x/) = ^ ( / ) , which implies C(c; (7i, (72) < V(c; cri, (72)

In general, this inequality is strict but in some cases it holds with

the equality sign Measures /x G F((71,(72) are called (feasible)

solu-tions and measures of the form /x/ with / G ^((71,(72) are called

Monge solutions to MKP{c\ai^a2)' If there exists an optimal solution

to Mi(rP(c;c7i,(72), which is a Monge solution /x/, then / is an

opti-mal solution to MP(c;(7i,(72) and C(c;(71,(72) = V(c;(71,(72) This is an

immediate consequence of the identity (c, /x/) = ?*(/)

Another type of MKP is the Monge-Kantorovich problem with a

fixed marginal difference It relates to the case Y = X and is formulated

as follows Given a (signed) Borel measure p(= cri — G2) on X such that

pX = 0 and a (not necessarily continuous) universally measurable cost

function (f : X x X ^^R^ the problem is to find the optimal value

A{ip; p) := inf I I ip{x, z) fi{d(x, z)) : fi> 0, TTI/X - 7r2/x = p\

UxxX )

Both types of MKP, with fixed marginals and with a fixed marginal

difference, were posed by L.V Kantorovich (see [9], [10], [11], [12]) who

examined the case where X — Y is a metric compact space with its metric as the corresponding cost function c = cp In such a case, two

problems are equivalent:

C((^; (71,0-2) = A{(p;ai - ( 7 2 ) ,

and there exists a measure ^ ET{ai,a2) which is an optimal solution to

both the problems Moreover, for each of two problems, a duality theorem holds true, that is optimal values of the corresponding original and dual infinite linear programs are equal These equivalence and duality remain

true when X is a (not necessarily metrizable) compact space and (p is a,

continuous (or merely a lower semi-continuous) function on X x X that satisfies the triangle inequality and vanishes on the diagonal [25] (Two variants of MKP cease to be equivalent when the triangle inequality

is not satisfied.) Generalizations of the duality theorems to MKPs on non-compact (or non-topological) spaces see [14], [17], [18], [19], [23] A crucial role in duality theory for general MKPs with a given marginal

difference is played by the reduced cost function

where x^ = x, x^ = z and (^(x, x) = 0 Vx E X , and by the constraint

set of the dual linear program

Q{(p) := {u e C{X) : u{x^) - ^/(x^) < (p{x^,x^), x^,x^ G X},

where C{X) stands for the space of bounded continuous real-valued tions on X; see Levin - Milyutin [25] and Levin [14] (Clearly (p^ = (p when (p satisfies the triangle inequality and vanishes on the diagonal.)

func-By analogy with Q((/?), we define a broader set

transportation, cyclically monotone operators, dynamic optimization, approximation theory, utility theory, demand theory); see [3], [13], [14],

[15], [18], [20], [21], [22], [23], [24\\ Many problems in those fields may

^ Most of the corresponding results with references to the original papers by the author may be found in a book [28], Chapter 5 Also in [28], Chapter

4, the duality theory from [25], [14] is expounded

be reduced to the single question of whether or not the set Qo(^) is

nonempty for some specific cost function ^

The following theorems are particular cases of more general results

that are contained in [13], [14], [16], [20], [21], [24]

Given a continuous cost function c on X xY and a map f : X —^Y,

we define on X x X the function

(^^(x, z) := c{z, f{x)) - c(x, / ( x ) ) (1.7)

Theorem A (Levin [21], [24]) Suppose that f e ^(cri,cr2) is continuous

and the support ofai,Z:= spt(cri), is compact The following statements

are equivalent:

(a)iif{ai) is an optimal solution to MKP{c\ai^a2) hence / is an

optimal solution to MP(c;cri,cr2);

(b)the set Qo{(p^\zxz) is nonempty

Theorem B (Levin [14], [16]) Given an abstract nonempty set X and

a function (^ : X x X —> R satisfying

the following statements are equivalent:

(ajQoi^f) is nonempty;

(h)for every positive integer I and every cycle x^,x^,

x^ in X, the inequality holds:

rj.1 ^Z + 1 _

Y,^{x\x^-^^)>Q (1.9)

k=i

(c)for every x £ X, (f^{x,x) = 0

If, in addition, X is a topological space and ip is continuous onXxX,

then every u G Qo{y^) is a continuous function on X

Theorem C (Levin [13], [20]) Let X be a domain in R^ Suppose that

if is C^ on an open set containing the diagonal P = {(x, x) : x G X} and

vanishes on D Then either Qo{^) is empty or there exists a C^ function

u{x), unique up to a constant term, that satisfies the equation

Vu(x) = -V^(^(x,z)U=^ (1.10)

In the latter case, Qo{^) = {u{')-\-a : a: G R} and</?*(x,z) = u{x)—u{z)

for all X, 2; G X

The next theorem generaUzes an earlier similar result by the author;

see [13], [20], where stronger regularity assumptions were imposed on (p

Theorem D Let X be a domain in W^ Suppose that: (i)ip vanishes on

the diagonally J ie.,(1.8) is satisfied, (ii)(f is C^ on an open neighborhood

ofD, and (Hi) on D the partial derivatives d^(p/dzidxj exist, and

d'^(fi(x,x) d^(p(x,x) „ r^ 1 /- -»-.N

^ ! ä ^ = ^ ä ^ foram,je{l, ,n} (1.11) The following statements hold then true:

(a) IfQoi^) is nonempty, then for every XEX, the matrix {d'^(p{x, x)/

dzidxj)ij is symmetric negative semidefinite

(b) Suppose, in addition to (i), (ii), (Hi), that X is convex and on D

there exist the partial derivatives d'^^p/dzidzj satisfying

^ ^ ; ä ^ = ^ ^ ä ^ V M G { l , , n } (1.12)

If for every x^z E X the inequality holds:

(1.13)

then Qo(^) is nonempty

(c) Suppose, in addition to hypotheses of (b), that the derivatives

dtdx ' dtd'z ^^^ continuous functions of x Then every u G Qoi^p)

isC^!

Proof, (a) Given u € QoC^)? we consider for every x E X the function

^^ on X as follows:

g'^iz) := u{z) + ^{x, z), zeX (1.14)

Since u G Qo{^), g^ is C^, and we have

p^(a:):=minp^(2:);

therefore Vg^{x) = 0 (this is the first order condition for z = a; to be the

minimum point oi g^) and the matrix {d'^g^{x)/dzidzj)ij is symmetric

positive semidefinite provided these second partial derivatives exist (this

is the second order condition for minimaUty of ^ = x) Taking into

account (1.10), we get

dg'^jz) ^ d(p{x,z) _ dip{z,z) dzi dzi dzi

hence Vg^{x) = 0 and the first order minimahty condition is thus

sat-isfied Moreover, (1.15), along with (iii), imphes that the derivatives

d'^g^{x)/dzidzj exist and

d^g^{x) _ d^ip{x,x) dzidzj dzidxj

and taking into account (1.11), the result follows

(b)Let us consider on X the vector field e{x) = ( e i ( x ) , ,en(a:)),

where

dcp{x,x) ei{x) = , z = l , , n

OZi

We have

dei{x) _ d^(p{x^x) d^ip{x,x) dxj dzidxj dzidzj

which, along with (1.11) and (1.12), implies

^ = ^ « - ^ ^ ( ' " > •

Then there is a C^ function u{x) satisfying (1.10)

It remains to verify that u G Qoi^)- To this end, fix x, z G X, denote

x{t) := tx-\- {1 — t)z, and consider the function

a{t):=g''{x{t)), 0 < t < 1, where g^ is given by (1.14) We have a(0) = g^{z) = u{z)-^ip{x^ z)^ a{l) =

(c) Continuity of the functions x i—> d^(p{x, x)/dzidxj and x i->

d^(p{x,x)/dzidzj implies that the right-hand side of (1.10) is C^, and

the result follows D

2 Concave-utility-rational demand maps

In what follows, P stands for a nonempty subset of

intR!|: = {p= ( p i , ,Pn) : pi > 0 , ,pn > 0},

and p • ^ := YliPiQi ^ r any p = ( p i , ,Pn),^ = (gi, "-^qn) e W^

The consumer's budget set is defined as

B{p):={q€Rl:p-q<I{p)}, (2.1)

where the income function / : P —> (0, +00) is assumed to be given

Definition 1 A nonempty-closed-valued multifunction D : P -^ 2^+ is

called a demand map if

D{p) C B{p) n intM!;: for every p e P (2.2)

A demand map D is called insatiate ifp-Qp = I{p) whenever p E P,qp £

Dip)

It follows from the above definition of a demand map that all

D{p),p G P, are compact subsets in B{p) fl intR!fi

Definition 2 A demand map D is said to be rationalized by a

util-ity function U : R^ -^ R U {-00} if for each p e P, D{p) C

Arg max C/15(p), i.e.,

D{p) c\qe B{p) : U{q) = max [/(g')} • (2.3)

A demand map D is said to be strictly rationalized by a utility function

C7 : E!^ -^ R U {-00} if for each peP, D{p) = ArgmaxC/|5(p), i.e.,

D{p) = \qe B{p) : U{q) = max C/(gO| • (2.4)

Definition 3 Given a utility function U : W^ -^ R U {—00} and a

budget map B : P ^^ 2^+, the function u(p) = sup^^^^p) U{q) on P is

called the indirect utility function associated with U

Let Z) be a demand map For any A : P —> R+ we consider on P x P

the cost functions

CA(P,PO := A(PO ( min p' • q - I{p')) (2.5)

and

c'x{p.p') := cx{p\p) = A(p) ( min p • q' - I(p)) (2.6) (both minima are attained because of compactness of D{p), p e P)

Recall that the superdifferential of a concave function U : W^ -^

R U {—00} at a point q e domU is the set

d'Uiq) :={p€R":p-iq- q') < U{q) - U{q') \/q' € M!^},

or, equivalently, d'U{q) = —d{—U){q), where d stands for the

subdiffer-ential of a convex function; elements p € d'U{q) are called supergradients

of U at q

Definition 4 Say a function U : W^ —^ R U {—00} is:

- non-decreasing if U{q + g') > U{q) whenever g, q^ G W^;

' increasing if U{q -h g') > U{q) whenever q € R!^, g' G int R!J:;

- strictly increasing if U{q + q') > U{q) whenever g, g' G R!J:, g' ^ 0

Theorem 1 Gi^en a demand map D, the following statements hold

true:

(a) Suppose D is rationalized by a concave utility function U : R!fi —>

R U { - o o } with domUD D{P) where domU := {g G R!^ : U{q) > - 0 0 }

Then there exists a function A : P —> R_|_ such that

X{p)(p • q — lip)) = 0 for every p e P and every q G D{p) (2.7) and that the indirect utility function u associated with U belongs to

Qo{cx) In such a case, X{p)p is a supergradient ofU at every q^ G D{p):

non-decreasing use concave utility function U : R!f: —> R U {—00} with

domU D D{P) such that D is rationalized by U and u is the indirect

utility function associated with U As such a utility function one can take

U{q) := inf {«(p) + A(p)(p • q - /(p))}

peP

(c) Let A and u be as in (b), and suppose, in addition, that u satisfies

(2.8) and that the set D{P) is either open or convex and closed Then

there exists a use concave utility function U such that domU D D{P),

u is the indirect utility function associated with f/, and D is strictly

rationalized by U

(d) Suppose that X: P -^ R+ satisfies (2.7) Then Qo{cx) is nonempty

if and only if, for every positive integer I and for every cycle p^^p^, -»p^

pi+i _ pi ^^ p^ ^^g inequality holds

I

fc=i

whenever q^ G D{p^)^ fc = 1 , , /

Proof, (a) Fix p e P and consider the constraint maximization problem:

U{q) -^ max, qeR^, P'q< I{p),

It is clear that each q e D{p) is an optimal solution to this problem,

and that the Slater condition is satisfied: there is an element ^o = 0 in

R!J such that p- qo < lip)- By the Kuhn-Tucker theorem, there exists a

Lagrange-Kuhn-Tucker multiplier A(p) > 0 such that, for every q e D{p),

(2.7) holds and

U{q)>U{q')-^X{p){I{p)-p-q') V^'€ R!^ (2.9)

It follows from (2.9) and (2.7) that U{q) = U{q') for q,q' e D{p);

therefore the equality holds

u{p) = U{qp) for all p e P, (2.10) where u is the indirect utility function associated with U and qp is taken

arbitrarily from D{p) Furthermore, (2.9) implies —u E (5o(c^); therefore

u e QO(CA)- Also, since I{p) > p-qp, (2.9) implies U{qp) > U{q')-\-\{p)p'

{% — q') whenever q' G R!J:, that is \{p)p G d'U{qp)

Suppose now that D is strictly rationalized by U and show that (2.8)

holds true To this end, take any qp G D{p) and any qp' G {B{p)\D{p))r\

D{p') We have u{p) = U{qp), u{p') = U{qpf) and, as g^/ G B{p) \ D{p),

it follows that U{qpf) < U{qp)^ i.e., u{p^) < u{p)

(b) Since u G Qo{cx), it follows from (2.5) and (2.7) that u{p) = u{p')

whenever D{p) fi D{p') ^ 0 Then the function U{q) := n(p), where

p e P and q = qp is an arbitrary element of D{p), is well-defined and

finite on D{P) The inclusion u G QO(CA) is equivalent to the inclusion

—u G Qo(c^)j which, in turn, can be rewritten as

Uiqp) > U{q) + A(p)(7{p) -p-q) (2.11)

for all Qp e D{p) and all q e D{P) Since, in view of (2.7), c^ vanishes

on the diagonal, we see from (2.11) that, for every q G D{P),

U{q) = inf {u{p) + X{p){p q - I{p))} (2.12) peP

We extend now U to the whole of W^ by formula (2.12) Obviously

the extended function is non-decreasing use concave, domU 2 D{P),

and D is rationalized by U Furthermore, in virtue of (2.11) and (2.7),

we have u{p) = U{qp) = max{C/(g) : q G B{p)}^ that is u is the indirect

utility function associated with U

(c) First suppose that D{P) is open and take [/ as in the proof of (b)

We have to verify that D is strictly rationalized by U Take q^ G D{p)^

q G B{p) \ D{p) U q e D{P), then q G D{p') for some p' G P , and

from (2.8) it follows that U{qp) > U{q) li q ^ D{P), then by (2.11)

U{qp) > U{q), and it remains to show that the inequality is strict Indeed,

if U{qp) = U{q), then, by concavity of [/,

C/((l - a)qp + aq) > U{qp) whenever 0 < a < 1 (2.13)

Clearly {l — a)qp-^aq G B{p) for every a, 0 < a < 1 Furthermore, since

D{P) is open, D{p) is a compact subset in D{P), q ^ D{p), and qp G

D{p), there exists a, 0 < a < 1, such that {l — a)qp-haq G D{P)\D{p)

Hence (1 - a)^p + aq e D{p') fl (5(p) \ D{p)) where p ' ^ P^p' ^ p,

and applying (2.8) yields [/(g'p) > C/((l — a)gp -I- aq)^ which contradicts

(2.13) Thus, in the case where D{P) is open the proof is complete

If now -D(P) is convex and closed, we define

t / ( g ) : = [ / i ( g ) - d i s t ( g , Z ? ( P ) ) , (2.14) where

UM ••= inf{«(p) + A(p)(p • q - I{p))} (2.15) Then U{q) < Ui{q) whenever q ^ D{P) Clearly [/ is use concave and

domU = domf/i 2 D{P) According with (b), u is the indirect utility

function associated with C/i, and D is rationalized by Ui Since Ui{q) =

[/(g) for every q G D ( P ) and f7(g) < Ui[q) for g ^ D{P), it follows that

1/ is the indirect utility function associated with U and D is rationalized

byC/

It remains to verify that D is strictly rationalized by U To this end,

take qp G D{p),q G 5(p) \ L)(p) If g G D{P), then g G D{p') for some

/ G P , and from (2.8) it follows that U{qp) = u{p) > u{p') = U{q), If

q ^ D{P) then, taking into account (2.14) and (2.15), we get U{qp) =

Ui{qp) = u{p) > u{p)+\{p){p'q-I{p)) > Ui{q) = U{q)+dist{q,D{P)) >

U{q)

(d) This is an easy consequence of Theorem B D

There is a close connection between nonemptiness of Qo{cx) and

strong axioms of revealed preference

Recall that Houthakker's strong axiom of revealed preference (SARP) [6] relates to a single-valued demand A traditional form of SARP is as

follows: if / > 2 is an integer, p\.,, ,p^ € P , p^+^ • {D{p') - D{p^-^^)) <

0, i = 1 , , / - 1, and D{p^) ^ D{p^), then p^ • {D{p^) - D{p^)) > 0

The next formulation is clearly an equivalent restating the axiom: given

a cycle p ^ , ,p^p^"*"^ = p^ in P such that, for at least one z € { 1 , , / } ,

D{p^) ^ £)(p*'^^), then the inequality holds

max{p2 (^1 _ ci\ ,,.J- {q'-' - q%p' • {q' ~ q')} > 0, (2.16) where q'' = D{p^), k = 1 , , / Notice that if D{p') ^ D{p'-^^) and D{p') ^ P(p^+^), then p'-^^'D{p') > /(p^+^) > p^+i •D(p^+i) and (2.16)

proves to be trivial Therefore SARP may be reformulated equivalently

as follows: if p ^ , ,p^p^"^^ = p^ is a cycle in P such that, for at least

one i e { 1 , , / } , D{p') ^ D{p'-^^) and D{p') e B{p'-^^), then for

q^ = D{p^), fc = 1 , , /, (2.16) holds true

This formulation of SARP is extended to a multivalued demand as follows

Definition 5 Say a demand D satisfies SARP if the following condition

is valid Given a cycle p ^ , ,p^p^"^^ = p^ in P such that, for at least one i G { 1 , ,Z}, {B{p'^^)\D{p'-^^))nD{p') ^ 0, then inequality (2.16) holds whenever q^ G D{p^)^ A; = 1 , , /

Definition 6 Following Varian [34], say a demand D satisfies the

generalized axiom of revealed preference (GARP) if, for every cycle

p ^ , ,p^p'•*•^ = p^ in P, the inequality

max{p2 (gl - q% J (g'-i - g ' ) , p i (g' - q^)} > 0 (2.17) holds whenever q^ G D{p^), A: = 1 , , /

Proposition 1 Given a demand map D : P -^ intR^J:, the following

statements hold true:

(I)If D is insatiate, A is strictly positive, and QQ{C\) is nonempty, then D satisfies GARP

(11)If there exist A and u G QO(CA) satisfying (2.7) and (2.8), then

whenever q^ G D{p^)^ k = 1, ,l^q^'^^ := q^ Summing up these

in-equalities yields

I

Yl A(/+i)p'^+i {q^ - q^+^) > 0,

k=i

and as p^+i = p \ g ' ^ ^ = q\ and all X{p^) > 0, (2.17) follows

(II) Taking into account (2.7) and (2.8) we have

« ( p ' = ) - u ( / + i ) < CA(p^p'=+^) < A(p'=+i)p'=+i-(g'=-g'^+i), k ^ i (2.18)

and

u(p'+^) - u{p') > 0 (2.19)

Summing up inequalities (2.18) and taking into account (2.19), we get

Yl A(p*+')p'=+i • (g* - q''+^) > w(p'+i) - u{p') > 0;

kik^i

hence

max A ( / + i ) / + ^ • (g'^ - q^-^^) > 0,

and, as all A(p^) > 0, (2.16) holds true D

Remark 1 Suppose P and D{P) are finite, and D is insatiate In such

a case, the relation u G Qo{cx) ioi X : P —> intM^ means positive

solvability of an appropriate Afriat's system of inequalities, and GARP

proves to be equivalent to the existence of a strictly positive A such that

Qo{cx) is nonempty; see the proof of (2)=>(3) in [34], p.969

Dejßnition 7 Given a utility function U : W^ —> Mu{—oo} rationalizing

a demand map D, we say a function X : P -^ M4 is compatible with U

if it is a Lagrange-Kuhn-Tucker multiplier with regard to U, i.e., if {2.7)

and (2.9) hold true whenever p G P and q G D{p)

The next result is an immediate consequence of statements (a), (b)

of Theorem 1

Corollary 1 Suppose D is rationalized by a concave utility function

U :Wl^ MU{-oo} with domC/ D D{P) LetX:P-^R^ be compatible

with Uj and let u be the indirect utility function associated with U Then

there exists a non-decreasing use concave utility function U' : W^ -^

R U {-00} such that U'{q) > U{q) for every q G M!;:, U'{q) = U{q)

for q G D{P) (hence dovaU' D doiaU), D is rationalized by U', X is

compatible with U', and u is the indirect utility function associated with

U' The function U' is as follows:

U'{q) = inijuip) + X{p){p • q - I{p))}

-Then D{P) = (0,2] It is easily seen that D is (strictly) rationaUzed

by a concave utility function C/, as follows:

is compatible with U An easy calculation shows that a non-decreasing

function t/' from Corollary 1 is as follows:

U\q) \q, i f O < g < l ,

11, i f 9 > l

Remark 2 It is clear that if D is rationalized by an increasing utility

function, then it is insatiate

Proposition 2 Suppose D is insatiate Let U be an increasing concave

utility function such that: (i)domU D D{P), (ii) for every p e P, U is continuous on domU f) B{p)j and (Hi) the interior of domU 0 B{p) is nonempty The following statements are then equivalent:

(a) D is rationalized by U;

(b)for every p G P, the inequality holds

sup U{q')-u{p) < inf u{p) - U{q')

92€R!f :p-92>/(p) p-q'^- lip) q^mi:p-g^<I(p) / ( p ) - p • q^ ' (2.20)

where u is the indirect utility function associated with U

If these equivalent statements hold true, then any number lying

be-tween the left-hand and the right-hand sides of (2.20) can be taken as a

multiplier X{p) compatible with U

Proof (a)=^(b) By the Kuhn-Tucker theorem, there is a

Lagrange-Kuhn-Tucker multipHer A : P —> R+ such that

U{qp) = u{p) (see (2.10)), (2.20) follows

( b ) ^ ( a ) Given p e P and q G B{p) we have to show that U{q) <

U{qp), where qp is some (any) element of D{p) We assume q G domC/,

otherwise the inequality is obvious Fix qp G D{p) and take g^ > g^;

then P'{q^ — qp) > 0, and as C/ is increasing, U{q'^) > U{qp) Taking into

account that u{p) = U{qp) (see (2.10) and p- qp = I{p), we get

„p £ ( 4 ^ > H Ö _ ^ > 0 (2.21)

q^eRl:pq^>I{p) P ' Q'^ ' I{p) P ' {Q^ - Qp)

li P' q < I{p) then, for q^ = q, from (2.20) and (2.21) it follows

that U{q) < U{qp) li p - q = I{p), then we find a convergent sequence

q^ G int (dom U H B{p)), q^ -^ q Since q^ G int B{p), we have p - q^ <

I{p); therefore U{q^) < U{qp), and as C/ is continuous on domC/fi 5 ( p ) ,

Uiq) = lim Uiq") < U{qp) D

k—^oo

The next result generalizes Proposition 2 to the case where U is not

supposed to be increasing

Proposition 3 Suppose D is insatiate Suppose also that U is a concave

utility function with domC/ D D{P) and, for every p e P, U is

contin-uous on domU f) B{p) and the interior of domC/ fl B{p) is nonempty

The following statements are then equivalent:

(a) D is rationalized by U;

(b)for every p G P the inequality holds

max < 0 sup ^^'i")-<P)

where u is the indirect utility function associated with U

If these equivalent statements hold true, then any number lying

be-tween the lefl-hand and the right-hand sides of (2.22) can be taken as a

multiplier X{p) compatible with U

Proof, (a)=^(b) This follows from the Kuhn-Tucker theorem

(b)=>(a) For every p E P we take X{p) from the condition

max < 0, sup ^ Ä Z ^ <A(p)< inf "(^)-^(^^)

g-GR?: P'Q^-HP) I q'eR^: I{p) - P ' 0^

pq^>I{p) J P-Q^<HP)

(2.23)

It follows from (2.23) that A(p) > 0 and

U{qp) = u{p) > U{q) + \{p){I{p) -p-q) (2.24) whenever qp G D{p),q G Wl;.,p • q ^ I{p)' If we show that (2.24) holds

true for all q € M!f:, then the implication will follow from the

Kuhn-Tucker theorem For q ^ dom U (2.24) is trivial If now p - q = I{p)

and q e domf/, then q G domU D B{p) and we can find a sequence

q^ G int {dom.Ur[B{p)) such that q^ converges to q Since q^ G intB{p)^

we have p - q^ < I(p)\ consequently, every q^ satisfies (2.24) Now, by

using continuity of U on dom [/ D 5 ( p ) , we get

U{qj,) > lim (C/(g'=) + A(p)(/(p) - p • q")) = U{q) + Xip){I(p) -p-q),

fc—>00

and the proof is completed D

Remark 3 Clearly (2.20) along with (2.10) implies (2.22); therefore

Proposition 2 proves to be a consequence of Proposition 3

3 Concave-utility-rational demand functions

In this Section, demand is considered as a single-valued function of prices

Definition 8 Given a set P C intW^ and a budget map B : P -^

2^+, B \py-^ B{p)j ^^ ^^2/ that f : P ^^ W^ is a demand function if

f{p) e Bip) n intM!J: for all p e P (3.1)

A demand function f is called insatiate if p • f[p) = I{p) for all p E P

Remark 4- Here / is not assumed to be continuous Moreover, any

f : P -^ intR!f can be considered as a demand function with regard to

a budget set B{p) = Bf{p) := {q e Wl : p - q < p - fip)} determined by

the income function I{p) = If{p) := p • f{p) In such a case, condition

(3.1) is satisfied automatically

Definition 9 We say a demand function f is rationalized by a utility

function U : W^ —^ R U {—oo} if for each p G P,

f{p)eAvgmaxU\B{p), (3.2) i.e.j U{f{p)) > U{q) for all q e B(p) We say f is strictly rationalized

by a utility function U : W^ —> E U {—oo} if for each p £ P,

f{p) = 8iTgmaxU\B{p), (3.3)

^.e U{f{p)) > U{q) for all q e B{p) \ {/(p)}

Let / be a demand function For any A : P —> R_|_ we consider on

P X P the cost function

<xip,p') ••= WW • m - lip'))- (3.4)

Theorem 2 Given a demand function f, the following statements hold

true:

(a) If f is rationalized by a concave utility function U : W\ -^ M U

{—oo} with domC/ D f{P), then there exists a function X : P -^ R4

such that

mip-m-Hp))=o ^pep (3.5) and that the indirect utility function u associated with U belongs to

Qo(Cx)' If^ ^^c/i a case J X{p)p is a supergradient ofU at f{p):

Xip)ped'U{f{p)) V p e P (3.6)

If in addition^ f is strictly rationalized by U, then an implication

holds as follows:

fip') 7^ m , fip') e B{p) ^ u(p) > u(p') (3.7)

(b) If X : P —^ R+ satisfies (3.5) and u G (5O(CA); then there exists a

non-decreasing use concave utility function U : W^ —> R U {—00} with

domU D f{P) such that f is rationalized by U, X is compatible with U,

and u is the indirect utility function associated with U As such a utility

function one can take

U{q) = inUuip) + X{p){p • q - /(p))} (3.8) peP

(c) Let A and u he as in (b), and suppose, in addition, that u satisfies

(3.7) and that the set f{P) is either open or convex and closed Then

there exists a use concave utility function U such that domU D f{P), u

is the indirect utility function associated with U, and f is strictly

ratio-nalized by U

(d) Suppose that X : P -^ R_f- satisfies (3.5) Then QO(CA) is nonempty

if and only if for every positive integer I and for every cycle p^^p^, "",p\

pZ+i _ pi j^rjri p^ the inequality holds

J2 xip'^')?'^' • ifip") - fip"^')) > 0

fc=l

Proof This is a direct consequence of Theorem 1 taking into account

that, for a single-valued multifunction D = f, the function CA, as given

by (2.5), turns into CA- D

Remark 5 If / is rationalized by a concave function C/, which is

differ-entiable at / ( p ) , p £ P, then (3.6) impUes

w V ^ ( / ( P ) ) - / ( P ) ^ ^ p

P' f{P)

In the next theorem we consider a class of concave functions U :

R^ -^ E with nonempty sets d'U{q)nmtR%, q G intR!^, and completely

describe those functions inside the class that rationalize a given insatiate

demand function

T h e o r e m 3 Suppose U : R!^ —^Risa use concave utility function

such that, for every q e intR!f., d'U{q) flintR!J: is nonempty Given an

insatiate demand function / : P —> int R!f:, the following statements are

CA'(P,P') := X'{p')p' • if'ip) - f'ip')) ^p,p' e P'

(Without loss of generality, one can take A'(p') = 1 for allp' ^ P' \ P.)

Proof (a)=^(b) For every q G int R!J:\/(P) we chose p^ € d'U{q)nmtWl,

define

P':=PU{p,:qemtRl\f{P)},

consider the multifunction D : P' -^ c^intR^ ^ where D{p) = f{p) for

p € P , D{p) = {q:p = pq} for p e P'\P, and take f'\P'-^ int W\ to be

an arbitrary selection of D Also we set \'{p) = A(p) for p G P,\\p) = 1

for p e P'\P, and define tz'(pO = U{f{p')) for all p ' G P ' Since,

by Theorem 2, A(p)p G d'U{f{p)) for every p e P and, by definition,

p ' G d'U{f'{p')) for every p ' G P ' \ P , we have

U{q)-U{f{p'))<\\p')p'-{q-f\p'))

whenever q G M!}:,p' G P ' This implies

C/(g) < if^,{u'{p') + A ' ( p ' y • {q - /'{p'))} Vg € Rl- (3.10)

It follows from [30], Theorem 10.2, that U and the right-hand side of

(3.10) are continuous functions of q Therefore, it suffices to prove (3.9)

for all q G intR!f: For q = f{p) G / ( P ) we have

U{q) = u'ip) = min Wip') + ^(^0^ • {q - f{p%

and (3.9) is thus satisfied If q e intRlJ \ f{P), then pg € d'U{q) D

d'Uif'ipg)) We get C/(/'(p,)) U{q) < p , • (fipg) q) and U{q)

-Uif'iPg)) <P,-iq- f'ip,)) hence C/(g) - C/(/'(p,)) = p , • (g - / ' ( p , ) )

account that u' G QoiCy)^ we get

Uif'ip)) < u'ip) < u'ip') + \'(p')p' • if'ip) - f'ip'))

whenever p ' € P' Since, again by (3.9),

infp'6p/{u'(p')+A'(p')p'-(/'(p)-f'ip'))} = Uif'ip)), we get

u'{p) = Uif'ip)) (3.12)

Now (3.11) together with (3.12) imply

t / ( 9 ) < C / ( / ( p ) ) + A ' ( p ) p ( ^ - / ( p ) ) < f / ( / ( p ) ) - f V ( p ) ( p g - / ( p ) ) < f / ( / ( p ) ) ,

hence / is rationalized by U and A = A'|P is strictly positive and

com-patible with U D Remark 6 Clearly / ' can be considered as an insatiate demand function

with regard to the price set P' and the income I'{p') := 'p''f'(p')^p' G

P'-Statement (b) impUes that / ' is rationalized by 17, A' is compatible with

C/, and u' is the indirect utility function associated with U

Definition 10 Given a concave function U : W^ —^ M U {—oo} and a

(not necessarily convex) open set M C dom U, we say that U is strictly

concave on M if, for every q £ M and for every supergradient p €

d'U{q)j the inequality holds:

Uiq)-Uiq')>p-{q-q') V9'€ R^ \ {g}

If M is convex, this definition turns into the standard one:

U{{l-t)q-\-tq')>{l-t)U{q)-htU{q') whenever 0<t<l, q,q'eM, q ^ q\

Given a function X : P -^ M-|_ satisfying (3.5), we consider the set

QI(CA) := {u e QO(CA) : u{p)-u{p') < Cx{p.p') whenever f{p) ^ f{p')},

(3.13)

Theorem 4 Suppose f{P) is open, the following statements hold then

true:

(a) If f is rationalized by a concave utility function U : R!f —^ R U

{—oo} such that domU D f{P) and U is strictly concave on f{P) (in

such a case, f is strictly rationalized by U), and if X : P -^ M^ is

compatible with U, then the indirect utility function associated with U

belongs to

QiiCx)-(bJIfXiP—^ R-i- satisfies (3.5) andQi{(^x) is nonempty, then for

ev-ery u G QI(CA) there exists a non-decreasing use concave utility function

U : R!f: -^ RU {—oo} such that: (i)domU 2 f{P) and U is strictly

con-cave on f{P), (ii)X : P -^ R+ is compatible with U, (Hi) f is (strictly)

rationalized by U, and (iv) u is the indirect utility function associated

with U

Proof (a) It follows from Theorem 2 that X{p)p e &U{f{p)) \/p e P

(see (3.6)), and as f/ is strictly concave on / ( P ) , we get

U{f{p)) > U{q) - X(p)p • {q - f(p)) (3.14)