Adv. Math. Econ. 7, 47-93 (2005) Advances in MATHEMATICAL ECONOMICS ©Springer-Verlag2005 A method in demand analysis connected with the Monge—Kantorovich problem Vladimir L. Levin* Central Economics and Mathematics Institute of Russian Academy of Sci- ences, 47 Nakhimovskii Prospect, 117418 Moscow, Russia (e-mail: vl_levin@cemi.rssi.ru) Received: February 18, 2004 Revised: August 6, 2004 JEL classification: C65, Dil Mathematics Subject Classification (2000): 91B42 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. Key words: demand function, budget set, insatiate demand, utility function, indirect utility function, rationalizing, strict rationalizing, inducing, strict in- 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 Introduction This article is devoted to concave-utility-rational demand functions. The problem of demand rationalizing is studied in mathematical 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. 48 V.L. Levin since 1886 (Antonelli); for history and references see [4], [7]. A substan- 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, an utility function rationalizing the demand function need not be con- cave. It will be automatically quasiconcave when the revealed preference relation is convex. If f/ is a quasiconcave utility function that rationalizes a given demand and if a function /i: R ^' R is (strictly) increasing, then the composition hoU rationalizes the demand and is quasiconcave as well. In some cases, such a composition proves to be not merely quasi- 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]. In [20], [22] we proposed a new method in demand analysis (and in an abstract rational choice theory). The method is based on duality re- 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 is rationalized by a given concave utility function if and only if the cor- 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 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 A method in demand analysis 49 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 = (pi, • ,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 par- 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 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: 50 V.L. Levin 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 prelim- inary information on the Monge-Kantorovich problem including condi- tions for Qo{(p) to be nonempty. Section 2 is devoted to concave-utility- rational demand maps. We give criteria for rationalizing (strict 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 concave- 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 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 pos- itive homogeneous utility functions. Here the cost function ^ is used. In Theorem 7, that generalizes the corresponding variant of the Afriat- 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 conditions for Qo(0 ^^ t)e nonempty are established for a smooth /, and A method in demand analysis 51 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(pi), , 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, 52 V.L. Levin {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 ai, (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 A method in demand analysis 53 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 -(72), 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 func- tions on X; see Levin - Milyutin [25] and Levin [14]. (Clearly (p^ = (p when (p satisfies the triangle inequality and vanishes on the diagonal.) By analogy with Q((/?), we define a broader set Qo((p) :={'a€R^ :u{x^)-u{x^) < (p{x^, x^), x^, x'^ G X}, (1.6) which is the constraint set of an infinite linear program dual to an ab- stract (non-topological) version of MKP with a fixed marginal difference [17]. Clearly Q{^) = Q{(p*) and Qo(^) = Qo(<^*)- Notice also that if (p is bounded, continuous and vanishes on the diagonal, then Qo{^) = Q{^). The multifunction Qo arises in a natural way in various topics (mass 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. 54 V.L. Levin 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 (p{x,x) = 0 \/XGX, (1.8) 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. A method in demand analysis 55 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^ VMG{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 [...]... 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 > equivalent: (a) U rationalizes f, and there is a strictly positive multiplier X{p),p G P, compatible with U; (b) U is represented in form U{q) = iniJu'ip') + \'{p')p' • {q - f'{p'))}, q € R!J:, (3.9) where P C P' C intM!J:, f : P' -^ intE!J:, f'\P = f, X' : P' -y intR+, u'... P ) , and u is the indirect utility function associated with U 76 V.L Levin Let us define MP):=^^,P^P (4.8) Taking into account (see (3.4) and Remark 7) that CA„(P',P) = ^^^ip-np')-np))=nip) (^^^ -1) vp,p' € p, we get a chain of equivalencies: u e QoiCxu) "^ [^ satisfies (4.1)] 4=^lnue QoiO^ clarifying a link between two characterizations of the indirect utility function as given in Theorems 2 and... ^=e^*(-')=infinfn4^^, u{p^) I li p^'fip^) where the inner infimum is taken over all p^, ,p^"^^ in P with p^ = p,p^-^^ =p\ Consequently, if f is rationalized by two such utility functions, Ui and U2y then the corresponding indirect utility functions, ui and U2, can differ by a positive multiplier only, so that ^ 4 ^ = c > 0 for allp E P Corollary 6 Let P and f be as in Corollary 5, and suppose, in addition,... of demand rationalizing, where budget constraint (2.1) is absent and the gain to be maximized by a consumer is utility minus expences Definition 12 We say that a function f : P ^»^ W^ is induced by a utility function U : W^ -^ RU {-00} if, for each p e P and each q £ W^, the inequality holds U{f{p))-p-f{p)>U{q)-p.q (5.1) //, in addition, for q ^^ /(p), (5.1) holds as the strict inequality, then we... 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... nonempty, then it consists of continuous functions In such a case, for every u G QO(CA) there exists a non-decreasing continuous concave utility function U : R!f: -^ R such that: (i) A is compatible with U, (ii) f is rationalized by U, and (Hi) u is the indirect utility function associated with U If, in addition, A is strictly positive on P (in such a case, (3.5) means that f is insatiate), then for every... D{p^)^ k = 1, ,l^q^'^^ := q^ Summing up these inequalities 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... Corollary 3 Theorem 8 Suppose P is a domain m R!|:, f : P ^»^ intR!f: is C^ and insatiate The following statements hold true: (i)If Qo{Cj ^s nonempty J then for every p E: P the matrix A{p) = {ciij{p))ij, where a.Ap) = P • m ^ -m t y - ^ ^ (4-9) is symmetric negative semidefinite (a) If P is a convex domain in W^, f isC'^, matrix (4.9) is symmetric for every p £ P, and the inequality holds for allp,p' € P,... 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 U{q')-u{p) sup 92€R!f :p-92>/(p) p-q'^lip) < inf u{p) - U{q') q^mi:p-g^ . Advances in MATHEMATICAL ECONOMICS ©Springer-Verlag2005 A method in demand analysis connected with the Monge—Kantorovich problem Vladimir L. Levin* Central Economics and Mathematics Institute. budget set, insatiate demand, utility function, indirect utility function, rationalizing, strict rationalizing, inducing, strict in- ducing, Monge-Kantorovich problem (MKP) with a fixed marginal difference,. cor- 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