the existence and computation of competitive equilibria in markets with an indivisible commodity

JOURNAL OF ECONOMIC THEORY 38, 118-136 (1986)

The Existence and Computation of Competitive Equilibria in Markets with

an Indivisible Commodity

MAMORU KANEKO AND YOSHITSUGU YAMAMOTO

Institufe of’ Socio-Economic Planning. University qf Tsukuba,

Sakura, Niihari, Ibaruki 305, Jupan

Received June 4, 1984; revised July 9, 1985

This paper considers an exchange economy called a generalized assignment market, in which sellers and buyers trade one indivisible commodity possibly with product differentiation for a perfectly divisible commodity. The existence of a competitive equilibrium in this economy is proved using Kakutani’s fixed point theorem. This existence theorem is applied to a production economy in which sellers are formulated as producers with convex cost functions. Two examples of housing markets are provided and their competitive equilibria are numerically calculated. Journal qf Economic Literalwe Classification Numbers: 021, 022, 930. ‘1. 1986 Academx Press. Inc

1. INTRODUCTION

Since the pioneering works of Gale and Shapley [4] and Shapley and Shubik [143, assignment games, assignment markets, and their variations have been intensively investigated.’ One of the most general models in the series of such studies was presented by Kaneko [6].’ His model of an assignment market is formulated as an exchange economy in which sellers and buyers trade one indivisible commodity possibly with product differentiation for a perfectly divisible commodity. A crucial assumption of his model is that every buyer wants to buy no more than one unit of the indivisible commodity and any seller does not. Kaneko proved the non-

’ Other than the subjects which will be discussed in this paper, typical variations are partitioning games, college admission games, and job-matching games presented in Kaneko and Wooders [S], Gale and Shapley [4], and Crawford and Knoer [l]. The models of markets with product differentiation of Gabszewicz and Thisse 123 and Shaked and Sutton [ 121 can also be considered as variations of assignment markets.

’ Another general model is given by Quinzii [ 111. Her model will be briefly discussed in Section 5.

0022-0531/86 $3.00 Copyright (0 1986 by Academic Press. Inc. All nghis of reproduction m any form reserved.

118

INDIVISIBLE COMMODITY 119

emptiness of the core of the assignment market by showing the balanced- ness of the associated characteristic function, and proved the existence of a competitive equilibrium showing the equivalence of the core to the set of all competitive equilibria.

An application of the assignment market model to a housing market is also presented in Kaneko [7]. He gave a recursive equation that deter- mined a competitive equilibrium in the housing market and investigated several properties of competitive equilibria, e.g., the shapes of competitive rents and several comparative statics, and numerically calculated competitive equilibria in several examples. However this was possible because of the restrictive assumptions imposed on the model, for example, it was assumed that every household had an identical utility function, which implied that the housing market had a unique central district. Of course, the existence theorem of a competitive equilibrium in [6] can be applied to housing markets without such assumptions. However the theorem is not so constructive as to be helpful for the consideration of behavior of competitive rents in general housing markets. Therefore numerical treatments will play a more important role in investigating the behavior of competitive equilibria in housing markets.

On the other hand. algorithms for locating a fixed point of a continuous function or an upper semi-continuous correspondence have been well developed in the field of “fixed point algorithm.” Therefore if the existence problem of a competitive equilibrium in an assignment market is reduced into a fixed point problem, we can efficiently calculate a competitive equilibrium by these algorithms. The purpose of this paper is to prove the existence of a competitive equilibrium in a generalized model of an assignment market using Kakutani’s fixed point theorem, and to demonstrate the applicability of fixed point algorithms in numerical examples.

The format of this paper is as follows. In Section 2, an exchange economy called a generalized assignment market is presented and the existence theorem of a competitive equilibrium in this market is stated. It is also shown that this existence theorem can be applied to a production economy in which sellers are formulated as producers with convex cost functions. In Section 3 competitive equilibria calculated by a fixed point algorithm are shown for two numerical examples of housing markets. In Section 4 the existence theorem of Section 2 is proved using Kakutani’s fixed point theorem. In the last section, it is pointed out that the existence of a competitive equilibrium of the model by Quinzii [ 113 can be proved in a similar manner, and a brief explanation of the fixed point algorithm is given.

120 KANEKOAND YAMAMOTO

2. A GENERALIZED ASSIGNMENT MARKET

Consider an exchange economy (M, N), where M = ( 1,2,..., m ) is the set of all sellers and N= { l’, 2’,..., n’} is the set of all buyers. In the economy one indivisible commodity is traded for a perfectly divisible commodity called “money.” We allow the presence of product differentiation in units of the indivisible commodity. According to product differentiation, the indivisible commodity is divided into s types. The set M of all sellers is partitioned into M,, M, ,..., MS, i.e., M, = {m,.. , + 1, mh- , + 2 ,..., mh> for all h = 1, 2 ,..., s, where 0 = m, < m, < ... cm, =m. Every seller iE M, is endowed with w’ units of the indivisible commodity of type h and with I’ ( > 0) amount of money, where wi is a nonnegative integer for all i E M. Every buyer j E N is endowed with Zi ( > 0) amount of money but with no unit of the indivisible commodity.

It is assumed that every seller ie M,, (h = 1, 2,..., S) is interested only in the indivisible commodity of type h and money. Therefore his utility function ui(k, m) is defined on Z, x R, , where Z, is the set of all nonnegative integers, and R, the set of nonnegative real numbers. We make the following assumptions on ui (i E M).

ASSUMPTION A. For each kEZ+, u’(k, m) is a continuous and monotonically nondecreasing function of m.

ASSUMPTION B. Zf d( k + 2, m) 2 u’(k + 1, m + A ), A 2 A’, and k 2 0, then u’(k+l,m+A)~u’(k,m+A+A’).

ASSUMPTION C. Either (C.l) or (C.2) holds:

(C.l) u’(o’, Ii) > u’(k, 0) for all kc Z, ;

(C.2) d(w’, Ii) = u’(k, Ii) for all k 2 0’.

ASSUMPTION D. u’(o’, I’) > u’(k, I’) for all k with 0 5 k < 0’.

Assumption A would not need any explanation. Assumption B is a convexity condition with respect to the indivisible commodity of type h and money. Assumption (C.l) means that the initial endowment is preferred to any state without consumption. Assumption (C.2) means that seller i does not want to have the indivisible commodity of type h more than his initial endowment. This implies that if Assumption (C.2) holds, then he does not act as a buyer. Assumption D means that any seller does not sell his indivisible commodity at zero price.

Remark 1. In fact, u’(wi, Ii) > u’(k, Ii) in Assumption D can be replaced by the weaker form ui(oi, I’) 1 u’(k, Ii) to establish the following existence theorem. However we use the stronger form to keep the clarity of its proof.


Every buyer je N has a utility function ui(x, m) on (0, el, e2 ,..., es} x R,, where eh is the hth unit vector of R”. We mean by (eh, m) that the buyer j consumes one unit of the indivisible commodity of type h and m amount of money. We make the following assumptions on the buyers’ utility functions uj (jE N), which correspond to Assumptions A and (C.1):

ASSUMPTION E. For each ?CE {0, e’, e?,..., es}, Uj(x, m) is a continuous and monotonically nondecreasing function of m.

ASSUMPTION F. Uj(0, Ii) > V(x, 0) for all x E (0, e’, e2 ,..., e.‘).

A competitive equilibrium is a triple (p, k, x) = (pl, p2 ,..., ps, k’, k2 ,..., k”, x1, x2,..., x”) such that

(a) p~R;,k’~z+ foralli~Mand~jE:(O,e’,e~,...,e”)forallj~N;

(b.M) for all iE&fh (h= 1, 2,..., s), u’(k’, Ii+ph(coi-ki))2ui(k, m) for all (k,m)EZ+ x R, with m~I’+p,(o’-k);

(b.N) for all jE N, U(xj,P-pxj)>=V(y,m) for all (y,m)E N e’, e2,...,eSj x R + with py + m 5 P;

(c) Xi=1 C,tMhkieh+Cj,N~~j=C~=l CzE,+,I1~ieh.

Condition (a) ensures the well-delinedness of each variable. Con- ditions (b.M) and (b.N) are the individuals’ utility maximization under the budget constraints. Condition (c) requires the equivalence of the total demands and the total supplies.

The main result of this section is the following theorem.

THEOREM. Under Assumptions A-F, there exists a competitive equilibrium.

We are now in a position to compare the above model with the preceding works of Shapley and Shubik [14] and Kaneko [6]. In their models, seller i’s initial endowment o’ of the indivisible commodity of type h is assumed to be 1 for all ie M, (h = 1,2,..., s). As was shown in [14], if the transferable utility assumption is made, i.e., ui and Uj are represented as u’(k, m) =f’(k) -t-m and u’(x, m) = g’(x) + m for some functions fi and g’, then the associated characteristic function can be formulated as assignment problems, which is the reason for the name “assignment market.” In [14], the existence of a competitive equilibrium was proved by showing the nonemptiness of the core and the equivalence of the core to the set of all competitive equilibria under the transferable utility assumption. In Kaneko

122 KANEKOAND YAMAMOTO

[6, Sect. 31 these results were proved without the restrictive assumption. Further his model was generalized in [6, Sect. 41 so that o’ might take an integer more than 1, but then the transferable utility assumption was again made on the sellers’ utility functions. The same game theoretical argument as in [ 141 was adopted in his proof. As will be shown in Section 4, we will directly prove the existence of a competitive equilibrium, using the fixed point argument. This will equip the theory of assignment markets with quite efficient methods of numerical calculation of competitive equilibria. Some numerical examples will be given in Section 3.

Our generalized assignment market is formulated as an exchange model, but sellers are often treated as producers in the literature of economics. Indeed, Gabszewicz and Thisse [Z] and Shaked and Sutton [ 123 formulated sellers as producers in very similar models with product differentiation. Therefore for future applications, it is convenient to reformulate our generalized assignment market as a production economy. In the following we show that Theorem 1 can be applied to the production economy.

Let N be the set of all buyers whose characteristics are described as before. Let M = { 1, 2 ,..., m} = U; =, M, be the set of all producers. Each i E M,, (h = 1,2,..., s) produces and sells the indivisible commodity of type h. The cost function of producer i is C’. That is, C’(k) is the cost (in terms of money) of producing k units of the indivisible commodity of type h. We make the convexity assumption:

ASSUMPTION G. For any ie M, Ci( 1) > t?(O), and C’(k + 2) - C’(k + l)z C’(k + 1) - C’(k) for all k E Z,.

A competitive equilihrium in this economy is defined to be a triple (p, k, X) satisfying conditions (a), (b.N), and

(b.M’) for all iEM,, (h= I, 2 ,..., s), p,,k’- C’(k’)zphl- C’(l) for all lEZ+;

(~‘1 Ct=, CrtMhk’eh=CltN.~I.

Condition (b.M’) is the producers’ profit maximization, and (c’) is the balance of total supplies and demands. This is an ordinary definition of competitive equilibrium for a production economy. The existence of a competitive equilibrium in this economy can be shown as a corollary of the Theorem.

COROLLARY. Under Assumptions E, F, and G, there exists a competitive equilibrium in the above production economy.

The proof of this corollary will be given in Section 4.


3. NUMERICAL EXAMPLES

In this section two examples of rental housing markets are considered and competitive rent vectors in these markets are numerically calculated.

EXAMPLE 1. Consider the rental housing market pictured on Fig. 1: The housing area has the unique central district CD and two railways with stations A through J. The sets of landlords (sellers) and of households (buyers) are given as

M= { 1, 2,..., 35) and N= .( 1’ 2’ 20”. 1 1. ., I,

respectively. The potential supplies of apartments are described in Fig, 1. For example, in the area of station C, apartments 4, 9, 13, and 16 are potentially supplied, which are l-, 2-, 3-, and 4-rooms apartments, respectively. We assume that each landlord owns only one apartment unit.

Every household has his office in the central district CD and goes there every day. It takes 10 minutes to go from each station to an adjacent station. For example, if a household rents apartment 13, it takes him

CD

10 min

( 4,1)-C H-(21,1) ( 9,Z) (26,2) (13,3) (30,3) (16.4) 10 min (33,4)

( 5.1). I-.(22,1) (10.2) (27,2) (14,3) (31.3) (17,4) 10 min (34,4)

I J (23,l) (28.2) (32,3) (35.4)

FIG. 1. (h, s). h: the apartment number; s: the number of rooms

124 KANEKO AND YAMAMOTO

TABLE I

j IJ .i I’ ($/month) ($/month)

1 800 11 1432 2 863 12 1495 3 926 13 1558 4 989 14 1621 5 1053 15 1684 6 1116 16 1747 7 1179 17 1811 8 1242 18 1874 9 1305 19 1937

10 1368 20 2000

30 minutes to get to CD. It is assumed that every household je N has, as his direct utility function,

V(s, t,m)=2JiG+5Jbict+3fi, (3.1)

where s and t are the number of rooms and the distance from CD to the apartment he rents measured in terms of minutes, respectively, and m is the amount of available money after paying the rent. The (indirect) utility function v’ (Jo N) is calculated by putting s and t into V(s, t, m). For example, if household 11’ rents apartment 13, then his utility function is U”‘(e13, m) = 2 ,/%+ 5 ,,/%+ 3 &.’ Household fs income (initial endowment of money) I’ is given as

I’= 12OO(j- 1)/19 + 800

for each jE N. That is, the initial endowments are distributed uniformly in the interval 800 to 2000. Table I gives concrete figures of P’s.

Each landlord i E A4 has the utility function u’(k, m) given as

u’(k, m) = k(200 + 4Os, - 2ti) + m, (3.2)

where k = 0 and k = 1 mean that he does and does not lease his apartment, respectively. For example, u13(k, m) = 260k + m because si3 = 3 and t,, = 30. This means that if the rent for apartment 13 is more than $260, then the landlord 13 prefers leasing his apartment to holding it. Since for each landlord in M, u’(k, m) is linearly separable, it is not necessary to specify his initial endowment I’.

To calculate a competitive equilibrium in this example, we use the fixed point algorithm (3” - 1 )-method proposed in Kojima and Yamamoto

3 Note that the utility value U/(0, m) should be set so that it satisfies Assumption F, e.g.. u/(0, m) = 3&l.

INDIVISIBLE COMMODITY

CD

125

( 4,160, /) ( 9,242, 2) (13,362,11) (16,488,19)

( 5,120, /) (10,188, /) (14,283, 5) (17,394,15)

( 6,130, /) (11,160, /) (15,216, /) (18,300, 7)

J

(20,179, /) (25,305,10) (29,444,18)

(21,140, /) (26,242. 3) (30,362,12) (33.488.20)

(22,120, /I (27,188, /) (31,283, 6) (34,394,16)

(23,130, /) (28,160, /I (32,216, /) (35,300, 8)

FIG. 2. (h,p,,j). ph: the rent of apartment h; j: the household renting apartment h

[lo]. Figure 2 gives the result of calculation. For example, in the area of station C, apartments 9, 13, and 16 are rented to households 2, 1 I ‘, and 19’ at rents $242, $362, and $488, respectively. Apartment 4 is not rented to any household. Since the housing area in this example is symmetric with respect to the railways, the rent vector is also symmetric.

In fact, if a housing area has the unique central district and if every household has the identical direct utility function as in the above example, then the recursive equation given in Kaneko [S] can also be applied to calculating a competitive rent vector. However, it can not be applied to the following example which has two central districts.

EXAMPLE 2. Consider the housing market (M, IV*) which is the same as Example 1 except for the appearance of new households 21’,..., 30’, that is, N* = ( l’,..., 20’, 21’ ,..., 30’ ). The oftices of households 1’ ,..., 20’ are located at the first central district CD, while the others’ are located at the second central district CD, = G.

Households 21’,..., 30’ have the same direct utility functions as (3.1).


However the distance from a station to CD2 is different from the distance to CD,. For example, if a household in 21’ to 30’ rents apartment 13, then it takes 50 minutes to get to his office, while it takes 30 minutes in the case of households l’,..., 20’, as was mentioned before. Thus the new households’ utility function U’(X, m) are identical, but different from those of households 1 I,..., 20’. The income of each new household is given as

I’ = 12OO(j - 2 1)/9 + 800,

i.e., I”s are uniformly distributed in the interval 800 to 2000. Table II gives concrete figures of Ps.

A competitive equilibrium is calculated in Fig. 3. For example, the rent for apartment 13 increases from $362 to $452. The new tenant is household 14’, who rented apartment 24 in Example 1. It can be seen from (3.1) or Figs. 2 and 3, that household 14’ prefers apartment 24 to 13. The appearance of new households compels him to move to a less preferred apartment 13 and to pay a higher rent. On the ohter hand, in Example 1 the rents for apartments 13 and 30 were identical because of the sym- metricity of the housing area. However in this example, apartment 30 is more demanded by new households because it is located nearer to CD2. Therefore the rent of this apartment is higher than that for apartment 13. Thus the appearance of new households raises the rents for the apartments in the areas of stations H, I, and J. However the apartments in the areas of stations F and G have the same rents as those in the areas of stations A and B. This phenemenon is probably due to the particular form of the utility function V(s, t, m) and the distribution of apartments.

Although the above examples are rather small and simple as compared with real housing markets, the calculation of a competitive equilibrium in

TABLE II

.i I’ ($/month 1

21 800 22 933 23 1067 24 1200 25 1333 26 1467 27 1600 28 1733 29 1867 30 2000


(19,294, 7)

3,237, 2) 8,392,12)

12,533,18)

4,181, 1) 9,321, 8) 3,452,14) 6,576,20)

5,127, /) (10,250, 4) (14,368,10) (17,484,17)

( 6,130, /I (11,179, /I (15,283, 5) (18,387,11)

'B G'CD2

C H

D I

E J

(20,237, 3 (25,392,13 (29,533,19

(21,183,21 (26,329.25 (30.461,28 (33,584,30

(22,154, /) (27,266,23) (31,390,26) (34,508,29)

(23,110, /) (28,206,22) (32.320.24) (35,429,27)

FIG. 3. (h. ~,~,j). ph: the rent of apartment h; .j: the household renting apartment h.

Example 2 required no more than 145 s on the FACOM M-200 computer. Therefore it will be promising to calculate competitive equilibria in larger and more complex housing markets. It will help us to find interesting phenomena in the problem of housing markets.

4. PROOFS.

First let us give a fixed point lemma. Put P = {p E R”: 02~~5~~ for all h = 1, 2 ,..., s}, where R” is the s-dimensional Euclidean space and ph > 0 for all h = 1, 2 ,..., S.

LEMMA 1. Let F(p) be a correspondence from P to some compact subset C of R”, i.e., F(p) c C for all p E P. Assume that

(i) F(p) is nonempty, convex-valued, and upper semi-continuous at each P E P.


Then there is a fi E P and a f E F(a) such that

f,SO if ph =o,

=o if O<$,<Ph, 20 if ph=ph.

If, further, F(p) satisfies

(ii) if p,, = 0, then z,, 2 0 for all z E F(p);

(iii) $p,,=ph, then z,lOfor allzEF(p),

then OEF(~).

(4.1)

Proof Let r(x) be the retraction from R’ to P that assigns p = r(x) defined by (4.2) to each XE R”:

ptl=O if X,50,

Yh if O<X,<Ph, (4.2 1

= Ph lf XhZPh.

Then r(x) is a continuous function. Let G(p) and H(x) be the correspondences defined by

G(P)=P+F(P) ,for each p E P;

H(x) = G(r(x)) for each x E R”.

Since IJP. P F(p) is included in the compact subset C of R”, U,, RS H(x) is included in the compact subset P+C=(p+x:p~P,x~c}. Let U be a compact convex subset of R” including P + C and consider the restriction of H(X) to U. By the assumption on F(p) and the continuity of r(x), H(x) is nonempty, convex-valued, and upper semi-continuous at each XE U. Therefore applying Kakutani’s fixed point theorem, there is an .f~ U such that i E H(z), i.e., 3 = r(2) + P for some i E F(r(i)). Let $ = r(i) E P. Then I E F(p) and i = z? -@ = .g - r(a) satisfies (4.1) by (4.2).

If F(p) satisfies (ii) and (iii), then t, =0 when ah =0 or Ph. Hence i=o. i

We are now going to define supply and demand correspondences. Let

D = {O, l,..., w};

P= {p~R~:O~p~~fiforallh=1,2 ,..., s},

where o is an integer with o > max{xi, Mh wi: h = 1, 2,..., s} and rii is a real number with fi > max{Z’: i E Mu N}. The supply and demand correspon-


dences S’(p) (REP, ieMh, h=1,2 ,..., s) and D’(p) (REP, ZEN) are defined by

s’(p)= ~(o’-k)eh:k~SZ,~hk~Z’+~h~iand

u’(k,Z’+p,(w’-k))~u’(Z,m)forall(I,m)~QxR+

with I’+p,o’zm+p,lJ;

D’(p)= {XE {O,e’,e’,..., e”j:px5Pand

Vj(x,I’-px)>=U’(~,m)forall(y,m)~{O,e’,e’,..., e”}xR+

withpy+msI/}.

LEMMA 2. For all i E M, (h = 1, 2 ,..., s), the following propositions hold:

(i) si( p) is nonempty-valued and upper semi-continuous at each p E P;

(ii) ph = 0 and leh E S’(p) imply ISO;

(iii) p,,=fi and le”ES’(p) imply 120;

(iv) $1~ I’ and leh, l’eh E S’(p), then t”eh E s’(p) for all integer I” with l<I”<I’.

Proof (i) The nonemptiness follows the definition of Si(p). We prove upper semi-continuity. Let pl’-+p’, (a’- k”) eh+z S’(p”) for all v and k” + k”. Since p;; k” SZ’ +p; w’ for all v, pi k” 5 Z’ +pi c.?. Suppose (o’- k”) eh$ s’(p’). Then there is some k’EZ+ such that

Z’+pz(w’-k’)zO and u’(k’,Z’+pi(o”-k’))>u’(k’,Z’+pi(w’-k’)).

(4.3)

Since u’(k”, Z’+p;l (w’-ky))zu’(wi, I’) for all v, it holds that u’(k’, I’ + p; (oi - EC’)) 2 ui(o’, I’). If Assumption (C. 1) holds, then I’+ pi (0’ - k’) > 0, because otherwise u’(u.#, Zi) > u’(k’, 0), which is a contradiction. If (C.2) holds, then we can choose k’ in (4.3) with k’s,+. In fact, if k’>o’, then u’(k’, I’+pi (0’ - k’))s u’(k’, ri) = u’(o’, Z’) by Assumptions A and (C.2) and so we can replace k’ by w’. Then we again have I’ + pi (0’ - k’) > 0 because I’ > 0. Since PI’ + p” and k” -+ k”, for a sufficiently large v,O<Z’+p;;(o’-k’) and u’( k’, I’ + p;; ( ui - k’)) > ~‘(k’, ri +p;; (o’- k”)) by Assumption A. This is a contradiction.

(ii) Let I= wi- k. If I> 0, i.e., k < wi, then by Assumption D u’(k, Ii) < ui(oi, Z’), which is a contradiction to the supposition that leh = (w’ - k) eh E S’(p).

(iii) Since ti>max(l’:iEMvNj and I’+p,i=Z’+&l~O, we have 120.

130 KANEKOANDYAMAMOTO

(iv) Let l=o’-k and l’=o’--k’. Then k>k’. Since (co-k)e”E S’(p), Zi+Ph(oi-k”)~I’+Ph(wi-k)~O and u’(k, Z’+Ph(oY-k))z u’(k”, Z’ +ph (0’ -k”)) for all k” with k” 5 k. For k” = k - 1 we have

u’(k, Z’+p,&&k))&(k- 1, Z’+p,(w’-k+ I)),

and by Assumption B

u’(k- l,Z’+p,(w’-k)+P,)zu’(k-2, Zi+ph(co-k)+2ph).

Repeating this argument, we obtain

u’(k, Z’+Ph(cui-k))&u’(k- 1, I’+P,(o’-k)+P,)

u’(k- 1, I’+ph(o’-k)+P,)zu’(k-2, I’+P,(w’-k)+2P,)

(4.4) u’(k’+ 1, Z’+P,(w’-k’- l))zu’(k’, Z’+P/,(oY-k’)).

However, u’(k, I’+P,(w’- k)) = u’(k’, I’+P,(o’- k’)) because (o’- k) eh, (IX’- k’) eh E S’(p). This together with (4.4) implies that (w’- k”) eh E S’(P) for all k” with k > k” > k’, i.e., l”eh E S’(p) for all I” with I< I” < 1’. 1

LEMMA 3. For any j E N, the following propositions hold

(i) D’(p) is nonempty-valued and upper semi-continuous at each PEPi

(ii) x’,zO,for all x/u/ED.(p) and h= 1, 2,..., s;

(iii) ph=m and x/~@(p) imp1.v xi==.

Proof. (i) The upper semi-continuity of H(p) can be proved in the ,same manner as in the proof of Lemma 2 (i).

(ii) Since D’(p) c (0, e’,..., e’}, we have .x; 2 0 for all .u’ E o’(p).

(iii) Since fi > max (I’: i E MU N 3 and (/ zpx’, we have pi = 0.

Define the excess demand correspondence E(p) by

E(p)= c D’(P)- c S(P) for each p E P. icN IE M

The excess demand correspondence E(p) satisfies the properties (i)-(iii) of Lemma 1 except convexity. Therefore we take the convex hull of E(p) and show that the convex hull of E(p) inherits the properties of E(p). 1

LEMMA 4. (i) Let A, B be subsets of R”. Then cov(A + B) = cov A + cov B, where cov A denotes the convex hull of A.

(ii) Let F(p) be a correspondence from a subset of R” to some finite set S = { y’, y2,..., y”} of a finite dimensional Euclidean space. Zf F(p) is


upper semi-continuous at p”, then cov F(p) is also upper semi-continuous at p”.

Proof We prove only the second assertion. Let p\’ + p”, xv E cov F(p”) for all v and XI’ --f x0. Since x” E cov F(p’) c cov S for all v, there is a sequence {A”} in the (k - 1)-dimensional simplex such that xv = C,,” I ,I;; yh and A;l >O only if y” E F(p”) for all v. Then there is a convergent sub- sequence of (A”}. Here we can assume without loss of generality that the sequence {A’} itself converges to 1’. If Ai > 0, then 2;; > 0 for suffkiently large v, which implies y” E F(p’) for sufficiently large v. Since F(p) is upper semi-continuous at 0 p , this implies yh E F(p’). Therefore, x”=C;=,,l;yh~cov F(p’). 1

LEMMAS. For the correspondence covE(p)=&,.covD’(p)-

Cie ** cov S’(p), the following propotitions hold:

(i) cov E(p) is nonempty, convex-valued, and the upper semi-continuous at each p E P;

(ii) p,, = 0 and -7 E cov E(p) imply ~~20;

(iii) ph=m and ZECOV E(p) imply ~~50.

Proof: (i) Clearly cov E(p) is nonempty and convex-valued. The upper semi-continuity of cov E(p) follows Lemmas 2 (i ), 3 (i), and 4 (ii).

(ii) Let p,, = 0. Then Lemma 2 (ii) implies ~$20 for all yie cov S’(p), and Lemma 3 (ii) implies xi 2 0 for all xj~ cov D’(p). Hence zh=~jENXjyCiEMy;~O.

(iii) Let ph = fi. Then Lemmas 2 (iii) and 3 (iii) imply yX>=O for all yi E cov S’(p) and xi 5 0 for all xj~ cov Dj( p). Hence zh 5 0. 1

Proof of the Theorem. The correspondence cov E(p) is included by a compact subset of R” for all p E P by the definition of Si( p) and o’(p), and Lemma 5 verifies that cov E(p) satisfies the assumptions (i)-(iii) of Lemma 1. Therefore there exists a price vector $ E P by Lemma 1 such that 0 E cov E(p). This implies that there are k’ (ie M) and .ui (Jo N) such that

(0’ - k’) eh E cov Si( jj) for all i E M,, h = 1, 2 ,..., s;

X’ECOV o’(d) for all ,j E N;

(4.5 1

(4.6)

0= C xl- i 1 (co'-k')eh, icN h= I iEMh

i.e.,

hc, i,C,, kieh +ik xi= i C Jeh. h=l r.zMh

(4.7 1


These k’ (in M) and xj (Jo N) make the total demands and supplies be balanced, but may not be “integers.” We have to find an integers solution with the same properties.

Since (w’-I?) eh ECOV S’(d), there are integers kf and ki such that

kf skiski 2 and (co’-kk;)eh, (wi--k;)eh~Si(d). (4.8)

Note that kl and k; may be identical. Consider the following system:

k’&I’k; for all i E M; (4.5’)

for all j E N; (4.6’)

c t+cjJh= c co’ for all h = 1,2,..., S; (4.7’) IEMh IEN IEMh

I’ER, foralliEM and .v’ E RS, for all j E N. (4.9)

Let y= ( JI’, y’,..., ~1”) and I= (I’, I’,..., I”). Then (4.6’) and (4.7’) are written in matrix form as follows:

< = =

=

where

A=

1 ‘.’ 1 I I

1 ... 1 I

x1 Co’ ic MI

c d

itM, _

I

1 . ..I t

_--- ------- 1’-------

1 1 1 ) I...1

., . . . . I 1 . . 1

1 1 1 I --

ml m2

-

0

,----.

1 1 . . .

rn,


Since the matrix A satisfies the condition of Hoffman and Kruskal [S, Theorem 31, A has the (totally) unimodular property, which is known as a sufficient condition for the system (4.5’)-(4.7’) and (4.9) to have the integral property. That is, every extreme point solution of the system is an integer solution. It is easy to see that (k’,..., k”, x1 ,..., x”) in (4.5))(4.7) is a solution of the system (4.5’))(4.7’) and (4.9). Therefore there are integer extreme point solutions (k”‘,..., k”“, xyl ,..., x’“) (v = 1, 2 ,..., ,u) of the system such that

(k’,..., k”, x1 ,..., x”) = i; ;l”(kvl ,..., k”“, xv’,..., Y”) I’ = I

for some I.‘,..., IP>O with Et=, iv= 1. We show now that (a, k”, x\‘) is a competitive equilibrium for all v. Since

(k”, xv) is an integer solution of the system (4.5’)-(4.7’) and (4.9), it holds that

i c kvieh + c x”‘= i 1 wieh for v = 1, 2 ,..., p. (4.10) h=l IEM~ itN h=l reM/,

Since (k”, x’) is an integer solution of (4.5’))(4.7’) and (4.9), k; ik”‘s k; for all in M. By (4.8) and Lemma 2 (iv) we have

(co’- kvi) eh E S(c) for all in M. (4.11)

Finally we show that

XV’ E D/( $) for all jE N. (4.12)

On the contrary, suppose xti 4 @v(d). Let xv # 0. Since (k”, x”) is an integer solution of (4.5’)-(4.7’) xvi = eh for some h with 1 Ih 5s. In this case xi > 0 because ;1’ > 0. Since xJ E cov o/(p) by (4.5), eh E o’(d). This is a contradiction. If x”j= 0, then 1; = 1 xi < 1 because xi = C:= 1 2”~‘~ and A” > 0. On the other hand, x1 E cov P( /j) and 0 = XI” F$ o’(d) imply Ci = 1 xi = 1. This is a contradiction.

We have shown that (J?, k”, x”) satisfies (4.10)-(4.12) for all v, which means that (p, k”, xv) is a competitive equilibrium for all v = 1, 2,..., ,u. m

Remark 2. Assumptions (C.l) and F can be replaced by weaker assumptions: Define s’(p) and D(p) (in M,, h = l,..., s,j~ N, and p E P) by

S’(p) = S’(p) ifZ’+p,(w’-k)>OforallkES’(P)

= {kEQ: Z’+p,(w’-k)LO) otherwise;

m)=wP) ifI’-px>OforallxELY(p)

= (XYE {O. e’,..., es),: I’-pxl0) otherwise.


Then s’(p) and &(p) satisfy all the properties of Lemmas 2 and 3, respectively. Therefore Lemma 5_holds for the correspondence cov E:(p) = CjsN cov gl(P)-Iilci~M cov S’(p). By Lemma 1, there exists a price vector p E P such that 0 E cov E(g). Therefore if Z’ +dh (w’ - k) > 0 for all k E S’(c) (REM) and I’-@>0 for all XELY($) (HEN), then covE(j)=cov~(@) has the zero vector. By the same argument as the above, we get an integer solution (R’)...‘P,.? ,...) .?)ES’(@)X ..’ xYy$)xD’(p)x ... xD”($) such that ~,.,.?j=C;=, CitMh (w’ - 6’) eh. Hence (p, k’,..., km, 1’,..., Y) is a competitive equilibrium. Therefore Assumptions (C. 1) and F are replaced by the condition that at any price vector given by Lemma 1,

Z’+j?h(c&k)>O for all k E s’(d) and in M,, h = l,..., s;

P$@x>O for all x E @(a) andj E N.

Proof of the Corollary. We show that the production economy can be regarded as a special case of the exchange economy. Let us associate producer i E M with the utility function u’(k, m) and an initial endowment (co’, Zi) given by (4.13):

u’(k, m) = m - C’(o’- k) if k<o’,

= m - Ci(0) if kzo’, (4.13)

0’2 n and I’ > 0.

Claim. The utility function u’(k, m) in (4.13) satisfies Assumptions A, B, C, and D of Section 2. It is easy to see that u’(k, m) satisfies Assumptions A and (C.2). Since Ci(Z)-Ci(Z- l)zCi(Z-- l)-C’(Z-2)z ... zCi(l)- C’(O) >O by Assumption G, C’(Z) > Ci(0) for all positive 1. Hence ui(oi, Ii) = I’- Ci(0) > Z’- C’(Z) = u’(w’ - 1, Zi) for all 0 < Igo’. This means Assumption D. We prove that u’(k, m) satisfies Assumption B. Define an extension of C’ by

Z;‘(k) = C’(k) if kEZ,

= C(O) if kEZ and k<O.

Then ci satisfies

c”(k+2)-?‘(k+ l)zc’(k+ l)- c”(k) for all k E Z. (4.14)

Hence u’(k, m) = m - c’(o’- k) for all (k, m) E Z, x R,. Suppose ui(k+2,m)2ui(k+1,m+A) and 424’. Then m-p(w’-k-2) 1 m+A--(G-k-l), i.e., Z”(o’-k-l)-p(o’-k-2)zAkA’. By (4.14) we have ~‘(~‘-k)-~‘(o’-k-1)~~(w’-k-1)-~(o’-k-2)~,4’. This implies m + A - C’(w’ - k - 1) 2 m + A + A’ - C’(O’ - k), i.e., u’(k+l,m+A)~u’(k,m+A+A’).


From the Theorem, the exchange economy has a competitive equilibrium (@, k, a). Let k = (k’,..., k”) be given by k’=o’-I? for all in iki. Then the statements u’(l’, P +fih(& - ffi)) 2 ~‘(1, m) for all (1,m)~Z+ xR+ and xi=, Cj~Mh~eh+~j~N~~=C;=, Ci,Mhmieh are equivalent to conditions (b.M’) and (c’), respectively. 1

5. CONCLUDING REMARKS

5.1. Quinziis Market

In the models presented by Shapley and Shubik [14] and Kaneko [6], an indivisible commodity is traded for money. These models are not symmetric in the sense that the agents are partitioned into two different groups-sellers and buyers-and their roles are completely different between two groups. On the other hand, Shapley and Scarf [13] gave a similar but symmetric model, in which only indivisible units are exchanged among agents without using money. Quinzii [ 111 unified these three models by proposing a model which was symmetric and also permitted the presence of money. In her model there are n agents 1,2,..., n with the initial endowment (w’, I’), where wi is the ith unit vector for 1 siss, wi= 0 for s + 15 is n, and P > 0 for all agents. This means that agent i is endowed with one unit of indivisible commodity for 1 s is s and the others are not endowed with the indivisible commodity. Each agent i has a utility function u’(x, m) defined on (0, e’, e* ,..., e.‘} x R,. Under the following assumptions the existence of a competitive equilibrium of the market can be proved by applying the first assertion of Lemma 1 with P= (pi R”:Ogp,_lti for h= I ,..., s) for some real number rfi > I;=, 1’. The reader might refer to Kaneko and Yamamoto [9] for the proof.

ASSUMPTION H. For each x E (0, el, e* ,..., e” 1, U’(.Y. m) is a continuous and monotonically nondeereasing function of m.

ASSUMPTION I. U’(W’, 1’) > u’(x, 0) for all .Y E {O, e’, e’,..., es}.

ASSUMPTION J. u’(0, m) 5 U’(eh, m) for h = I,..., s and all m E R +

Note that Quinzii [ 111 and Svensson [ 151 imposed additional assumptions to prove the existence of a competitive equilibrium in this market, for example, Quinzii assumed the Archimedian property of lJ/‘, i.e., u’(x, m) + + 00 as m + + cc. Gale [3] made assumptions directly on demands instead of the utility functions and proved the existence of a competitive equilibrium by generalizing the lemma of Knaster, Kuratowski, and Mazurkewicz. It is not diflicult to see that Assumptions H-J imply this assumptions.


5.2. Fixed Point Algorithm

In the preceding section it is shown that an equilibrium price vector is given as a fixed point of the correspondence H of Lemma 1. In Section 3, we used the fixed point algorithm called (3” - 1 )-method of Kojima and Yamamoto [lo] to find competitive rent vectors in the numerical examples. The applicability of the (3” - 1 )-method to the correspondence H can be verified by checking Theorem 6.9 in [lo]. The reader should refer to [lo] and also to Todd [ 161 for a survey of the fixed point algorithms.

REFERENCES

1. V. CRAWFORD AND E. M. KNOER, Job matching with heterogeneous firms and workers, Econometrica 49 (1981), 437450.

2. J. J. GAJBZEWICZ AND J.-F. THISSE, Price competition, quality and income disparities, J. Econ. Theory 20 (1979). 34&359.

3. D. GALE, Equilibrium in a discrete economy with money, Inf. J. Game Theory 13 (1984), 61-64.

4. D. GALE AND L. S. SHAPLEY, College admissions and the stability of marriage, Amer. Math. Monthly 69 (1962), 9-15.

5. A. J. HOFFMAN AND J. B. KRUSKAL, Integral boundary points of the convex polyhedra, in “Linear Inequalities and Related Systems,” (H. W. Kuhn and A. W. Tucker. Eds.), Annals of Mathematical Studies No. 38, Princeton Univ. Press, Princeton, N. J., 1956.

6. M. &N!iKO, The central assignment game and the assignment markets, J. Math. Econ. 10 (1982), 205-232.

7. M. KANEKO, Housing markets with indivisibilities, J. Urban Econ. 13 (1983), 22-50. 8. M. KANEKO AND M. WCIODERS. Cores of partitioning games, M&h. Sot. Sci. 3 (1982),

313-327. 9. M. KANEKO AND Y. YAMAMOTO. The existence and computation of competitive equilibria

in markets with indivisible commodities (the original of this paper), DP No. 225, Institute of Socio-Economic Planning, University of Tsukuba, 1984.

10. M. KOJIMA AND Y. YAMAMOTO, A unilied approach to the implementation of several restart fixed point algorithms and a new variable dimension algorithm, Math. Progrum- ming 28 (1984), 288-328.

11. M. QUINZII, Core and competitive equilibria with indivisibilities, In!. J. Ganle Theory 13 (1984), 41-60.

12. A. SHAKED AND J. SUTTON. The self-regulating profession. Rev. Econ. Stud. 48 (1981). 2 17-234.

13. L. S. SHAPLEY AND H. SCARF, On cores and indivisibilities, J. Math. Econ. 1 (1974). 23-37. 14. L. S. SHAPLEY AND M. SHUBIK, The assignment game I: The core, Int. J. Game

Theory 1 (1972), 111-130. 15. L.-G. SVENSON, “On the existence of competitive equilibria in an economy with large

indivisibles,” Univ. of Lund, Lund, Sweden, 1983. 16. M. J. TODD, An introduction to piecewise-linear homotopy algorithms for solving systems

of equations, in “Topics in Numerical Analysis,” (A. Dold and B. Eckmann, Eds.), Lec- ture Notes in Mathematics 965, Springer-Verlag, New York, 1981.

the existence and computation of competitive equilibria in markets with an indivisible commodity

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