Duality between direct and indirect utility functions under minimal hypotheses

Journal

of Mathematical

Economics

20 (1991)

199-209.

North-Holland

Duality between direct and indirect utility functions under minimal hypotheses J.-E. Martinez-Legaz* Universidad de Barcelona, 08071 Barcelona, Spain Submitted

December

1988, accepted

December

1989

We give a characterization of those functions which can be obtained as the indirect utility function associated with the utility function of a consumer. This permits to formulate the duality between direct and indirect utility functions in the most general possible setting, which exhibits a perfect symmetry.

1. Introduction

The duality between the utility function of a consumer and the corresponding indirect utility function has been studied extensively [see Diewert (1982) and the references contained therein]. Crouzeix (1977) established a quite symmetric duality for continuous functions and, later, for the differentiable case [see Crouzeix (1983)]. Some abstract approaches to the duality between direct and indirect utility functions, based on generalized conjugation theory, can be found in the works of Volle (1985) and Singer (1986, 1987). The latter author has introduced a general notion of conjugation ‘of type Lau’, the name being due to one of the economic theorists who first studied these duality relations [see Lau (1969)]. The aim of this paper is to give such a symmetric duality under the weakest possible assumptions. In doing this, we obtain a simple characterization of indirect utility functions and, symmetrically, of those utility functions which can be recovered from its associated indirect function. *I wish to express my gratitude to Professor Manuel Santos for his careful reading of the manuscript and for his useful suggestions for improving its presentation. Likewise, I thank Professor Jean Pierre Crouzeix for his remarks, especially for supplying the proof of the fact that (1) holds on int K* for any non-increasing function II (in the first version of the paper, I proved it only for the much smaller class of non-increasing evenly quasiconvex functions). Thanks are also due to an anonymous referee, who suggested to discuss the case of finite valued utility functions, thus leading me to obtain Theorems 2.6 and 2.7 and Corollary 2.8. Financial support from the Direccibn General de Investigacibn Cientifica y Tbcnica (DGICYT), under project PS89-0058, is gratefully acknowledged 0304-4068/91/$03.50

0

1991-Elsevier

Science Publishers

B.V. (North-Holland)

200

J.-E. Martinez-Legaz, Duality between direct and indirect utility functions

In order to present the mathematical results in a suitable abstract framework, we will place ourselves in a locally convex topological vector space X with dual X*. A preorder S is given on X, defined by x1 Ix, if x2-x1 EK (xi, x1 EX), where K is a closed convex cone (the positive cone) in X. From an economic viewpoint, K may be interpreted as the set of possible consumption vectors of an economy. Of course, in standard applications, X = R” and K is the non-negative orthant. The positive cone in X* is K*=(x*EX*l( x,x*) 20 for all XEK), where (x,x*) denotes the value of the continuous linear functional x* at x. If X* is equipped with the weak* topology, which we will assume throughout the paper, then K* is also a closed convex cone. It induces a preorder on X*, which we will also denote by S, defined by XTS xt if xt -XT E K* (x7, xt E X*). It is well known that, dually, K={xeXl( x, x*) 20 for all x* E K*} [see, for instance, Ponstein (1980, p. 39)]. In the economic setting, the elements of K* may be regarded as price vectors, so that (x,x*) indicates the value of the consumption represented by x under the price vector x*. When X=R”, X* can be identified with R” in such a way that (s, *) becomes the usual Euclidean scalar product. In this case, if K is the non-negative orthant then K* = K. The viewpoint of a consumer is represented by his utility function u:K+R (R denotes the extended real line), which reflects his preference structure with respect to possible consumptions. The indirect utility function associated with v is u:K*+R, defined by u(x*) = sup {u(x) 1(x, x*> s l}

(x* E K*).

It assigns to any price vector the greatest utility which the consumer can achieve when he is constrained to spend no more than one unit of money. It is natural to assume that v is non-decreasing, i.e., that u(xl) S u(xJ whenever x1, X,EK satisfy x1 5x2. Analogously, any indirect utility function u is non-increasing, which means that u(x:)z~(xT) for any XT,xr EK* such that x7 Sxf. Although a utility function v usually takes only finite values, its associated indirect utility function may take the value + co, since it is defined as a supremum. Therefore, a more realistic assumption should be to suppose that v is real valued and that u takes values in R u { + m} but, for the sake of symmetry, we adopt the point of view that all functions are extended real valued. Nevertheless, we shall also give a simple set of necessary and sufficient conditions for an indirect utility function u to be such that the greatest direct utility function with which u is associated is finite valued. We will employ the symbols cl, bd and int to denote the topological closure, boundary and interior, respectively, of sets either in K or in K*. The lower semicontinuous hull of a function u:K*+R (i.e., its greatest lower semicontinuous minorant) will be represented by U; similarly, u will denote

J.-E. Martinez-Legar,

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201

the upper semicontinuous hull of u:K+R (defined as its smallest upper semicontinuous majorant). We recall that an extended real valued function f defined on a convex subset of a locally convex topological vector space is said to be quasiconvex if its level sets S,(f) = f- ‘([ - co, A]), 2 E R, are convex; if -f is quasiconvex, then f is called quasiconcave. When the level sets of f are evenly convex, in the sense of Fenchel (1952), i.e., intersections of open halfspaces, f is called evenly quasiconvex [see Martinez-Legaz (1983), Passy and Prisman (1984)] and -f evenly quasiconcave. Since, as a consequence of the Hahn-Banach theorem, any open or closed convex set is evenly convex, the class of evenly quasiconvex functions includes those quasiconvex functions which are either upper semicontinuous or lower semicontinuous. Moreover, any quasiconvex function of one real variable is evenly quasiconvex. The class of evenly quasiconvex functions is, in a sense, the most appropriate to develop a conjugation theory in quasiconvex analysis [see Martinez-Legaz (1988) and the references contained therein].

2. Characterization

of indirect utility functions; duality results

The symmetric character of the duality between direct and indirect utility functions has been made evident by Crouzeix (1977, 1983) who has obtained conditions which, when imposed to one of them in conjunction with continuity or with differentiability, ensure the corresponding properties of the other function. Our purpose in this section is, rather than giving new sets of assumptions under which a symmetric duality must hold or which guarantee some desired property of the indirect function, to identify those conditions which are both necessary and sufficient for these duality relations to be satisfied or, equivalently, to give an axiomatic characterization of indirect utility functions. By the symmetry of the duality relations between a class of direct functions and that consisting of their corresponding indirect ones we mean that, taking the opposites of the functions in one class and exchanging the roles of K and K*, one obtains the other class. It is quite easy to show that any indirect function is non-increasing and evenly quasiconvex, but these properties are not characteristic for indirect functions. However, we will prove (Theorem 2.2) that, imposing an additional relation [see (1) below] between the function and its lower semicontinuous hull, one obtains a set of necessary and sufficient conditions for being an indirect utility function. We will need the following result: Lemma

2.1.

Let

u: K*+R

be non-increasing.

Then ii is also non-increasing.

Proof. We shall prove that the level sets of U satisfy the condition x* +S,(U)cS,(ii) for any AER and x* EK *. According to Crouzeix (1982, p.

202

J.-E. Martinez-Legaz,

Duality between direct and indirect utility functions

11 l), one has S,(u) = nF,l cl S,(u). Let xz E S,(U), p>il and let V be a neighborhood of the origin in X *. Since x,*~clS,,(u), we have (xg*+ V)n S,(u) # 4, whence, using that u is non-increasing, we obtain (x*+x0*+ V)nS,(u)=~(x*+x$

+

V)n(x* +S,(u))

=x* +((xg* + V) n S,(u))

# 4.

This proves that x* +x0* E cl S,(u) for any x0*E cl S,(u), i.e., that x* + cl S,(u) c cl S,(u). Therefore, x*+S,(ii)=x*+n,,,

cl S,(u) = n,,,

cn,,,clS,(u)=S~(IS).

cl (x* + S,(u)) Cl

Our main result establishes a symmetric duality between direct and indirect utility functions under the weakest possible assumptions, as well as a characterization of those functions which can be obtained as indirect utility functions: Theorem 2.2. Let u:K*-+R. There exists v:K-+i? such that u is the indirect utility function associated with v if and only tf u is non-increasing, evenly quasiconvex and satisfies the condition

u(xo*)~ lim U(crxo*) u+1-

In this case, satisfying

(x; E bd K*).

v can be taken non-decreasing,

v(xO)>= lim z~(ctx,)

(x,, E bd K).

(1) evenly quasiconcave

and

(2)

a-+1-

Under these conditions, v is unique, namely, v is the greatest function with which u is associated: furthermore, it satisfies

v(x) =inf (u(x*) 1(x,x*)

5 1)

(x E K).

(3)

Proof: We first prove the ‘only if’ statement. It is well known, and easy to check, that u is non-increasing. Let us prove that any level S,(u) is evenly convex. If xt E K*\S,(u), since

Duality between direct and indirect utility

J.-E. Martinez-Legaz,

there exists x,,EK x* E S,(u), by

such that

(x0, x;l;) 5 1 and u(x,)>I.

functions

203

Hence, for any

we must have (x,, x*) > 12 (x,, xz). This proves that, indeed, S,(u) is evenly convex. Therefore, u is evenly quasiconvex. To prove that (1) holds, let x0*EK* and cc~(O, 1). Suppose that ti(axg) 12 (x,,, xt), whence (x,, x*) 2 1 for any x* E cl S,(u). In particular, (x,, ax:) 2 1 and we obtain (x,, xt) 2 l/a> 1, which yields a contradiction. Therefore, we must have U(ax,*)zu(x,*) for any ae(0, l), whence, using that U is non-increasing (see Lemma 2.1), we arrive at lim U(axo*)= a+1-

inf .E(O.

ti(axt) zu(x,*). 1)

Conversely, let us assume that u is non-increasing, evenly quasiconvex and satisfies (1). Then, the function u defined by (3) is, clearly, non-decreasing and, similarly to the proof of the ‘only if’ part (namely, ‘reversing signs’), it can be shown to be evenly quasiconcave and to satisfy (2). We shall prove that u is the indirect utility function associated with u i.e., that, for any x; E K*,

u(xg) = sup inf {u(x*)l (x, x*) s l} 1(x, xc) 5 1 . x i X* I

(4)

The inequality 2 in (4) is immediate (and does not require any assumption on u). Moreover, the right-hand side of (4) is, obviously, not less than infu(K*). Hence, it sufftces to prove the inequality 5 in (4) under the hypothesis that u(xg) > inf u(K*). Let AE (inf u(K*), u(x5)). Since x0*#S,(u) and this level set is evenly convex, there exists xOeX such that (x0,x*) > (x0, xg) for every x* ES~(U). From S,(u) #+ and the fact that u is nonincreasing, one can derive, using a standard argument, that (xo,y*) 20 for any y* E K*, i.e., that x0 E K. Then (x0, x;l;) 20. If x5 E int K*, this inequality holds strictly, since x0 #O. If x$ E bd K* n cl S,(u), by (l), Lemma 2.1 and the formula of Crouzeix (1982, p. 111) for the lower semicontinuous hull of any function, we have u(2xo*)5 lim U(2ax,*)= a-11-

U(2axg)

inf ae(O,

1)

204

J.-E. Martinez-Legaz,

Duality between direct and indirect utility functions

5 U(xo*)= inf {p 1x0*E cl S,(U) 5 A, whence 2x,* E S,(U). Therefore, (x,, 2x,*) > (x,, xt), i.e., (x,, x$) > 0 also in this case. If x~$clS,(u), by the classical separation theorem between closed convex sets and points, x0 can be chosen so as to satisfy the stronger condition inf ((x,, x*) lx* ESJU)} > (x,, xc). Take any t between these numbers and define x1 = (l/t)x, if x0*# cl S,(u), [l/(x0, xt )]x, otherwise. Then, for any x* ESJLJ), we have (xi, x*) > 11 (xi, x;F), whence

sup inf {u(x*)~(x,x*)~l}~(x,xg*)~l x i x*

1

~inf{u(x*)I(xi,x*)~l)~jl; X*

since 1 can be taken arbitrarily close to u(xg), this proves (4). We shall now show that the function u defined in (3) is the greatest function with which u is associated. Let 0”:K -+R be any function satisfying (x* E K*) and let XE K and x* E K* be such that (x,x*) Therefore,

5 1. Then,

(5) by (5), u(x*) zt?(x).

v”(x)5 inf {u(x*) 1(x, x*) 5 l} = u(x), which proves that i750. Finally, we have to check that, among the functions with which u is associated, u, given by (3), is the only one being non-decreasing, evenly quasiconcave and satisfying (2). Indeed, let v be any function with these properties and let u be the indirect utility function associated with u. Then, -u is a non-increasing evenly quasiconvex function and satisfies (1) (on bdK). Moreover, since u is the indirect utility function associated with v, (3) holds with u and u replaced by --u and -0, respectively. Thus, by the proof of the existence statement, after a change of signs we obtain (3). 0 The necessity of u being evenly quasiconvex admits the following intuitive explanation and proof: each level set S,(U) consists of those vector prices x* for which any consumption vector x satisfying the budget constraint (x,x*)2 1 must belong to S,(u) or, equivalently, for which the budget constraint is not satisfied for any consumption vector whose utility exceeds ;1; in other words, SJu) is the intersection of the open halfspaces defined by (x,x*) > 1, where x runs over the set X\S,(o). Concerning condition (l), we have proved that it holds for any x0*EK* (not only for xg E bd K*). Actually, any non-increasing function satisfies it on

J.-E. Martinez-Legaz,

Duality between direct and indirect utility functions

205

int K*, no matter whether it is an indirect utility function or not. Indeed, let x0*E int K*, u < 1 and p E R be such that ctxg ECIS,(U). Then, the set I/= (x* E K* Ix* 5 xa} is a neighborhood of crx& whence Vn S,(u) # 4, i.e., there exists x* E K* with x* Ix,* and u(x*) sp. Since u is non-increasing, this implies that u(x~) 5~. Therefore, by Lemma 2.1 and the lower semicontinuous hull formula of Crouzeix (1982, p. 11 l), we obtain lim G(axo*)= inf ti(axt) = inf inf {,u1ax: E cl S,(u)) 1 u(xt). a-1-

a<1

a<1

PER

We have already observed (see the Introduction) that any lower semicontinuous quasiconvex function is evenly quasiconvex. Also, for a non-increasing function u:K+i?, condition (1) is weaker than the lower semicontinuity of u at xg. Indeed, if u is lower semicontinuous at x$, by Crouzeix (1982, p. 113) and Lemma 2.1, u(x$) =ii(xz) sti(axg) (O 1, and K is the non-negative orthant, since in this case the above-mentioned examples can be easily found). Note also that any continuous function u satisfies (1); thus, in the general case, (1) does not imply neither that u is non-increasing nor that it is quasiconvex. Corollary 2.3.

Let u:K*+R

and let uO:K*+l?

uo(x*) = sup {u(x) 1(x, x*> 5 l}

(x*

be defined by E

K*),

where v is given by (3). Then, uO is the greatest quasiconvex minorant of u satisfying

u,(xt)5

lim U,(axX)

(x5 E bd K*).

(6) non-increasing

evenly

(7)

U-l-

Proof It is easy to see that u0 is a minorant of u and, by Theorem 2.2, u0 is non-increasing, evenly quasiconvex and satisfies (7). Let u,:K*-+R be any

206

J.-E. Martinez-Legaz,

Duality between direct and indirect utility functions

minorant of u with these properties. xg E K* we have

By Theorem

2.2, (3) and (6), for any

ui(xo*)=sup inf {ul(x*)~(x,x*)~l}~(x,xg*)~l x ix*

1

Ssup inf {u(x*)~(x,x*)~1}J(x,xg*)~l X i .X* = sup {V(X)I (x, x0*) 5 l} = uJxo*).

1 0

From the preceding result, it follows that, starting with an indirect utility function 24[i.e., a non-increasing evenly quasiconvex function satisfying (l)], deriving u by Theorem 2.2, and then passing to u,, by Corollary 2.3, we end up with uO= u. In fact, this is just the meaning of (4), which was the basis for the proof of Theorem 2.2. Theorem 2.2 and Corollary 2.3 admit their corresponding counterparts in terms of direct utility functions. Theorem 2.4. Let v:K+R. There exists u:K*+i? such that (3) holds if and only zfv is non-decreasing, evenly quasiconcave and satisfies the condition

u(xO)z lim ~(ux,)

(x0 E bd K).

a+1-

Moreover, u can be taken non-increasing, evenly quasiconoex and satisfying (I). Under these conditions, u is unique, namely, u is the smallest function for which (3) holds: furthermore, u is the indirect utility function associated with v. Corollary 2.5.

Let v: K-ri?

~~(x)=inf{u(x*)I(x,

and let v”: K+B

x*) s l>

be defined by

(x E K),

where u is the indirect utility function associated with v. Then v” is the smallest non-decreasing evenly quasiconcave majorant of v satisfying

u”(xo)g lim ~‘(clx,)

(x0 E bd K).

12-l-

Analogously to the observation we have made after Corollary 2.3, the preceding statement implies that, starting with a non-decreasing evenly

J.-E. Martinez-L.egaz,

Duality between direct and indirect utility functions

207

quasiconcave function v:K+R having property (2), constructing its associated indirect utility function u:K*+R and then deriving v” by Corollary 2.5, we finally obtain v0 =v. Up to now, we have considered that utility functions are extended real valued, but they usually take only finite values. Note that if v is bounded then clearly u is also bounded, namely, if v(K) c [a, b] for some a, be R then also u(K*) c [a, b]. Conversely, if u(K*) c [a, b] then the function v defined by (3) satisfies v(K) c [a, b]. Therefore, our results remain valid when we replace R by any closed real interval [a, b]. The case when v is finite but unbounded is slightly more complicated: Theorem 2.6. Let u:K*+R be an indirect utility function and let v:K--+R be the greatest direct utility function [or, equivalently, the unique non-decreasing evenly quasiconcave function satisfying (2)] with which u is associated, namely, that given by (3). Then v is finite valued if and only if u is bounded from below and O~clu-r(R). Proof Let us first assume that u is finite valued. Then, u is obviously bounded from below [by v(O)]. Since u is quasiconvex (Theorem 2.2) its domain u-‘(R) is convex. Hence, if we had 0 $cl u-‘(R) then, by the classical separation theorem between closed convex sets and points, there should exist x0 E X and t >O such that (x,, x*) 2 t for any x* E u-‘(R). As in the proof of Theorem 2.2, one can easily deduce that x0 E K. Without loss of generality, we can assume that t > 1 (dividing, e.g., by t/2). Then, if X*E K* satisfies (x,,, x*) 5 1
+aI,

which contradicts the finiteness of v. Conversely, assume that u is bounded from below and that 0~~1 u-‘(R). Then, by (3), it is obvious that v is also bounded from below; in particular, v(x) > - co for any x E K. Suppose that v(x,,) = + cc for some x0 E K. This, by (3), implies that u(x*) = + co whenever (x0, x*) 2 1 or, in other words, that X*EU-‘(R) only if (x,,x*)> 1. But then we deduce that clu-r(R) is contained in the closed halfspace defined by the inequality (x0,x*) 2 1, which contradicts the assumption that 0 E cl u-‘(R). 0 Symmetrically, the finiteness ing characterization:

of indirect

utility

functions

admits

the follow-

Theorem 2.7. Let u and v be as in Theorem 2.6. Then, u is finite valued if and only zf v is bounded from above and 0~~1 v-‘(R).

208

J.-E. Martinez-Legaz,

Duality between direct and indirect utility functions

Using either Theorem 2.6 or 2.7, one easily obtains the following extremely simple symmetric result. Corollary 2.8. Let u and v be as in Theorem statements are equivalent: u and v are finite v is bounded. (iii) u is bounded.

2.6.

Then,

the following

valued.

Proo$ If (i) holds then, by Theorem 2.7, v is bounded from above. Since it is non-decreasing (Theorem 2.2), v is also bounded from below [by v(O)]. This proves the implication (i) *(ii), while (ii) *(iii) and (iii)*(i) follow from the already observed fact (see the comments preceding Theorem 2.6) that, for any a, b E R, u(K) c [a, b] if and only if u(K*) t [a, b]. 0

Finally, let us mention that the results obtained in this paper can be interpreted in the framework of the generalized conjugation theory of Moreau (1970), following a method described in Martinez-Legaz (1982), Singer (1984, 1986) and Volle (1985). Namely, our results show that, for the coupling function c: K* x K+R defined by 0

if

(x, x*) 5 1

-cc

if

(x,x*)>1

(x* E K*, x E K),

c(x*, x) =

the set T(K*,K) of suprema of elementary functions c(.,x)+/?;K*+R, with x E K and BER, coincides with the set of non-increasing evenly quasiconvex functions u:K*+l? which satisfy (1). Note added in proofs. Under the additional assumption that int K* =a (resp., int K # @), it is easy to prove that the condition 0~~1 u-‘(R) [resp., OECIv- ‘(R)] in Theorem 2.6 (resp., 2.7) can be equivalently replaced by the inclusion int K*cu-‘(R) [resp., int Kcv-l(R)].

References Crouzeix, J.P., 1977, Contributions a l’btude des fonctions quasiconvexes. Ph. D. thesis (Universitb de Clermont II, Aubiere). Crouzeix, J.P., 1982, Continuity and differentiability properties of quasiconvex functions on R”, in: S. Schaible and W.T. Ziemba, eds., Generalized concavity in optimization and economics (Academic Press, New York) 109-130. Crouzeix, J.P., 1983, Duality between direct and indirect utility functions. Differentiability properties, Journal of Mathematical Economics 12, 149-165.

J.-E. Martinez-Legaz, Duality between direct and indirect utility functions

209

Diewert, W.E., 1982, Duality approaches to microeconomic theory, in: K.J. Arrow and M.D. Intriligator, eds., Handbook of mathematical economics, Vol. II (North-Holland, Amsterdam) 535-599. Fenchel, W., 1952, A remark on convex sets and polarity, Comm. Sim. Math. Univ. Lund (Medd. Lunds Univ. Math. Sem.) Tome Supplem., 82-99. Lau, L.J., 1969, Duality and the structure of utility functions, Journal of Economic Theory 1, 374396. Martinez-Legaz, J.E., 1982, Conjugacibn asociada a un grafo, in: Actas IX Jornadas Matematicas Hispano-Lusas (Universidad de Salamanca, Salamanca) 837-839. Martinez-Legaz, J.E., 1983, A generalized concept of conjugation, in: J.B. Hiriart-Urruty, W. Oettli and J. Stoer, eds., Optimization, theory and algorithms (Marcel Dekker, New YorkBasel) 45-59. Martinez-Legaz, J.E., 1988, Quasiconvex duality theory by generalized conjugation methods, Optimization 19, 603-652. Moreau, J.J., 1970, Inf-convolution, sous-additivite, convexite des fonctions numeriques, Journal de Mathimatiques Pures et Appliquees 49, 109154. Passy, V. and E.Z. Prisman, 1984, Conjugacy in quasiconvex programming, Mathematical Programming 30, 121-146. Ponstein, J., 1980, Approaches to the theory of optimization (Cambridge University Press, Cambridge). Singer, I., 1984, Generalized convexity, functional hulls and applications to conjugate duality in optimization, in: G. Hammer and D. Pallaschke, eds., Selected topics in operations research and mathematical economics (Springer-Verlag, BerlinHeidelberg-New York) 49-79. Singer, I., 1986, Some relations between dualities, polarities, coupling functionals and conjugations, Journal of Mathematical Analysis and Applications 115, l-22. Singer, I., 1987, Inhmal generators and dualities between complete lattices, Annali di Matematica Pura ed Applicata 148, 289-352. Voile, M., 1985, Conjugaison par tranches, Annali di Matematica Pura ed Applicata 139, 279-312.