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A system of inequalities arising in

mathematical economics and connected with

the Monge-Kantorovich problem

G. Carlier

, I. Ekeland

, V.L. Levin

, A.A. Shananin

§

19th March 2002

Abstract

In this article, we study a class of functions defined by a system of linear inequalities which arises naturally in at least four different economic issues: redistribution, spatial economics, theory of incentives and utility theory. The main tool used to study solutions of those inequalities is the duality theory for the Monge-Kantorovich problem with a given marginal difference.

R´esum´e

Dans cet article, nous ´etudions des classes de fonctions d´efinies par un en-semble d’in´egalit´es lin´eaires intervenant naturellement dans au moins qua-tre domaines diff´erents de l’analyse ´economique : les politiques redistribu-tives, l’´economie spatiale, la th´eorie des incitations et la th´eorie de l’utilit´e. L’argument essentiel est de remarquer que ces classes de fonctions sont ´ etroite-ment li´ees `a la th´eorie de la dualit´e dans le probl`eme de Monge-Kantorovich `

a diff´erence de marginales fix´ee.

Acknowledgement

The first three authors are supported by the INTAS foundation (project 97-1050). Also, V.L.L. and A.A.S. are supported by the Russian Foundation for Humanitarian Sciences (projects 00-02-00159, 01-02-00481).

Universit´e Bordeaux 1, MAB, and Bordeaux 4, GRAPE, Avenue L´eon Duguit, 33608, PESSAC

Guillaume.Carlier@math.u-bordeaux.fr

Universit´e Paris IX Dauphine, Ceremade, Place de Lattre de Tassigny, 75775 Paris Cedex

16.Ivar.Ekeland@dauphine.fr

CEMI, Nakhimovskii Prospect 47, 117418 Moscow. vl levin@cemi.rssi.ru §Moscow State University, Vorobievy Gory, 119899 Moscow. shan@ccas.ru

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1

Introduction

The aim of this article is to give direct economic applications of the dual-ity theory for Monge-Kantorovich problems. We believe that this theory has many powerful applications in very different fields of mathematical economics (see for instance Levin (1990), Levin (1997a), Carlier (2001)). In the present work, we study a class of functions defined by a system of linear inequali-ties which appears in various economic contexts by using the duality theory for Monge-Kantorovich mass transportation problem with a given marginal difference (see Levin (1990, 1991, 1992, 1996, 1997a, 1997b)).

More precisely, given some domain X of Rn and a function ϕ : X × X →

(0, +∞] we are interested in the class of positive functions satisfying the inequalities:

v(x) ≤ v(y)ϕ(x, y) for all (x, y) ∈ X × X. (1) Let A(ϕ) denote the set of positive functions v satisfying (1) and A1(ϕ)

its subset consisting of functions that satisfy the additional normalization condition:

Z

X

v(x)dx = 1. (2)

Clearly, one has:

A(ϕ) = [

λ>0

λA1(ϕ).

In section 2, economic examples where the classes A(ϕ) and A1(ϕ) play

an important role are given. In section 3, the main results (necessary and sufficient conditions for A(ϕ) to be nonempty) are stated in the case where ϕ is smooth (section 3.1) as well as in the nonsmooth case (section 3.2). Finally, a generalization of the Afriat-Varian Theorem is given (section 3.3).

2

Economic motivations

In this section, we show that the classes of functions A(ϕ) and A1(ϕ) arise

naturally in at least four different economic areas. More precisely, we shall consider in the following sections problems arising in redistribution theory, spatial economics, incentive-compatibility issues and utility theory. In all of those topics, the class A(ϕ) plays an important role and admits specific interpretations.

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2.1

Redistribution

Given a set of economic agents, X ⊂ Rn, a Lebesgue integrable function

v : X → R++ := {a ∈ R : a > 0} is called an allocation if

Z

X

v(x) dx = 1. (3)

Here, dx stands for the integration with respect to the n-dimensional Lebesgue measure and X is supposed to be a domain (a connected open set or its clo-sure) of finite Lebesgue measure. Suppose a function ϕ : X × X → (0, +∞] is given and let A1(ϕ) denote the set of allocations v satisfying:

v(x) ≤ ϕ(x, y)v(y) ∀x, y ∈ X. (4) Let us give an interpretation of (4). Suppose a unit of some resource (or an amount of money) is to be distributed between the economic agents. Let v(x) be the fraction of the resource allocated to x. For example, one can assume that there is an allocation v0supplying the profit v0(x) to agent x, and

the allocation v is its redistribution obtained by means of taxes (penalties) on some agents (v(x) < v0(x)) and payments (subsidies) to others (v(x) >

v0(x)), so that

Z

X

(v(x) − v0(x)) dx = 0. (5)

Realizing such a redistribution is based on some ideas of social fairness and pursues the goal of limiting unbounded superiority of some agents’ profits to others: v(x) may exceed v(y) no more than ϕ(x, y) times. As is known, such kind of social ideas is often in contradiction with ideas of economic efficiency. Being a restriction on mutual values v(x) and v(y) for pairs x, y ∈ X with ϕ(x, y) < +∞ only, (4) seems to be a rather weak constraint. Nevertheless, we will show that in some cases (4) implies total equalizing agents’ profits.

2.2

Spatial economics

In this section, following Lucas and Rossi-Hansberg (2001), we assume that X represents a city so that elements x of X are locations in that city. Assume that each agent living in X has a total amount of time available (normalized to 1 in the sequel) that he spends in working and commuting from his living location to his working place.

The wage per unit of working time at a job location x is denoted v(x). An agent commuting from y to x spends a fraction α(x, y) ∈ [0, 1) of his total time in transportation so that agents living at location y choose job

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locations that maximize their income i.e. the function x 7→ v(x)(1 − α(x, y)). It is natural to assume α(y, y) = 0 for all y ∈ X.

Defining then the function ϕ by

ϕ(x, y) := 1 1 − α(x, y)

the class of wage functions A(ϕ) consists exactly of those wages for which it is optimal for every agent to work where he lives.

Note that in this problem, it is natural to assume that ϕ is symmetric, equals 1 on the diagonal D of X × X and is greater than 1 outside D.

Given a transportation network or technology (the function ϕ), a natural question is : how big is A(ϕ) ? Indeed if the wage v does not belong to A(ϕ) some agents will choose job locations that are different from their living place. As we shall see in the next section, A(ϕ) consists of constant wage functions only. This means that, if wages are not constant in the whole city, some agents will necessarily work at a location different from where their house is.

2.3

Incentive-compatibility, falsification and corruption

In this section, we consider some principal-agent models, in which agents have the possibility at some cost (either by costly falsification or by colluding with a third party) to misrepresent some relevant information to the principal.

This kind of problems has two different interpretations in terms of cor-ruption or fraud (sabotage). We will distinguish two cases : the case of costly falsification and the case of collusion. We illustrate those situations by the examples of insurance fraud and taxation. We will see that in that framework, for some natural function, ϕ, A(ϕ) can be interpreted as the set of incentive-compatible or non-manipulable mechanisms.

An agent called the principal (firm, social planner...) specifies a mecha-nism with a population of agents as a payoff function x 7→ u(x) depending on some variable or information x ∈ X. For instance, one can think of x being the realization of some random variable (see examples below). Each agent is then characterized by the true value of the state x.

Falsification or costly fraud

We assume here that when state y occurs, it is possible for the agent at the cost c(x, y) to misrepresent the real state into the state x without being detected. Note that it is realistic here to assume that c(y, y) = 0 ∀y ∈ X and c may take the value +∞. This is the case when it is impossible for the agent to falsify y into x when x is too far from y.

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In that case, agents for which state y occurs choose to declare any state x that maximizes the quantity x 7→ u(x) − c(x, y) or equivalently x 7→ v(x) exp(−c(x, y)) where v := exp(u). We will therefore say that a pay-off function or mechanism u is incentive-compatible or non-manipulable if it is optimal for every agent to announce his true state and not to cheat. It is easy to check that u is incentive-compatible if and only if v := exp(u) ∈ A(ϕ) where ϕ(x, y) := exp(c(x, y)).

Collusion with a third party

A variant of the situation depicted above takes place when agents have the opportunity to collude with a third party (expert, supervisor...). Let us assume now that the actual state cannot be observed directly by the principal but only by the agent and a third party (expert, supervisor...). If the third party truthfully reports the actual state y to the principal, he gets a payoff normalized to 0 and the agent gets the payoff u(y). Now if the supervisor may misreport without being detected, the state x at the cost c(x, y) and there exists x such that u(x) − c(x, y) > u(y), some colluding behavior will take place between the agent and the supervisor. Indeed, in that case they would both agree to misreport a state x∗ 6= y maximizing their total income u(y) − c(x, y) and would share the surplus u(x∗) − c(x∗, y) − u(y) > 0. A mechanism given by the payoff function u is therefore collusion-proof, in the sense that collusion never takes place between the agents and the supervisor if and only if exp(u) ∈ A(ϕ) where ϕ is defined as in the previous paragraph.

Example 1 (insurance fraud) :

An insurance contract specifies a premium P paid before random losses occur and a payoff function x = (x1, ..., xn) 7→ u(x) where x = (x1, ..., xn) is

the realization of some random variable of losses and u(x) is the wealth of the insured agents when x occurs.

It is realistic in many applications to assume that the insured are able at some cost to misrepresent their real losses either themselves (sabotage) or by colluding with a third party (doctor, garage owner, expert...). The insurance company has therefore to impose some restrictions on its contracts which are characterized by a certain class A(ϕ).

Example 2 (taxation and corruption):

A tax schedule maps individual vector of profits y = (y1, ..., yn) (or

in-comes before taxes and redistributions) into a net income u(y) (after taxes and redistribution). As in the insurance example it is realistic to assume that agents often have the possibility to falsify their profits by their own or more realistically by corrupting some tax officer.

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Let us finally mention that links between the Monge-Kantorovich the-ory and principal-agent problems were already noticed in Levin (1997a) and Carlier (2000), see also Rochet (1987).

2.4

Utility theory (rationality of consumer behavior)

Here we describe consumer’s behavior by means of the reversed demand function

p(x) = (p1(x), · · · , pm(x))

where x ∈ Rm

+ is a vector of products and p(x) is a vector of prices on those

products (or reversed demand function). By definition, the function p(·) is said to be rationalized by a utility function U on a set X ⊂ Rm

+ if for all

x ∈ X one has

x ∈ ArgmaxU (y) s.t. p(x) · y ≤ p(x) · x, y ∈ Rm+ . (6) The verification that the reversed demand function can be rationalized in the class of positively homogeneous utility functions is an important part of the nonparametric method in the consumer demand analysis (see for instance Shananin, 1993).

Assume that X is a cone and p is continuous with nonnegative compo-nents such that:

(i) p(y) · x > 0, for all (x, y) ∈ X × X, (ii) for all λ > 0, 1 ≤ i, j ≤ m and x ∈ X

pi(λx)

pj(λx)

= pi(x) pj(x)

.

It is known that if X consits of a finite number of rays

X =

K

[

k=1

{λxk: λ > 0}

then rationalizability of p is equivalent to both the strong axiom of revealed preferences and the existence of a positive continuous solution to the system of linear inequalities

for all (x, y) ∈ X × X, v(x) ≤ v(y)p(y) · x

p(x) · x. (7) This result is known as Afriat-Varian Theorem, see Afriat (1967) and Varian (1983). For X = Rm+ a similar theorem is proved in Pospelova and Shananin

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Note that the set of positive functions satisfying (7) is exactly the set A(ϕ) with ϕ given by:

ϕ(x, y) := p(y) · x p(x) · x.

3

Main results

3.1

The smooth case

Theorem 1. I. Suppose ϕ(x, x) = 1 for all x ∈ X and ϕ is finite and smooth (C1) on some open set G in X × X containing the diagonal D := {(x, x) : x ∈ X}. Then A1(ϕ) is empty or consists of the single allocation v given by

v(x) = exp u(x), where u satisfies

∇u(x) = ∇xϕ(x, y)|y=x = −∇yϕ(x, y)|y=x. (8)

II. If, in addition, ϕ(x, y) ≥ 1 for all (x, y) ∈ X × X, then A1(ϕ) is

non-empty and consists of the single allocation v(x) ≡ 1/mesX ∀x ∈ X where mes stands for the n-dimensial Lebesgue measure.

Remark. A necessary condition for A1(ϕ) to be non-empty is thus the

va-lidity of the equation

∇xϕ(x, y)|y=x = −∇yϕ(x, y)|y=x.

Let us state further conditions (necessary and sufficient) for non-emptiness of A1(ϕ).

Theorem 2 (necessary conditions). Suppose ϕ is C2 on an open set G ⊃ D

and equals 1 on D. For any (x, y) ∈ G we consider the quadratic form

B(s; x, y) := n X i,j=1 βij(x, y)sisj, s = (s1, . . . , sn) (9) with coefficients βij(x, y) = − ∂x∂ϕ i(y, y) ∂ϕ ∂xj(y, y) + ∂2ϕ ∂xi∂xj(y, y) − ∂ϕ ∂xi(y, y) ∂ϕ ∂yj(y, y) + ∂2ϕ ∂xi∂yj(y, y) − ϕ−2(x, y)∂ϕ ∂yi(x, y) ∂ϕ ∂yj(x, y) + ϕ −1(x, y) ∂2ϕ

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If A1(ϕ) is non-empty, then on the diagonal D the equality holds ∂2ϕ ∂xi∂yj = ∂ 2ϕ ∂xj∂yi ∀i, j ∈ 1, n (10) and, for every x ∈ X, the quadratic form B(s; x, x) is positive semi-definite. Theorem 3 (sufficient conditions). Suppose X is convex, ϕ is C2 on X × X

and equals 1 on D. Also suppose that (10) holds on D and, for all (x, y) ∈ X × X, the quadratic form (9) is positive semi-definite. Then A1(ϕ) is

non-empty.

The above theorems are connected closely with some results relating to the Monge-Kantorovich problems with smooth cost functions; see Levin (1992, 1997a).

Given a set X and a cost function c : X × X → R ∪ {+∞} we consider the set

Q0(c) := {u ∈ RX : u(x) − u(y) ≤ c(x, y) ∀x, y ∈ X}.

This is the constraint set for the infinite linear program which is dual to a non-topological version of the Monge-Kantorovich problem with a given marginal difference; see Levin (1997b). If X is a topological space, c is continuous and vanishes on the diagonal, then any u ∈ Q0(c) is continuous

(see lemma 4 in Levin (1992) or theorem 3.3 in Levin (1997a)). We have v ∈ A(ϕ) ⇔ u = ln v ∈ Q0(ln ϕ).

Taking into account the equalities ∇xc(x, y) =

∇xϕ(x, y)

ϕ(x, y) , ∇yc(x, y) =

∇yϕ(x, y)

ϕ(x, y)

for the cost function c(x, y) = ln ϕ(x, y), theorem 1 is derived from theorem 3.5 in Levin (1997a) (see also theorem 1 and remark 3 in Levin (1992)), while theorems 2 and 3 follow from theorems 3.6 and 3.7 in Levin (1997a) (see also theorems 2 and 4 in Levin (1992)). One more sufficient condition for A1(ϕ)

to be non-empty can be derived from theorem 5 in Levin (1992) (see also theorem 3.8 in Levin (1997a)). We omit its formulation.

3.2

The nonsmooth case

Now consider the case where ϕ is not supposed to be smooth. Given a cost function c : X × X → R ∪ {+∞} vanishing on D, the reduced cost function is defined by

c∗(x, y) := lim

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Properties of such functions are discussed in Levin (1990, 1991, 1992, 1996, 1997a, 1997b). Mention here the equality

Q0(c) = Q0(c∗).

Suppose ϕ(x, x) = 1 ∀x ∈ X and associate with ϕ the function

ϕ#(x, y) := lim n→∞inf (n+1 Y i=1 ϕ(zi−1, zi) : z1, . . . , zn ∈ X )

where z0 := x, zn+1 := y. Notice that c∗(x, y) = ln ϕ#(x, y) for c(x, y) =

ln ϕ(x, y). The next result follows at once from corollary 3.3 in Levin (1997a). Theorem 4. Suppose ϕ#(x, y) < +∞ for all (x, y) ∈ X × X. The following

statements are equivalent:

(a) there is a unique allocation v in A1(ϕ);

(b) for any (x, y) ∈ X × X the equality

ϕ#(x, y) =

v(x) v(y) holds.

If A1(ϕ) contains more than one allocation, then we can choose an

alloca-tion v in A1(ϕ) with nice additional properties. Given probability measures

σ1 and σ2 on X we consider the extremal problem: maximize

Z X ln v(x)σ1(dx) − Z X ln v(x)σ2(dx)

subject to constraints (1), (2). The next result follows from the duality the-orem for the Monge-Kantorovich problem with a given marginal difference; see Levin (1990).

Theorem 5. Suppose ϕ is bounded continuous on X × X and equals 1 on D. (I) The following assertions are equivalent:

(a) the set of allocations v ∈ A1(ϕ) is non-empty and any such allocation is

a bounded continuous function;

(b) the function ϕ# is continuous and ϕ#(x, x) ≥ 1 for all x ∈ X. In such a

case,

ϕ#(x, y) = sup v∈A(ϕ)

v(x) v(y).

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(II) Suppose in addition that σ1 ∧ σ2 = 0. An allocation v ∈ A1(ϕ) is an

optimal solution for the above extremal problem if and only if there exists a probability measure µ on X × X such that

Z X×X ϕ#(x, y) dµ = Z X ln v(x)σ1(dx) − Z X ln v(x)σ2(dx)

and π1µ = σ1, π2µ = σ2 where π1µ and π2µ denote the marginals of µ: for

any Borel set B ⊂ X, π1µ (B) = µ(B × X), π2µ (B) = µ(X × B).

Remarks

1. If σ1∧ σ2 6= 0, then statement (II) remains true if one replaces equalities

π1µ = σ1, π2µ = σ2 by π1µ = (σ1−σ2)+, π2µ = (σ1−σ2)−where (σ1−σ2)+=

(σ1− σ2) ∨ 0 and (σ1− σ2)− = (σ2 − σ1) ∨ 0 are the elements of the Jordan

decomposition of σ1− σ2.

2. If σ1 ∧ σ2 = 0 and σ1 is absolutely continuous with respect to the

n-dimensional Lebesgue measure, then for some classes of cost functions c(x, y) = ln ϕ(x, y) the probability measure µ is unique and given by µ = (idX × f )(σ1) where f : (X, σ1) → (X, σ2) is a measure preserving map and,

for any Borel set M ⊂ X × X,

(idX × f )(σ1)M := σ1{x : (x, f (x)) ∈ M }

(cf. theorems 6.1 and 6.2 in Levin (1999)).

3. If ϕ is bounded and continuous on X × X and ϕ# ≥ 1 on the diagonal

then ϕ# is continuous on X × X as well. This follows from lemma 4 in Levin

(1992).

3.3

An application to utility theory

Now we give a direct application of Theorem 5 to utility theory. The next re-sult generalizes earlier theorems of Afriat (1967), Varian (1983) and Pospelova and Shananin (1998).

Theorem 6

Suppose X ⊂ Rm+ is a (not necessarily convex) cone and p(·) : X → Rm+

is a continuous function such that:

(i) p(y) · x > 0 for all (x, y) ∈ X × X, (ii) for all λ > 0, 1 ≤ i, j ≤ m and x ∈ X

pi(λx)

pj(λx)

= pi(x) pj(x)

.

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(a) there exists a positively homogeneous concave continuous, positive on Rm++ utility function U that rationalizes p(.) on X,

(b) there exists a positive continuous function v on X such that for all (x, y) ∈ X × X

v(x) ≤ v(y)p(y) · x p(x) · x;

(c) homogeneous strong axiom of revealed preferences is satisfied, that is for all n ∈ N and for any cycle (x1, · · · , xn, xn+1) ∈ Xn+1 with x1 = xn+1:

n Y i=1 p(xi) · xi+1≥ n Y i=1 p(xi) · xi.

Proof. (b)⇒(a). Let v satisfy (b) and define for all x ∈ X U (x) := inf

z∈Xv(z)p(z) · x

so that U is positively homogeneous concave continuous and positive on Rm++.From (b), we get for all x ∈ X:

U (x) = v(x)p(x) · x.

Then v(x)p(x) · (y − x) ≥ U (y) − U (x), which proves that x ∈ Argmax{U (y) s.t. p(x) · y ≤ p(x) · x, y ∈ Rm+}.

(a)⇒(c). Let U satisfy (a) and define:

F (p) := inf  1 U (x)p · x : x ∈ R m +, U (x) > 0 

so that F is positively homogeneous concave continuous and positive on Rm++.

From (a) we have, for all (x, y) ∈ X × X:

F (p(x))U (x) = p(x) · x, (11) F (p(y))U (x) ≤ p(y) · x, (12) hence

F (p(y))p(x) · x ≤ F (p(x))p(y) · x. (13) Now taking a cycle in X, (x1, · · · , xn, xn+1 = x1), and multiplying out

in-equalities (13) with y = xi, x = xi+1, i = 1, . . . , n we get: n Y i=1 F (p(xi))p(xi+1) · xi+1≤ n Y i=1 F (p(xi+1))p(xi) · xi+1,

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so that n Y i=1 p(xi+1) · xi+1≤ n Y i=1 p(xi) · xi+1.

Finally, (c)⇒(b) follows from Theorem 5 and Remark 3 in subsection 3.2.

4

Conclusion

The class of positive functions defined by the system of inequalities (1) has many applications in economics, some of which are described in the present paper. Thanks to the duality theory for the Monge-Kantorovich mass trans-fer problem with given marginal diftrans-ference, criteria for existence, uniqueness and regularity of solutions of (1), (2) can be given.

More generally, we hope to develop a more systematic line of research using mass transportation techniques in economics, especially in demand analysis, spatial economics and theory of incentives. Indeed, in those topics the use of mass transfer problem (or its dual) is very natural and offers a wide range of powerful tools.

REFERENCES

Afriat S.N. (1967), The construction of utility functions from expenditure data. - International Economic Review, n. 7, 129–150.

Carlier G. (2000), Duality and existence for a class of mass transportation problems and economic applications, downloadable at: http://www.ceremade. dauphine.fr/cadrepub.html.

Carlier G. (2001), A general existence result for the principal-agent problem with adverse selection. - Journal of Mathematical Economics, v. 35, pp. 129-150.

Levin V.L. (1990), General Monge-Kantorovich problem and its applications in measure theory and mathematical economics. - In: Functional Analysis, Optimization and Mathematical Economics (A collection of papers dedicated to the memory of L.V. Kantorovich), (L.J. Leifman, ed.), N.Y.—Oxford, Oxford University Press, pp.141–176.

Levin V.L. (1991), Some applications of set-valued mappings in mathematical economics. - Journal of Mathematical Economics, v. 20, 69–87.

Levin V.L. (1992), A formula for optimal value of Monge-Kantorovich prob-lem with a smooth cost function and characterization of cyclically monotone

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mappings. - Math. USSR—Sbornik, vol. 71, no.2, 533–548. (English trans-lation of a paper published in Russian in Matem. Sbornik (1990), v. 181, no.12, 1694–1709.)

Levin V.L. (1996), A superlinear multifunction arising in connection with mass transfer problems. - Set-Valued Analysis, v. 4, 41–65.

Levin V.L. (1997a), Reduced cost functions and their applications. - Journal of Mathematical Economics, v. 28,155–186.

Levin V.L. (1997b), On duality theory for non-topological variants of the mass transfer problem. - Sbornik: Mathematics, v. 188, no.4, 571–602 (En-glish translation).

Levin V.L. (1999), Abstract cyclical monotonicity and Monge solutions for the general Monge-Kantorovich problem. - Set-Valued Analysis, v. 7, 7–32. Lucas R. Jr., Rossi-Hansberg E. (2001), On the Internal Structure of Cities, to appear in Econometrica.

Pospelova L. Ya., Shananin A. A. (1998), The value of nonrationality for consumer behavior and generalized nonparametric method. - Mathematical Modelling, v. 10, no. 4, 105–116 (in Russian).

Rochet J.-C. (1987), A necessary and sufficient condition for rationalizabilty in a quasi-linear context. - Journal of Mathematical Economics , v. 16, 191-200.

Rochet J.-C. (1985), The taxation principle and multi-time Hamilton-Jacobi equations. - Journal of Mathematical Economics , v. 14, 113-128.

Shananin A. A. (1993), Nonparametric methods for the analysis of the con-sumer demand structure. - Mathematical Modelling, v. 5, no. 9, 3–17 (in Russian).

Varian H. (1983), Non-parametric tests of consumer behavior. - Review of Economic Studies, v. 1, no. 160, 99–110.

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