Market allocation of indivisible goods

Journal of Mathematical Economics 32 Ž1999. 457–466 www.elsevier.comrlocaterjmateco

Market allocation of indivisible goods Tatsuro Ichiishi b

a,)

, Adam Idzik

b,1

a Department of Economics, Ohio State UniÕersity, Columbus, OH 43210-1172, USA Institute of Computer Science, Polish Academy of Sciences, ul. Ordona 21, 01-237 Warsaw, Poland

Accepted 21 November 1998

Abstract Market for indivisible goods is modelled, in which a financial intermediary plays the role as an income redistributor and each consumer can demand as many goods as he wants subject to his budget constraint. The existence of a competitive equilibrium is proved. The proof is based on a dual version of the extension of Gale’s covering lemma, recently established by the authors. q 1999 Elsevier Science S.A. All rights reserved. PACS: D3; D4; D5 Keywords: Covering of a simplex; Market allocation; Indivisible good

1. Introduction Shapley and Scarf Ž1974. constructed a model of an exchange market, in which each consumer is initially endowed with one unit of an indivisible good. They established the existence of a core allocation and then, in collaboration with David Gale, the existence of a competitive equilibrium. Quinzii Ž1984. introduced a divisible good, called money, into the Shapley–Scarf model and established the

)

Corresponding author. Tel.: q1-614-292-0762; Fax: q1-614-292-3906; [email protected] 1 Phone: q48-22-36-28-41; Fax: q48-22-37-65-64; E-mail: [email protected]. 0304-4068r99r$ - see front matter q 1999 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 8 . 0 0 0 6 5 - 2

E-mail:

core equivalence Žequivalence of a core allocation and a competitive allocation. and the existence of a competitive equilibrium. Gale Ž1984. provided an extension of the K–K–M covering lemma and derived Quinzii’s existence result from his extended K–K–M lemma. These three works are instances of the assignment game broadly defined, in light of their basic postulate that each consumer supplies his indivisible good and demands one unit of another. On the other hand, in the price-guided economy of the neoclassical paradigm, a consumer can demand seÕeral goods as long as these goods are within his budget constraint. The purpose of the present paper is to consider a modified version of the Shapley–Scarf model in which each consumer can demand several goods subject to his budget constraint, thereby bringing together the assignment game and the neoclassical paradigm. Let n be the number of consumers in the economy. While the total demand for each good can be any integer between 0 and n in disequilibrium, it has to be equal to 1 in equilibrium, since the total supply is 1. Therefore, an assignment emerges as a consequence of equilibrium even in our modified Shapley–Scarf model. We introduce to the model a particular role of financial intermediaries and establish the existence of a competitive equilibrium. The covering lemma of Gale Ž1984. quoted above concerns n covers of an Ž n y 1.-dimensional simplex, each satisfying the K–K–M type boundary condition. In our earlier paper ŽIchiishi and Idzik, forthcoming., we considered more general theorems on n covers of a simplex. There, we established ŽA. an extension of Gale’s lemma, which allows for covers that satisfy the boundary condition of Shapley Ž1973., ŽB. a dual version of our extended Gale lemma, which allows for covers that satisfy the boundary condition studied by Alexandrov and Pasynkov Ž1957. and Scarf Ž1967., and ŽC. yet another dual version of the extended Gale lemma, which allows for covers that satisfy the boundary condition of Ichiishi Ž1988.. It is this second dual version ŽC. that we apply to establish our aforementioned existence result for the modified Shapley–Scarf model. For another recent work on market allocation of indivisible goods, see van der Laan et al. Ž1997. and Konishi et al. Ž1997.; for normative allocation of indivisible goods, see, e.g., Alkan et al. Ž1991. and Thomson Ž1995. and the references therein; for a formulation which also encompasses the asymmetric roles of the sellers and the buyers, see, e.g., Kaneko and Yamamoto Ž1986.. Section 2 constructs our modified Shapley–Scarf model and presents an existence theorem. Section 3 provides a proof of this theorem.

2. A segmented housing market with a financial intermediary This section first constructs a model of indivisible goods that are traded in a competitive market. Let N be a set of n consumers, n - `. There is a financial

intermediary besides these n consumers. Each consumer j initially holds one unit of an indivisible good Žsay, a house., called here the jth good. A consumer can obtain a loan from, or make an investment in, a financial intermediary, but his initial balance at the financial intermediary is zero. The loanrinvestment is a special form of money, so its price is equal to 1. He can buy as many indivisible goods Žcalled henceforth simply goods. as he wishes subject to his budget constraint, but knowing that there is one and only one unit of each good available in the economy, he demands at most one unit of each good. Denote by P Ž N . the family of nonempty subsets of N Ž[ 2 N _œ4. 0 . Each consumer’s consumption set is R = P Ž N .; an element Ž t,S . g R = P Ž N . means that he obtains a loan t and holds the set of goods S. A negative loan t means a positive investment < t <. In the following, the phrase ‘to receive a loan t’ will be used synonymously with the phrase ‘to make an investment yt’. Implicit in our formulation of a consumption set is the postulate that a consumer has to hold at least one good. Consumer j’s preference relation is summarily represented by a price-dependent continuous N utility function u j : R = P Ž N . = Rq ™ R. Here, function u j incorporates both consumer j’s taste and the financial intermediary’s behavior; this formulation will be discussed in detail in the next six paragraphs. Consumer j’s initial endowment is Ž0, j4. g R = P Ž N .. The financial intermediary takes prices of goods as given, but has a monopoly power over the loanrinvestment market, so sets the lending interest rate r L G 0 and the deposit interest rate r D G 0 as the monopolist ŽStackelberg leader.. Given these two interest rates, each consumer j’s loan t j determines j’s future debt leÕel d j as dj s

½

Ž 1 q rL . tj , Ž 1 q rD . tj ,

in case t j G 0, in case t j - 0.

The number d j is j’s debt in case t j ) 0 and the number < d j < is his asset in case t j - 0. The consumer is concerned about his future debt and his present consumption of goods, so his underlying primitiÕe utility function is given as u˜ j : R = P Ž N . ™ R, whose domain is the space of pairs Ždebt, set of goods.. N of the goods prevails, consumer j sells his initial When price vector p g Rq endowment in the market, thereby receives the sale value pj . He also decides the amount of a loan t he receives from the financial intermediary. Let S ; N be the set of goods he purchases. His total expenditure on goods is then Ý i g S pi and he has to satisfy his budget constraint,

Ý pi F pj q t. igS

Given the financial intermediary’s strategy Ž r L ,r D . and the prevailing price vector p, consumer j ŽStackelberg follower. chooses his commodity bundle Ž t j ,S j .

so that it solves: u˜ j Ž d,S .

Maximize Ž t ,S .

subject to

Ý pi F p j q t , igS

d s Ž 1 q r L . t if t G 0, d s Ž 1 q r D . t if t - 0. The solution Ž t j ,S j . depends upon Ž r L ,r D , p .. The financial intermediary as the monopolist in the loanrinvestment market knows these responses of the consumers, in particular the total investment Ý j:t j - 0 < t j < and the total loan Ý j:t j ) 0 t j . The latter has to be funded from the former,

Ý j:t j)0

tj F

Ý

< tj <,

j:t j-0

that is,

Ý t j F 0, jgN

which constitutes the constraint on the intermediary’s behavior. Subject to this constraint, the intermediary chooses the two interest rates Ž r L ,r D . to maximize its future net return,

Ý

rL tj y

j:t j)0

Ý

rL < tj <.

j:t j-0

The optimal interest rates depend upon the prevailing market price vector p, hence the notation Ž r LŽ p .,r D Ž p ... Given market prices of goods p and the intermediary’s optimal interest rates Ž r LŽ p .,r D Ž p .., a loan level t determines the debt level dŽ t, p ., so a commodity bundle Ž t,S . determines consumer j’s utility level as u˜ j Ž dŽ t, p .,S .. This is the price-dependent utility function u j Ž t,S, p ., introduced at the outset of this section; formally, u j Ž t ,S, p . [ u˜ j Ž d Ž t , p . ,S . , with dŽ t , p. [

½

Ž1 q r L Ž p . . t , Ž1 q r D Ž p . . t ,

if t G 0, if t - 0.

The preceding paragraphs presented the intermediary’s net-return maximizing behavior. But our assumptions on the model Žto be introduced later. impose only fairly mild conditions on the intermediary’s behavior, so a wide range of its alternative behavior is consistent with the present analysis.

A pure exchange economy with indiÕisible goods and a financial intermediary Žcalled henceforth simply an economy . is a specified list of data  R = P Ž N .,u j ,Ž0, j4.4j g N of consumption set R = P Ž N ., utility function u j : R = P Ž N . N = Rq ™ R and initial endowment Ž0, j4. for every consumer j g N. Consumer j’s demand behavior is summarized by his inÕerse demand correN spondence from P Ž N . to the subsets of Rq , S ¨ C jS. Here, p g C jS means that j demands goods S if p is the prevailing market price vector of goods; in light of the budget constraint, he is also obtaining a loan of t j G Ý i g S pi y pj . His behavior comes from utility-maximization, so u j Ž t j ,S, p . G u j Ž tXj ,SX , p . for all Ž tXj ,SX . for which Ý i g SX pi F pj q tXj . In competitive equilibrium, the total demand for good i is equal to its total supply and the latter is equal to 1, i g N. Each consumer demands at least one good. An equilibrium is achieved, therefore, if weach consumer demands one and only one good, and each good is demanded by some consumerx. Formally, a competitiÕe equilibrium of an economy is a pair Ž pU ,p U . ofU a price vector N and a bijection p U : N ™ N such that pU g l j g N C jp Ž j.4. pU g Rq Markets of indivisible goods were considered by a pioneering paper, Shapley and Scarf Ž1974.. They do not introduce any financial intermediaries, but make the postulate that each consumer demands one indivisible good. This is contrasted with our setup that a consumer can obtain a loan or make an investment, and can hold seÕeral indivisible goods at the same time Žprovided that his budget constraint is satisfied.. So, while the consumption set of each consumer is N in the Shapley–Scarf setup, the consumption set of each consumer in our setup is R = P Ž N .; recall that element j g N is identified with one unit of the jth good, and element S g P Ž N . is identified with set S of goods. We have followed Quinzii Ž1984. and Gale Ž1984. in our formulation of an economy, but our model differs from theirs in two important respects. First, while the divisible commodity that Quinzii and Gale introduced is interpreted as money as a store of value, the divisible commodity that we introduce is interpreted as a loanrinvestment, which essentially functions as a channel for income redistribution. The price domain both N Ž in the Quinzii–Gale setup and in our setup is  14 = Rq here, the price of money is always equal to 1.. Second, while Quinzii and Gale postulate that each consumer can hold a pair of money and one indivisible good, we postulate that he can hold a pair of a loan and seÕeral indivisible goods. Thus, while the consumption set of each consumer is Rq= N in the Quinzii–Gale setup, the consumption set of each consumer in our setup is R = P Ž N .. We point out that although each consumer is allowed to hold seÕeral indivisible goods in our setup, he ends up holding one indivisible good in a competitive equilibrium. In short, an assignment of goods emerges in equilibrium even in our setup. The purpose of this paper is to establish an equilibrium existence theorem ŽTheorem 2.3.. The first assumption is the following monotonicity condition on a consumer’s preference relation, which says that goods affect his utility positively and a loan affects his utility negatively; the latter assumption is justified because a

loan creates commitment to future payments and an investment yields future returns to the consumer. N be any price Õector of goods. Assumption 2.1. Let p g Rq Ži. For eÕery t g R, u j Ž t,S, p . ) u j Ž t,SX , p . for all S,SX g P Ž N . for which S > SX and S / SX . Žii. For eÕery S g P Ž N ., u j Ž t,S, p . ) u j Ž tX ,S, p . for all t,tX g R for which t - tX .

Assumption 2.1 Žii. guarantees that given any p g C jS, consumer j demands goods S by obtaining the exact amount of loan t j s Ý i g S pi y pj . Without loss of generality, therefore, his constrained maximization problem becomes: Maximize S

uj

žÝ

/

pi y pj ,S, p ,

igS

subject to

SgPŽ N . ,

given

N p g Rq .

Since P Ž N . is a finite set, a solution to this problem always exists. In other N words, for each consumer j, the family  C jS 4S g P Ž N . is a coÕer of Rq . The next assumption says that as long as the average price of goods is high, the intermediary can always set the lending and deposit interest rates so that its constraint Žthe total loan is funded from the total investment. is met. This is justified as follows. A loan is typically demanded by relatively low-income consumers, i.e., by those consumers whose initially endowed goods have low prices. Some other consumers must have high incomes, in view of the high average price. A high-income consumer opts to sell his high-priced good, buys low-priced goods and invests the surplus for high future returns. Thus, whenever there is demand for a loan, the two interest rates can be set to create supply of N be the jth unit vector, j g N. Given a positive number investment. Let e j g Rq M, define the simplex,

D N Ž M . [ co  Me j < j g N 4 , where co A denotes the convex hull of set A ; R N. A price vector p has the average price Mrn, iff p g D N Ž M .. Assumption 2.2. There exists a positiÕe real number M such that for any p g l j g N C jS j l D N Ž M ., it follows that Ý j g N t j F 0, where t j [ Ý i g S j pi y pj . Assumption 2.2 may be viewed as Walras’ law within the markets for the goods, provided that the average price is Mrn. Indeed, when p g F j g N C jS j , the

total demand for good i is the number of the consumers who demand i, a j g N < S j 2 i4 , so the value of the total excess demand is:

Ý pi Ž a  j g N < S j 2 i 4 y 1 . igN

sÝ

žÝ

pi y p j

jgN igS j

/

s Ý tj jgN

F 0. The main result of this paper is the following existence result. Theorem 2.3. Let  R = P Ž N .,u j ,Ž0, j4.4j g N be an economy which satisfies Assumptions 2.1 and 2.2. Then, there exists a competitiÕe equilibrium of the economy. We present a quantitative example in order to illustrate the role of the financial intermediary and Assumption 2.2. Let N s  1,2,34 . Let M be the positive number given in Assumption 2.2, let d be a positive number such that everybody demands all goods i for which pi F d . Such a number d exists due to the monotonicity assumption; this will indeed be proved in Lemma 3.2 in the Section 3. Consider price vector p s Ž M y 2 d , d , d .. Each consumer demands the second and the third houses Žlow-priced houses., so his total expenditure is 2 d . Consumer i g  2,34 receives his sale value d , so he needs to receive a loan of d . Consumer 1 receives his sale value Ž M y 2 d ., so he can invest value Ž M y 4d . in the financial intermediary. The value of the total excess demand is, therefore, p 1 Ž 0 y 1 . q p 2 Ž 3 y 1 . q p 3 Ž 3 y 1 . s yM q 6 d , which is nonpositive because d is very small, hence Walras’ law within the markets for the goods. 3. Proof of the existence of a competitive equilibrium Our proof of the main result is crucially based on Theorem 3.1C of Ichiishi and Idzik Žforthcoming.; we reproduce its special case as Theorem 3.1 below. Let D N be the unit simplex in R N Ž[ D N Ž1... Theorem 3.1. For each i g N, let  CiS 4S g P Ž N . be a closed coÕer of D N satisfying

DT ;

D

CiS , for every T g P Ž N . .

S>N _ T

Then there exists a function p : N ™ P Ž N . such that

FCip Ž i. /œ0 and Dp Ž i . s N. igN

igN

In order to prove Theorem 2.3, we need to establish two lemmas; the first says that each consumer demands all the goods whose prices are substantially low. N , there exists a positiÕe number d Lemma 3.2. For each compact subset C of Rq S such that for eÕery p g C j l C it follows that S >  i g N < pi F d 4 .

Proof. Suppose the contrary. Then for each positive integer k, there exist k S k g P Ž N ., p k g C jS l C and i k g N such that pikk F 1rk and i k f S k . By passing through a subsequence if necessary, one may assume without loss of generality that p k ™ pU g C and S k s SU , i k s iU for every k. Then, pUiU s 0, and iU f SU . Define t jk [ Ý i g SU pik y pjk , tUj [ Ý i g SU pUi y pUj . Clearly, t jk ™ tUj By the monotonicity assumption, u j Ž tUj ,SU , pU . - u j Ž tUj ,SU j  iU 4 , pU .. By continuity of u j , there exists a neighborhood U of Ž tUj , pU . in R = C and a positive number t such that ; Ž t j , p . g U: u j Ž tUj ,SU , pU . q t - u j Ž t j ,SU j  iU 4 , p . . But for all k sufficiently large, u j Ž t jk ,SU , p k . - u j Ž tUj ,SU , pU . q t and Ž t k q pikU , p k . g U. Thus, under price vector p k , the commodity bundle Ž t jk q pikU ,SU j  iU 4., which satisfies the budget constraint, yields a higher utility than Ž t jk ,SU . for all k sufficiently large; this contradicts the choice of Ž t jk ,SU . as a maximizer of utility u j ŽP, p k . in the budget set given p k .I The next lemma says that the demand correspondence is upper semicontinuous and closed-valued: the properties equivalent in the present setup to closedness of N the graph of the demand correspondence, Ž p,Ý i g S pi y pj ,S . g Rq = R = P Ž N .< S4 p g Cj . N Lemma 3.3. For each j g N and each S g P Ž N ., the set C jS is closed in Rq .

Proof. Upper semicontinuity of the demand correspondence follows from the standard argument which uses the maximum theorem. We only need to show that N the budget-set correspondence Bj from the price-domain Rq to the subsets of the consumption set R = P Ž N ., defined by

½

Bj Ž p . [ Ž t ,S . g R = P Ž N .

Ý pi F p j q t igS

5

.

N is lower semicontinuous. Let  p k 4k be any sequence in Rq which converges to U U U U o p , and let Ž t ,S . be any point in Bj Ž p .. Choose t g R so that for all k sufficiently large,

Ý

igS U

pik - pjk q t o .

For each k, define a k by

½

a k [ max a g w 0,1 x

Ý igS

U

5

pik F pjk q a tU q Ž 1 y a . t o .

Clearly, a k ™ 1. Define t k [ a k tU q Ž1 y a k . t o . Then, Ž t k , SU . g Bj Ž p k . for all k sufficiently large, and Ž t k ,SU . ™ Ž tU ,SU ..I Proof of Theorem 2.3 Given the positive numbers M of Assumption 2.2 and d of Lemma 3.2 applied to C s D N Ž M ., define the trimmed simplex,

DdN Ž M . [  p g D N Ž M . < ; i g N : pi G d 4 . Its faces are defined by

DdT Ž M . [  p g DdN Ž M . < ; i g N _T : pi s d 4 , T g P Ž N . . Lemma 3.2 says that for each consumer j, ;T ; N : DdT Ž M . ;

D

C jS ,

S>N _ T

and Lemma 3.3 says that each C jS is closed. By Theorem 3.1 applied to the covers of DdN Ž M .,  C jS l DdN Ž M .4S g P Ž N . , j g N, there exists p : N ™ P Ž N . such that

F

C pj Ž j. l DdN Ž M . /œ 0,

jgN

D p Ž j . s N. jgN

This means that ' pU g F C pj Ž j . l DdN Ž M . ,

Ž 1.

jgN

; i g N : a  j g N < p Ž j . 2 i 4 G 1.

Ž 2.

On the other hand, by Assumption 2.2,

Ý pUi Ž a  j g N < p Ž j . 2 i 4 y 1. F 0.

Ž 3.

igN

By strict positiveness of pU and Ž2., inequality Ž3. holds true only if a j g N < p Ž j . 2 i4 y 1 s 0 for every i g N. This means that each p Ž j . is a singleton and function p may be regarded as a permutation on N. The theorem is established in view of Ž1..I

Acknowledgements We are grateful to Marcin Malawski, Julide Yazar and anonymous referees for ¨ their helpful comments and suggestions. The research reported in this paper is supported by the NSF under Award No. INT-9121247 and the KBN Grant No. 2 P03A 01711.

References Alexandrov, P., Pasynkov, B., 1957. Elementary proof of the essentiality of the identical mapping of a simplex Žin Russian.. Uspekhi Matematiceskikh Nauk 12, 175–179. ˇ Alkan, A., Demange, G., Gale, D., 1991. Fair allocation of indivisible goods and criteria of justice. Econometrica 59, 1023–1039. Gale, D., 1984. Equilibrium in a discrete exchange economy with money. International Journal of Game Theory 13, 61–64. Ichiishi, T., 1988. Alternative version of Shapley’s theorem on closed coverings of a simplex. Proceedings of the American Mathematical Society 104, 759–763. Ichiishi, T., Idzik, A., forthcoming. Equitable allocation of divisible goods. Journal of Mathematical Economics. Kaneko, M., Yamamoto, Y., 1986. The existence and computation of competitive equilibria in markets with an indivisible commodity. Journal of Economic Theory 38, 118–136. Konishi, H., Quint, T., Wako, J., 1997. On the Shapley–Scarf economy: the case of multiple types of indivisible goods Žmimeo.. Quinzii, M., 1984. Core and competitive equilibria with indivisibilities. International Journal of Game Theory 13, 41–60. Scarf, H., 1967. The approximation of fixed-points of a continuous mappings. SIAM Journal of Applied Mathematics 15, 1328–1342. Shapley, L.S., 1973. On balanced games without side payments. In: Hu, T.C., Robinson, S.M., ŽEds.., Mathematical Programming, Academic Press, New York, pp. 261–290. Shapley, L.S., Scarf, H., 1974. On cores and indivisibility. Journal of Mathematical Economics 1, 23–37. Thomson, W., 1995. The replacement principle in economies with indivisible goods. Rochester Center for Economic Research Working Paper No. 403. van der Laan, G., Talman, D., Yang, Z., 1997. Existence of an equilibrium in a competitive economy of indivisibilities with money. Journal of Mathematical Economics 28, 101–109.