a generalized linear exchange model

ISSN 1990-4789, Journal of Applied and Industrial Mathematics, 2008, Vol. 2, No. 1, pp. 125–142. c© Pleiades Publishing, Ltd., 2008.Original Russian Text c© V.I. Shmyrev, 2006, published in Diskretnyi Analiz i Issledovanie Operatsii, Ser. 2, 2006, Vol. 13, No. 2, pp. 74–102.

A Generalized Linear Exchange Model

V. I. Shmyrev*

Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090 RussiaReceived May 30, 2005

Abstract—The linear exchange model is studied in which alongside the consumers there are firmsminimizing expenses to guarantee some minimal level of the total cost of production. Some finitealgorithm for finding an equilibrium is proposed and justified in the case of the fixed budgets ofconsumers. This algorithm develops the original approach of polyhedral complementarity which wasproposed by the author for the classical exchange model.

DOI: 10.1134/S1990478908010134

INTRODUCTION

In the article we study the linear model of economics which we can consider as a natural general-ization of the classical linear exchange model. As in the case of the exchange model, the model understudy can be considered with the variable or fixed budgets of participants. For simplicity, we will confineto the detailed consideration of the model with fixed budgets. In this case, we propose a finite procedureof finding an equilibrium. This procedure can also be used for the model with variable budgets which willbe mentioned at the end of the article.

The basic idea of this article is the general approach proposed in [4, 6] for the linear exchange modelswhich can be described as polyhedral complementarity. The approach consists in the following:

Linearity of the target functions of participants generates a partition of the price simplex into somepolyhedrons defining the zones of stable preference of various products by the participants. Each zonecorresponds there to the set of price vectors which makes a balanced product exchange possible withinthe structure of preferences of the participants. This is an equilibrium zone which is also a polyhedron.We thus come to a point-to-set mapping of the price simplex into itself possessing some typicalproperties of the mappings of the problems of linear complementarity. We can say that in this case wedeal with polyhedral complementarity. The fixed points of the above-described mapping determine theequilibria prices of the model.

It is well known that the problem of finding an equilibrium in the linear exchange model can bereduced to the problem of linear complementarity [9]. The approach under consideration [4, 6] followsa principally different scheme and, while yielding an algorithm for finding an equilibrium, allows usto discover the monotonicity property of the model. We can describe this property as follows: the arisingproblem of polyhedral complementarity reduces locally to the problem of linear complementarity with thematrix of constraints from the class P (the matrix with positive principal minors; see [11]).

The model under consideration is a generalization of the model of exchange in the sense thatalongside the participant-consumers there are participant-firms. Each firm is characterized by sometotal financial liability of the production which is to be kept providing a minimal discontent with theproduction plan. As the measure of discontent we can take either the expenses of the realizationcalculated by some a priori fixed prices, or labor-intensiveness, or the costs of some resource, etc. Thepurchase and sale of products for all participants (consumers and firms) are carried out by commonprices. The concept of an equilibrium for this model is introduced by analogy with the exchange model.

The interest in consideration of the model originated in connection with the study of a more generalmodel of the Arrow–Debre type in [8].

*E-mail: [email protected]

125

126 SHMYREV

1. DESCRIPTION OF THE MODEL

Consider the model with n products, m participant-consumers, and l participant-firms. Let J =1, . . . , n, I = 1, . . . ,m, and K = m + 1, . . . ,m + l be the sets of the numbers of products,consumers, and firms. Note that the numeration of the participants is common for the consumers andfirms, i.e., S = I ∪ K = 1, . . . ,m + l is the set of numbers of all participants. Each consumer i ∈ I isdescribed by the two vectors ci, di ∈ Rn

+. The vector ci consists of the coefficients of the ith consumer’starget function. The components ci

j determine a comparative scale of the worth of different products forthe ith participant; choosing a purchase vector xi = (xi

1, . . . , xin) of the products, the consumer tends

to maximize (ci, xi). The vector di = (di1, . . . , d

in) determines the initial stock of the products of the ith

consumer.In the market, the exchange of products among the consumers is possible at some nonnegative

prices pj , j ∈ J . Let p = (p1, . . . , pn) be the price vector. We assume the scale of prices to be fixed sothat

n∑

j=1

pj = 1. (1)

Thus, the price vector ranges over the simplex

σ =

p ∈ Rn+ :

∑

j∈J

pj = 1

.

Some products are also supplied to the market by the participant-firms. At that, the kth firm plansto deliver to the market the products to a total sum of at least λk, where λk is a given positive value.If xk = (xk

1 , . . . , xkn) denotes a plan of the kth firm to manufacture different goods then the total cost

of such a supply at the prices pj equals (p, xk). The quality of the plan xk is estimated by the firmitself in tending to minimize (ck, xk). Here ck = (ck

1 , . . . , ckn) are some fixed nonnegative vectors whose

components determine a comparative scale of the “undesirability” of various products for the firm (forexample, their relative production costs).

Thus, the kth firm makes its choice according to a solution of the optimization problem:

(ck, xk) → min (2)

under the conditions

(p, xk) λk, (3)

xk 0. (4)

To buy all products in the market including the products supplied by the firms, the participant-consumers need some additional resources to those obtained for the sale of their own products. Letαi 0 be the initial money stock of the ith consumer. His total budget after selling his stock of theproduct di is equal to αi + (p, di). Thus, the ith consumer will choose the purchase vector xi looking foran optimal solution to the following problem:

(ci, xi) → max (5)


(p, xi) αi + (p, di), (6)

xi 0. (7)

An equilibrium is described by a price vector p and a collection of vectors xi and xk, i ∈ I and k ∈ K,representing some solutions to the optimization problems (5)–(7) and (2)–(4) for p = p and satisfyingthe balance of products:

∑

i∈I

xi =∑

k∈K

xk +∑

i∈I

di. (8)

JOURNAL OF APPLIED AND INDUSTRIAL MATHEMATICS Vol. 2 No. 1 2008

A GENERALIZED LINEAR EXCHANGE MODEL 127

In the sequel, we assume for simplicity that the unit account of the products is chosen so that∑

i∈I

dij = 1, j ∈ J ; (9)

i.e.,∑

i∈I di = e for e = (1, . . . , 1) ∈ Rn. Then (8) takes the form

∑

i∈I

xi =∑

k∈K

xk + e.

It is clear that, for vectors xi and xk, inequalities (6) and (3) hold as equalities for p = p. Moreover,it follows from (8) that

∑

i∈I

(p, xi) =∑

k∈K

(p, xk) +∑

i∈I

(p, di). (10)

Thereafter and since (p, xi) = αi + (p, di) and (p, xk) = λk, we obtain∑

i∈I

αi =∑

k∈K

λk. (11)

Thus, if the set of firms K is empty then αi = 0 for each i ∈ I, and the model becomes the classicallinear exchange model.

2. POLYHEDRAL COMPLEXES OF THE MODEL

For simplicity, we suppose below that all vectors ci and ck are positive; i.e., ci > 0 for each i ∈ I andck > 0 for each k ∈ K. This means that the consumers have no products with zero value, and the firmshave no worthless products.

By analogy with [4, 6], to the model under consideration with a fixed price vector p we put incorrespondence the following net transportation problem:

∑

i∈I

∑

j∈J

zij ln cij −

∑

k∈K

∑

j∈J

zkj ln ckj → max (12)


−∑

j∈J

zij = −αi − (p, di), i ∈ I, (13)

∑

i∈I

zij −∑

k∈K

zkj = pj, j ∈ J, (14)

∑

j∈J

zkj = λk, k ∈ K, (15)

zij 0, zkj 0, i ∈ I, j ∈ J, k ∈ K. (16)

Constraints (13), (15), and (16) are obtained by introducing the variables zsj = pjxsj , s ∈ I ∪ K and

j ∈ J , from the conditions (6) and (3) which, as was observed, we can consider as equalities, and from (4)and (7). Writing down the balance of products componentwise, we find

∑

i∈I

xij =

∑

k∈K

xkj + 1 for j ∈ J.

Multiplying the jth condition by pj , we obtain (14).

Thus, (13)–(16) arise from the constraints of the problems of participants and the balances of eachof the products.


128 SHMYREV

We consider the above-introduced transportation problem as a parametric problem of linear program-ming with parameters pj , j ∈ J , on the right-hand side of the constraints. Summing up (13)–(15), weinfer

0 = −∑

i∈I

αi +∑

k∈K

λk −(

p,∑

i∈I

di

)+

∑

j∈J

pj.

By (9) and (11), this holds for the arbitrary values of pj . In other words, the transportation problem isalways balanced.

Using the theory of transportation problems, it is easy to describe the set of vectors p such that theproblem has an admissible solution. Without detailed computations, we indicate only that this set isgiven by the system of linear inequalities

∑

j∈Q

pj + α 0 for Q ⊂ J, (17)

where, by (10), α is a common value of∑

k∈K λk and∑

i∈I αi.

Let Ω be the set given by (17) and (1). We have σ ⊂ Ω. But σ = Ω only if λk = 0 for all k ∈ K; i.e.,when firms are absent, and the model is the usual exchange model.

It follows from (13), (15) and (16) that, for p ∈ Ω, the set of admissible solutions to the transportationproblem is bounded. Indeed, summing up (13) and using (9) and (1), we obtain

∑

i∈I

∑

j∈J

zij =∑

i∈I

αi + 1.

Consequently, using (16), we conclude the boundedness of zij . Similarly, the boundedness of zkj followsfrom (15) and (16).

Thus, for p ∈ Ω, the transportation problem (12)–(16) is solvable. Denote the optimal value of thetarget function (12) by f(p). By the theory of the parametric problems of linear programming, f isa piecewise-linear concave function on Ω, and the domains of linearity of f are generated by the dual-admissible basis sets of the problem.

For the dual problem to (12)–(16), the system of constraints has the form:

−ui + vj ln cij, for i ∈ I, j ∈ J, (18)

uk − vj − ln ckj , for k ∈ K, j ∈ J. (19)

We assume the standard dual nondegeneracy condition; for every solution of (18)–(19), the numberof inequalities holding as equality is at most m + n + l − 1.

Put S = I ∪ K. Let B be the collection of all dual-admissible basis sets B ⊂ S × J of the problem(12)–(16) and all their possible subsets B ⊂ B possessing the property of s-covering:

for every s ∈ S, there exists j ∈ J such that (s, j) ∈ B. (20)

Every basis set B to within a constant summand generates the collection of us and vj , where s ∈ S andj ∈ J , for which (18)–(19) hold as equality for (s, j) ∈ B. If, moreover, B ∈ B then, under the conditionof dual nondegeneracy, this means that, for the collection of us and vj , all other conditions of (18)–(19)hold as strict inequality; i.e., for (s, j) /∈ B.

Assign to each B ∈ B the polyhedral set Ω(B) ⊂ Ω described as follows:

Let Γ(B) be the oriented graph with the set of vertices J ∪ n + s : s ∈ S and the two sets of arcs(n + i, j), where (i, j) ∈ B and i ∈ I, and (j, n + k), where (k, j) ∈ B and k ∈ K.

It is well known from the theory of transportation problems that the basis sets of (12)–(16) are exactlythe sets B, for which the graphs Γ(B) are trees. The proper subsets of the basis sets are the graphsconsisted of the trees connected with some components.



Let τ be the number of these components, and the vertices of the νth component are the verticesn + i for i ∈ Iν ⊂ I, n + j for j ∈ Jν ⊂ J , and n + k for k ∈ Kν ⊂ K. The balance condition for thetransported νth component has the form:

∑

i∈Iν

(αi + (p, di)

)=

∑

k∈Kν

λk +∑

j∈Jν

pj, ν = 1, . . . , τ. (21)

Under the conditions (21), the system (13)–(15) is also compatible with the additional constraints of theform

zsj = 0, zsj /∈ B. (22)

Moreover, since B ∈ B, the variables zsj , (s, j) ∈ B, are uniquely determined as linear functions of pj ;i.e., zsj = zsj(p). The inequalities

zsj(p) 0, (s, j) ∈ B, (23)

together with condition (1) determine Ω(B) that represents the set of all p ∈ Ω such that the optimalsolution to (12)–(16) satisfies (22).

Thus, for B ∈ B, the set Ω(B) is described by (21) and the system of inequalities (23); i.e., Ω(B) isa polyhedron.

By linear programming, the consequence of the dual nondegeneracy condition is the uniqueness ofthe optimal solution to the primal problem (if it is solvable). With reference to the transportation problem(12)–(16), we find that, for every p ∈ Ω, this problem has the unique solution z∗ij(p).

If B = (i, j) : z∗ij > 0 then B ∈ B and p ∈ Ω(B) (there Ω(B) is the set of relative interior pointsof Ω(B); i.e., the interior points relative to the affine hull of Ω(B)).

Consequently, we find that the sets Ω(B), B ∈ B, have no common relatively interior points and coverthe whole Ω. Moreover, it follows from B1 ⊂ B2 that Ω(B1) ⊂ Ω(B2), and the polyhedron Ω(B1) is a faceof the polyhedron Ω(B2). Using the terminology of combinatorial topology, we say that polyhedronsΩ(B), B ∈ B, constitute a polyhedral complex ω and call Ω(B) the cells of this complex.

Summarizing, we can say that the complex ω arises as a consequence of the structure of the subgraphof the polyhedral concave function f generated by the parametric transportation problem in the modelunder consideration.

We now introduce another polyhedral complex, in some sense dual to ω, and denote it by ξ.The appearance of this complex can be explained informally as follows:We say that each set B ⊂ S × J determines some structure of production-consumption, i.e., for

(i, j) /∈ B, i ∈ I, the jth product is not used by the ith consumer; and, for (k, j) /∈ B, k ∈ K, the kthfirm does not produce the jth product. The structure B for the price vector q ∈ σ is preferable for allparticipants of the model, if for the defined below values yi and yk, i ∈ I and k ∈ K, the following hold:

yi = maxh∈J

cih

qh=

cij

qjfor (i, j) ∈ B, i ∈ I, (24)

yk = minh∈J

ckh

qh=

ckj

qjfor (k, j) ∈ B, k ∈ K. (25)

Condition (24) means that, for (i, j) ∈ B, the jth product has the maximal profit per unit of the spentmoney for the ith consumer, i ∈ I, and for prices q. Similarly, condition (25) means that, for (k, j) ∈ B,for the firm k ∈ K, and for the prices q, the jth product is characterized by the minimal costs per unit ofthe firm’s revenue. We can write down (24) and (25) as

yiqj cij for i ∈ I, j ∈ J,

yiqj = cij for (i, j) ∈ B, i ∈ I,

ykqj ckj for k ∈ K, j ∈ J,

ykqj = ckj for (k, j) ∈ B, k ∈ K.


130 SHMYREV

Finding the logarithm of the left-hand and the right-hand sides of these conditions, we obtain

ln yi + ln qj ln cij for i ∈ I, j ∈ J,

ln yk + ln qj ln ckj for k ∈ K, j ∈ J,

ln ys + ln qj = ln csj for (s, j) ∈ B.

Therefore,

ui = − ln yi, uk = − ln yk, vj = ln qj (i ∈ I, k ∈ K, j ∈ J)

constitute an admissible solution to the dual system (18)–(19); with that, the inequalities correspondingto (s, j) ∈ B hold as equality. Moreover, for each s ∈ S, there is a pair (s, j) ∈ B.

Under the dual nondegeneracy condition, the above means that B ∈ B. Correspond the set B withthe set of all q ∈ σ satisfying (24)–(25). This set is denoted by Ξ(B). It is clear that if B1 ⊂ B2 thenΞ(B1) ⊃ Ξ(B2), and Ξ(B2) is a face of the polyhedral set Ξ(B1). It is easy to convince that, under thedual nondegeneracy condition, the sets Ξ(B), B ∈ B, have no common relatively interior points andcover σ. The collection of sets Ξ(B), B ∈ B, constitutes a polyhedral complex ξ.

Theorem 1. A price vector p ∈ σ is an equilibrium if and only if p ∈ Ω(B) ∩ Ξ(B) for someB ∈ B.

Proof. By p ∈ Ξ(B), it follows that, for p = p, the production-consumption structure B, as wasdemonstrated above, is preferable for all participants of the model. From this and p ∈ Ω(B), it followsthat, for zsj uniquely obtained as solutions to (13)–(15) with (22), xs

j = zsj/p determine the optimalvalues of xs

j to the problems of participants; with that, the balance conditions (8) hold. Therefore, p is anequilibrium price vector of the model.

Conversely, if p is an equilibrium then p corresponds to the collection xs, s ∈ S, which are thesolutions to the corresponding problems of participants (2)–(4), (5)–(7) and satisfy the balancecondition (8). If we take

B = (s, j) ∈ S × J : xsj > 0

then it is easy that p ∈ Ω(B) ∩ Ξ(B). Theorem 1 is proved.

Thus, the problem of finding an equilibrium of the price vector leads to the following problem: Finda pair of corresponding cells Ω(B) ∈ ω and Ξ(B) ∈ ξ that have the nonempty intersection.

It is well known that the linear complementarity problem (see [12]) leads to the similar problem withthe only singularity that the polyhedral complexes appeared in this case consist of the polyhedral cones.

It follows from the above that the problem of finding an equilibrium of the price vector in themodel under consideration leads to the polyhedral complementarity problem generalizing the linearcomplementarity problem.

We can also rewrite the problem in the form of a fixed point problem. To this end, we introduce thepoint-to-set mapping F : Ω → 2σ

on the set Ω as follows: Recall that each point p ∈ Ω is in the relativeinteriority of exactly one set Ω(B), B ∈ B. Denoting the corresponding B by Bp, we put F (p) = Ξ(Bp).Consequently, Theorem 1 can be rewritten as follows:

Theorem 2. A vector p ∈ σ is an equilibrium if and only if p is a fixed point of the mapping F ;i.e., p ∈ F (p).

Since p ∈ F (p) is equivalent to p ∈ F−1(p), we can replace the mapping F by Φ = F−1 in Theorem 2.The mapping Φ is defined on σ and relate to each point q ∈ σ the cell Ω(Bq) ∈ ω as the image Φ(q),where Bq is uniquely determined by q if q belongs to the relative interiority of the cell Ξ(Bq).



3. THE MODEL WITH FIXED BUDGETS

Among the classical linear exchange models, a particular case is well known that was considered byD. Gale [1] and called the model with fixed budgets. We mean the model where the consumers haveno initial stocks of products, i.e., the products are already on the market in necessary quantities. Theconsumers have only some initial stocks of money.

The corresponding analog of the exchange model with fixed budgets is obtained if the right-hand sideαi + (p, di) in (6) is replaced with λi under the assumption that λi and λk, i ∈ I and k ∈ K, are relatedas

∑

i∈I

λi =∑

k∈K

λk + 1, (26)

which is an analog to (11). Condition (13) of the transportation problem is changed analogously to theform

−∑

j∈J

zij = −λi, i ∈ I. (27)

It is well known that there is always an equilibrium in the exchange model with fixed budgets, whereas,in general case, there my be no equilibrium [10]. The proof of existence of an equilibrium in the modelwith fixed budgets in [1] bases on reducing the problem of finding an equilibrium to the optimizationproblem of the form: Maximize

∑i∈I λi ln(ci, xi) under the conditions

∑i∈I xi = e, xi 0, and i ∈ I.

The approach below bases on a principally different optimization problem.As was observed, the transportation problem for the model is solvable for each p belonging to the

set Ω that is described by (1) and the inequalities∑

j∈Q

pj +∑

k∈K

λk 0 for Q ⊂ J. (28)

Consequently, the concave function f(p) arises that sends p ∈ Ω to the optimal value of the targetfunction in (12). For p /∈ Ω, put f(p) = −∞. As observed above, f(p) is piecewise linear, and the linearitydomains of this function coincide with Ω(B), B ∈ B; i.e., are defined by the cells of the polyhedralcomplex ω. The peculiarity of the model with fixed budgets consists in the fact that the cells Ξ(B) ofthe dual polyhedral complex ξ allow us to describe the subdifferential mapping G : p ∈ Ω → ∂f(p).

For a concave function f , the set of its subgradients at the point p is understood to be ∂f(p); i.e., theset of vectors g such that

f(q) f(p) + (g, q − p) for all q. (29)

It is easy that, for a convex function (−f), the vector (−g) is a subgradient in the conventional sense,and its subdifferential ∂(−f)(p) is related with ∂f(p) by the equality ∂(−f)(p) = −∂f(p).

Since in the case under study the function f(p) is generated by the linear programming problemwith varying right-hand sides of the constraints, the set ∂f(p) can be described by means of the optimalvectors of the dual variables corresponding to the optimal solution to (12)–(16) at p (see [7]); i.e., if W (p)is the set of the optimal dual vectors w = (u1, . . . , um+l, v1, . . . , vn) at p then ∂f(p) is a projectionof W (p) onto the space of variables vj , j ∈ J , which can be written down as ∂f(p) = V (p). Show thatthe mapping G = ∂f is related with the introduced mapping F : p ∈ Ω(B) → Ξ(Bp) as follows:

∂f(p) = ln q + te : q ∈ F (p), t ∈ R1, (30)

where ln q = (ln q1, , . . . , ln qn) and e = (1, . . . , 1) ∈ Rn.

Indeed, for p from the relative interior Ω(B) of Ω(B), B ∈ B, the set W (p) is defined as the set of allsolutions of the inequalities (18)–(19) provided that those corresponding to (s, j) ∈ B hold as equality:

−ui + vj = ln cij for i ∈ I, (i, j) ∈ B,

uk − vj = − ln ckj for k ∈ K, (k, j) ∈ B.


132 SHMYREV

It is clear that, adding the same summands to us and vj (s ∈ I ∪K and j ∈ J), we do not change us − vj .Hence, we have the set of corresponding vectors v = (v1, . . . , vn); i.e., V (p) is such that if v ∈ V (p) then(v + te) ∈ V (p) for every t.

Introducing qj > 0 and ys > 0 by the formulas us = − ln ys and vj = ln qj , we find that (24)–(25)hold for these values; and q = (q1, . . . , qn) can be multiplied by a positive number. Therefore, we canassume that q ∈ σ. Consequently, q ∈ Ξ(B) = F (p). We also obtain the converse; i.e., taking q ∈ Ξ(B)B ∈ B, we can define ys using (24)–(25) and show that us = − ln ys and vj = ln qj determine theoptimal vector of the dual variables in (12)–(16) for p ∈ Ω(B).

Equality (30) allows us to state Theorem 2 in the case of the model with fixed budgets:Theorem 3. A point p ∈ σ is an equilibrium price vector of the model with fixed budgets if and

only if ln p ∈ ∂f(p).Remark. In the above proof of this assertion, we use the polyhedral complexes w and ξ introduced

under the assumption that the dual nondegeneracy condition holds. However, we can show thatTheorem 3 is also valid without this assumption. We will not prove this fact here.

Introduce h(p) by putting h(p) =∑n

j=1 pj ln pj for p ∈ σ and extent continuously it on the bound-ary ∂σ of the simplex σ as follows: h(p) = 0 for p ∈ ∂σ. Given p /∈ σ, put h(p) = +∞.

The function h(p) is strictly convex on σ. The concave function f(p) is defined on σ ⊂ Ω. Thereby,the strictly convex function ϕ(p) = h(p) − f(p) is defined on σ. This function is obviously continuousand, hence, has a minimum point on the compact set σ. By the strict convexity of ϕ, this point isunique. Moreover, by the properties of h and f , the minimum point cannot belong to the boundary ∂σ.To demonstrate this, we consider the points p ∈ σ tending to some point p ∈ ∂σ along a definitedirection g. Then it is easy that ∂h

∂g (p) → +∞, and ∂f∂g (p) does not change for p sufficiently close to p

(by the piecewise linearity of f ). Eventually, we find

∂ϕ

∂g(p) → +∞, asp → p.

Thus, the minimum point of ϕ on σ lies inside σ (i.e., belongs to σ).Extent ϕ to the whole R

n by putting ϕ(p) = +∞ for p /∈ σ.Theorem 4. A vector p is an equilibrium price vector in the model with fixed budgets if and only

if p is a minimum point of ϕ(p).

Proof. Both an equilibrium price vector of the model and a minimum point of ϕ belong to σ. Thenecessary and sufficient condition for p ∈ σ be a minimum point of ϕ is as follows: 0 ∈ ∂ϕ(p). By theMoreau–Rockafellar Theorem,

∂ϕ(p) = ∂h(p) + ∂(−f)(p) = ∂h(p) − ∂f(p).

Therefore,

∂h(p) = ln p + te : t ∈ R1.

Moreover, ∂f(p) is such that if g ∈ ∂f(p) then (g + te) ∈ ∂f(p) for every t. Thereby, the condition0 ∈ ∂ϕ(p) is equivalent to ln p ∈ ∂f(p). This and Theorem 3 complete the proof of Theorem 4.

Corollary. Provided that all cs > 0, s ∈ S, there is a unique equilibrium price vector in themodel with fixed budgets.

Remark. The requirement of positivity of the vectors ci for i ∈ I can be weakened. We elaborate thisin Section 6.

The starting point of proving Theorem 4 is Theorem 2 relating the equilibria price vectors with thefixed points of F . Since, as was observed, the fixed points of F coincide with the fixed points of the inversemapping Φ = F−1; basing on Φ, we can state the dual equilibrium criterion in some sense. To this end,we should pass to the conjugate functions. For concave functions, this notion is introduced as follows:If f is a concave function then we let f∗ stand for the conjugate function of f which is defined as

f∗(y) = infx(x, y) − f(x).



It is easy that f∗ is a concave function and f∗(y) = −(−f)∗(−y), where (−f)∗ is the conventionalconjugate of a convex function (−f). Moreover, as in the case of convex functions, the subdifferentialmappings ∂f and ∂f∗ are mutually inverse. Recall that a subgradient of a concave function is determinedby (29).

Theorem 5. p ∈ σ is an equilibrium price vector of the model if and only if p is a minimumpoint of the function ψ(p) = −f∗(ln p) on σ.

Proof. As follows from the proof of Theorem 4, p ∈ σ is an equilibrium if and only if ln p ∈ ∂h(p) ∩∂f(p). Since the subdifferential mappings are inverse to one another, this condition is equivalent to thefollowing property of the conjugate functions:

p ∈ ∂h∗(ln p) ∩ ∂f∗(ln p).

The last containment means that, for the convex function η = h∗ − f∗, we have 0 ∈ ∂η(ln p).Consequently, p is a minimum point of η(ln p) on the set of positive vectors p.

Thus, the equilibrium at p ∈ σ is equivalent to the fact that p is a minimum point of

η(ln p) = h∗(ln p) − f∗(ln p) on Rn++.

We are left with using the fact (see [2], p. 166) that h∗(x) can be represented as

h∗(x) = lnn∑

j=1

exj .

Therefore, h∗(ln p) = ln∑n

j=1 pj for every p > 0. Now, if p is an equilibrium price vector then p ∈ σ;and since p is a minimum point of η(ln p) on Rn

++; p is a minimum point of this function on σ. However,∑nj=1 pj = 1 on σ. Consequently, h∗(ln p) = 0. Therefore, p is a minimum point of ψ(p) = −f∗(ln p)

on σ.

Check the validity of the converse. If p is a minimum point of ψ(p) on σ then p is a minimum pointof η(ln p) on σ, since these functions coincide on σ. However, it is easy that η(ln p) is a positivehomogeneous function of zero degree in the variable p, i.e., η(ln(tp)) = η(ln p) for all t > 0. Indeed,for q /∈ σ, we have h(q) = +∞. Consequently,

h∗(ζ) = supq∈Rn

((q, ζ) − h(q)

)= sup

q∈σ

((q, ζ) − h(q)

).

Therefore, for ζ = ln(tp), we obtain

h∗(ln(tp)) = supq∈σ

(ln t)

n∑

j=1

qj + (q, ln p) − h(q)

.

Since∑n

j=1 qj = 1 for q ∈ σ, we thus infer

h∗(ln(tp)) = ln t + supq∈σ

(q, ln p) − h(q) = ln t + h∗(ln p).

Similarly, since∑n

j=1 qj = 1 and f(q) = −∞ for q /∈ Ω; therefore,

f∗(ln(tp)) = ln t + infq∈Ω

(q, ln p) − f(q) = ln t + f∗(ln p).

Consequently,

η(ln(tp)) = ln t + h∗(ln p) − (ln t + f∗(ln p)) = h∗(ln p) − f∗(ln p) = η(ln p).

Using the homogeneity of η(ln p), we conclude that p is a minimum point of η(ln p) on Rn++. Hence, p is

the equilibrium price vector of the model. Theorem 5 is proved.


134 SHMYREV

4. A SUBOPTIMIZATION ALGORITHM FOR FINDINGAN EQUILIBRIUM PRICE VECTOR

4.1. Description of the Algorithm

The procedure generalizing the scheme of the method of successive improvement (the simplex-method) for the linear programming problems was proposed in [3] for the problem of minimization ofa quasiconvex function on a convex polyhedron given by a linear system of equations and inequalities.This procedure called the suboptimization method consists in the goal-directed choice of the faces ofthe initial polyhedron. Considering a plane, we solve a problem of finding a minimum of the function onthe affine hull of a face, which explains the name of the procedure.

We can consider the above procedure of finding an equilibrium price vector as an analog to the subop-timization method for the problem of the minimizing the function ψ(p) on σ introduced in Section 3. Thecells Ξ(B), B ∈ B, play the role of the faces of the initial polyhedron. This procedure is a generalizationof similar constructions for the classical exchange model exposed in [5, 14].

A step of procedure is the following: To the beginning of step (t + 1), there are some Bt ∈ B andqt ∈ Ξ(Bt). Let L(Bt) be the affine hull of Ξ(Bt) ∈ ξ, and let M(Bt) be the affine hull of Ω(Bt) ∈ ω.We will demonstrate that, for every B ∈ B, there is a unique intersection point r of the correspondingaffine hulls L(B) and M(B); i.e., r = L(B) ∩ M(B). Let rt be this point generated by Bt. The twocases are possible:

(i) qt = rt. In this case, the validity of rt ∈ Ω(Bt) is checked. To this end, insert p = rt into (13) and(14); and, requiring additionally that zsj = 0, (s, j) /∈ Bt, we define zsj(rt) from (13)–(15). The conditionrt ∈ Ω(Bt) is equivalent to zsj(rt) 0 for all (s, j) ∈ Bt. If it is true then p = rt is an equilibrium pricevector of the model. If it is not then choose an arbitrary zsj(rt) < 0 and proceed to the next step withqt+1 = qt and Bt+1 = Bt \ (s, j).

(ii) qt = rt. In this case consider the varying point q(ε) = qt + ε(rt − qt) and define the maximal valueε = ε∗ for q(ε) ∈ Ξ(Bt) and ε 1.

The condition q(ε) ∈ Ξ(Bt) leads (according to the description of Ξ(Bt)) to q(ε) > 0 and the systemof inequalities (see (24)–(25))

cij

qj(ε) ci

h

qh(ε)for (i, j) ∈ Bt, h ∈ J, i ∈ I, (31)

ckj

qj(ε) ck

h

qh(ε)for (k, j) ∈ Bt, h ∈ J, k ∈ K. (32)

We demonstrate below that the maximal value ε = ε∗ exists always.

If ε∗ = 1 then we proceed to the next step with qt+1 = rt and Bt+1 = Bt. In this event we already havecase (i).

If ε∗ < 1 then the choice of ε > ε∗ is limited by some of the inequalities (31) or (32). If this limitinginequality is, for example, from (31) and corresponds to (i, h) /∈ Bt then we put Bt+1 = Bt ∪ (i, h)at the next step. Similarly, if the limiting inequality is from (32) and corresponds to (k, h) /∈ Bt thenBt+1 = Bt ∪ (k, h). In both cases, put qt+1 = q(ε∗) and proceed to the next step.

4.2. Well-Posedness of the Algorithm

Show that, owing to the above-presented description, the procedure of a single step of this algorithmis always possible.

4.2.1. Show first that, for every B ∈ B, the intersection point r of the affine hulls of Ξ(B) and Ω(B)is determined uniquely. The affine hull L(B) of Ξ(B) is given by (1) and

psj

csj

=ps

h

csh

for (s, j), (s, h) ∈ B. (33)



To define the affine hull M(B) of Ω(B), as was observed above, we should consider the balanceconditions (21) on each connected component of Γ(B). Applying in case under study the problem withfixed budgets, we obtain

∑

i∈Iν

λi =∑

k∈Kν

λk +∑

j∈Jν

pj , ν = 1, . . . , τ. (34)

Here Iν ⊂ I, Jν ⊂ J , and Kν ⊂ K are the sets defining the vertex set of the νth connected componentof the graph. Thus, for finding the point r, we need to consider the systems of equations (33) and (34)simultaneously. Since B ∈ B, the graph Γ(B) does not contain any cycles. Therefore, (33) is compatibleand determines some proportions between the unknown pj , j ∈ Jν . This means that pj on the νthconnected component are determined uniquely to within a factor; i.e., pj = tνgj , where gj are somepositive numbers defining the proportion. The factors tν are uniquely determined from the correspondingequations of (34) after inserting the representation for pj .

Thus, the solution of (33)–(34) is always unique, and so the point r is determined uniquely.

4.2.2. We show now that, under the conditions q(ε) ∈ Ξ(Bt) and ε 1, the maximal value ε =ε∗ always exists. This requires a proof, since not all cells Ξ(Bt) are closed by the condition q > 0in description of these sets. We need to show that this condition cannot be limiting as ε increases.

Since qt ∈ Ξ(Bt), we have qt > 0. If also rt > 0 then, for q(ε) = qt + ε(rt − qt), we see that q(ε) > 0for all t ∈ [0, 1].

Let rt≯ 0; let, for some ε ∈ (0, 1], q(ε) ∈ Ξ(Bt) for ε ∈ (0, ε); and let the zero exist among qj(ε).

Given this j, we infer

csj

qj(ε)−→ +∞ as ε → ε − 0 for every s ∈ I ∪ K.

This means that (k, j) /∈ Bt for all k ∈ K. At the same time, among these j there is j such thatat least one pair (i, j), i ∈ I, is in Bt; for example, the pair, corresponding to the maximal ci

j/qj(ε) for ε

sufficiently close to ε. Let ν be the number of connected components of the graph Γ(Bt) to which thearc belongs corresponding to (i, j). We have Kν = ∅ and Iν = ∅. Moreover, rt

j 0 for every j ∈ Jν .However, it follows from (34) that

∑

i∈Iν

λi =∑

j∈Jν

rtj;

the left-hand side of this is positive, and the right-hand side is not.

The contradiction obtained implies that the condition q > 0 cannot be limiting as ε increases.Therefore, some maximal value ε = ε∗ exists.

4.2.3. To show the well-posedness of the algorithm, we should check that Bt+1 ∈ B. To this end,observe that if we add a new pair (i, h) or (k, h) to Bt then no new cycle appears in the graph Γ(Bt);since, in each connected component of Γ(Bt), the mutual proportions between the values qj(ε) donot change, and, therefore, the added arc joins two different connected components. Consequently, allconnected components of Γ(Bt+1), as well as of Γ(Bt), are the trees. As was observed in Section 2, thismeans that Bt+1 is a subset of some basis set of (12)–(16).

The dual feasibility conditions of Bt+1, equivalent to the conditions of preferable connections for(s, j) ∈ Bt+1, obviously hold by the choice of the extra pair (i, h) or (k, h).

Finally, the elimination of (s, j) from Bt in the case (i) does not lead to violation of the s-coveringcondition (20), since (15) and (27) hold for zsj = zsj(rt), and all λi, λk are positive. This means thatthere also must be positive values among zsj(rt); they correspond to the pairs (s, j) = (s, j), sincezsj(rt) < 0. It is clear that such (s, j) belongs to Bt ⊂ Bt+1.

Thus, we always have that Bt+1 ∈ B. The well-posedness of the algorithm is completely justified.


136 SHMYREV

4.3. Start of the Process

We can take an arbitrary point from σ as a starting point q. The corresponding set B such thatq ∈ Ξ(B) is composed of the pairs (i, j) and (k, j) such that

maxh∈J

cih

q0h

=cij

q0j

for i ∈ I, minh∈J

ckh

q0h

=ckj

q0j

for k ∈ K.

Moreover, we should guarantee the absence of any cycles in the corresponding graph Γ(B). This canbe reached by inessentially varying, if need be, the chosen point q so that, for each s ∈ S, the only pair(s, j) belongs to B (i.e., for each i ∈ I and k ∈ K, the above max and min are attained for the only h).It is clear that in this case B ∈ B.

5. FINITENESS OF THE PROCESS

The goal of the below considerations is to prove the followingTheorem 6. Under the dual nondegeneracy condition, the suboptimization process allows us

to obtain an equilibrium price vector of the model with fixed budgets in finitely many steps.

Proof. Since, in the case qt = rt, the value ψ(qt) is uniquely determined by Bt; henceforth, such Bt

cannot repeat if ψ decreases at the current point qτ . We will demonstrate that if qt+1 = qt thenψ(qt+1) < ψ(qt). Thereafter to conclude the proof, we need only to show that the process cannot “getstuck” at some point; i.e., it is impossible that qt = qτ for all t τ .

In more detail, let qt = rt at the next step of the process. According to description of the algorithm,the translation from qt in the direction to rt over the interval [qt, rt] within the Ξ(Bt) follows.

Lemma 1.If h = rt − qt = 0 then

∂ψ

ψh(qt) < 0. (35)

Proof. Since ψ(q) = −f∗(ln q), we have to consider the function f∗(ln q) in detail. The function f∗ isconcave, proper and closed. For these functions, the Fenchel inequality (see [2], p. 121) holds, which inthe case under study has the form

f(p) + f∗(ln q) (p, ln q) for p, q ∈ Rn++. (36)

It is also known that this inequality turns into equality if and only if ln q ∈ ∂f(p) or, equivalently,p ∈ ∂f∗(ln q). We are interested in q ∈ Ξ(Bt). As was observed, for these q, we have ln q ∈ ∂f(p) forp ∈ Ω(Bt). This follows from (30). Thus,

f∗(ln q) = (p, ln q) − f(p) for p ∈ Ω(Bt), q ∈ Ξ(Bt). (37)

Introduce the functions ψp(q) = f(p)− (p, ln q). It follows from the above that ψ(q) coincides with ψp(q)for p ∈ Ω(Bt) on Ξ(Bt). In turn, we can pass from ψp(q) to the functions

ψp(q) = ψp(q) + lnn∑

j=1

qj

which coincide with ψp(q) on σ, since∑n

j=1 qj=1. However, for p ∈ Ω(Bt) ⊂ Ω, condition (1) holds;i.e., (p, e) = 1. Consequently, ψp(q) is a positive homogeneous function of zero degree, and ψp(tq) =ψp(q) is valid for all t > 0. Indeed,

ψp(tq) = f(p) − (p, ln(tq)) + lnn∑

j=1

(tqj) = f(p) − (p, ln q) − (p, e) ln t + ln t + lnn∑

j=1

qj = ψp(q).



Therefore, instead of ψ(q) on Ξ(Bt) we can consider ψp(q) on Rn++. Thus, for the derivative of ψ(q)

at qt in direction h not leaving the affine hull of Ξ(Bt), we obtain

∂ψ

∂h(qt) =

(∇ψp(qt), h

), p ∈ Ω(Bt), (38)

where h = rt − qt and rt is the unique intersection point of the affine hulls of the cells Ω(Bt) and Ξ(Bt).It is easy that

∇ψp(qt) =

⎛

⎜⎜⎜⎝

−p1/qt1

...

−pn/qtn

⎞

⎟⎟⎟⎠ +1

n∑j=1

qtj

⎛

⎜⎜⎜⎝

1...

1

⎞

⎟⎟⎟⎠ =

⎛

⎜⎜⎜⎝

−p1/qt1

...

−pn/qtn

⎞

⎟⎟⎟⎠ +

⎛

⎜⎜⎜⎝

1...

1

⎞

⎟⎟⎟⎠

since∑n

j=1 qtj = 1 for qt ∈ Ξ(B) ⊂ σ. Henceforth, we have

(∇ψ(qt), h

)=

(∇ψp(qt), rt − qt

)=

(∇ψp(qt), rt

)−

(∇ψp(qt), qt

).

But(∇ψp(qt), qt

)= −

n∑

j=1

pj +n∑

j=1

qtj = −1 + 1 = 0,

since p ∈ Ω(Bt) ⊂ Ω and∑n

j=1 pj = 1 for p ∈ Ω. Therefore,

(∇ψp(qt), h

)=

(∇ψp(qt), rt

)= −

n∑

j=1

pj

rtj

qtj

+n∑

j=1

rtj = −

n∑

j=1

pj

rtj

qtj

+ 1,

since∑n

j=1 rtj = 1. Now, by (38), the proof of (35) reduces to checking that

n∑

j=1

pj

rtj

qtj

> 1 for p ∈ Ω(Bt). (39)

Let the graph Γ(Bt) consist of τ connected components, and let the νth connected componentcorrespond to Jν ⊂ J . Then

n∑

j=1

pj

rtj

qtj

=τ∑

ν=1

∑

j∈Jν

pj

rtj

qtj

. (40)

The proportions given by the equalitiesqj

csj

=ql

csl

, (s, j), (s, l) ∈ Bt

must hold for qtj , j ∈ Jν . Let the positive gj , j ∈ Jν , determine these proportion; i.e.,

qtj = tqνgj , j ∈ Jν , (41)

where tqν is some positive factor. We can suppose that∑

j∈Jνgj = 1 for each ν = 1, . . . , τ . For rt

j , j ∈ Jν ,the proportions are the same, since rt and qt belong to the affine hull of Ξ(Bt):

rtj = trνgj , j ∈ Jν . (42)

In the general case, the vector rt need not be strictly positive, unlike qt. This is valid for trν either. Theyare determined from the condition that rt belongs to the affine hull of Ω(Bt); i.e., they are obtained fromthe balance conditions on the connected components of Γ(Bt):

∑

i∈Iν

λi =∑

j∈Jν

rtj +

∑

k∈Kν

λk, (43)


138 SHMYREV

where Iν ⊂ I and Kν ⊂ K are the sets corresponding to the νth connected component. We haven∑

j=1

pj

rtj

qtj

=τ∑

ν=1

trνtqν

∑

j∈Jν

pj . (44)

Since p ∈ Ω(Bt), the balance conditions similar to those for rtj must hold for pj :

∑

i∈Iν

λi =∑

j∈Jν

pj +∑

k∈Kν

λk.

Hence,∑

j∈Jν

pj =∑

i∈Iν

λi −∑

k∈Kν

λk =∑

j∈Jν

rtj = trν

∑

j∈Jν

gj = trν .

Consequently, by (44), we obtainn∑

j=1

pj

rtj

qtj

=τ∑

ν=1

(trν)2

tqν, (45)

and the sought inequality (39) equivalent to (35) takes the formτ∑

ν=1

(trν)2

tqν> 1. (46)

As regards tqν of (41), we haven∑

j=1

qtj =

τ∑

ν=1

tqν∑

j∈Jν

gj =τ∑

ν=1

tqν .

Since qt ∈ σ, we have∑n

j=1 qtj = 1. Consequently,

∑τν=1 tqν = 1. Similarly, it follows from (42) and∑n

j=1 rtj = 1 that

∑τν=1 trν = 1.

As was observed, by qt > 0, we obtain tqν > 0 for every ν = 1, . . . , τ , and there can be some negativevalues among trν .

To prove (46), check the following

Lemma 2.For some fixed βν > 0, ν = 1, . . . , τ , such that∑τ

ν=1 βν = 1, the minimal value of thefunction γ(α) =

∑τν=1 α2

ν/βν under the condition

τ∑

ν=1

αν = 1 (47)

is equal to 1 and attained at αν = βν .

Proof. Introduce λ, the Lagrange multiplier of (47). Then we have for the optimal values αν :

2αν

βν= λ, ν = 1, . . . , τ,

τ∑

ν=1

αν = 1.

The solution of this system is λ = 2 and αν = βν , ν = 1, . . . , τ . For the minimal value γ(α), we obtain

γ(α) =τ∑

ν=1

β2ν

βν=

τ∑

ν=1

βν = 1.

Lemma 2 is proved.



Thus, in the conditions of Lemma 2, it follows from α = β that γ(α) > 1.Continue the proof of Lemma 1. Applying Lemma 2 for αν = trν and βν = tqν , we can claim that if

tr = tq (i.e., rt = qt) then (46) holds, and (35) is valid as well. Lemma 1 is proven.

Show now that, under the dual nondegeneracy condition for (12)–(16), the process cannot “getstuck”; i.e., after finitely many steps, a nonzero translation from the current point qt is realized bynecessity. In other words, ε∗ > 0, qt+1 = qt, and hence, ψ(qt+1) < ψ(qt).

The reason for the zero translation is the presence of a pair (s, h) /∈ Bt such that

csj

qtj

=csh

qth

, (s, j) ∈ Bt, (48)

and, at the same time, the corresponding inequality of (31)–(32) turns out to be limiting for the definitionof ε∗. Consequently, ε∗ = 0.

The pair (s, h) /∈ Bt for which (48) holds appears during realization of case (i) at the current step,while eliminating from Bt the pair (s, j) such that zsj(rt) < 0. In this case

Bt+1 = Bt \ (s, j), qt+1 = qt.

Therefore,

csj

qt+1j

=csj

qt+1j

, (s, j) ∈ Bt+1, (s, j) /∈ Bt+1. (49)

We demonstrate below that rt+1 = qt+1 holds at step (t + 1) of the algorithm; i.e., case (ii) takes place. Inthis case, if at step (t + 1) of the algorithm, the corresponding inequality in (31)–(32) generated by Bt+1

turns out to be limiting for the definition of ε∗ then this would return to the previous set Bt, i.e.,

Bt+2 = Bt+1 ∪ (s, j) = Bt.

Let us show that this is impossible.Lemma 3. During the run of the algorithm, case (ii) always follows case (i). If, in case (i)

at step t, the pair (s, j) is eliminated from Bt then the corresponding inequality in (31)–(32)generated by Bt+1 does not prevent increasing ε at step (t + 1).

Proof. Consider the graph Γ(Bt). Let, for definetness, (s, j) correspond to the arc belonging to theconnected component with the number ν = 1. Let the vertex set of this component consist of the verticesj ∈ J1 ⊂ J and the vertices of the form n + s for s ∈ S1 ⊂ S. After eliminating the arc corresponding to(s, j), the connected component splits into two components. The sets J1 and S1 split respectivelyinto J1 = J ′

1 ∪ J ′′1 and S1 = S′

1 ∪ S′′1 ; the sets J ′

1 and S′1 correspond to one part of the component,

while J ′′1 and S′′

1 , to the other. Let s ∈ S′1. Then j ∈ J ′′

1 . Write down the equations of (13)–(15) forp = rt corresponding to the vertices of the part of the component which corresponds to J ′

1 and S′1. Put

S′1 = I ′1 ∪ K ′

1.

−∑j∈J

zij = −λi, i ∈ I ′1

∑i∈I

zij −∑

k∈K

zkj = qtj, j ∈ J ′

1,

∑j∈J

zkj = λk, k ∈ K ′1.

In the above sums, the variables zsj are to be omitted for (s, j) /∈ Bt, since zsj(rt) = 0, (s, j) /∈ Bt. If wesum up these equations then all variables, but zsj , will cancelled, which gives an explicit formula forzsj(rt). Let s ∈ I ′1; i.e., s = i. Summing up, we obtain

−zij = −∑

i∈I′1

λi +∑

k∈K ′1

λk +∑

j∈J ′1

qtj


140 SHMYREV

or

zij =∑

i∈I′1

λi −∑

k∈K ′1

λk −∑

j∈J ′1

qtj. (50)

Recall that qt = rt and zij = zij(rt) < 0. Therefore, zij(rt+1) = 0, since after the above-indicatedsplit of the connected component into the two parts, the equation

∑

i∈I′1

λi =∑

k∈K ′1

λk +∑

j∈J ′1

ptj

appears in system (34) describing the affine manifold M(Bt+1). Hence, rt+1 = rt(= qt+1). Therefore,case (ii) already takes place at step (t + 1). It is simple that J ′

1 = ∅, for if J ′1 = ∅ then we obtain

K ′1 = ∅, I ′1 = i, and zij = λi > 0. Moreover, J ′′

1 = ∅, since j ∈ J ′′1 . By (50), it is easy to see

how to vary qj , deflecting them from qtj , in order to make the value zij increase and finally become zero

that means getting of the point q into L(Bt+1); namely, the values qj , j ∈ J ′1, must be decreased, and, to

keep∑

j∈J qj = 1, the values qj for j ∈ J ′′1 must be increased simultaneously. Moreover, if we preserve

the mutual proportions between qj in each part of the connected component (i.e., between qj , j ∈ J ′1, as

well as between qj , j ∈ J ′′1 ) then q ∈ M(Bt+1). Hence, as q gets into L(Bt+1), it coincides with rt+1.

Thus, the above-described changes of the current point q correspond exactly to the motion fromqt+1 = rt to rt+1 executed by the algorithm at step (t + 1). If j1 ∈ J ′

1 then qj1 decreases and qj in-creases. It follows from (49) that

cij1

qt+1j1

=cij

qt+1j

.

Consequently, for the changes of q (under the condition qj1 > 0), we obtain

cij1

qj1

>cij

qj. (51)

This means that the constraint from (31) corresponding to (s, h) = (i, j) is not limiting for thedefinition of ε∗.

Consideration of the case s = k ∈ K1 is similar except the only fact that the change of q isaccompanied by some increase of qj for j ∈ J ′

1 and some decrease of qj for j ∈ J ′′1 . Thus, qj1 increases,

qj decreases, and the inequality similar to (51) changes the sign. Hence, the constraint of (32)corresponding to (s, h) = (k, j) holds for an arbitrary translation from qt+1 in the direction to rt+1

and, therefore, cannot be limiting for the definition of ε∗. Lemma 3 is proven.

Remark. We should observe that it is exactly in the above fact that there appears the monotonicityproperty mentioned in the Introduction which determines the class of models of the polyhedral com-plementarity problem under consideration and relates it with a linear complementarity problem of theclass P .

We now pass to the final part of the proof of Theorem 6. Observe first that the implementing case (ii)of the algorithm, i.e., the case of qt = rt, some extension of current set Bt takes place. Consequently,this case can be repeated only finitely many times, and, afterwards, case (i) takes place, and qt = rt.Let us show that, between every two successive implementations of case (i), the current value of ψ(qt)decreases strictly.

Let case (i) take place for the set Bt and qt ∈ Ξ(Bt); i.e., rt = qt; and let zsj(rt) < 0 for zsj(rt)obtained at p = rt.

Let Bt+1 = Bt \ (s, j) and qt+1 = qt. By Lemma 3, for finding ε∗ at step (t + 1), the inequalitiesof (31)–(32) corresponding to (s, h) = (s, j) do not limit the increase of ε. Let the inequality corre-sponding to (s, h) = (s1, j1) be limiting; we proceed the next step putting

Bt+2 = Bt+1 ∪ (s1, j1).



If it terns out that ε∗ = 0, i.e., qt+2 = qt+1; then

cs1j1

qt+1j1

=cs1j

qt+1j

, (s1, j) ∈ Bt+1. (52)

We have Bt+2 =(Bt ∪ (s1, j1)

)\ (s, j). Introduce B′

t+1 = Bt ∪ (s1, j1). By the dual nonde-generacy condition, Bt cannot be a basis set of (12)–(16). Moreover, the arc corresponding to (s1, j1)cannot close a cycle in Γ(Bt), for otherwise the corresponding inequality in (31)–(32) cannot be limiting.Thus, B′

t+1 ∈ B, and (52) means that qt+1 ∈ Ξ(B′t+1). At the same time, M(B′

t+1) ⊃ M(Bt) and,hence, qt+1 = rt ∈ M(B′

t+1), since rt ∈ M(Bt). Thus, the intersection point rt+1 of L(B′t+1)∩M(B′

t+1)is qt+1. In other words, if at step (t + 1) we start with B′

t+1 and qt+1 then case (i) arises. Moreover,

defining zB′

t+1

sj (rt+1) by solving (13)–(15) with the additional condition zsj = 0, (s, j) /∈ B′t+1, we obtain

the previous values zBtsj (rt). Consequently, according to the description of the algorithm, we have

B′t+2 = B′

t+1 \ (s, j) = Bt+2, qt+2 = qt+1.

By the above arguments, the situation at step (t + 2) is qualitatively the same as at step (t + 1) anddiffers only in the replacement of Bt by B′

t+1 = Bt ∪ (s1, j1). Therefore, the inequality in (31)–(32)corresponding to the excluded pair (s, j) again cannot be limiting.

It is clear that the successive implementations of case (ii) with zero translation can be repeated onlyfinitely many times, since the current set Bt only expands with that and, by the assumption of the dualnondegeneracy, when Bt becomes a basis set, for Bt+1 \ (s, j) there is no pair (s1, j1) such that (52)holds, and the value of the translation is nonzero. By Lemma 1, for a nonzero value of translation thevalue of ψ decreases; i.e., ψ(qt+1) < ψ(qt).

Thus, the sets Bt arising in the implementation of case (i) does not appears again. The process isfinite. Theorem 6 is proven.

6. POSSIBLE GENERALIZATIONS

The exposition of the algorithm for the model with fixed budgets is presented above under theassumption that ci > 0 for all i ∈ I. We can weaken this condition by replacing it with the traditionalmaxi∈I ci

j > 0 for j ∈ J. The appearance of zero cij leads to the fact that, in the transportation problem

(12)–(16), the corresponding zij are eliminated from consideration. This influences the effective domaindom f = Ω. However, we can show that Ω ∩ σ = ∅. All assertions and description of the algorithmremain valid. Some necessary adjustments of exposition can easily be made by analogy with [14], wherethis is done for a usual linear exchange model.

The above algorithm essentially uses the potentiality of F . Indeed, the cells

F (p) = Ξ(B) ∈ ξ

corresponding to the points p of the relative interiors of Ω(B) ∈ ω are obtained by the subdifferentialmapping ∂f : p → ∂f(p) of the function f that defines the optimal value for the transportation problem(12)–(16) given a price vector p ∈ Ω. This fact is false without the fixed budgets and passage to thegeneral case of the model as exposed in Section 1. However, the constructions of Section 2 remain validconcerning the polyhedral complexes ω and ξ generated by the model. To obtain the algorithm in thiscase, we can use some more general polyhedral complementarity scheme that was considered in [6] forthe classical exchange model.

In this case it is also possible to apply an iterative procedure and use the model with fixed budgets asan approximation to the initial model at each step of the process; from the price vector pk given at step k

of the process, the consumers budgets are calculated, and an equilibrium price vector pk is found in themodel with fixed budgets which is taken as pk+1. This procedure was studied in [14] for the classicalexchange model.


142 SHMYREV

ACKNOWLEDGMENTS

The author expresses his sincere gratitude to the anonymous referee for their attentive perusal of themanuscript and substantial remarks that contribute to improving the exposition.

The author was supported by the State Maintenance Program for the Leading Scientific Schools ofthe Russian Federation (project no. NSh–4999.2006.6).

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a generalized linear exchange model

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