UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Money as a transaction technology : a game-theoretic approach Wiens, Elmer G. 1975

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Notice for Google Chrome users:
If you are having trouble viewing or searching the PDF with Google Chrome, please download it here instead.

Item Metadata

Download

Media
831-UBC_1975_A1 W53.pdf [ 4.71MB ]
Metadata
JSON: 831-1.0100098.json
JSON-LD: 831-1.0100098-ld.json
RDF/XML (Pretty): 831-1.0100098-rdf.xml
RDF/JSON: 831-1.0100098-rdf.json
Turtle: 831-1.0100098-turtle.txt
N-Triples: 831-1.0100098-rdf-ntriples.txt
Original Record: 831-1.0100098-source.json
Full Text
831-1.0100098-fulltext.txt
Citation
831-1.0100098.ris

Full Text

MONEY AS A TRANSACTION TECHNOLOGY: A GAME-THEORETIC APPROACH by ELMER GERALD WIENS B . S c , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1967 M . S c , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Depa r t m e n t o f ECONOM ICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e re q u i r e d s t a n d a rd THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1975 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agr e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and stud y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia 20 75 Wesbrook Place Vancouver, Canada V6T 1W5 Date ABSTRACT A barter economy and a monetary economy are modelled using the cooperative game approach. The feature that d is t inguishes the two economies is the manner in which exchange a c t i v i t i e s are organized in the face of t rasnact ion cos ts . While d i v i s i o n of labour or s p e c i a l i z a t i o n is exploi ted in the monetary economy's technology of exchange, i t is not exploited in that of the barter economy. The presence of a medium of exchange in the monetary economy permits i t s spec ia l i zed traders to operate e f f i c i e n t l y . The cooperative game approach admits group r a t i o n -a l i t y along with the usual assumption of ind iv idua l r a t i o n -a l i t y . Group r a t i o n a l i t y means that ind iv idua ls are able to perceive the i r interdependence. Money is explained as the product of in teract ions between ind iv idua l r a t i o n a l i t y ( u t i l i t y maximizing consumers and p r o f i t maximizing traders) and group r a t i o n a l i t y (the a b i l i t y to perceive the benef i ts of monetary exchange versus barter exchange). Consequently, money is viewed not as an object , but as an i n s t i t u i o n . Its value r e f l e c t s the re la t ive super io r i ty of a monetary economy over a barter economy. i i TABLE OF CONTENTS Page ABSTRACT i i ACKNOWLEDGMENTS i v C h a p t e r 1 INTRODUCTION 1 2 THE "BARTER" ECONOMY 11 3 THE "MONETARY" ECONOMY 37 4 THE RELATIVE EFFICIENCY OF A "MONETARY" VERSUS A "BARTER" ECONOMY 59 FOOTNOTES 78 BIBLIOGRAPHY 81 APPENDICES A MATHEMATICAL CONCEPTS 87 B THE EXISTENCE OF COMPETITIVE PRICE EQUILIBRIA 92 i i i ACKNOWLEDGMENTS I wish to thank the members of my d isser ta t ion committee, Charles E. Blackorby, W. Erwin Diewert, and e s p e c i a l l y my chairman, Keizo Nagatani, for the i r advice and encouragement during the preparation of th is t h e s i s . The comments of Louis P. Cain , Richard G. H a r r i s , R.A. Restrepo, and Gideon Rosenbluth have also been h e l p f u l . I g r a t e f u l l y acknowledge f inanc ia l assistance from H.R. MacMillan Family Fel lowships, 1972-74, and from the Canada Counc i l , 1974-75. I am also indebited to G.C. A rch ib a ld , W.E. Diewert, and T . J . Wales for employment as research a s s i s t a n t . F i n a l l y , I am happy to thank Sharon Hal ler for her expert typing of the f ina l manuscript. iv Chapter 1 INTRODUCTION A. In th is thesis I shal l attempt to bui ld a sensible model of a monetary economy. The model wi l l include some important features of monetary economics which I bel ieve have not been given enough at tent ion in the l i t e r a t u r e . To bring out these features , I shal l compare my monetary economy with a barter economy. By doing so, I am able to examine the st ructura l d i f ferences between monetary and barter exchange. The purpose of this exercise is to understand better how a monetary economy funct ions . Neoclassical economic theory does not provide an adequate explanation of the importance of money in modern economics. Persumably, money serves some useful purpose in the Arrow-Debreu models. But these models f a i l to bring out money's role because they do not describe how goods are exchanged between agents. Recently a number of authors have attempted to prove the usefulness of money ([25, [49] and [64]). While the de ta i l s of the models d i f f e r from one author to 1 2 another, they a l l postulate money as something which is inherent ly u s e f u l . In a way, therefore , the conclusion is assumed at the outset . What I propose to do instead is to bui ld models of a monetary economy and a barter economy which are p lausib le in the l i g h t of economic h is to ry . Then I shal l examine the condit ions under which money is in fact u s e f u l . I bel ieve that there is a s t ructura l d i f ference between monetary and barter exchange. This s t ructura l d i f ference wi l l be modelled r igorously in Chapter 2 and 3. The d i f ference between monetary and barter exchange wi l l be developed by recognizing that real resources are used up when ind iv idua ls (agents) in an economy exchange goods. The real resources used up when goods are exchanged are ca l l ed transact ion cos ts . Examples of transact ion costs are the cost of moving goods from one agent to another and the legal cost of t ransfer r ing ownership. There are various ways of organizing the exchange of goods in the presence of t ransact ion c o s t s . The s t ructura l d i f ference referred to above is based on the d i f f e ren t manner in which goods are exchanged in my monetary and barter economies. I shal l argue that the usefulness of money in monetary exchange in contrast to barter exchange is the resu l t of th is d i f ference in organizing exchange a c t i v i t i e s . Let me be a b i t more s p e c i f i c about how transact ion costs are handled in my model. I assume each agent is endowed with a cer ta in degree of e f f i c i e n c y in exchanging goods. An agent's a b i l i t y at performing exchanges is represented by his 3 t ransact ion technology. A transact ion technology is s imi la r to the more conventional production technology. While the l a t te r describes feas ib le outputs for each set of inputs , the former describes a l l f eas ib le exchanges and the i r at ten-dant real resource cos ts . An indiv idual who exchanges the vector y of goods for the vector x of goods w i l l use up real resources as represented by some vector z. The magnitude of the transact ion cost vector z depends on the i n d i v i d u a l ' s e f f i c i e n c y at exchanging goods. In other words, z depends on an i n d i v i d u a l ' s t ransact ion technology. In my model an i n d i v i d u a l ' s transact ion technology is taken as a p r i m i t i v e . I choose not to invest igate the source and nature of t rans-act ion costs because i t is not necessary for my purposes. j3. In Chapter 2 I shal l bui ld a model of an economy which I c a l l a "barter economy." In this economy I require that each i n d i v i d u a l ' s exchanges be constrained both by goods in his possession and by his own transact ion technology. I do not permit any agent to execute exchanges on behalf of another agent. Each agent in the economy w i l l have some idea about the ra t ios at which goods are being exchanged. When an agent wants to exchange goods with one or more agents, his desired exchanges wi l l be based on his ex is t ing stock of goods, his be l i e fs about the preva i l ing exchange ra t ios between goods, and his transact ion technology. 4 An agent is permitted to engage not only in d i rec t exchange but also in ind i rec t exchanges. He can use some good as an intermediary or l ink in exchange i f i t is to his advantage. Furthermore, he is not l imited to b i l a t e r a l exchanges. He can involve himself in mu l t i l a te ra l exchange to the extent permitted by his personal c r e d i b i l i t y . Con-sequently, my notion of a barter economy is much wider than that which is general ly used. More w i l l be said about this in Chapter 2. General ly , i t does not make much sense to postulate the presence of pr ices in a barter economy. H i s t o r i c a l l y , barter economies were not highly u n i f i e d , but consisted of a number of iso la ted markets. While an indiv idual might engage in arbi trage in a local market, the scope of his a c t i v i t i e s was l imi ted by his informat ion, tas tes , i n i t i a l endowment, and transaction technology. Thus, e s p e c i a l l y between iso la ted markets, there would probably have been no high degree of consistency in the ra t ios at which goods were exchanged. Although my analys is of a barter economy deals with a pr ice vector in connection with existence proofs , my descr ip t ion of barter exchange does not depend on the presence of p r i c e s . The main feature of barter exchange that I want to bring out is the absence of s p e c i a l i z a t i o n among agents in carrying out exchanges. At each stage in the process of exchanging goods, an i n d i v i d u a l ' s planned exchanges are constrained both by goods in his possession and by his 5 t r a n s a c t i o n t e c h n o l o g y . The b a r t e r economy can be c h a r a c -t e r i z e d by t h e s t a t e m e n t t h a t d i v i s i o n o f l a b o u r i s n o t e x p l o i t e d i n t h e manner i n w h i c h goods a r e e x c h a n g e d between i n d i v i d u a l s . C_. In C h a p t e r 3, on t h e o t h e r hand, I s h a l l b u i l d a model o f an economy w h i c h I c a l l a " m o n e t a r y economy." The c r u c i a l d i f f e r e n c e between t h i s economy and t h a t o f C h a p t e r 2 i s t h a t I now remove t h e r e s t r i c t i o n t h a t e a c h a g e n t must o n l y e x c h a n g e goods on h i s own b e h a l f . H e r e , an i n d i v i d u a l i s p e r m i t t e d t o e x e c u t e e x c h a n g e s on b e h a l f o f o t h e r s . An a g e n t may a c t as a t r a d e r by b u y i n g goods f r o m some i n d i v i d u a l s and r e s e l l i n g them t o o t h e r s . By a c t i n g as a t r a d e r , an a g e n t hopes e v e n t u a l l y t o consume a more d e s i r a b l e b u n d l e o f goods t h a n he c o u l d have i f he o n l y e x c h a n g e d goods on h i s own b e h a l f , o r , i f he p e r m i t t e d some o t h e r a g e n t t o e x e c u t e h i s e x c h a n g e s . On t h e o t h e r h a n d , i f an a g e n t i s n o t p a r t i c u l a r l y e f f i c i e n t a t e x c h a n g i n g g o o d s , i t may be t o h i s a d v a n t a g e t o have a n o t h e r a g e n t e x e c u t e h i s e x c h a n g e s . C o m p e t i t i o n between t r a d e r s w i l l e n s u r e t h a t o n l y t h e r e l a t i v e l y e f f i c i e n t a g e n t s a c t as t r a d e r s . I f t h e a g e n t s w i t h s u p e r i o r t r a n s a c t i o n t e c h n o l o g i e s a r e a c t u a l l y p e r f o r m i n g t h e e x c h a n g e t a s k s , t h e n t h e manner i n w h i c h goods a r e e x c h a n g e d i n t h i s economy i s more e f f i c i e n t t h a n t h a t i n t h e b a r t e r economy. Thus t h e c r u c i a l s t r u c t u r a l d i f f e r e n c e between t h e two e c o n o m i e s i s t h a t t h e m o n e t a r y economy e x p l o i t s d i v i s i o n o f l a b o u r i n t h e way goods a r e e x c h a n g e d between i n d i v i d u a l s . 6 Let me now explain why a medium of exchange has an important role to play in this monetary economy. A t rader 's act of buying some goods from an indiv idual is separate from his act of s e l l i n g these same goods to another indiv idual . If the t rader 's customers always demand spot payment in goods whenever the trader wants to buy goods, the trader may not be able to meet the i r demands from his inventory. A t rader 's i n i t i a l endowment of the goods may have been smal l , or e l s e , recent trades may have depleted his stock of pa r t i cu la r goods. The trader has a problem. He must buy goods in order to s e l l goods, but this is d i f f i c u l t i f ind iv idua ls always demand spot payment in goods. In a monetary economy, the presence of a medium of exchange solves the traders problem. As we shal l see in the next s e c t i o n , his customers w i l l accept money in exchange for goods. They know that they can buy or order any goods they desire from any trader and pay for them with money. Through the use of a medium of exchange, the trader is no longer constrained at each point in time by goods in his possession. Thus the universal a c c e p t a b i l i t y of money allows the t raders ; in the economy to use the i r t ransact ion technologies with maximum e f f i c i e n c y . p_. In conc lus ion , l e t me describe some of the important features of the monetary economy that emerge from my a n a l y s i s . 7 F i r s t , I have seen the essent ia l d i f ference between a barter economy and a monetary economy in the manner in which exchange a c t i v i t i e s are organized in the two economies. To put i t b r i e f l y , a monetary economy makes use of d i v i s i o n of labour or s p e c i a l i z a t i o n in i t s technology of exchange while a barter economy does not. On the basis of th is s t ruc-tural d i f f e r e n c e , I have shown that the set of f eas ib le a l l o -cat ions in a monetary economy is larger than that in a barter economy, and hence, that the former is po ten t ia l l y (but not necessar i ly ) more e f f i c i e n t than the l a t t e r . The potent ia l benef i t of monetary exchange is thus es tab l ished . Reliance on spec ia l i zed traders means separation between the act of sale and the act of purchase of a good both in time and p lace . For the reasons given below, traders and the i r customers wi l l accept money in exchange for goods. Consequently, each par t ic ipant in a transact ion is no longer constrained by goods in his possession and/or by his tas tes . A medium of exchange cuts the act of sale and the act of purchase loose from the requirements of double coincidence of wants. Second, in formulating the two economies, I have employed a cooperative game or core theoret ic approach. The ra t ionale for this choice l i e s in the pecul iar nature of money. Money has convent ional ly been introduced into general equi l ibr ium models as an addi t ional good and made to work on the strength of i n d i v i d u a l s ' demand for i t . Unlike ordinary 8 goods, however, one cannot leg i t imate ly derive an i n d i v i d u a l ' s demand for money from his physio logica l needs. But to ensure the pos i t ive exchange value of money, one ends up assuming the usefulness of money. This is e s s e n t i a l l y what Starr [64], Hahn [25] and others have done. Given the d i f f i c u l t y of deducing the usefulness of money from indiv idual t as tes , one is natura l ly led to a soc ie ty -oriented approach in which society and i ts members perceive the usefulness of money. But i t is equal ly d i f f i c u l t to explain the process of such percept ion. Core theory enables us to deal systemat ica l ly with the two types of economies at both the indiv idual and at the group l e v e l . Core theory reta ins the usual assumption that ind iv idua ls maximize their u t i l i t y . It also assumes, however, that ind iv idua ls are able to perceive the i r interdependence and that any group of ind iv idua ls wi l l carry out acts which are of mutual benef i t . In other words, core theory recognizes group r a t i o n a l i t y along with indiv idual r a t i o n a l i t y . In fact the core is a set of a l loca t ions which are rat ional from the point of view of both ind iv idua ls and groups. I stated above that monetary exchange is po ten t ia l l y more e f f i c i e n t than barter exchange and that money permits the traders in a monetary economy to operate e f f e c t i v e l y . I have explained money as something which emerges as a product of in teract ions between indiv idual r a t i o n a l i t y (p ro f i t maxi-mizing on the part of traders) and group r a t i o n a l i t y ( a b i l i t y 9 t o p e r c e i v e b e n e f i t s o f m o n e t a r y e x c h a n g e t h r o u g h o u t t h e e c o n o m y ) . T h u s , i n an e q u i l i b r i u m o f t h e m o n e t a r y economy, any g r o u p o f i n d i v i d u a l s i s f r e e t o b r e a k away and use a l t e r n a t i v e means of e x c h a n g e . However, no g r o u p w i l l do so b e c a u s e t h e r e i s n o t any g r o u p w h i c h c a n o f f e r a l l i t s members g r e a t e r u t i l i t y t h a n t h e y c a n g e t by r e m a i n i n g i n t h e m o n e t a r y economy. T h u s , an i n d i v i d u a l must j o i n t h e m o n e t a r y economy t o e x c h a n g e g o o d s . To e x c h a n g e g o o d s , he must use money. F o r t h i s r e a s o n , i t can be s a i d t h a t i n d i v i d u a l s ' demand f o r money i s d e r i v e d b o t h f r o m t h e i r c h o s e n e n v i r o n m e n t , n a m e l y , t h e r e q u i r e m e n t s o f m o n e t a r y e x c h a n g e , and t h e i r d e s i r e t o e x c h a n g e some o f t h e i r i n i t i a l endowments. In s h o r t , I have c h a r a c t e r i z e d money as s o m e t h i n g t h a t r e f l e c t s i n i t s v a l u e t h e r e l a t i v e s u p e r i o r i t y o f a m o n e t a r y economy o v e r a b a r t e r economy. E_. T h e p p r o g r a m o f t h i s t h e s i s p r o c e e d s as f o l l o w s . In C h a p t e r 2 I s h a l l p r e s e n t a model o f a b a r t e r economy. I f i r s t model t h e economy u s i n g t h e c o o p e r a t i v e game a p p r o a c h . Then I deduce t h e s t r u c t u r e o f p r i c e s needed so t h a t c o m p e t i -t i v e b e h a v i o u r on t h e p a r t o f a g e n t s i s e q u i v a l e n t t o c o o p e r a -t i v e b e h a v i o u r . In o t h e r w o r d s , I d e r i v e a c o m p e t i t i v e b a r t e r economy f r o m t h e c o o p e r a t i v e economy. In C h a p t e r 3 I s h a l l p r e s e n t a model o f a m o n e t a r y economy. I a g a i n s t a r t f r o m a c o o p e r a t i v e economy. Here a g e n t s f o r m c o a l i t i o n s f o r t h e p u r p o s e o f e x p l o i t i n g t h e d i v i s i o n o f l a b o u r i n e x c h a n g e . B e c a u s e I s t a r t w i t h a 10 c o o p e r a t i v e economy r a t h e r t h a n a c o m p e t i t i v e economy, I am b e t t e r a b l e t o model t h e p r o c e s s by w h i c h a g e n t s a r e a s s i g n e d t o p a r t i c u l a r e x c h a n g e t a s k s . I a g a i n deduce t h e s t r u c t u r e o f p r i c e s needed so t h a t c o m p e t i t i v e b e h a v i o u r i s e q u i v a l e n t t o c o o p e r a t i v e b e h a v i o u r . I f i n d t h a t a s e t o f b u y i n g and a s e t o f s e l l i n g p r i c e s i s now needed i n s t e a d o f t h e one s e t o f p r i c e s o f t h e b a r t e r economy. C h a p t e r 4 e x p a n d s t h e model o f C h a p t e r 3. Here p e r m i t a g e n t s t h e c h o i c e between m o n e t a r y and b a r t e r e x c h a n g e . The c o o p e r a t i v e a p p r o a c h i s p a r t i c u l a r l y s u i t e d f o r t h i s p u r p o s e b e c a u s e i t a l l o w s g r o u p r a t i o n a l i t y . In o t h e r w o r d s , a g r o u p o f a g e n t s can c o n s i d e r t h e a d v a n t a g e s o f b r e a k i n g away f r o m t h e m o n e t a r y economy and o f u s i n g a l t e r n a t i v e ways o f e x c h a n g i n g g o o d s . In t h i s c h a p t e r I p r e s e n t a s u f f i c i e n t c o n d i t i o n f o r m o n e t a r y e x c h a n g e t o d o m i n a t e b a r t e r e x c h a n g e . By t h i s I mean t h a t a l l a g e n t s w i l l c h o o s e t o use m o n e t a r y e x c h a n g e r a t h e r t h a n b a r t e r e x c h a n g e . F i n a l l y , t h e a p p e n d i x c o n t a i n s t h e p r o o f s w h i c h e s t a b l i s h t h e e x i s t e n c e o f c o m p e t i t i v e p r i c e e q u i l i b r i a i n t h e b a r t e r and m o n e t a r y e c o n o m i e s . T h i s i s n e c e s s a r y t o e n s u r e t h e l o g i c a l c o n s i s t e n c y o f my m o d e l s . Chapter 2 A "BARTER" ECONOMY A. Before I present my descr ip t ion of a barter economy, l e t us f i r s t consider how a barter economy is usual ly descr ibed. Jevons [37] has described a barter economy as one where exchange requires the "double coincidence of wants." By this he means that i f two agents are to exchange goods, each agent must have something that the other agent wants. In a recent paper, Starr [64] has attempted to model double coincidence of wants. According to S t a r r , in a barter economy exchange of goods must s a t i s f y two requirements. F i r s t , the exchange of goods between a pair of agents must be "pr ice cons is ten t . " This means that for any given agent the value of goods supplied to any other agent must equal the value of goods received from him at the current p r i c e s . In other words, trades between any two agents must always be cleared between them. A th i rd agent cannot be involved. Thus S t a r r ' s "pr ice consistency" assumption r e s t r i c t s trade to b i l a t e r a l exchange of goods. 11 12 S t a r r ' s second r e s t r i c t i o n on exchange is that i t be "monotone excess demand d imin ish ing ." This r e s t r i c t i o n on exchange ensures that i t is voluntary. By th is Starr means that an agent wi l l not give up a good unless he has an excess supply of th is good. Conversely, an agent wi l l not accept a good unless he has an excess demand for this good. To put i t b r i e f l y , the exchange of goods is said to be "monotone excess demand diminishing" i f trade between any pair of agents does not increase the excess demand for any good by e i ther agent, or increase the excess supply of any good by ei ther agent. In e f f e c t , this r e s t r i c t i o n prevents the use of an intermediary in exchange. It is easy to construct simple examples in which S t a r r ' s two r e s t r i c t i o n s on exchange prevents some agent from at ta in ing his desired bundle of goods. This can happen even though the price vector at which goods are being exchanged would be an equi l ibr ium price vector i f one of the r e s t r i c t i o n s was l i f t e d . Starr shows that th is d i f f i c u l t y of barter can be circumvented i f a good — ca l led money — is appended to the ex is t ing l i s t of goods. Starr designates as money that good which is always acceptable in exchange. An agent w i l l accept money even though he does not wish to consume i t . The use of money is found to be s o c i a l l y desirable because i t s use enlarges the set of f eas ib le t rades. A l loca t ions of goods which were impossible to achieve through barter exchange can 13 now be achieved because money's universal a c c e p t a b i l i t y over-comes the absence of double co inc idence.of wants. J3. While S t a r r ' s formal izat ion of double coincidence of wants is an important contr ibut ion to the theory of money, his paper leads to some serious problems. F i r s t , double coincidence of wants cer ta in ly is a d i f f i c u l t y of barter exchange. However, I do not bel ieve that i t is a h i s t o r i c a l l y va l id descr ip t ion of barter exchange. In my op in ion , double coincidence of wants is too narrow a d e f i n i t i o n . It r e s t r i c t s trade to b i l a t e r a l exchange, i t precludes a l l debt cont rac ts , and i t does not permit the use of even a l imi ted intermediary in exchange. H i s t o r i c a l l y , there is evidence that mul t i l a te ra l exchange, c red i t between i n d i v i d u a l s , and the local use of intermediate goods in exchange occurred in barter economies [20]. Furthermore, Starr does not explain his use of pr ices in conjunction with his double coincidence of wants requirements. In a barter economy there wi l l be £ ( £ - l ) / 2 exchange ra t ios between goods. To be able to reduce these exchange ra t ios to a set of £-1 r e l a t i v e p r i c e s , someone must be engaged in a rb i t rage . However, in S t a r r ' s economy no agent can use a th i rd good as an intermediate l ink between two goods. Nor can any agent act as a th i rd party to a i; t ransact ion between a pair of agents. C lear ly arbitrage is ruled out in S t a r r ' s economy, and thus double coincidence of 14 wants is inconsistent with S t a r r ' s use of a set of £-1 r e l a -t ive p r i c e s . My model of a barter economy is more general than S t a r r ' s . It permits agents to engage in arb i t rage . Agents are permitted to use ind i rec t exchange, e i ther through a th i rd good or through a th i rd party. However, the i r arbi trage operations are l imi ted by the i r i n i t i a l endowments, t as tes , and transact ion technologies. Because I consider the s t ruc-ture of trade in an economy along with the presence of a medium of exchange, I am able to reta in a meaningful d i s t i n c t i o n between barter and monetary exchange, in spi te of the gener-a l i t y of my barter economy. C_. In Chapter 1 I b r i e f l y described my model of a barter economy. The important feature that d ist inguished i t from the monetary economy was that agents were not permitted to execute exchanges for other agents. Each agent's exchanges were con-strained both by goods in his possession and by his t ransact ion technology. I shal l now be a b i t more exact and rigorous in explaining what I mean by this statement. Let A represent the set of agents in the barter economy. By a e A we mean that the indiv idual a is a member of the economy. With each agent a we associate the vector w(a), agent a 's i n i t i a l endowment of goods. The vector w(a) has dimension I, where I is the number of goods in the economy. We write 03(a) e R+, where R^  is the non-negative orthant of 15 the Euclidean space R of dimension l. An agents preferences are represented by s , . The statement x « y , where x and y a a are vectors in R + , means that agent a prefers the bundle of goods y to the bundle x, or else he is ind i f f e ren t between the two bundles. In the las t chapter we said that an agent's e f f i c i -ency at exchanging goods is expressed by his t ransact ion technology. An agent's t ransact ion technology is modelled by his t ransact ion set . Suppose agent a e A wants to exchange the bundle of goods y(a) for the bundle x(a) . The vectors y(a) and x(a) are elements of R^, the commodity or good space. An agent's t ransact ion set w i l l indicate whether the exchange of y(a) for x(a) is technolog ica l ly f e a s i b l e . If exchange is f e a s i b l e , the transact ion set w i l l also indicate the real resources required to execute the exchange. Agent a 's t ransact ion set is given by S(a) , where S(a) is a subset of the Euclidean space of dimension equal to three times the number of goods in the economy or R|^. If agent a e A wants to exchange y(a) e R^  for x(a) e R^, then this exchange is technolog ica l ly feas ib le i f and only i f there ex ists a vector z(a) e R^  such that x(a) , y (a ) , z(a) e S(a) . (1) The vector z(a) represents the quant i t ies of real resources needed by a to exchange y(a) for x(a) . 16 Even though an exchange of y(a) for x(a) is tech-no log ica l l y feas ib le for agent a , he may not be able to execute the exchange because his i n i t i a l endowment is i n s u f f i c i e n t . The exchange y(a) for x(a) with transact ion costs z(a) is compatible with a's i n i t i a l endowment co(a) i f 0) (a) + x(a) - y(a) - z(a) > 0. (2) This is the material balance condit ion for agent a. Relations (1) and (2) express mathematically the c ruc ia l r e s t r i c t i o n that I place on exchange in the barter economy. Each agent's exchanges are constrained both by his transact ion technology and by goods in his possession. I s t i l l need one more re la t ion to ensure that material balance is maintained for the ent i re economy. Suppose that each agent a e A wants to exchange some bundle y(a) for x(a) and that there is a vector z(a) such that x(a) , y (a ) , z(a) e S(a) Let f(a) = w(a) + x(a) - y(a) - z(a) be the d i s i r e d consumption vector for every a e A , where f(a) > 0. Then the material balance condit ion for the ent i re economy is given by I f(a) = I u(a) - I z(a). a e A a e A a e A 1 7 It reads that the total quantity of each good consumed must equal the total i n i t i a l endowment of this goods minus the total quantity of the good used up in the process of exchanging goods. D_. I have already explained why I use the cooperative approach to model l ing. Let me now explain more f u l l y what I mean by the cooperative game approach. The core theoret ic or cooperative game approach assumes that economic agents in a soc ia l exchange economy wi l l enter into re la t ions with others . Unlike the competitive approach, the cooperative approach assumes that ind iv idua ls w i l l form c o a l i t i o n s or associat ions which are of mutual benef i t to the i r members. It is assumed that every possible c o a l i t i o n of agents forms and considers the p o s s i b i l i t y of r e d i s t r i b u t i n g the i r a v a i l -able goods. An indiv idual w i l l not jo in a c o a l i t i o n , unless he is offered a more desirable bundle of goods than his i n i t i a l endowment. Furthermore, he wants to jo in that c o a l i t i o n which of fers him the most desi rable bundle of goods. Thus indiv idual r a t i o n a l i t y is admitted in cooperative economies just as i t is in competitive economies. The main d i f ference is that group r a t i o n a l i t y is also admitted. Suppose some rea l loca t ion of goods throughout the economy is proposed. Group r a t i o n a l i t y means that th is proposed a l loca t ion of goods wi l l not be accepted by a c o a l i t i o n i f each member of the c o a l i t i o n can get a more desi rable 18 bundle of goods from some possible r e d i s t r i b u t i o n of the c o a l i t i o n ' s resources. Rather than exchange goods with the rest of the economy, the members agree to exchange goods only within the c o a l i t i o n . An a l loca t ion of goods to ind iv idua ls which resu l ts from a r e d i s t r i b u t i o n of goods among a l l agents is said to be a core a l loca t ion i f i t does not v io la te e i ther the i n d i -vidual or group r a t i o n a l i t y c r i t e r i o n . The set of a l l core a l loca t ions is said to be the cove of the economy. C l e a r l y , a core a l loca t ion is also a Pareto optimal a l l o c a t i o n , although the converse need not be t rue. The core is an equi l ibr ium concept of cooperative economies which can be compared with the equi l ibr ium price vector of t rad i t iona l general equi l ibr ium a n a l y s i s . It has been demonstrated that each competitive a l l o c a t i o n , i . e . the d i s t r i b u t i o n of goods among agents af ter trading at equi l ibr ium p r i c e s , belongs to the core . For economies with a f i n i t e number of agents, the core is general ly larger than the set of competitive a l l o c a t i o n s . As the number of agents "gets large" the core "shrinks" to the set of competitive a l l o c a -t ions [4] , [15], [31] and [70]. The equivalence, for large economies, of the core with the set of competitive equi l ibr ium a l loca t ions wi l l be exploi ted throughout this t h e s e s . 1 The cooperative game aspects used in th is thesis re ly heavily on and borrow f ree ly from the papers by Aumann [4] and [6] and Hildenbrand [31], [32], [33] and [34] with 19 respect to both the d e f i n i t i o n s of concepts used in models and the techniques used in the proofs of p ropos i t ions . Other papers of importance to the appl ica t ion of the theory of cooperative games to economics include those by Cornwall [12], Debreu and Scarf [15], Schmeidler [55], Sondermann [62] and Vind [70]. E_. The conventional theory of the core assumes, however, that no resource costs are incurred in e f fec t ing a r e d i s t r i -bution of goods among members of a c o a l i t i o n . By incorporat ing transact ion costs in a cooperative economy, i t is possible to consider formal ly , d i f fe ren t ways of organizing an economy's exchange process. Later in Chapter 4 the use of core theory permits me to model) the choice between barter and monetary exchange. We have been consider ing an economy whose i n i t i a l state is described by i t s agents' preferences, endowments and transaction technologies. To give i t the f lavour of a barter economy, we also added the constra int that each agent must use his own transact ion technology in performing exchanges. Let us now set up the economy in i ts cooperative context. The set of agents in the economy was given by A. Now le t fi be the set of admissible c o a l i t i o n s of agents. fi consists of those c o a l i t i o n s which are permitted to form. If the number of agents in the economy is f i n i t e , fi is usual ly the set of a l l subsets of A. Formally, fi is required to be 20 a a - f i e l d , w h i c h m e a n s t h a t c o u n t a b l e u n i o n s a n d f i n i t e i n t e r s e c t i o n s o f i t s e l e m e n t s a r e a l s o a d m i s s i b l e c o a l i t i o n s . F o r e a c h c o a l i t i o n E e Q, t h e r e i s d e f i n e d a r e a l n u m b e r v ( E ) w h i c h r e p r e s e n t s t h e f r a c t i o n o f t h e t o t a l i t y o f a g e n t s b e l o n g i n g t o t h e c o a l i t i o n E . A n allocation of commodities, d e n o t e d b y f , i s a d i s t r i b u t i o n o f g o o d s a m o n g t h e a g e n t s , w h e r e f ( a ) i s t h e v e c t o r a s s i g n e d t o a g e n t a . C o a l i t i o n s o f a g e n t s f o r m f o r t h e p u r p o s e o f r e a l l o c a t i n g i n i t i a l e n d o w m e n t s a m o n g t h e i r m e m b e r s . T o m o d e l b a r t e r e x c h a n g e , a n y r e a l l o c a t i o n o f g o o d s m u s t b e i n a c c o r d a n c e w i t h e a c h a g e n t ' s t r a n s a c t i o n s e t a n d i n i t i a l e n d o w m e n t . A g i v e n a l l o c a t i o n f i s s a i d t o b e a t t a i n a b l e f o r a c o a l i t i o n E e Q, u s i n g b a r t e r e x c h a n g e , i f f o r e a c h m e m b e r a g e n t a e E t h e r e e x i s t v e c t o r s x ( a ) , y ( a ) a n d z ( a ) i n R + s u c h t h a t i ) ( x ( a ) , y ( a ) , z (a ) ) e S(a) , i i ) f ( a ) = 03 ( a ) + x ( a ) - y ( a ) - z ( a ) , a n d 1 1 1 ) I f (a) = I 03(a) - I z (a ) . aeE aeE aeE T h e f i r s t t w o c o n d i t i o n s s t a t e t h a t t h e b a r t e r e x -c h a n g e p a t t e r n w h i c h r e s u l t s i n a l l o c a t i o n f m u s t be b o t h t e c h n o l o g i c a l l y f e a s i b l e f o r e a c h a g e n t a n d c o m p a t i b l e w i t h e a c h a g e n t ' s i n i t i a l e n d o w m e n t . T h e l a s t c o n d i t i o n i s c o a l i t i o n E ' s m a t e r i a l b a l a n c e e q u a t i o n . 2 1 An a l l o c a t i o n w h i c h i s a t t a i n a b l e by t h e c o a l i t i o n c o n s i s t i n g o f all a g e n t s i n t h e economy i s s a i d t o be a state of the economy. A s t a t e f o f t h e economy i s s a i d t o be b l o c k e d by t h e c o a l i t i o n E e n i f t h e c o a l i t i o n can r e -d i s t r i b u t e i t s i n i t i a l endowments among i t s members i n s u c h a way t h a t t h e r e s u l t i n g a t t a i n a b l e a l l o c a t i o n g i s p e r f e r r e d t o f by a l l members o f t h e c o a l i t i o n E. The core i s t h e n d e f i n e d as a s e t o f s t a t e o f t h e economy w h i c h c a n n o t be b l o c k e d by any a d m i s s i b l e c o a l i t i o n . F_. As was m e n t i o n e d a b o v e , t h e c o r e i s an e q u i l i b r i u m c o n c e p t f o r c o o p e r a t i v e e c o n o m i e s w h i c h can be compared w i t h t h e e q u i l i b r i u m p r i c e v e c t o r o f c o m p e t i t i v e e c o n o m i e s . I f p e R + i s a v e c t o r o f p r i c e s i n t h e c o m p e t i t i v e economy, t h e n t h e b u d g e t s e t f o r an a g e n t a e A c a n be g i v e n by a) ( x , y , z ) e S ( a ) b) co(a) + x - y - z > 0, and • c) p • x < p • y The b u d g e t s e t o f o u r " b a r t e r " economy can be compared w i t h t h e u s u a l b u d g e t s e t o f an a g e n t i n an A r r o w - D e b r e u economy, i . e . B ( a , p ) = ( x , y , z ) e R 3 £ 22 | s e I p • s < p • oj(a)| In th is chapter I am interested in discover ing under what condit ions the a l loca t ions of-.goods, resu l t ing from competit ive behaviour coincides with the core . i-o prbvesithatea-ilds t to fit rel a t i v.eopr ii-ees , oneopriiee per good, is s u f f i c i e n t for competitive behaviour to achieve the same resu l t as cooperative behaviour. The competit ive version of th is economy is just the t rad i t iona l pure exchange economy with t ransact ion cos ts . This is in contrast to the model in the next chapter where a set of buying and s e l l i n g pr ices and c o a l i t i o n traders are required to achieve the same r e s u l t . These statements are establ ished by proving the fol lowing propos i t ions . Proposition 1. A competitive a l loca t ion is also a core a l l o c a t i o n . Proposition 2. In a "per fect ly competit ive" economy —where each agent has only a neg l ig ib le inf luence on any f i n a l a l l o c a -t ion of commodities — i t is possible to derive from a given core a l l o c a t i o n a set of equi l ibr ium r e l a t i v e pr ices such that the quasi-competi t ive a l l o c a t i o n corresponding to these pr ices is the given core a l l o c a t i o n . 23 Proposition 3. Under cer ta in cond i t ions , a quasi-competi t ive a l l o c a -t ion is also a competitive a l l o c a t i o n . Let me explain b r i e f l y why i t is necessary to work with a quasi-competi t ive a l l o c a t i o n . Below, I shal l assign a consumption set X(a) to each agent a e A . For each agent a, X(a) is a subset of and i t consists of a s possible con-sumption vectors . The quasi-competi t ive equi l ibr ium concept, described by Debreu [14], was introduced because an agent's i n i t i a l wealth may not be s u f f i c i e n t to ensure him a consump-t ion vector in his consumption set a f ter exchanging goods at a given price vector . When this happens, the demand corre -spondence used to es tab l ish the existence of an equi l ibr ium is d iscont inuous. If an agent cannot exchange any goods and s t i l l remain inside his consumption se t , his choice of a consumption vector w i l l be sui table r e s t r i c t e d to ensure that the demand correspondence is in fact continuous. G_. In the remainder of this chapter I shal l prove in a rigorous manner the proposit ions made above. But f i r s t i t is necessary to give precise d e f i n i t i o n to the concepts introduced. 1. The Measure Space of Agents, (A, fi, v) The economy consists of a measure space of agents (A, fi, v) where A is the set of economic agents, fi is a 24 a - f i e l d of subsets of A and consists of the admissible set of c o a l i t i o n s and v is a countably addi t ive funct ion on Q, to R + . The function v is ca l led a measure > 3 2. The consumption set correspondence, X The consumption set correspondence X is a v-measur-able mapping from A to the subsets of R + , minorized by a v - in tegrab le func t ion . The non-empty, closed convex set X(a) associated with agent a consists of his possible con-sumption vectors; 3. The set of a l l o c a t i o n s , L„ An a l loca t ion is a v - in tegrab le function f from A to such that f (a) e X(a) , a .e . a in A J h °T h"e= >s e^t<" o=f' a l l a l l o -cations is denoted by L . A 4. The i n i t i a l endowments, to The i n i t i a l d i s t r i b u t i o n of goods among the agents oo is a v - in tegrab le function from A to R + such that w(a) e X(a) , where the i n i t i a l endowment of agent a is co(a). 5. Agent's preferences, ~ For every a e A there is defined a quasi -order on X(a) — denoted by ~ a and ca l l ed p r e f e r e n c e - o r - i n d i f f e r e n c e . 25 This relation is t ransi t ive, reflexive and complete. From the relation 2 we also define the relation called pref-a a erence by: s « t i f s 2 t but not t 2 s. The two rela-J a a a tions have the following properties: i) ~ is continuous, i .e . i f s e X(a) a then the set {t e X(a) | s 2 t} is a closed. i i ) J- exhibits local nonsati ati on , ^ i .e . a for every s e X.(a) and every open,set Uheontaining s,Xthere if:s a t enXf.a) n U such that s <* t. s a Furthermore, the preference function <* mapping A into R x R' is v-measurable (Hildenbrand [31]). 6. Agent transaction technologies, S The transaction technological correspondence S maps 3 £ elements of A into subsets of R +. I assume S has the follow-ing properties. i) S ( a ) i s closed for al l a e A. i i ) (x-(a), y(a), z(a)) e S(a) and x'(a) < x(a), y'(a) < y(a) and z'(a) > z(a) then (x 1 (a) , y ' (a ) , z'(a)) e S(a). i i i ) 0 e S(a) for al l a e A. 26 iv) for any f e L and any a e A there A exists x(a) , y ( a ) , z(a) e such that (x(a) , y ( a ) , z(a)) e S(a) and u(a) + x(a) - y(a) - z(a) > f ( a ) . v) S is a v-measurable correspondence. The f i r s t three propert ies need l i t t l e comment. Condition i ) is usual ly assumed in the l i t e r a t u r e , condit ion i i ) admits free disposal and condit ion i i i ) allows for the p o s s i b i l i t y of no exchange. Condit ion iv0 is a technological f e a s i b i l i t y cond i t ion . Because the total resources ava i lab le in the economy are unbounded from an ind iv idua ls point of view, this condit ion implies that given enough resources an ind iv idual agent can a t ta in any vector in his consumption se t . Property v) is s imi la r to the assumption that the preference function £ be v -measurable. 7. At ta inable a l l o c a t i o n s , K 1 to The a l loca t ion f e L is said to be at ta inable A for c o a l i t i o n E e ft i f and only i f there ex is t v - i n t e g r a b l e functions x, y , z from A to R + such that 27 i ) (x(a) , y(a) z(a)) e S(a) i i ) f (a) = w(a) + x(a) - y(a) - z (a ) , and i i i ) fdv = todv -• zdv E J E J E These condit ions have already been discussed above. Condi t ion i i ) together with condit ion i i i ) implies that iv) x (•) dv y( • )dv, That i s , the total quantity of each commodity received in exchange by a l l members of the c o a l i t i o n must equal the tota l quantity of each commodity given up in exchange. For a given c o a l i t i o n E e the set of a l 1 7 a t t a i n -able a l loca t ions is denoted by K W (E)-8. A state of the economy is defined as an a l l o c a t i o n which is at ta inable by the c o a l i t i o n consis t ing of a l l agents (a.e. agents) of the economy. ^ ( A ) i s t n e s e t o f a 1 1 states of the economy. g 9 . The "barter" economy, 5 The descr ip t ion of our barter economy is now complete. We denote the economy by H B = [(A, fi, v ) , X, S, u ] . 28 10. The core of the economy, C(S ) D A state f of the economy 5 is said to be blocked by the coalit ion E e A i f there exists an attainable al loca-tion g e K (E) such that i) f(a) « a g(a), a.e. in E i i) v(E) > 0. The core C(E ) is the set of states of the economy which cannot be blocked by any coal i t ion. 11) Competitive allocations The price vector of this economy should be inter-preted as a l i s t of relative prices, one price per commodity, and is denoted by p e R .^ Following Hildenbrand [31], three P basic states of the economy H are defined. P Let f(= u + x - y - z) be a state of the economy 5. p a) Competitive al locat ion, W(5 ) f is called a competitive allocation or Walras a allocation i f there exists a price vector p e R + , p f 0 such that p • x(a) < p • y(a), for al l a e A and i f s(= w(a) + x' - y' - z') e X(a), (x 1 , y \ z') e S(a) with f(a) s , 29 t h e n p • x * > p • y 1 . p L e t W(H ) b e t h e s e t o f a l l c o m p e t i t i v e a l l o c a t i o n s p b ) Q u a s i - c o m p e t i t i v e a l l o c a t i o n , Q(5 ) f i s c a l l e d a q u a s i - c o m p e t i t i v e a l l o c a t i o n o r a q u a s i - W a l r a s a l l o c a t i o n i f t h e r e e x i s t s p e R + , p f 0 s u c h t h a t : p • x ( a ) < p • y ( a ) , f o r a l l a e A a n d i f s ( = u (a) + x.' - y ' - z 1 ) e X ( a ) , ( x 1 , y ' , . z ' ) e S ( a ) w i t h w° f ( a ) « s , a a n d i f . " {p ° x : ° y. < C p:)«)cx S^ sp^  • yl < >Q i n % x .. y s-/ \ / \ / \ x , y , z ) e S ( a ) CO ( a ) +i(-sy}-ieX<(a ) y ? e X ( a } t h e n p • x ' > p • y ' . P L e t Q(H ) b e t h e s e t o f a l l q u a s i - c o m p e t i t i v e a l l o c a t i o n c ) E x p e n d i t u r e m i n i m i z i n g a l l o c a t i o n , E (5 ) f i s c a l l e d a n e x p e n d i t u r e m i n i m i z i n g a l l o c a t i o n o r a p s e u d o - c o m p e t i t i v e a l l o c a t i o n i f t h e r e e x i s t s p e , p f 0 s u c h t h a t 30 p • x (a ) < p • y(a) and i f s(= w(a) + x' - y' - z') e X(a), ( x \ y ' , x') e S(a) with f(a) <* s then p • x' > p • y 1 . Let E(S ) be the set of a l l expenditure minimizing al locations. From the definitions and comments made above i t is clear that W(5 B) C Q(5 B) C E ( » B ) . Expenditure minimizing allocations are introduced to fac i l i ta te the proofs of the following theorems. The propositions made above wil l be established by provdnggthe following Theorems. Theorem 1 establishes Proposition 1, Theorem 2 and its corollary establish Proposi-tion 2. F ina l ly , an example of an economy in which Proposi-tion 3 is true will be given in 'Appeinrd i x5 .B. Theorem 1 Every competitive allocation is also a core a l loca-t ion, i .e . W(HB) c C ( « B ) . Theorem 2 If the measure space of agents is non-atomic, then 31 every core allocation is also a pseudo-competitive al locat ion, i .e . C ( H B ) c E(~ B ) . CoroIlary Every core allocation is also a quasi-competitive al locat ion, i .e . C(~B) c Q(~ B). J<. The following proofs follow closely the proofs by Hildenbrand [31] for coalit ion production economies. B B Proof of Theorem 1. W(H ) C C(H ) . R B Let f e W(5 ) but suppose f £ C(E ). Then there exists E e fi with v(E) > 0 and h e K (E) such that h = u> + x1 - y' - z ' , (x'(a), y ' (a ) , z'(a)) e S(a) and f(a) =a h(a) for a.e. a a in E. But f e W (H B ) + p • x'(a) > p • y ' (a ) , a.e. a in E p • x'(•)dv > P • y'(•)dv x' (-)dv f . y ' (•.)dv. But this contradicts the material balance requirement that h e K (E). Thus f e C ( H B ) . 32 R R Proof of Theorem 2. C(H ) C E(H ). Let f e C(H ). From the nonsatiation assumption 5. i i ) assumption 6.iv) for the transaction technologies, the set p(a) x(a), y(a), z(a) e S(a)|f(a) « oi(a) + x(a) - y(a) - z(a) is non-empty for a.e. a in A. Now define the correspondence 6 from A to R£ by 6(a) = - x(a) - y(a)| x(a), y(a), z(a) e p(a) Let L g be the set of v-measurable function g from A to such that g(a) e 6(a) for a.e. a in A. Since f is v-integrable and S is a v-measurable correspondence L^  + $[31, p. 448]. Define the set U c R£ by U = u {E e fi|v(E) > 0} L 6dv = 4 gdv [geL I claim 0 t U. Suppose 0 e U. Then there exists E e fi with v(E) > 0 and a function g : A + Rl such that 33 a) g e L^ + lett ing x(a) = g(a) T , y(a) = g(a) ' , where g 1 (a ) + = g 1(a)~ = g 7(a) i f g^a ) > 0 0 otherwise - g 1 (a) i f g 1 (a) < 0 0 otherwise then^gxxyyaarid?simee 6 is v-smeasurabl e by Theorem B in [ 3 3 ] there exists integrable z : A -* R such that (x(a),. y(a), z(a)) e S(a) and f(a) « w(a) + x(a) - y(a) - z(a) = h(a) b) 0 = gdv (x(-) - y(-))dv x(•)dv -J E y(•)dv, But a) and b) imply h e K ^ ( E ) and h is a blocking allocation for coalit ion E contradicting f e C(E ) . Therefore 0 t U. Because U is the integral of a set correspondence with respect to a non-atomic measure v , U is convex (see Vind [ 7 0 ] ) . Using a separating hyperplane theorem, i t is possible to show that there exists a vector p e R , p f 0 such that u e U implies p • u > 0. It is now possible to show that f is a pseudo-competitive allocation for the price vector p. Let M = {a e A | p • x(a) > p • y(a), for al l (x(a), y(a), z(a)) e p(a)>. It is possible to show that M e fi. In fact , 34 v(M) = v(A). If not, there exists B e fi with v(B) > 0 such that for every a e B there exists a point (x'(a), y ' (a ) , z'(a)) e p(a) such that p • x'(a) < p • y ' (a ) . Without'loss of generality I can assume that the functions x ' , y 1 , x' from B to are measurable (see Theorem B [33, p. 621]). But p • x'(a) < p • y'(a) a.e. a in B J B x'(•)dv < p y' (-)d P * (x1 (•) - y 1 ( - ) )dv < 0, But since SB ( x ' ( » ) - y'(*))dv e U by construction, we have a contradiction and thus v(M) = v(A). It just remains to demonstrate that f = w + x - y - z sat isf ies each agents budget constraint. Since (x(a), y(a), z(a)) e closure of p ( a ) by continuity of preferences we have p « x ( a ) > p « y ( a ) a.e. a in A Suppose there exists C e fi with v(C) > 0 such that p • x(a) > p • y (a ) , a l l a e C The last two equations imply that A^ p • x(*)dv > P • y(*)dv 35 or x(-)dv f y(•)dv. But this contradicts the material balance constraint that f e K (A). Therefore, CO p » x ( a ) = p # y ( a ) a.e. a in A. and thus f is a pseudo-competitive al locat ion, i.e f e E ( S B ) Proof of Collary C ( H B ) C Q(H B) To prove that f is also a quasi-competitive a l locat ion, i t is necessary to show that in the case where •itn-f : - p 'x x- -p p • y-k < '0 x",y;,z')e^(a); co (a ) + x-y-zeX (a ) co (a ) + x - y i e X {a ) then f(a) is a maximal element in the budget set. Since f is a pseudo-competitive al locat ion, i f s e X(a) where s = co(a) + x' - y' - z' , (x1 , y 1 , z' ) e S(a) and p • x' < p • y' then f(a) <* s a Let s be in the budget set of a e A , i .e . p • x' < p • y' . s can be obtained as a l imit of a sequencers } where s n = co(a) + x ' n - y ' n - z ' n , ( x ' n , y ' n , z ' n ) e S(a) with P • x ' h < p • y ' n . Then by continuity of preferences and assumption 6 . i ) on S(a), we get f(a) s. Thus f(a) is a maximal element a the budget set. C h a p t e r 3 THE "MONETARY" ECONOMY A. I s t a t e d i n C h a p t e r 1 t h a t I want t o b u i l d a s e n s i b l e model o f a m o n e t a r y economy. The s t r a t e g y was t o f o c u s on t h e s t r u c t u r e o f e x c h a n g e . W h i l e t h e l a s t c h a p t e r d e a l t w i t h t h e s t r u c t u r e o f b a r t e r e x c h a n g e , h e r e I s h a l l e x a m i n e t h e s t r u c t u r e o f m o n e t a r y e x c h a n g e . The n e x t c h a p t e r w i l l b r i n g t o g e t h e r b o t h m o n e t a r y and b a r t e r e x c h a n g e and w i l l c o n s i d e r t h e c h o i c e between t h e two methods o f e x c h a n g e . The b a r t e r economy was f i r s t m o d e l l e d by u s i n g t h e c o o p e r a t i v e a p p r o a c h . I t was assumed t h a t e a c h a g e n t p o s s e s s e d a t r a n s a c t i o n t e c h n o l o g y . A c o a l i t i o n o f a g e n t s f o r m e d f o r t h e p u r p o s e o f e x c h a n g i n g goods among i t s members. However, e a c h i n d i v i d u a l was r e s t r i c t e d t o e x e c u t i n g o n l y h i s own e x c h a n g e s . From a g i v e n c o r e a l l o c a t i o n , I d e r i v e d t h e s t r u c t u r e o f p r i c e s r e q u i r e d f o r c o m p e t i t i v e b e h a v i o u r t o r e p l i c a t e c o o p e r a t i v e b e h a v i o u r . We d i s c o v e r e d t h a t a s£t o f I e q u i l i b r i u m p r i c e s , one p r i c e p e r g o o d , was s u f f i c i e n t . 37 38 In competitive behaviour, individuals accept these prices as parametric. They attempt to obtain the most desirable consump-tion bundle in their budget set by exchanging goods at these prices, using their own transaction technology. In Appendix B, I have proven that a set of equilibrium prices does in fact exist . Since each agent is constrained at each stage of the exchange process by goods in his possession and by his transaction technology, a general medium of exchange is not necessary for the barter economy to function ef f ic ient ly within its given constraints. B^  We will again use the cooperative approach to model the monetary economy. However, in the monetary economy coalitions of agents do not just form for the purpose of exchanging goods. They also form for the purpose of exploit-ing division of labour in order to reduce transaction costs. I assume that a coalit ion assigns exchange tasks to its members. Its efficiency at exchangingigoods depends on its s k i l l at allocating members to tasks according to their a b i l i t i e s . To capture this idea of division of labour or special izat ion, I begin by assigning a transaction technology to every coal i t ion. I assume that the transaction technology assigned to a coalit ion consists of the most ef f ic ient subset of its member's transaction technologies. In other words, I assume exchange tasks have been allotted as ef f ic ient ly as 39 possible. The following chapter will describe a method for obtaining a coal i t ion's transaction technology from its member's transaction technologies. Furthermore, in this chapter I do not allow individuals to "barter," i .e . use their own transaction technologies to effect exchanges. This assumption will be relaxed in the next chapter when I give agents the choice of barter or monetary exchange. However, here an agent must join a coalit ion i f he wants to obtain more desirable goods. An agent requires access to a coal i t ion's transaction technology in order to trade with its members. His potential trades depend on his environment, i .e . on the coalit ion to which he belongs. The coal i t ion, on the other hand, wants to admit those individuals whose exchange ab i l i t ies enhance its tech-nology of exchange and whose in i t i a l endowments complement the coal i t ion's in i t i a l endowments. I shall again derive the structure of prices needed for competitive behaviour to replicate cooperative behaviour. I shall show that a l i s t of equilibrium buying and sell ing prices will do the job. In the competitive version of this model, coalitions act as profit maximizing traders; individuals act as u t i l i t y maximizing consumers. As a profit maximizing trader, the coalit ion is wil l ing to buy goods from its members or to sell goods to its members. To cover transaction costs, the trader must establish a differential between his buying and sel l ing prices. 40 The idea of a coalit ion trader may seem a bit strange at f i r s t . In fact however, the coalit ion trader consists of a set of individual traders who are individually trying to maximize prof i ts , given their transaction technologies and the parametric l i s t of buying and sel l ing prices. The coal i -tion traders represent the commercial sector of the economy. In Chapter 1 I discussed the importance of money when some individuals specialize in trade. There I assumed some "institution" was present to provide the needed medium of exchange function. In the cooperative approach that I am using, this "institution" is provided by the coal i t ion. A coalit ion could set up different, although conceptually equivalent, arrangements to play the role of a medium of exchange. An account could be maintained for each member agent. An agent's account would be credited when he sold goods to one of the coalit ion traders and debited when he bought goods from a trader. Some clearing arrangements would also be necessary among the traders. An agent's budget constraint would be satisf ied i f his account was nonnegative at the end of al l trading. Alternatively, the coalit ion could issue f ia t money to its consumers and traders. By agreement, the f ia t money would always be acceptable in exchange for goods. The precise institution used by a coalit ion will depend, of course, on the transaction costs incurred in setting up and running the inst i tut ion. The use of different inst i tut ions" would be reflected in the coalitions efficiency at executing 41 exchanges, i .e . in its transaction technology. Throughout this thesis I assume that a coal i t ion's transaction tech-nology embodies the optimal configuration of such inst i tut ions. C_. Let me be a bit more expl ic i t about the monetary model. For every coalit ion ME e fi, there is specified a subset T(E) of R x R - the transaction technological set. A coalit ion can only reallocate goods among its members in accordance with the coal i t ion's transaction technology. For a given vector (x,-y) e T ( E ) , where x,y e R ,^ the vector x denotes the total quantities of goods delivered to member agents by the coal i t ion; the vector y denotes the total quantities of goods obtained from member agents by the coal i t ion. The vector y-x which must belong to R+ represents the real resources used up in effecting the reallocation of goods. I assume that the transaction set corresponding to each admissible coalit ion is closed, admits free disposal of resources and allows for the possibi l i ty of no exchange within a coal i t ion. These properties need l i t t l e comment as 42 t h e y a r e s t a n d a r d a s s u m p t i o n s f o r p r o d u c t i o n s e t s . I a l s o assume t h a t t h e c o r r e s p o n d e n c e T mapping e l e m e n t s o f fi t o s u b s e t s o f R i s c o u n t a b l y a d d i t i v e . T h i s means t h a t f o r e v e r y c o u n t a b l e f a m i l y { E ^ J ^ j o f p a i r w i s e d i s j o i n t c o a l i -t i o n s i n fi, we have T ( U i e I E^ ) = I. j T ( E ^ ) . The c o n c e p t o f a p r o d u c t i o n s e t f o r a c o a l i t i o n o f a g e n t s i n a measure t h e o r e t i c c o n t e x t was f i r s t i n t r o d u c e d by Hi 1 d e n b r a n d [ 3 1 ] . As I have m e n t i o n e d , t h e d e r i v a t i o n o f T f r o m t h e t r a n s a c t i o n a b i l i t i e s o f a c o a l i t i o n ' s members w i l l be c o n s i d e r e d i n t h e n e x t c h a p t e r . D_. An Allocation of commodities, d e n o t e d by f , i s a g a i n d e f i n e d as a d i s t r i b u t i o n o f goods among t h e a g e n t s where f ( a ) - t h e v e c t o r o f c o m m o d i t i e s a s s i g n e d t o a g e n t a i s an e l e m e n t o f a g e n t a's c o n s u m p t i o n s e t X ( a ) . I f t h e i n i t i a l endowment i s to, t h e n an a l l o c a t i o n f i s s a i d t o be a t t a i n a b l e f o r c o a l i t i o n E i f and o n l y i f , l e t t i n g x(a) = [ f ( a ) - c o ( a ) ] + , y ( a ) = [ f ( a ) - u(a)]" x ( • ) d v , y y ( • ) d v , t h e n ( x , - y ) e T ( E ) 43 The vectors x(a) and y(a) are respectively the quantities of goods received in exchange and the quantities of goods given up in exchange by agent a. The vectors x and y are respectively the total quantities of goods received from the coalit ion by member.agents and the total quantities of goods given up to the coalit ion by members agents. The definit ion implies that material balance is maintained for both individual agents and coalit ions of agents, since, for every a e A f(a) - (.(a) = [f(a) - 03(a)] + - [ f (a) 03 (a)]", whence, and f(a) = u(a) + x(a) - y(a) , 7, / f(-)dv = 03 (•)dv + x(•)dv - y(-)d^ 03 (•)dv - (y - x) A state of the economy is again defined as an allocation which is attainable by the coalit ion consisting of a l l agents in the economy. Similar ly, the cove is defined as the set of states of the economy which cannot be blocked by any admissible coal i t ion. 44 E_. In this chapter I am again interested in discovering under what conditions the allocations of goods resulting from competitive behaviour coincide with the core al locations. I establish the fact that a l i s t of equilibrium buying and sel l ing prices are required. The difference between the buying and sel l ing prices ref lect the transaction costs incurred by the coalit ion while transporting goods from one agent to another. The prices represent contracts between the coalit ion of the entire economy and its member agents regarding the terms of acquiring commodities and the al loca-tion of transport tasks to agents. Because the act of buying and the act of sel l ing a good are separate with respect to both time and pilace, debt contracts between the coalit ion as a set of specialized traders and its member consumers are also required to ensure that agent's budget constraints and coali t ion material balance requirements are sat is f ied . These statements are established by proving the following propositions. Proposition 1: A competitive allocation is also a core al location. Proposition 2: In a "perfectly competitive" economy — where each agent has only a negligible influence on any final allocation 45 o f c o m m o d i t i e s — i t i s p o s s i b l e t o d e r i v e f r o m a g i v e n c o r e a l l o c a t i o n a s e t o f e q u i l i b r i u m b u y i n g and s e l l i n g p r i c e s s u c h t h a t t h e q u a s i - c o m p e t i t i v e a l l o c a t i o n c o r r e s p o n d i n g t o t h e s e p r i c e s i s t h e g i v e n c o r e a l l o c a t i o n . Proposition S: Under c e r t a i n c o n d i t i o n s , a q u a s i - c o m p e t i t i v e a l l o -c a t i o n i s a l s o a c o m p e t i t i v e a l l o c a t i o n . F. T hese p r o p o s i t i o n s w i l l be p r o v e n i n a r i g o r o u s manner a f t e r g i v i n g p r e c i s e d e f i n i t i o n t o some c o n c e p t s n o t r e q u i r e d i n t h e l a s t c h a p t e r . 1. C o a l i t i o n T r a n s a c t i o n T e c h n o l o g i e s . T^ The c o a l i t i o n a l t r a n s a c t i o n s c o r r e s p o n d e n c e T maps 2 0 e l e m e n t s o f ft i n t o s u b s e t s o f R . I assume T has t h e f o l l o w -i n g p r o p e r t i e s . i ) T ( E ) i s a c l o s e d * c o n v e x s e t f o r a l l E e ft. i i ) i f ( x , - y ) e T ( E ) t h e n x 1 < x and y 1 > y i m p l i e s ( x ' ,-y') e T ( E ) . i i i ) 0 e T ( E ) f o r a l l E e ft. i v ) T i s d o m i n a t e d by t h e measure v , i . e . E e ft w i t h v ( E ) = 0 i m p l i e s T ( E ) ='{0}. 46 v) T is- countably additive on fi. vi) T possesses a Radon-Nikodym derivative [3], [5], [17] and [18]. That i s , there exists a correspondence x mapping A into subsets of R^  x R^  such that for every .L dv , where L = {t I t JE T x 1 E e fi, T(E) is a v-integrable function from A to R£ x RZ such that t(a) <* x(a), a.e. a in A v i i ) T(E) is a compact set for al l E e fi. The f i r s t three conditions need l i t t l e comment and are similar to the properties for agent's transaction sets. Condition iv) indicates that only "significant" coalitions are capable of production. Conditions v) and vi) imply constant returns to scale with respect to the nonmarketed factors owned by coal i t ions, i .e . with respect to member agent's ab i l i t i es used in the operation of coal i t ions' transaction technologies. Properties i ) , i i i ) - v) plus v i i ) imply property vi) [31, p. 447]. 2. Attainable al locations, K . 2 W The allocation f e L is said to be attainable for X coalit ion E e fi i f and only i f , letting 47 x ( a ) = [ f ( a ) - co (a ) ] \ y(a) = [f(a) - u ) ( a ) ] \ x = x(«)dv, y = y ( * ) d v then (x,-y) e T(E) The set of al l attainable allocations for coalit ion E is denoted.by K^(E). A state of the economy is an element of K (A). CO 3 . The "monetary" economy, 5 M The description of the monetary economy in its cooperative game context is now complete and is denoted by = [ ( A , ft, v ) , X, 2 , T, u ] 4 . Prices, Profits and the Radon-Nikodym Derivative A coalit ion buys commodities at one set of prices and resells commodities at another set of prices. The dif -ferential in the prices pays for the cost of transporting goods from one agent to another. This process could result in a profit or loss for the coal i t ion. The assumption was made that the specialized traders operating the transaction technology of a coalit ion were profit maximizers. Denote the price vector by p = (p^, pg) £ R x R where p s and p^  48 represent the l i s t of prices the traders respectively pay when buying commodities and receive when sel l ing commodities, The profit function for coalit ion E e fi for a price vector p is defined by n(p,E) = max (p. • x - p_ • y) . (x,-y) £ T(E) b S That i s , the coal i t ion's profit function is equal to the maximum difference between the value of commodities sold and the value of commodities purchased, for a l l feasible combina-tions of quantities bought and quantities sold. It is possible to verify that the mapping n(p, •) of fi into R u {°°} has the following properties and therefore is a measure. i) n (p, *) = 0, i i ) n(p>*) is countably additive on fi, i i i ) n (p»* ) is dominated by the measure v. From the Theorem of Randon-Nikodym [60], there exists a function TT(p, •) of fi into R u {«>}, where TT is v-measurable and for every E e fi, n(p,E) = TT (P , • )dv E The function TT evaluated at a e A, i .e . T (p ,a) can be inter-preted as agent a's share in a coalit ions profit (or loss) . 49 An a g e n t ' s s h a r e o f p r o f i t s i s i n d e p e n d e n t o f t h e p a r t i c u l a r c o a l i t i o n he j o i n s b e c a u s e o f t h e c o u n t a b l e a d d i t i v i t y o f Q t h e t r a n s a c t i o n t e c h n o l o g y c o r r e s p o n d e n c e . F u r t h e r m o r e , s i n c e T i s a compact mapping TT i s c o n t i n u o u s i n p. 5. The C o r e and C o m p e t i t i v e A l l o c a t i o n s M L e t f be a s t a t e o f t h e economy H and d e f i n e x ( a ) = [ f ( a ) - w ( a ) ] + , y ( a ) = [f (a ) - u>(a)I x ( * ) d v , y = J E J y ( - ) d v E da) The c o r e , C ( E M ) f i s c a l l e d a c o r e a l l o c a t i o n i f i t c a n n o t be b l o c k e d by any c o a l i t i o n E e ft. M L e t C(H ) be t h e s e t o f a l l c o r e a l l o c a t i o n s . M b) C o m p e t i t i v e a l l o c a t i o n s , W(B ) f i s c a l l e d a c o m p e t i t i v e a l l o c a t i o n o r W a l r a s 2 £ a l l o c a t i o n i f t h e r e e x i s t s a v v e c t o r p = ( p b , p^) e R , p f 0 s u c h t h a t 50 i ) p b • x(a) ^ P S • y(a) + Wp,a) and s e X(a) with f(a) « s a impl ies p b • [s - u)(a)]+ > p s • [ s - u(a)]- + TT (p , a ). i i ) p b • x - p s • y = ^ _ m a x ( x , - y ) e T ( a ) • x - Pc * y L e t W ( H ) be t h e s e t o f a l l c o m p e t i t i v e a l l o c a t i o n s . c ) Q u a s i - c o m p e t i t i v e a l l o c a t i o n s , Q(5 ). f i s c a l l e d a q u a s i - c o m p e t i t i v e a l l o c a t i o n o r a q u a s i - W a l r a s a l l o c a t i o n i f t h e r e e x i s t s a v e c t o r p = ( P b » P s ) e R 2 i , p f 0 s u c h t h a t i )) p b • x ( a ) < p s • y ( a ) + 7 r ( p , a ) and s e X ( a ) w i t h f ( a ) « s and a i n f {p. • [ r - c o ( a ) ] + - p c • r e X ( a ) b s [ r - w ( a ) ] " } < 7 T ( p , a ) , i m p l i e s Pb • C ; TT!(»P ,a) P B • [ s - u ( a ) ] + > p s • [ s - o o ( a ) ] - + i i ) same as above . 51 L e t Q ( H ) be t h e s e t o f a l l q u a s i - c o m p e t i t i v e a l l o c a t i o n s . M d) E x p e n d i t u r e - m i n i m i z i n g a l l o c a t i o n s , E ( 5 ) f i s c a l l e d an e x p e n d i t u r e - m i n i m i z i n g a l l o c a t i o n o r a p s e u d o - c o m p e t i t i v e a l l o c a t i o n i f t h e r e e x i s t s a v e c t o r p = ( p ^ , p s ) e RzSj, p f 0 s u c h t h a t : i ) • x ( a ) * p s • y ( a ) + T r ( p , a ) and s e X ( a ) w i t h f ( a ) « s i m p l i e s a P b • [s - o ) ( a ) ] + > p s • [ s - w ( a ) ] ~ + T r ( p , a ) . i i ) same as a b o v e . L e t E ( 5 ) be t h e s e t o f a l l e x p e n d i t u r e - m i n i m i z i n g a 1 1 o c a t i o n s . The d e f i n i t i o n s o f t h e c o m p e t i t i v e a l l o c a t i o n s c o r r e s p o n d to t h o s e used by H i l d e n b r a n d [ 3 1 ] . As I m e n t i o n e d t h e c o n c e p t o f a qua s i - e q u i 1 i b r i u m was f i r s t i n t r o d u c e d by Debreu [ 1 4 ] t o cope w i t h t h e " b a s i c m a t h e m a t i c a l d i f f i c u l t y t h a t t h e demand c o r r e s p o n d n e n c e o f a consumer may n o t be u p p e r s e m i - c o n t i n u o u s when h i s w e a l t h " — a t a g i v e n p r i c e v e c t o r — " e q u a l s t h e minimum c o m p a t i b l e w i t h h i s c o n s u m p t i o n s e t . " From t h e d e f i n i t i o n s we a g a i n g e t W ( H M ) c Q ( H M ) c E ( H M ) 52 G_- The propositions made above will be established by proving the following theorems. Theorem 1 establishes Proposition 1, Theorem 2 and its corollary establish Proposi-tion 2. An example of an economy in which Proposition 3 is true will be given in (Append i x5 B . Theorem 1 Every competitive allocation is also a core alloca^ t ion, i .e . W(5M) c C(5 M). Theorem 2 If the measure space of agents is non-atomic, then every core allocation is also a pseudo-competitive a l locat ion, i .e . C(H M ) c E ( H M ) . CoroIlary Every core allocation is also a quasi-competitive M M al locat ion, i .e . C(«") c Q ( s " ) . I_. The following proofs are again based on those by Hildenbrand [31] for coalit ion production economies. M M Proof of Theorem 1. W(S ) c C(S ). Let f e W(5M) but suppose f t C(5 M). Then there exists E e Q with v(E) > 0 and h e K ; i(E) such that 53 f (a ) • h ( a ) a . e . a i n E and a [ h ( - ) - o ) ( - ) ] + dv, - f [ h ( . ) - u ) ( . ) ] ' d v ] e T ( E ) M But f e W(H ) i m p l i e s P K * [ h ( a ) - w ( a ) ] + > P s * L~h(a) - w ( a ) ] " + T r ( p , a ) a.e. a i n 'b o r P K ' [ h ( . ) - o ) ( - ) ] dv - p, [h (a ) - co(a ) ]"dv > T r ( p , ' ) d v = n ( p , E ) C o n t r a d i c t i n g t h e d e f i n i t i o n o f t h e c o a l i t i o n p r o f i t f u n c t i o n Q.E.D. M M Proof of Theorem 2. C ( s ) C E(E ). L e t f e C ( H M ) . S i n c e f ( a ) i s a n o n s a t i a t i o n con-s u m p t i o n v e c t o r f o r a l m o s t a l l a g e n t s a e A , t h e s e t ij,(a) = {s e X ( a ) . | f ( a ) « a s } i s non-empty f o r a.e. a i n A Now d e f i n e t h e c o r r e s p o n d e n c e p-mappiing A ' i n t o s u b s e t s o f R £ x R £ by 54 p(a) = {([s - uj (a ) ] + , - [s - w ( a ) ]~ ) | s e ^(a)} £ £ Let L be the set of measurable functions g from A to R x R P such that g(a) e p(a) a.e. a in A. It is possible to show that L p f 0 (see Hi 1denbrand [31]). Define the set U c R£ x R£ by U = u -* L dv - I dv -> 0} E p 1 E T J where T is the Radon-Nikodym derivative of the correspondence T w.r. t . v. I claim that 0 £ U. Suppose 0 e U. Then there £ £ exists E e ft with v(E) > 0 and a function g:A -> R x R such that a) g e L p + lett ing g(a) = (x(a), -y(a)) where x(a), y(a) e R+, and h*(a) = u ( a ) + x(a) - y(a) then f (a) « h'(a) , a £ E. a b) J E gdv £ T(E) L dv. T But, gdv J E g + dv g~dv (x(-), 0)dv - (0,y(-))dv (x,-y) £ T(E) 55 But a) and b) i m p l y h e K^(E) and t h e s t a t e f i s b l o c k e d by c o a l i t i o n E u s i n g a l l o c a t i o n h. T h i s c o n t r a d i c t s f e C(H ) and t h e r e f o r e 0 t U. The s e t U i s c o n v e x b e c a u s e i t i s t h e u n i o n o f i n t e g r a l s o f a s e t c o r r e s p o n d e n c e w . r . t . t h e n o n - a t o m i c m e asure v ( V i n d [ 7 0 ] ) . From M i n k o w s k i ' s s e p a r a t i n g h y p e r p l a n e t h e o r e m t h e r e e x i s t s a v e c t o r p = ( p b , p g ) e x s u c h t h a t u e U i m p l i e s (Of i f E e fi, t h e n p • u > 0 +P t e ' x - p • y | ( x , - y ) e T ( E ) f < n ( p , E ) x ( - ) d v - p, y(«)dv | ( x ( a ) , • y ( a ) e p ( a ) a e E-f ( 1 ) I now show t h a t f i s an e x p e n d i t u r e - m i n i m i z i n g a l l o -c a t i o n f o r t h e p r i c e - v e c t o r p = (p br? p g ) . L e t M = {a £ A | p b • x ( a ) > p g • y ( a ) + 7 T ( p , a ) , ( x ( a ) , - y ( a ) e p ( a ) } . I t i s p o s s i b l e t o show t h a t M e fi; In f a c t v(M) = v ( A ) t c I f n o t , ..there exsiists t a c q 3 l i t i <3n ~ B £ fi-.with v ( B ) > 0 s u c h t h a t , - f o r a l l a ^ e a B , t ( i e r e e x i s t s (x\(a) , ty,(a ) ) £ p ( a ) s u c h t h a t 56 p b • x ( a ) < p s • y ( a ) + T r ( p , a ) W i t h o u t l o s s o f g e n e r a l i t y I can assume, x, y a r e m e a s u r a b l e 0 f u n c t i o n s f r o m B t o R +. I n t e g r a t i n g t h e l a s t e q u a t i o n we g e t x ( j ) d v - p, y ( • ) d v < TT(P , • ) d v , o r P b * x - p c • y < n ( p , B ) c o n t r a d i c t i n g r e l a t i o n (1) .above. To show t h a t f ( a ) b e l o n g s t o e a c h a g e n t ' s b u d g e t sse.t l e t x ( a ) = [ f ( a ) - o ) ( a ) ] + , Y ( a ) = [ f ( a ) - w ( a ) ] " . Then s i n c e ( x ( a ) , - y ( a ) ) e c l o s u r e o f p ( a ) f o r a.e. a i n A we know f r o m (1) t h a t P b * x ( a ) > p s • y ( a ) + i r ( p , a ) , a.e. a i n A Suppose t h e r e e x i s t s C e Q, v<(c) >00ssuch1:that p b • x ( a ) > p s • y ( a ) + u ( p , a ) , a e C Then c l e a r l y 57 Ph * A x(«)dv - p( y(• )dv > (p, • )dv or P b • x - p s • y > Il(p,A) But s i n c e (x,ry) e T(A) by a s s u m p t i o n , we have a c o n t r a d i C ' t i o n of the d e f i n i t i o n o f n. Therefore p b • x(a) = p s • y(a) + Tr(p,a) Integrating the last equation we get P b • x - p s • y = n(p,A) i .e . f maximizes profits on T(A) M Therefore f e E(H ) Q.E.D M M Proof of C o r o l l a r y . C(E ) c Q(» ) To prove that f is also a quasi-competitive alloca t ion, i t is necessary to show that in the case a e A where inf reX(aO p b • [r - u(a)]+ - p s • [s - oj (a)]"- < Tr(p,a) then f(a) is a maximal element in a's budget set. Since f is an expenditure-minimizing a l locat ion, i f 58 s e X(a) and p b • [s - co(a)]+ < p • [s - co(a)]~ + ir (p, a) then f(a) t s. Let s be in the budget set of a e A, i .e . p b • [s - o)(a)]+ < p s • [s - o>(a)]~ + i r ( p , a ) . Then s can be obtained as a l imit of a sequence'{s } where P b • [s.-n - u)(a)] + < p s • [s - w (a) ]~ + Tr(p,a). and thus f (a ) 5* s n . Then by continuity of preferences we get f(a) £ s. Thus M f(a) is a maximal element in the budget set, i .e . f e Q(H ) . Q. E . D C h a p t e r 4 THE RELATIVE EFFICIENCY OF A "MONETARY" VERSUS A "BARTER" ECONOMY A. The p r e s e n c e o f d i v i s i o n o f l a b o u r c h a r a c t e r i z e d t h e t r a n s a c t i o n t e c h n o l o g y o f t h e m o n e t a r y economy i n C h a p t e r 3; t h e a b s e n c e o f d i v i s i o n o f l a b o u r c h a r a c t e r i z e d t h a t o f t h e b a r t e r economy i n C h a p t e r 2. In t h i s c h a p t e r , I s h a l l be i n v e s t i g a t i n g t h e s e e c o n o m i e s e f f i c i e n c y i n t h e a l l o c a t i o n o f c o m m o d i t i e s t h r o u g h c o m p e t i t i v e t r a d i n g and I s h a l l d e v e l o p a p r o c e d u r e f o r d e r i v i n g a g g r e g a t e t r a n s a c t i o n t e c h n o l o g i e s f r o m t h e t r a n s a c t i o n a b i l i t i e s o f i n d i v i d u a l a g e n t s . R e c a l l t h a t an i n d i v i d u a l i n t h e m o n e t a r y economy o f C h a p t e r 3 had t o use a c o a l i t i o n ' s a g g r e g a t e t r a n s a c t i o n t e c h n o l o g y t o o b t a i n more d e s i r a b l e g o o d s . The use o f t h e b a r t e r e x c h a n g e p r o c e s s , w h i c h u n d e r l i e s t h e m o n e t a r y economy, was n o t a v a i l a b l e t o a g e n t s . In t h i s c h a p t e r I s h a l l remove t h i s r e s t r i c t i o n and I s h a l l a l l o w a g e n t s t h e c h o i c e between m o n e t a r y and b a r t e r e x c h a n g e . I t w i l l be s a i d t h a t t h e 59 60 m o n e t a r y economy d o m i n a t e s i t s u n d e r l y i n g b a r t e r economy i f no g r o u p o f a g e n t s w ants t o b r e a k away f r o m t h e m o n e t a r y economy and use b a r t e r t o a l l o c a t e goods w i t h i n t h e g r o u p . I s h a l l e s t a b l i s h a s u f f i c i e n t c o n d i t i o n , b a s e d o n l y on an economy's a g g r e g a t e m o n e t a r y and b a r t e r t r a n s a c t i o n t e c h n o l o g i e s , w h i c h e n s u r e s t h a t a m o n e t a r y economy d o m i n a t e s i t s u n d e r l y i n g b a r t e r economy. B.. I t i s g e n e r a l l y b e l i e v e d t h a t s o c i e t y b e n e f i t s f r o m t h e use o f money. R e c e n t l y , some a u t h o r s have a t t e m p t e d t o e s t a b l i s h t h e b e n e f i t s o f m o n e t a r y e x c h a n g e by d e m o n s t r a t -i n g t h a t money's p r e s e n c e i m p r o v e s t h e a l l o c a t i o n o f r e -s o u r c e s i n an economy. The a p p r o a c h used by t h e s e a u t h o r s i s v e r y s i m p l e . F i r s t , t h e y s e t up two e c o n o m i e s w h i c h a r e i d e n t i c a l i n a l l d e t a i l s e x c e p t t h a t one economy u s e s "money" w h i l e t h e o t h e r does n o t . Then t h e y show t h a t w h i l e c o m p e t i t i v e e x c h a n g e w i t h "money" i s e f f i c i e n t i n a l l o c a t i n g c o m m o d i t i e s , e x c h a n g e w i t h o u t "money" may f a i l t o be e f f i c i e n t . Money's r o l e i n p r o m o t i n g e f f i c i e n t e x c h a n g e i n t h e s e m o d e ls depends on e a c h a u t h o r ' s c o n c e p t o f t h e d i s t i n g u i s h i n g f e a t u r e o f m o n e t a r y e x c h a n g e . 61 In Chapter 2 I discussed in some detail Starr's paper [64], where the presence of a medium of exchange overcomes possible ineff iciencies which result from the absence of double coincidence of wants. Ostroy [50], in a similar paper, claims that the presence of "money" in a decentralized economy is capable of improving the efficiency of the trading process. In Ostroy's model, trade occurs as the result of a sequence of simultaneous encounters, between pairs of agents. The trading decision of each pair of agents must be based only upon the agents^ knowledge of the state of the economy, i .e . onthe prevai l -ing equilibrium prices and the agents' tastes, endowments and trading histor ies. Ostroy's measure of the efficiency of a trading process is the number of simultaneous bilateral meetings required to move an economy from a state of zero aggregate excess demands to a state of zero individual excess demands. Every competitive trading process requires some mechanism to ensure that each agent lives within his budget. Budget balance can be ensured under decentralized trade i f , for every agent, the value of goods given up equals the 62 value of goods received at each bilateral encounter. However, this rather stringent restr ict ion on trade, called bilateral balance by Ostroy, may confl ict with the desire for an eff ic ient trading process. A more accomodating method of ensuring budget balance involves the use of "money." Suppose an account is maintained for each agent and that a l l violations of bilateral balance are recorded. Whenever the value of goods that is exchanged by a pair of agents is unequal, one agent's account would be credited while the other's would be debited. At the end of trading, budget balance will have been achieved i f each agent's account is nonnegative. Ostroy cal ls this record keeping device "money" and demonstrates the efficiency of the monetary exchange process. Other papers which use a similar research strategy, although in a different context, are those by Starett (an asset called "money" permits ef f ic ient intertemporal a l loca-tion of resources [66]), Feldman (rotating sequences of b i -lateral trade moves lead to a Pareto optimal allocation i f "money" is present [21]) and Ostroy and Starr (the presence of a medium of exchange reduces the information required to coordinate exchange and therefore permits eff ic ient decen-tral izat ion of the trading process [51]). C_. While these authors succeed in establishing the benefits of monetary exchange, given that money has a role 63 to play, they do not establish the superior efficiency of monetary versus barter exchange. In Chapter 2 and 3 I argued that the mere absence of money was not the proper characteri-zation of a barter economy. I claimed that there is a struc-tural difference in the organization of trade between a barter and a monetary economy, based on the presence of division of labour in the lat ter 's transactions technology. Therrole of money and the benefits of monetary exchange cannot be established just from an analysis of a monetary economy, but must be established in relation to a barter economy. Using this test of the superior efficiency of mone-tary exchange, I shall demonstrate the somewhat start l ing result that a monetary economy need not be more ef f ic ient than its underlying barter economy. The superiority of monetary exchange* depends on the proper assignment of agents in the operation of a monetary economy's transactions tech-nology. If tasks are allotted ineffectively to agents, a monetary economy may be less eff ic ient than its associated barter economy. Later in this chapter I shall construct an example to demonstrate this point. ID. While the cited art ic les deal with the social benefits of "money," they do not provide an adequate explana-tion of the presence of "money." The emergence of "money" in its role as a dominant medium of exchange has been analyzed 64 by Brunner and Metzler [9] and Nagatani [48]. In both models, some existing commodity achieves the status of a universal intermediary in exchange as the result of unconcerted u t i l i t y maximizing behaviour on the part of individuals. Because agents do not know the identity of potential trading partners with certainty, direct exchange involves the expenditure of real resources on search behaviour. Indirect exchange may reduce these research costs i f there exist goods acceptable as intermediaries in exchange. To an individual agent, the acceptability of a particular good as an intermediary in exchange depends on his information about the goods qualit ies and properties and about its acceptability to potential trading partners. Through a gradual process of learning by agents, some favourite intermediary in exchange becomes the dominant medium of exchange in these models. This " individual ist ic" approach, whose roots l ie in the works of Menger [47] and von Mises [67], is in con-trast to the "social" or cooperative approach that I am using in this thesis. While these authors' have concentrated on the presence of the object serving as a medium of exchange as the distinguishing feature of a monetary economy, I am concentrating on the structural difference between a monetary and barter economy. 65 E_. The cooperative game approach that I am using in this thesis does not require the usual assumption, that money has positive exchange value, in the existence proof of a monetary equilibrium. Usually, while the demand for any other good is based on the u t i l i t y agents derive from its consumption, or on its use as a productive agent, the demand for money is based on its objective exchange value (von Mises [67]; Kurz [41]; Nagatani [48]). Agents will use and hold money only i f i t has positive exchange value, that i s , only i f they believe that other agents will accept i t in exchange for more desirable goods. Unfortunately, the possibi l i ty exists that the equilibrium price of money is not positive (Hahn [24]; Kurz [41]; Starr [65]). When this happens i t must be concluded that no trade takes place in the economy, since the demand for money is zero and the use of money is necessary for trade. This problem has been circumvented by Starr [65], who shows that "suff iciently exacting" taxes payable in money will ensure the existence of equi l ibr ia with a positive price of money. Starr's approach, which is based on a suggestion by Lerner [45], uses taxes "to create a demand for money independent of i ts usefulness as a medium of exchange" [65, p: 46]. However, the imposition of taxes upon a pure exchange model, to ensure money's use, appears somewhat ad hoc. Using the cooperative game approach, I have shown that money's 66 u s e f u l n e s s i s r e l a t e d t o t h e s t r u c t u r e o f an economy's t e c h -n o l o g y o f e x c h a n g e . T h e r e f o r e , t o e x h i b i t a m o n e t a r y e q u i l i b r i u m , I o n l y need t o show t h a t an e q u i l i b r i u m e x i s t s f o r t h e economy o f C h a p t e r 3. T h i s i s done i n A p p e n d i x B. F_. The c h o i c e s t h a t a r e a v a i l a b l e t o an a g e n t i n S t a r r ' s model a r e v e r y l i m i t e d . An a g e n t must e i t h e r consume h i s i n i t i a l endowment, o r e l s e , he must use t h e m o n e t a r y e x c h a n g e p r o c e s s t o o b t a i n more d e s i r a b l e goods a n d / o r t o o b t a i n money f o r t a x e s . S t a r r u s e s t a x e s p a y a b l e i n money t o f o r c e p a r t i c i p a t i o n i n t h e m o n e t a r y economy. He does n o t g i v e an a g e n t an a l t e r n a t i v e t o m o n e t a r y e x c h a n g e . I t i s my b e l i e f t h a t t h e use o f t h e m o n e t a r y e x -change p r o c e s s s h o u l d n o t be a c o n s t r a i n t i m p o s e d on t h e t r a d i n g b e h a v i o u r o f a g e n t s . R a t h e r , i t s h o u l d r e s u l t f r o m t h e i r m a x i m i z i n g b e h a v i o u r . I t i s o b v i o u s t h a t an a g e n t , who b r e a k s away f r o m t h e m o n e t a r y economy by h i m s e l f , has no c h o i c e b u t t o consume h i s i n i t i a l endowment o f g o o d s , b e c a u s e he w i l l have no t r a d i n g p a r t n e r s . T h e r e f o r e , t o p r o v i d e an a l t e r n a t i v e t o m o n e t a r y e x c h a n g e , i t i s n e c e s s a r y t o c o n s i d e r t h e p o s s i b i l i t y t h a t some g r o u p s o f a g e n t s w i l l b r e a k away f r o m t h e m o n e t a r y economy and w i l l use an a l t e r n a t i v e method t o e x c h a n g e goods w i t h i n t h e g r o u p . T h i s p o s s i b i l i t y can be a n a l y z e d w i t h i n 67 t h e f r a m e w o r k o f t h i s t h e s i s b e c a u s e t h e c o o p e r a t i v e a p p r o a c h a d m i t s g r o u p r a t i o n a l i t y a l o n g w i t h i n d i v i d u a l r a t i o n a l i t y . In C h a p t e r 3 I assumed t h a t e a c h c o a l i t i o n ' s t r a n s a c t i o n t e c h n o l o g y c o n s i s t e d o f t h e most e f f i c i e n t s u b s e t o f i t s members' t r a n s a c t i o n t e c h n o l o g i e s . L a t e r i n t h i s c h a p t e r I s h a l l d e v i s e a way o f c o n s t r u c t i n g t h i s e f f i c i e n t t r a n s a c t i o n t e c h n o l o g y f r o m members' t r a n s a c t i o n t e c h n o l o g i e s . A c o a l i t i o n who i s g i v e n t h e c h o i c e between b a r t e r e x c h a n g e and t h e use o f t h i s e f f i c i e n t t e c h -n o l o g y w i l l c l e a r l y c h o o s e t h e l a t t e r . However, i n g e n e r a l i t i s p o s s i b l e t h a t some c o a l i t i o n s a r e n o t v e r y s k i l l e d a t a s s i g n i n g e x c h a n g e t a s k s . The n e x t s e c t i o n d e s c r i b e s an economy i n w h i c h a g e n t s a r e n o t g i v e n e x c h a n g e t a s k s a c c o r d i n g t o t h e i r a b i l i t i e s . C o n s i d e r f o r e x a m p l e a t r a d i t i o n a l s o c i e t y where t h e e l d e s t son a l w a y s t a k e s up h i s f a t h e r ' s t r a d e . The p o i n t I a m ' t r y i n g t o make i s t h a t f o r some c o a l i t i o n s , b a r t e r e x c h a n g e m i g h t be more e f f i c i e n t t h a n m o n e t a r y e x c h a n g e . In o t h e r w o r d s , t h e s e t o f f e a s i b l e a l l o c a t i o n s a t t a i n a b l e t h r o u g h b a r t e r i s l a r g e r t h a n t h e s e t o f f e a s i b l e a l l o c a t i o n s t h r o u g h m o n e t a r y e x c h a n g e , a t t h e g i v e n a s s i g n m e n t o f e x c h a n g e t a s k s t o a g e n t s . However, I s h a l l p r o v e i n s e c t i o n I b e l o w t h a t f o r mone-t a r y e x c h a n g e t o d o m i n a t e b a r t e r e x c h a n g e , i t i s s u f f i c i e n t t h a t f o r t h e c o a l i t i o n c o n s i s t i n g o f t h e e n t i r e economy, t h e s e t o f f e a s i b l e a l l o c a t i o n s a t t a i n a b l e t h r o u g h m o n e t a r y e x c h a n g e c o n t a i n t h e s e t o f f e a s i b l e a l l o c a t i o n s a t t a i n a b l e t h r o u g h b a r t e r . Mone-t a r y e x c h a n g e w i l l d o m i n a t e b a r t e r e x c h a n g e even t h o u g h t h e r e a r e s m a l l e r c o a l i t i o n s f o r whom b a r t e r e x c h a n g e i s more e f f i c i e n t t h a n m o n e t a r y e x c h a n g e . 68 G i . In t h i s s e c t i o n I s h a l l d e v i s e an e x a m p l e t o i l l u s t r a t e t h e a d v a n t a g e s and d i s a d v a n t a g e s o f m o n e t a r y e x c h a n g e and t h e d e r i v a t i o n o f a g g r e g a t e t r a n s a c t i o n s t e c h n o l o g i e s . C o n s i d e r an economy w i t h a f i n i t e number o f a g e n t s i n w h c i h t r a n s a c t i o n c o s t s a r e l i n e a r i n amounts e x c h a n g e d a n d , f o l l o w i n g N i e h a n s [ 4 9 ] , c o n s i s t s i m p l y i n a s h r i n k a g e by some p e r c e n t a g e i n t h e amount o f a good t h a t i s e x c h a n g e d . The t r a n s a c t i o n c o s t can be i n t e r p r e t e d as t h e c o s t i n c u r r e d i n t r a n s p o r t i n g t h e good t o o r f r o m t h e m a r k e t p l a c e , w i t h d i f f e r e n t a g e n t s h a v i n g d i f f e r i n g t r a n s p o r t a b i l i t i e s . I f i n some t r a d i n g p a t t e r n , a g e n t a e x c h a n g e s t h e v e c t o r y ( a ) o f goods f o r x ( a ) , the;:eomponents z ( a ) o f r e s o u r c e c o s t s t h a t a r e i n c u r r e d c a n be g i v e n by z.{a) = z\(a)(x...(a) + y ^ a ) ) , j = l , and a g e n t a's t r a n s a c t i o n s e t S ( a ) i s g i v e n by S(a) = « x ( a ) , y ( a ) , z ( a ) | x ( a ) , y ( a ) e z ( a ) z ( a ) x ( a ) + y ( a ) F o r any good j e i t h e r x - ( a ) o r y ^ ( a ) w i l l be z e r o , d e p e n d i n g on w h e t h e r good j i s r e c e i v e d o r g i v e n up i n e x c h a n g e by a g e n t a. The v e c t o r z o f r e s o u r c e c o s t s w h i c h a r e i n c u r r e d by t h e e n t i r e s o c i e t y i n t h e b a r t e r e x c h a n g e p r o c e s s i s t h e sum o f t h e i n d i v i d u a l a g e n t s ' r e s o u r c e c o s t s . 69 i . e . z = I z(a), aeA where z, = I z.(a) x,(a) + I z.(a) y,(a) J aeA J J aeA J J This trading process is a barter exchange process because each agent must transport his own goods. Suppose that agents are now allowed to s p e c i a l i z e in the transportation of certain goods. To capture this idea, for each good j select a(j) e A such that z.(a(j)) = min z.(a). This i s , agent a(j) is the most e f f i c i e n t of aeA J a l l agents in transporting good j . If for each j , agent a(j) is a l l o t t e d the task of transporting good j , the economy's transactions technology is exploiting the d i v i s i o n of labour and therefore i t is of the type described in Chapter 3. The jth component of the total resource cost vector for an arbitrary exchange pattern x, y is given by J aeA J a(j) x.(a) + yj(a) Clearly, this monetary exchange process is more e f f i c i e n t than the barter process for any trading pattern. On the other hand, suppose that tasks are a l l o t t e d by selecting a ' ( j ) e A such that z.(a'(j)) = max z.(a). J aeA J 70 T h a t i s , a g e n t a ' ( j ) i s t h e l e a s t e f f i c i e n t o f a l l a g e n t s i n t r a n s p o r t i n g good j . The economy where a g e n t a 1 ( j ) i s a l l o t t e d t h e t a s k o f t r a n s p o r t i n g good j i s a l s o a l e g i t i m a t e m o n e t a r y economy, h o w e v e r , i t s e x c h a n g e p r o c e s s i s c l e a r l y l e s s e f f i c i e n t t h a n t h e b a r t e r e x c h a n g e p r o c e s s . T h e r e f o r e , i t i s p o s s i b l e t h a t s o c i e t y does n o t b e n e f i t f r o m m o n e t a r y e x c h a n g e . H_. I now s h a l l c o n s i d e r t h e p r o b l e m o f d e r i v i n g t r a n s -a c t i o n t e c h n o l o g i e s f r o m a g e n t s ' t r a n s a c t i o n a b i l i t i e s . To b e g i n , c o n s i d e r a b a r t e r economy o f t h e t y p e d e s c r i b e d in. C h a p t e r 2 g i v e n by H B = C ( A , fi, v ) , X, 5, S, O J ] . R e c a l l t h a t t h e t r a n s a c t i o n s s e t S ( a ) o f a g e n t a c o n s i s t s o f a l l f e a s i b l e c o m b i n a t i o n s o f e x c h a n g e s and a t t e n d a n t r e s o u r c e c o s t s . An e x c h a n g e o f t h e b u n d l e o f goods y ( a ) f o r t h e b u n d l e x ( a ) i s t e c h n o l o g i c a l l y f e a s i b l e i f t h e r e e x i s t s a v e c t o r z ( a ) o f r e s o u r c e c o s t s s u c h t h a t ( x ( a ) , . y ( a ) , z ( a ) e S ( a ) . I f to(a) r e p r e s e n t s a g e n t a's i n i t i a l endowment, t h e r e s u l t i n g c o n s u m p t i o n b u n d l e i s g i v e n by f ( a ) = w.(a) + x ( a ) - y ( a ) - z ( a ) . 71 It is important to note that y(a) is the vector of quantities that is actually given up in exchange to other agents, while x(a) is the vector of quantities actually received from other agents. The transaction cost vector z(a) consists of goods from agent a's in i t i a l endowment and/or goods obtained from others during the process of exchange. Therefore any vector z(a) of resource costs can be decomposed into two components , z(a) = z i (a ) + z2 (a), where Zi ( a ) consists of goods obtained during the process of exchange and z 2 (a ) consists of goods obtained from a's in i t i a l endowment. This distinction was not necessary i n the"~d i scuss ion barter economy because each agent had to bear directly any resource costs which were incurred during the trading process. However, i f agent a's transaction ab i l i t ies are employed in the operation of an aggregate transactions technology, both Zi ( a ) and z 2 (a ) must be obtained from other agents. Now define x(a) = x(a) - £i (a) , y(a) = y(a) + z 2 (a ) . 72 Then, i f a coalit ion uses the transaction ab i l i t i es S(a) of agent a, i t must obtain y(a) of goods to deliver x(a) of goods. Therefore, the transaction ab i l i t ies of agent a as perceived by a coalit ion can be given by S'(a) x ( a ) , - y(a) |x(a) = x(a) - z i ( a ) , y(a) = y(a) + z 2 ( a ) , where x ( a ) , y ( a ) , z(a) = z i ( a ) + z 2 ( a ) e S(a) From the properties of the correspondence S, i t is easy to show that the correspondence S ' , which maps elements of A 0 0 into subsets of R x R , has the following properties. i) S 1 (a) is closed for a l l a e A. i i ) (x(a), -y(a)) e S'(a) and x'(a) < x(a), y 1 (a) > y(a) then (x'(a), - y'(a)) e S'(a) . i i i ) 0 e S 1 (a) for al l a e A. iv) S' is a v-measurable correspondence. I shall now use the correspondence S1 to construct aggregate transactions technologies for each coalit ion E e ft, * £ Z Define the correspondence T : ft -> R x R by 73 "k 0 T (E) ='{(x ,-y) | x,y e R+ and such that there exist v-integrable functions x ' , y ' : A -> R+ with (x'(a), - y'(a)) e S'(a) for a l l a e E and x = x 1 (• )dv; y = E y 1(-)dv} E T (E) is the integral of the correspondence S' with respect to the measure v over the set E. Using the notation of Chapter 3, i t can also be written as - T * ( E ) r L\,dv, E 3 k The set T (E) incorporates al l possible ways of organizing coalit ion E's transactions technology by al lott ing i ts member agents to various tasks. Since v is an atomless •k f in i te measure, T (E ) is convex. Furthermore, from the properties of S ' , i t is possible to show that the correspon-dence T sat isf ies properties i) through v) of Chapter 3 for coalit ion transactions technologies. If for each E e fi, T ( E ) is bounded by the total quantity of resources i n i t i a l l y * available to coalit ion E, then T is also a compact corre-spondence and therefore also sat isf ies property v i ) . Thus the economy given by [(A, fi, v ) , X, § ,T , oo] 74 is a legitimate monetary economy whose underlying barter economy is E B . Let T be any correspondence from ft to R x R with T(E) <s- T (E) for al l E e ft and that sat isf ies the conditions of Chapter 3 . Then the economy given by M 5 = [ ( A , ft, v ) , X , =, ' .T, oo] can also be interpreted as a monetary economy whose under-lying barter economy is E . However, the agents who operate the aggregate transactions technology T are not as e f f ic -iently specialized as those that operate T*. I_. Recall that the aggregate transactions set of the 2 0 monetary economy is a subset of the space R . While the transaction ab i l i t i es of an individual agent can be repre-2 0 sented in the space R , the barter economy's transactions technology cannot. However, i t is possible to derive an implicit aggregate transactions set for the barter economy 2 0 in R . This implicit transactions set can then be compared with the monetary economy's transactions set. p For al l E e ft, define the correspondence T : ft + R^  x R£ by 7.5 T B ( E ) = { ( x , - y ) | x , y e R* and s u c h t h a t t h e r e e x i s t s f e K (E) o f 5 B 03 w i t h [ f - w ] + dv; y [f-u]"dv} / \ M L e t f e K (E) o f E . Then t h e r e e x i s t v - i n t e g r a b l e f u n c t i o n s x', y', z ' = z[ + z\ : A ->• s u c h t h a t f o r a l l a e A x'(a), y ' ( a ) , z ' ( a ) . e S ( a ) and f(a) = 03(a) + x ' ( a ) - y ' ( a ) - z'(a) S u b s t i t u t i n g z ' ( a ) = z j ( a ) + z ^ ( a ) as d e f i n e d above i n t o t h e l a s t e q u a t i o n and r e a r r a n g i n g we g e t f(a) - 03(a) x'(a) - z ; ( a ) + y ' ( a ) + z ^ ( a ) But t h e n [ f ( a ) - 0 3 ( a ) ] + = x'(a) - zJU) , [f(a) - 03(a)]- = y ' ( a ) + z ' ( a ) i m p l i e s , s i n c e ( x ' ( a ) - z j ( a ) , - ( y ' ( a ) + z 2 ( a ) ) ) e S ' ( a ) , t h a t T B ( E ) c T * ( E ) 76 Now I shall demonstrate the condition which ensures that a monetary economy will dominate its underlying barter economy. Theorem: Consider the monetary economy given by H" = [(A, ft, v ) , X, « , T, u] B and i ts underlying barter economy 5 whose implicit aggregate p p transactions technology is given by T . If T (A) c T(A), M B then the monetary economy H dominates the barter economy 5 . p Proof: Let f be a core allocation of 5 . Then since TB(A) c T(A), f is also a state of the economy » M . Let f* M be any core allocation of H . I claim that f(a) « f*(a) a M for a.e. agents in the economy H . Otherwise f would be a blocking allocation for some E e ft contradicting the choice M of f*. Thus any agent who is given a choice between 5 and B M H will choose H . Furthermore, i t will not be to the advantage of any group of agents to break away from the monetary economy. P If g is an attainable allocation for any E e ft in 5 , with v(E) > 0, then by definit ion of f, g(a) cc f(a) for a.e. ~ a agents in E. By the t ransi t iv i ty of preferences, i t is also true that g(a) f*(a) for a.e. agent in E. ~ a Q.E.D. 77 I t i s i m p o r t a n t t o n o t e t h a t t h e c o n d i t i o n T (A) c T ( A ) a p p l i e s o n l y t o t h e c o a l i t i o n o f t h e e n t i r e economy. No r e s t r i c t i o n i s r e q u i r e d on t h e a g g r e g a t e t r a n s a c t i o n s e t s o f s m a l l e r c o a l i t i o n s . In o t h e r w o r d s , t h e t h e o r e m h o l d s even t h o u g h b a r t e r e x c h a n g e i s "more e f f i c i e n t " t h a n m o n e t a r y e x c h a n g e f o r some o f t h e a d m i s s i b l e c o a l i t i o n s i n t h e economy. T h a t i s , t h e r e m i g h t be c o a l i t i o n s E e Q, E f A, su c h t h a t T ( E ) c T B ( E ) , and m o n e t a r y e x c h a n g e w i l l s t i l l d o m i n a t e b a r t e r e x c h a n g e . FOOTNOTES 'I assume that no real resource costs are incurred in the hypothetical formation of a coal i t ion , hypothetical reallocation of goods within the coalit ion and dissolution of the coalit ion in the cooperative economy. The analogous assumption in the competitive version of the economy is that no real resource costs are incurred in the determination and dissemination of the equilibrium vector of prices. A more complete analysis would consider the structure of these inst i tut ions, the costs incurred in their operation and the efficiency of one institution relative to another. The papers by Feldman [21], Howitt [36] and Ostroy [50] are attempts to analyze the role of money in the operation of these inst i tut ions. The purpose of this thesis, however, is to show that the usefulness of money depends on the structure of an economy's transaction technology. To achieve this result , i t is suff icient to consider only the transaction costs resulting from the transportation of goods from one agent to another. See AppehdiixnAifor definitions of mathematical concepts unusual to economics. Generally, it is assumed that A is a f in i te set. For some proofs, however, i t is necessary to assume that the measure space is non-atomic. In this case, A must be of the cardinality of the continuum. nThe local nonsatistion assumption on <* is weaker a than the usual assumption that «• is monotonic, i .e . s , t , e X(A) a with s < t implies s °c t. Montonic preferences are assumed a in Chapter 5 to ensure that a quasi-competitive price equilibrium is also a competitive price equilibrium. 78 79 3In Kurz's [41] barter economy, the "market" provides the proof of resources required to effect exchanges. These resources can be contracted by individuals for the purpose of carrying out their exchanges. An agent does not bear directly the resource costs incurred in effecting exchanges, as is the case in my barter economy. Consequently, the barter economy of this chapter is more "primitive" in the degree of commercial development than that of Kurz. From condition 6 i i ) , y(a) and z(a) can always be chosen so that xi(a) > 0 implies yi(a) = 0. That i s , an agent need not buy and sell a good at the same time. Therefore, f(a) e X(a) c R + =* f ( a ) = w(a) + x(a) - y(a) - z(a) > 0 •=» co(a) + x(a) > y(a) + z(a) =*• to(a ) 5; y (a ), and co(a) + x(a) > z(a). The inequality oj(a) = y(a) states that an agent cannot sell more than his in i t i a l endowment, while the inequality co(a) + x(a) = z(a) states that the resource costs incurred by an agent in effecting an exchange cannot exceed his in i t i a l endowment plus the quantities of goods acquired in exchange. The aggregate transaction technology described in this chapter differs from Foley's [22] in that he combines both production and exchange ac t iv i t i es . It differs from Kurz's [41] in that he uses separate "buying" and "sel l ing" technologies linked by a medium of exchange. Furthermore, to my knowledge, aggregate transaction technologies have never been studied either in a core theoretic or a measure theoretical context. °Sondermann [62] has obtained "stable" profit distributions in the case certain productive factors show increasing returns to scale for coalit ion production economies. To incorporate increasing returns to scale intfche context of coalit ion transaction economies, we would have to let T be superadditive on Q. That i s , for every pair of disjoint coalit ion Ei and E 2 , T(Ex) + T(E 2) c T (Ei u E 2 ) . 80 If A is a f in i te set, then T * ( E ) = I S(a) aeA where Z indicates the set theoretic sum. Toeensure T* is convex, we must also assume that S(a) is convex for each a e A . Kurz [41] and [43] has investigated the existence of an equilibrium in barter and monetary economies under the more reasonable assumption: co( • )dv >> 0. A BIBLIOGRAPHY [1] K.J. Arrow and G. Debreu, "Existence of an equilibrium for a competitive economy," Eoonometrioa (22), July , 1954. [2] K.J. Arrow and F.H. Hahn, General Competitive Analysis, Holden-Day, San Francisco, 1971. [3] Z. Artstein, "Set-valued measures," Transactions of the American Mathematical Society (165), March, 1972. [4] R.J. Aumann, "Markets with a continuum of traders," Eoonometrioa (32), January-April , 1964. [5] R.J. Aumann, "Integrals of set-valued functions," Journal of Mathematical Analysis and A p p l i c a t i o n s (12), August, 1965. [6] R.J. Aumann, "Existence of competitive equilibra in markets with a continuum of t raders , "'Eoonometrioa (34), January, 1966. [7] T .F. Bewley, "The equality of the core and the set of equilibra in economies with in f in i te ly many com-modities and a continuum of agents," I n t e r n a t i o n a l Economic Review (14), June, 1973. [8] P. B i l l i n g s l y , Convergence of P r o b a b i l i t y Measures, John Wiley, New York, 1968. [9] K. Brunner and A. Metzler, "The uses of money: money in the theory of an exchange economy," American Economic Review (61), December, 1971. [TO] R.W. Clower, "A reconsideration of the microfoundations of monetary theory," Western Economic Journal (5), December, 1967. 81 82 [11] R.W. Clower, ed . , "Introduction," Monetary Theory, Penguin Books L t d . , England, 1969. [12] R.R. Cornwall, "The use of prices to characterize the core of an economy," Journal of Eoo,nomio Theory (1), December, 1 969 . [13] G. Debreu, Theory of Value, Yale University Press, New York, 1959. [14] G. Debreu, "New concepts and techniques for equilibrium analysis," International Economic Review (3), September, 1962. [15] G. Debreu and H. Scarf, "A l imit theorem on the core of an economy," International Economic Review (4), September, 19631 [16] G. Debreu, "Preference functions on measure spaces of economic agents," Econometrica (35), January, 1967. [17] G. Debreu, " I ntegrati on of ^cor»=espo ndenees "PProceedings of the F i f t h Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, 1967. [18] G. Debreu and D. Schmeidler, "The Radon-Nikodym deriva-tive of a correspondence," Proceedings of the Sixth Berkeley Symposium on Mathematical S t a t i s t i c s and Probability, University of California Press, 1 972. [19] F.Y. Edgeworth, Mathematical Psychics, Paul Kegan, London, 1881. [20] P. Einzig, Primitive Money, 2nd ed . , Pergamon Press, Oxford, 1966. [21] A.M. Feldman, "Bilateral trading processes, pairwise optimality, and Pareto optimality," Review of Economic Studies. (40), October, 1 973. [22] D..K. Foley, "Economic equilibrium with costly marketing," Journal of Economic Theory (2), September, 1970. 83 [23] J . Gabszewicz and J . Mertens, "An equivalence theorem for the core of an economy whose atoms are not 'too' big," Eoonometrioa (39), September, 1971. [24] F.H. Hahn, "On some problems of proving the existence of equilibrium in a money economy," in Hahn and F. Brechling, eds. , The Theory of Interest Rates, London, 1967. [25] F.H. Hahn, "Equilibrium with transaction costs," Eoonometrioa (39), May, 1971. [26] F.H. Hahn, "On transaction costs, inessential sequence economies and money," Review of Economic Studies (40), October, 1973. [27] J . C . Harsanyi , "A general theory of rational behaviour in game situations," Econometrica (34), July , 1 966. [28] W.P. Heller, "Transactions with set-up costs," Journal of Economic Theory (4), 1972. [29] J . Hicks, Value and Capital, Oxford University Press, London, 1939. [30] J . Hicks, Capital and Growth, Oxford University Press, London, 1965. [31] W. Hildenbrand, "The core of an economy with a measure space of economic agents," Review of Economic Studies (35), October, 1968. [32] W. Hildenbrand, "Pareto optimality for a measure, space of economic agents," International Economic Review (10), October, 1969. [33] W. Hildenbrand, "Existence of equilibra with production and a measure space of consumers," Econometrica (38), September, 1970. [34] W. Hildenbrand, "Metric measure spaces of economic agents," Proceedings of the Sixth Berkeley Sympo-sium on Mathematical S t a t i s t i c a and P r o b a b i l i t y , University of California Press, 1972. 84 [35] J . Hirshle i fer , "Exchange theory: the missing chapter," Western Economic Journal (11), June, 1973. [36] P. Howitt, "Walras and monetary theory," Western Economic Journal (11), December, 1973. [37] W.S. Jevons, Money and the Mechanism of Exchange, Kegan Paul, Trench, Trubner & Co. , Ltd.* London, 1899. [38] Y. Kannai, "Continuity properties of the core of a market," Econometrica^ (38), November, 1970. [39] J.M. Keynes, A Treatise on Money, The Pure Theory of Money, Volume I, Macmillan, London, 1930. [40] G.E. Knapp, The State Theory of Money, trans. H.M Lucas and J . Bonar, Macmillan and Co. , London, 1924. [41] M. Kurz, "Equilibrium with transaction cost and money in a single market exchange economy," Journal of Economic Theory (7), A p r i l , 1974. [42] M. Kurz, "Equilibrium in a f in i te sequence of markets with transaction cost," Econometrica (42), January, 1974. [43] M. Kurz, "Arrow-Debreu equilibrium of an exchange economy with transaction cost," International Economic Review (15), October, 1974. [44] M. Kurz and R. Wilson, "On the structure of trade," Economic Inquiry (12), December, 1974. [45] A.P. Lerner, "Money as a creature of the state," Proceedings of the American Economic Association (37), May, 1947. [46] L.W. McKenzie, "On the existence of general equilibrium for a competitive market," Econometrica (20), 1 959. [47] C. Menger, "On the origin of money," Economic Journal (2), June, 1892. \ 85 [48] K. Nagatani, A Survey of Theories of Money, mimeo-graphed lecture notes in Monetary Theory, U . B . C , 1972 and 1974. [49] J . Niehans, "Money and barter in general equilibrium with transaction costs," American Economic Review (61), December, 1971 . [50] J.M. Ostroy, "The informational efficiency of monetary exchange,"'American Economic Review (63), September, 1973. [51] J.M. Ostroy and R.M. Starr, "Money and the decentrali-zation of exchange," Econometrica (42), November, 1 974. [52] R. Radner, "Existence of equilibrium of plans, prices, and price expectations in a sequence of markets," Econometrica (40), March, 1972. [53] M.K. Ri chter, "Coal i tions.,. core , and competition," Journal of Economic Theory (3), September, 1971. [54] H. Scarf, "The core of an N person game," E c o n o m e t r i c a (35), January, 1967. [55] D. Schmeidler, "Competitive equilibrium in markets with a continuum of traders and incomplete preferences," Econometrica (37), October, 1969. [56] D. Schmeidler, "Fatou's lemma in several dimensions," Proceedings of the American Mathematical Society [24],-January-April , 1 970. [57] A. Schotter, "Core allocations and competitive equilibrium - a survey," Z e i t s c h r i f t fur Nalionalo-konomie (33), December, 1973. [58] L.S. Shapley and M. Shubik, "Pure competition, coa l i -tional power and fa i r d iv is ion ," International Economic Review (10), October, 1969. [59] B. Shitovitz , "Oligopoly in market with a coninuum of traders," Econometrica (41) , May, 1 973. 86 [60] M. Sion, Introduction to the Methods of Real Analysis, Holt, Rinehart and Winston, Inc., New York, 1968. [61] A. Smith, The Wealth of Nations, Modern Library, New York, 1973. [62] D. Sonderman, "Economies of scale and equilibra in coalit ion production economies," Journal of Economic Theory (8), July, 1974. [63] R.M. Starr, "Quasi.i=equi fin bra'-in marketsewith non-convex preferences," Eoonometrioa (37), January, 1969. [64] R.M. Starr, "Exchange in barter and monetary economies," Quarterly Journal of Economics (86), May, 1972. [65] R.M. Starr, "The price of money in a pure exchange monetary economy with taxation," Eoonometrioa (42), January, 1974. [66] D. Starrett , "Inefficiency and the demand for 'money' in a sequence economy," Review of Economic Studies (40), October, 1973. [67] L. von Mises, The Theory of Money and Credit, trans. H.E. Batson, Jonathan Cape, London, 1934. [68] J . von Neumann and 0. Morgenstern, Theory of Games and Economic Behaviour, John Wiley & Sons, 1964. [69] E. Veendorp, "General equilibrium theory for a barter economy," Western Economic Journal (8), March, 1970. [70] K. Vind, "Edgeworth-al1ocations in an exchange economy with many traders," International Economic Reviw (5), May, 1964. [71] A . J . Weir, General Integration and Measure, Cambridge University Press, London, 1974. APPENDIX A MATHEMATICAL CONCEPTS The fol lowing mathematical concepts are unusual to economic theory and so the i r d e f i n i t i o n s are gathered here. T;. Measure, Space : A measure space is a tripcle ( A , ft, v ) where A i s a se t , ft is a a - f i e l d of subsets in A, and v is a count-ably a d d i t i v e , non-negative set funct ion on ft with v ( A ) .= 1 2. a - f i e l d : ft is a o - f i e l d in A i f for every countable sequence {E n> of s u b s e t s .E n e ft, u E e ft, and n n E i - E 2 e ft 87 88 Furthermore, u E = A . Eefi Measurable set A set is ca l led measurable i f i t is an element of fi. 4. Measure R~ is ca l led a measure i f i t A funct ion v : fi is countably addi t ive on fi. That i s , for any countable sequence { E n > of d i s j o i n t sets in fi, EE 5 . Almost every a e A (a .e . a e A ) : A re la t ion is said to hold for almost every element of A (a .e . a e A ) i f the set of those elements for which the re la t ion is not true has measure zero. That i s , i f E e fi is the set for which the re la t ion is not t rue , then v ( E ) = 0 6 . Measurable funct ion: A function f : A -»- R i s ca l l ed measurable i f for every in terva l a c R, f - 1 [ a ] e fi. f is sometimes ca l led v -measurable or fi^measurable. A vector-valued function f : A R is called measurable i f each component f 1 is measurable. 7. v-integrable function: A function f : A -* R is called v-integrable i f i t is v-measurable and i f the Lebesque-Stieltjes integral of f with respect to v over A, denoted by, f ( - )dv, A exists. 0 A vector-valued function f : A -* R is called v-integrable i f each component f'1 is v-integrable. 8. Lebesque-Stieltjes integral and properties: See Si on [60]. 9. Correspondence: 0 A correspondence ip from A to R associates with every element a of A a subset ifi(a) of R . Its graph is (a,r) e A x R^  | r e. T/J (a) 90 10. Inverse of a correspondence: The inverse ty'1 of the correspondence is defined 0 as fo l lows: le t r be a family of subsets of R , then i p - H f ] = -la e A f I|J (a) e r 11 . Strong- inverse of a correspondence: p If X is a subset of R , then the st rong- inverse tys of the correspondence ty is given by 4>S[X] = --a e A \tytyU) -CXX 1 2. Upper semicontinuous correspondence: A correspondence ty : A -* R is upper semi conti nuous i f i t s graph is c losed . That i s , for every sequence {a n , rn> in G, with lim (a , r ) = ( a , r ) , then (a,r) e G , . 13. Measurable correspondence: The correspondence ty : A -> R^, where A is part of the measure space (A, fi, v ) , is said to be measurable (v-measurable or fi-measurable) i f for every open set X in R , 4>S[X] e fi. . See Debreu [17] for a l te rnat ive d e f i n i t i o n s . 91 14. Integral of a correspondence: Consider the correspondence ty : A -> R . Let L be the set of al l point-valued f : A -> R£ such that f is v-integrable over A and f(a) e ^(a) for al l a e A . The integral of the correspondence ty over A is defined by: See Aumann [ 5 ] , Debreu [17] and Artstein [ 3 ] for properties of the integral of a correspondence. 15. Non-atomic measure space: The set E e fi is called an atom of the measure space (A, fi, v) i f v(E) > 0 and E D F e fi implies v(F) or v(E-F) = 0. The measure space is called non-atomic i f i t has no atoms. The integral ^(•)dv is also written as L, dv. APPENDIX B THE EXISTENCE OF COMPETITIVE PRICE EQUILIBRIA A. In this appendix I shall provide the proofs that establish the existence of price equi l ibr ia for both the "barter" and "monetary" economies. In view of Theorems 2.1, 2.2, 3.1 and 3.2, I am also establishing the conditions under which the cores of these economies are non-empty. The proofs are based on similar existence proofs by Debreu [13], Aumann [6], Schmeidler [55] and Hildenbrand [33], [34]. The required mathematical tools can be found in Artstein [3], Aumann [5], Debreu [17], Debreu and Schmeidler [18], Schmeidler [56] and Sion [60]. F i rs t , I shall demonstrate the existence of a quasi-competitive price equilibrium for the "barter" economy, under the conditions of Chapter 2, in Theorem 1 and for the "mone-tary" economy, under the conditions of Chapter 3, in Theorem 2. At the end of the appendix, I shall l i s t the additional assump-tions required so each quasi-competitive price equilibrium is also a competitive price equilibrium. 92 C o n s i d e r t h e " b a r t e r " economy = [ ( A , fi, v ) , X, *, S, co] as d e s c r i b e d i n C h a p t e r 2. L e t A = 1 P £ R + 1 I P 1 = 1 = 1 be t h e u n i t p r i c e s i m p l e x . D e f i n e t h e b u d g e t c o r r e s p o n -dence 6 : A x A -»- R + by B ( a , p ) = < ( x , y , z ) e R + 3 £ a) ( x , y , z ) e S ( a ) , b) to(a) + x - y - z e X ( a ) , and c ) p • x < p • y 3 £ and t h e demand c o r r e s p o n d e n c e <f> : A x A R + by c f)(a,p) = < ( x , y , z ) e B(.a,p) | f o r e v e r y ( x ' . y ' . z 1 ) e 3 ( a , u ( a ) + x ' - y ' - z ' 2 f l u ( a ) + x - y - z f F i n a l l y , d e f i n e t h e q u a s i - d e m a n d c o r r e s p o n d e n c e 6:A x A -* by 94 <5(a,p) f ( a , p ) i f i n f . {p • x - p • y } < 0, . ( x , y , z ) e S ( a ) w(a) + x - y - z e X ( a ) B ( a , p ) o t h e r w i s e . Definition I. Q u a s i - c o m p e t i t i v e p r i c e e q u i l i b r i u m o L e t p be a p r i c e v e c t o r i n t h e u n i t s i m p l e x A c R + 0 and f be a v - i n t e g r a b l e f u n c t i o n f r o m A t o R +. The p a i r ( p , f ) i s c a l l e d a q u a s i - c o m p e t i t i v e p r i c e e q u i l i b r i u m o f t h e economy B I ~ i f t h e r e e x i s t v - i n t e g r a b l e f u n c t i o n s x , y , z : A -> R + s u c h t h a t i ) f ( a ) = u)(a) + x ( a ) - y ( a ) - z ( a ) , a.e. a e A , i i ) x ( a ) , y ( a ) , z ( a ) e 6 ( a , p ) , a e A , and i i i ) f ( * ) d v < (•)dv z ( • ) d v , C o n d i t i o n s i ) and i i ) s t a t e t h a t f ( a ) must be maximal w i t h r e s p e c t t o - i n a g e n t a's b u d g e t s e t w h e n e v e r a t h e minimum w e a l t h s i t u a t i o n does n o t o c c u r , w h i l e c o n d i t i o n i i i ) i s t h e m a t e r i a l b a l a n c e e q u a t i o n f o r t h e e n t i r e economy T h i s l a s t r e l a t i o n can a l s o be w r i t t e n as i i i ' ) 'A x ( • ) d v < y ( • ) d v , 95 D e f i n i t i o n 2: C o m p e t i t i v e p r i c e e q u i l i b r i u m T h e p a i r ( p , f ) i s c a l l e d a c o m p e t i t i v e p r i c e p e q u i l i b r i u m o f t h e e c o n o m y E i f i t f o r m s a q u a s i - c o m p e t i t i v e p r i c e e q u i l i b r i u m a n d i f t h e s e t o f a g e n t s f o r whom t h e m i n i m u m w e a l t h s i t u a t i o n o c c u r s h a s m e a s u r e z e r o . P • y ) ^ o • t h e n v ( E * ) = 0. I n o t h e r w o r d s , f o r t h e e q u i l i b r i u m p r i c e v e c t o r p e A " m o s t " a g e n t s h a v e s u f f i c i e n t w e a l t h t o e x c h a n g e s o m e g o o d s a n d s t i l l r e m a i n i n s i d e t h e i r c o n s u m p t i o n s e t . C_. T h e p r o o f o f t h e f o l l o w i n g t h e o r e m i s p a t t e r n e d a f t e r t h e e x i s t e n c e p r o o f s b y H i l d e n b r a n d [33] a n d [34]. T h e m a i n d i f f e r e n c e i s t h a t my m o d e l p o r t r a y s a b a r t e r e c o n o m y w i t h t r a n s a c t i o n c o s t s a n d i n d i v i d u a l s p e c i f i c t r a n s a c t i o n t e c h n o l o g i e s , w h i l e i n [34] H i l d e n b r a n d m o d e l s a p u r e e x c h a n g e e c o n o m y a n d i n [33] h e m o d e l s a c o a l i t i o n p r o d u c t i o n e c o n o m y . My e c o n o m y d i f f e r s f r o m t h e b a r t e r e c o n o m y p o r t r a y e d b y K u r z [43] i n t h e s p e c i f i c a t i o n o f t h e t r a n s a c t i o n t e c h n o l o g i e s ( s e e f o o t n o t e 3) a n d i n i t s m e a s u r e t h e o r e t i c c o n t e x t . i . e . i f a e A | i n f {p ( x , y , z ) e S ( a ) a ) ( a ) + x - y - Z £ X ( a ) 96 In part a) of the proof, to ensure that each agent's budget set is bounded and thus to ensure that his demand set is non-empty, a sequence of "truncated economies" is con-structed. In part b) i t is shown that each truncated economy has a quasi-competitive price equilibrium by showing its total quasi-demand correspondence sat isf ies the properties of Debreu's lemma [13, p. 82]. F ina l ly , in part c) i t is shown the existence of a sequence of quasi-equi1ibria for the sequence of truncated economies implies the existence of a quasi-competitive price equilibrium for the original economy. Theorem 1: If the measure space of agents of the "barter" g economy E is non-atomic, then there exists a quasi-competi-tive price equilibrium. Proof: Part a) In an economy with a continuum of agents, an agent of measure zero has only an infinitesmal portion of the goods of the entire economy. As Aumann [ 6 ] points out, the possibi l i ty exists that for a given price vector p e A the budget set $(a,p) for some agent a e A is unbounded, and hence the demand set cf) (a, p) may be empty. To circumvent this poss ib i l i ty , for every positive integer k consider the truncated consumption set 97 X k ( a ) s e X ( a ) s < k u>(a) + 1 and t h e t r u n c a t e d t r a n s a c t i o n s s e t S ( a ) = < ( x , y , z ) e S ( a ) | ( x , y , z ) < kfto(a) +• 1 , w(a) + 1 , w(a) + 1 D e f i n e t h e k - t h t r u n c a t e d b u d g e t c o r r e s p o n d e n c e , k k 8 , demand c o r r e s p o n d e n c e , <j> , and q u a s i - d e m a n d c o r r e s p o n d e n c e , 6 K , by r e p l a c i n g X(a) and S(a) w i t h X k ( a ) and S k ( a ) i n t h e d e f i n i t i o n s o f s e c t i o n B a b o v e . F i n a l l y , d e f i n e t h e t o t a l q u a s i - d e m a n d c o r r e s p o n d e n c e k & : A -* R f o r t h e " k - t h t r u n c a t e d economy by k f l i> (p) = - s e R | t h e r e e x i s t v - i n t e g r a b l e f u n c t i o n s x , y , z : A -> R + s u c h t h a t x ( a ) , y ( a ) , z ( a ) 6 ( a , p ) f o r a l m o s t e v e r y a £ A and s = x(-) - y ( - ) dv • * • A k 3 2, I f we d e f i n e t h e c o r r e s p o n d e n c e a : A -> R + by 98 a K ( p ) = J <5 K(-,p)dv i . e . a k ( p ) i s t h e i n t e g r a l o f t h e q u a s i - d e m a n d c o r r e s p o n d e n c e i/ w i t h r e s p e c t t o t h e measure v, t h e n ty can a l s o be d e f i n e d by ty (p) = * ( x - y ) | ( x , y , z ) e o (p) P a r t b) I c l a i m t h a t t h e c o r r e s p o n d e n c e ty has t h e f o l l o w i n g p r o p e r t i e s : i ) t h e r e i s a compact s e t N c R £ s u c h k t h a t ty ( p ) c N f o r e v e r y p e A, i i ) t h e g r a p h o f ty^ i s c l o s e d , i i i ) f o r e v e r y p e A, ^ ( p ) i s non-empty and c o n v e x , and i v ) f o r e v e r y p e A, p • ty ( p ) < 0. To p r o v e p r o p e r t y i ) l e t N H s e R s | < k[ (u>( •) + 1 ) d v ] Then by c o n s t r u c t i o n o f ty , f o r e v e r y p e A and v - i n t e g r a b l e n k f u n c t i o n s x , y , z : A -* R + w i t h ( x ( a ) , y ( a ) , z ( a ) ) e 5 ( a , p ) we have 0 | x ( a ) s k ( to(a) + 1), 0 < y ( a ) < k ( to(a) + 1) and t h u s 99 x ( a ) - y ( a ) | < k 00(a) + 1 , a.e. a i n A (1) I n t e g r a t i n g we g e t 'x-(-) - y ( - ) dv < • A (• A x ( - ) - y ( - ) dv < k [ 0) ( a ) + 1 d v ] o r ip ( p ) c N f o r e v e r y p e A, To p r o v e p r o p e r t y i i ) l e t (p ,s ) be a s e q u e n c e i n G, k = 4 ( p , s ) e A x R* I s e / ( p ) r , t h e g r a p h o f ij> , w i t h l i m ( p , s n ) = ( p , s ) . T h a t i s f o r e v e r y n-*-°° p o s i t i v e i n t e g e r n t h e r e e x i s t v - i n t e g r a b l e f u n c t i o n s x , y n , z n : A - R j s u c h t h a t ( x n ( a ) , y n ( a . ) , z p ( a ) ) e <5 k(a,p) and *„<•> - y„ ( . ) d v , l i m f fx (•) - y (•) dv = 1im s From (1) a b o v e , t h e s e q u e n c e o f v - i n t e g r a b l e f u n c t i o n s ' { ( x -y )} 0 f r o m A -* R i s bounded p o i n t w i s e i n a b s o l u t e v a l u e by t h e 1 00 v - i n t e g r a b l e f u n c t i o n k(w + 1 ) : A -* R^. U s i n g Theorem E [ 3 3 * P- 622] (a v e r s i o n o f F a t o u ' s lemma), t h e r e e x i s t s a o v - i n t e g r a b l e f u n c t i o n t : A -* R s u c h t h a t t ( • ) d v = s, and ( 2 ) A t ( a ) e e l o s u r e ••< x n (a)-y n (a) j- f o r a l m o s t e v e r y a e A . ( 3 ) S i n c e e a c h a g e n t i s e i t h e r p a r ' n e t s p u r c h a s e r " o r a n e t s u p p l i e r o f a p a r t i c u l a r g o o d , b u t n o t b o t h , i f we l e t x ( a ) = t ( a y ( a ) = t ( a ) - f o r a l l a e A , t h e n x ( a ) e c l o s u r e ' { x ( a ) } and y ( a ) e c l o s u r e {y ( a ) } . B e c a u s e S k ( a ) i s c o m p a c t , f o r e v e r y a e A t h e r e e x i s t s z ( a ) s u c h t h a t z ( a ) e c l o s u r e ' { z ( a ) } and ( x ( a ) , y ( a ) , z ( a ) ) e s ' " ( a ) . From t h e m e a s u r a b i 1 i t y o f t h e c o r r e s p o n d e n c e S , z ( a ) can be c h o s e n f o r ea c h a e A suc h t h a t t h e f u n c t i o n z : A -> R^ i s v - i n t e g r a b l e (Theorem B [ 3 3 , p. 6 2 1] I a l s o c l a i m t h a t f o r f i x e d a e A , 6 k(a,«) : A -* i s an up p e r s e m i - c o n t i n u o u s c o r r e s p o n d e n c e . F o l l o w i n g S c h m e i d l e r [ 5 5 , p,.' 5 8 2 ] , l e t -Cp > a n d ' { ( x , y , z m ) } be s e q u e n c e s s u c h t h a t l i m p m = p, l i m (x , y , z ) = ( x , y , z ) m-><» m->oo w i t h p m e A, ( x m , y m , z m ) e 6 (a,p m ) . Then we must have P • x < p • y , Km m = Hm Jm ' ( 4 ) 101 ( x m , y m , z m ) e S k ( a ) , and (5) w ( 5 ) + xm " ^ ~ zm £ X k ( g ) ( 6 ) Suppose t h a t i n f ( x . y . z ) e S * ( i ) ( x - y ) /\ S\ y\ CO ( a ) + x - y - z e X * ( a ) k - k -Then s i n c e S ( a ) and X ( a ) a r e c l o s e d and i n e q u a l i t i e s a r e p r e s e r v e d u n d e r l i m i t s , we g e t a f t e r t a k i n g l i m i t s on ( 4 ) , ( 5 ) and (6) t h a t ( x , y , z ) e 3 ( a , p ) (7) On t h e o t h e r h a n d , i f i n f / \ / \ / \ ( x , y , z ) e S k(5) ( x - y ) < 0 , ( a ) + x - y - z e X - ( a ) CO k - k -t h e n s i n c e S ( a ) and X ( a ) a r e c o m p a c t , t h e r e e x i s t s ( x 1 , y ' , z ' ) e g k ( a , p ) s u c h t h a t p • x' < p • y' (8) 102 B u t l i m p = p i m p l i e s t h e r e e x i s t s a p o s i t i v e i n t e g e r m i m - K » s u c h t h a t m > m{' p • x < p • v Hm Hm y ( x ' , y \ z ' ) e B k ( a , p m ) ( , ( 5 ) + x ' - y ' - z ' 5 5 o ) ( 5 ) + x m - y m - z m By c o n t i n u i t y o f ~ Q we g e t a f t e r t a k i n g l i m i t s t h a t to(a) + x ' - y ' - z ' ~- co(a) + x - y - z . (9) a I f we have ( x , y , z ) E 6 k ( a , p ) w i t h p • x = p • y - - - k -t h e n ( x , y , z ) i s t h e l i m i t o f p o i n t s i n 8 ( a , p ) w i t h p r o p e r t i e s (8) and ( 9 ) . A f t e r t a k i n g l i m i t s we g e t co(a) + x - y - z ~- u ( a ) - x - y - z . (10) a B u t s i n c e (7) h o l d s h e r e as w e l l , we have ( x , y , z ) E i j ; ( a , p ) . C o m b i n i n g t h e two c a s e s we g e t ( x , y , z ) E 6 k ( a , p ) and t h e r e f o r e 6 i s u p p e r s e m i - c o n t i n u o u s i n p. 103 Then l i m p n = p, ( x n ( a ) , y n ( a ) , z n ( a ) e 6 ( a , p n ) and ( x ( a ) , y ( a ) , z ( a ) ) e c l o s u r e ' { ( x p ( a ) , Y n ( a ) , z ( a ) ) } i m p l i e s i, by t h e u p p e r s e m i - c o n t i n u i t y o f 6 i n p t h a t ( x ( a ) , y ( a ) , z ( a ) ) e 6 ( a , p ) . T h e r e f o r e , s = x ( - ) - y( - ) dv e ty (p) and c o n s e q u e n t l y ( p , x ) e G^ k, i . e . t h e g r a p h o f ty^ i s c l o s e d . To show t h a t ty ( p ) f 0 f o r e v e r y p e A , i t i s s u f f i c i e n t t o show t h a t 6 ( a , p ) f 0 f o r a l m o s t e v e r y a e A and t h a t t h e c o r r e s p o n d e n c e 6 k ( « , p ) : A -»- R^ i s v - m e a s u r a b l e, k k S i n c e t h e b u d g e t s e t 6 ( a , p ) i s c o m p a c t , to(a) e X ( a ) f o r a l m o s t e v e r y a e A , and S i s c o n t i n u o u s f o r a l l a e A , a <5 ( a , p ) f 0 f o r p e A . The b u d g e t s e t c o r r e s p o n d e n c e can be w r i t t e n as B(a,p) = S ( a ) n p( a) n y( a) where p ( a ) ( x , y , z ) e | u ( a ) + x - y - z e X k ( a ) and R ( a ) = 3 P ( x , y , z ) e R | P • x g p • y (•• We know t h a t S i s a v - m e a s u r a b l e c o r r e s p o n d e n c e . The measur-a b i l i t y o f p f o l l o w s f r o m t h e m e a s u r a b i 1 i t y o f ca and X w h i l e 1 04 y(a) is t r i v i a l l y v-measurable. By Lemma 5.3 of Artstein [3, p. 109], 3(*,p) is also v-measurable and thus clearly so is. B k ( * ,p) . By Proposition 4.5 of Debreu [17, p. 360], for fixed p e A the set M = a e A | inf - P • (x -y) i < 01 (x,y,z)eS k(a)np( a) bel.ohgs to fin? since S ^ p is a v-measurable correspondence and since the function p • (x-y) is both continuous on S (a) n p(a) and is also t r i v i a l l y v-measurable. From Theorem B [33» P- 621], there exists a sequence'{(x n, y , zn)} of v-measurable functions of A into R5, such that'{x (a), y n (a ) , zn(a)} is dense in 3 k(a,p) for every a e A . Defi ne e n(a) = l(x ,y,z) e 3k(a,p) | w(a) + x n(a) - y n ( a ) - z n(a) oj(a) + x-y-zj- for a e M. Clearly, <5k(a,p) c © n (a ) for a e M. Suppose we have (x,y,z) e 00 1/ n © n ( a ) , but (x,y,z) £ 6 (a,p). That i s , there exists n=l n ( x ' . y ' . z 1 ) e 3 k(a,p) such that co(a) + x-y-z < x a co(a) + x ' - y ' - z ' Since is continuous and { (x (a ) , y „ ( a ) , z (a))} is dense ct n n n in 3 k (a,p), there is an integer n such that 105 <d(a) + x-y-z oc 00(a) + x-(a) - y-(a) - z-(a) a n n n Contradicting the fact that (x,y,z) e ©~(a) . Thus 6 (a,p) = n 0_(a) for every a e M. n=l n But since the correspondence 8*(*,p), the function x n , y , z n and the set M are v-measurable, 0 is v-measurable for a l l n positive integers n. By Lemma 5.3 of Artstein [3] we again have that 6 (*,p) is a v-measurable correspondence. Using Theorems 1 and 2 of Aumann [5, p. 2], we get that 6 (p) is non-empty and convex!foraeveryppeeaA. CCTearTy, this implies that I(J (p) is non-empty for every p e A, and i t is easy to show that ^ (p) is also convex for every p e A. F ina l ly , property iv) holds since s e \\> (p) implies 0 the.fceeexist v- integrable functions x,y,z : A ->- R+ such that s = x(•) - y(•) dv, and x(a), y(a), z(a) e 6 (a,p) for a.e. a e A . But since 6 (a,p) c B (a,p) we also have P • x(a) < p • y(a). 1 0 6 Integrating the last inequality we get A p • x(•)dv < p • y(•)dv, or p • s = x(-) - y(-) dv < o Part C We now can apply Debreu's lemma [13, p. 82], which is based on Kakutani1s fixed point theorem,ttotthe .correspon-dence ij> . It states that there is a p e A such that ^ k(p) n R£ f 0 That i s , there exist v-integrable functions x,y,z : A -> such that x(a), y(a), z(a) e 6 (a,p), and (11) A x(-) - y(-) dv < 0 . (12) But condition (!!2) is equivalent to x(• )dv < y(*)dv. (13) and condition (11) implies that 107 x(a), y(a), z(a) e SK(a) c S(a) Therefore, i f we let f = to + x-y-z then the pair (p,f) form a quasi-competitive equilibrium for the "k-th truncated economy." I have shown that for every positive integer k there is a price vector p e A and v-integrable functions k k k % k k x , y , z : A -> R+ such that the pair (p ,f ) form a quasi-equilibrium for the "k-th truncated economy" where f k k k to + x - y y - z . Since A is compact we can'assume without loss of generality that the sequence {p } converges to the price vector p* e A . From the material balance requirement we have for each k that 0 < I f * ( « ) d v < A to A ( • ) d v A z K(-)dv Thus we get immediately that and 0 < 0 < f k ( - )dv < f (o(-)dv, f z k ( . )dv < A to (•)dv. Since an agent cannot sell more than his in i t i a l endowment, we must have, 108 0 < y (a) < oo(a), for every a e A and therefore we must have using (13), 0 < x k(-)dv < [ y k ( - )dv < A " JA A u) (•) dv, Hence, - x k ( - )dv -9 » A J A y K ( - )dvk 4 z k ( - ) d v l and A i - J f k ( « ) d v - | are al l bounded sequences in R £ and by the Bolzano-Weierstrass Theorem each has a convergent subsequence. Without loss of generality, there exist x, yt z, f e R£ such that i m x' 1 k-*-°° (•)dv = x 1 im | < ->oo J y (')dv = y , 1 im k->°° J z K ( . )dv = z , lim A k-*°° J A f K ( - )dv = f . By Schmei.dl er' s E'56] version of Fatou's lemma, there exist v-integrable functions x * , - y * , z * , f* : A R £ such that 1 rx*(a), y*(a) , z*(a), f*(a) is a cluster point of the sequence - ) s y k ( a ) , z k ( a ) 9 • • k ' a i ' ' r 0 r t 3 a x k (a ) , y k (a ) , z k (a ) , f k (a) J » for a.e. a e A, and 109 x*(•)dv < x A y*(•)dv < y , A z*(•)dv < z , A f*(-)dv < f (15) Since inequalities are preserved under l imi ts , the material balance equation must also hold, i . e . f < A 03 (•) dv - z From (15) we get that f*(-)dv < o)(.)dv -A JA A z*(-)dv, (16) However, there is a subsequence {k1} of the positive integers such that lim I k'+°° x k , ( a ) , y ^ ( a ) , z k ' ( a ) , f k ' ( a ) x*(a),y*(a),z*(a),f*(a) But for k e {k1} we have f k (a) = 03(a) + x k(a) - y k (a) - z k(a) a.e. aeA, and taking limits we get f*(a) = oj(a) + xt(a) - y*(a) - z*(a) a.e. aeA. (17) Furthermore, x k ' ( a ) , y k ' ( a ) , z k ' ( a ) j e S k ' ( a , p k ' ) - c ( S k , ( a , p k ' ) S(a) and X(a) closed and the preservation of inequalities under limits implies x*(a), y*(a) , z * (a ) e B (a,p*) Thus f* is an attainable al locat ion. Suppose for fixed a e A we have i nf • jp* (x,y ,z)eS(a) co(a )+x-y-zeX (a) (x-y)j- < 0 Then, there exists ( x ' . y'jZ ' J - e B(a,p*) such that p* • x 1 < p* • y 1 But since lim p k' ->°° k' p*, for k e {k1} large enou we have p k • x1 < p k • y' , ( x ' , y ' , z ' ) £ Sk(a),> and w(a) + x ' - y ' - z ' £ X k(a) . 111 But (x k (a), y k ( a ) , z k(a)) e <5 k(a,p k) implies that o)(a) + x ' - y ' - z ' « 03(a) + x k(a) - y k (a) - z k (a ) . a Taking l imi ts , by the continuity of ~ we get a oi(a) + x ' - y ' - z 1 S a u ( a ) •+ x* (a )-y* (a )-z* (a ) = f*(a) a In the case (x,y,z) e 6(a,p*) with p* . x = p* • y (x,y,z) is the l imit of vectors (x^y^.z ) e B(a,p*) with and p* . x n < p* • y n , 03(a) + x n - y n - z n " a 03(a) + x * ( a ) - y*(a) - z * ( a ) Taking limits again we get u ( a ) - x-y-z = 03(a) .+ x*(a) - y * ( a ) - z * ( a ) a Thus, x*(a), y*(a) , z*(a) e <5(a,p*) (18) 112 C o r o l l a r y . I f t h e m e asure s p a c e o f a g e n t s o f t h e " b a r t e r " B economy H i s n o n - a t o m i c , o r i f f o r e v e r y a e A w i t h v ( { a } ) > 0, b o t h t h e p r e f e r e n c e - o r - i n d i f f e r e n c e r e l a t i o n ~ Q and t h e t r a n s a c t i o n s e t S ( a ) a r e c o n v e x , t h e n t h e r e e x i s t s a q u a s i -c o m p e t i t i v e p r i c e e q u i l i b r i u m . Proof. The p r o p e r t y t h a t v i s n o n - a t o m i c was r e q u i r e d i n t h e t h e o r e m t o show a ( p ) , t h e i n t e g r a l o f t h e c o r r e s p o n d e n c e <5 w i t h r e s p e c t t o v , i s c o n v e x . S i n c e v i s a f i n i t e m e a s u r e , t h e measure s p a c e (A, fi,vv) has a t most a c o u n t a b l e number o f atoms [ 3 3 , p. 6 1 5 ] . The s e t A, t h e r e f o r e , can be decomposed i n t o two s u b s e t s A = A i u A 2 where v i s n o n r a t o m i c on A i and A 2 i s c o u n t a b l e , a e A 2 => v ( a ) > 0. cj>'(a,p*) i s c o n v e x o r empty and t h u s <5(a,p) i s c o n v e x . C l e a r l y o%(ij?) 1 :i s a a i l socconv.exss.inee But a e A 2 i m p l i e s 3(a,p')} i s c o n v e x s i n c e S ( a ) and X ( a ) a r e c o n v e x . C o n s e q u e n t l y , ~, a c o n v e x as w e l l i m p l i e s a k ( p ) . = 6 k ( . , p ) d v + I 6 ( a , p ) . a e A 2 The r e s t o f t h e t h e o r e m goes t h r o u g h as b e f o r e . Q . E . D . 113 Consider the monetary economy [(A, ft, v ) , X, 5, T 5 v] as described in Chapter 3. Let A = p = (p b ,p s ) e R f | J (Pb + P )^ = 1 i = l be;th?v»unitcprice.xsimpTex , i n n R - v S L}efji ned|he budget corre-spondence 3 <2A XyA -> R 2 £ by, and 3(a,p) = «-([s - o)(a)] + s- [s - 03(a)]") |ss e'.X(a) Pb • [s - 03(a)] + < P s • [s - oj(a)]~ + Tr(p,a) • , 2 £ and the demand correspondence $> : A x A R bby <j)i(a,p) = - (x,-y) e 3(a,p) | for every (x ' , -y ' ) e 3(a,p) oj(a ) + x ' -y 1 5 03(a ) + x-y-d Final ly , define the quasi-demand correspondence 6 : A x A •+ R by 2 £ 1 1 4 ' f (a,p) i f inf - p. • [r-co(a)]+ - P • [r-u>(a)]~- < Tr(p,a) reXfaH D s J 6(a,p) ( ) 8(a,p) otherwise Definition 3. Quasi-competitive price equilibrium Let p be a price vector in the unit simplex A c R+ x R+ and f be a v-integrable function from A to R . The pair (p,f) is called a quasi-competitive price equi l ibr i M for the "monetary" economy E i f um a) | [ f (a ) - o)(a)] +, - [f(a) - o)(a)]-for almost every a e A , e <5(a,p) b) f [f(a) - to(a)] + dv , - f [f( .) - co(>)]av] e T ( A ) , U A J A J and c) P, C f ( - ) - o)(.)] + dv - b • [ f ( 0 - u ) ( - ) ] " d v max ( x . - y ) e T ( A ) P b ' x - p s • yj •'•'c.?. Condition a) implies that f(a) e X(a) and that f(a) must be maximal with respect to * in agent a's budget set whenever the minimum wealth situation does not occur. Condition 115 b) ensures that f is an attainable a l locat ion, while condi-tion c) is the profit maximizing relation for the coalit ion traders. Definition 4. Competitive price equilibrium. The pair (p,f) is called a competitive price equi l -M ibrium for the "monetary" economy S i f i t forms a quasi-competitive price equilibrium and i f the set of agents for whom the minimum wealth situation occurs has measure zero. i .e . i f a e A | inf Jp. • [r-oo(a)]+ - p • L>-w(a)]~} > ir(p,a) reX(a) l b S ] " then v ( E * ) = 0 £_. The strategy used in the proof of the next theorem follows closely that of Theorem 1. In part a) the properties of the total quasi-demand correspondence are investigated for a "truncated economy"; in part b) the properties of the supply correspondence f o r the coalit ion traders is invest i -gated; in part c) i t is shown that the excess quasi-demand correspondence has the properties required by Debreu's lemma. F inal ly , the existence of a sequence of quasi-equil ibria for the sequence of truncated economies is shown in part d) to imply the exitence of a quasi-equi1ibrium for the original economy. 116 Theorem 2: If the measure space of agents for the "monetary" M economy E is non-atomic and i f the economy's aggregate transaction set T(A) is compact, then there exists a quasi-competitive price equilibrium. Define the k-th truncated budget correspondence, demand k k correspondence, ty , and quasi-demand correspondence, 6 , by replacing X(a) with X (a) in the definitions of section D above. F ina l ly , define the total quasi-demand correspon-dence tyk : A -* R2Z by Proof Part a For every positive integer k we again define the truncated consumption set by X k(a) = |s e X(a) | s < k(u>(a) + 1 )} A = ~(x,-y) | x,y e R* and there exist v-integrable functions x,y : A -> such that 117 and x ( a ) , - y ( a ) e S ( a , p ) , a.e. a e A x = A x ( . ) d v , y y ( - ) d v | , A i I c l a i m t h a t t h e c o r r e s p o n d e n c e i> has t h e f o l l o w -i n g p r o p e r t i e s : i ) t h e r e i s a compact s e t N c R s u c h t h a t i p k ( p ) c N f o r e v e r y p e A, i i ) t h e g r a p h o f ^ i s c l o s e d , and i i i ) f o r e v e r y p e A, i ^ k ( p ) i s nonempty and c o n v e x . u The p r o o f t h a t has t h e s e p r o p e r t i e s f o l l o w s c l o s e l y t h e p r o o f i n Theorem 1 a b o v e . T h e r e f o r e we w i l l o n l y s k e t c h p a r t s o f i t . To p r o v e p r o p e r t y i ) , l e t N = s = ( s i , s 2 ) e R £ + R £ < k[ A (oo(-) + 1 ) d v ] I f ( x , - y ) e ^ (p) f o r p e A t h e n by c o n s t r u c t i o n t h e r e e x i s t it v - i n t e g r a b l e f u n c t i o n s x,y : A + R + s u c h t h a t ( x ( a ) , - y ( a ) ) 5 k ( a ,p) and x = (S; (a 5p ) i m p l i es x ( * ) d v , y = y ( ' ) d v . But ( x ( a ) . - y ( a ) ) A 118 o)(a) + x(a) - y(a) e X k(a) =*-co'Oa) + x(a) - y(a) < k 0) (a) + 1 x(a) < (k-l)to(a) + k < k OJ (a) + 1 Furthermore, since x^a) > 0 .=> y^a ) = 0 and X k(a) c R^ we have y(a) < oj(a) < k w(a) + 1 Therefore x( • )dv < k co(a) + 1 |dv y = y(•)dv < k A 'A to(a) + 1 dv Hence ^K(p) c N for every p e A. To prove property i i ) , let (p n , (x n , -y n ) ) be a sequence in G k_ with 1 im(.p' , ' x ( x ) ) ='p (p ', x ( x - y ) ) . Then ip n-*-°° there exist v-integrable functions x n , y n : A R+ such that A x n ( - )dv, y n A y n ( « ) d v , and xn ( a ) , - y n (a) e <5k(a,pn) for every a~e A 119 However the sequence of v-integrable funct ions ' { (x n > -y ) } is bounded pointwise in absolute value by the v-integrable function k(co+1 ). Applying Theorem E [33, p. 622] there exist v-integrable functions x,y : A -* R£ such that x ( » ) , - y(&) dv = (x,-y) (1) and x(a), - y(a) e closure - x„{.) for almost every a e A (2) It is possible to show, as before, fo^tf i-xed a e A , that the correspondence 6 k (a ,* ) is upper semi-continuous . Then lim p = p, (x (a), -y (a)) e 6 k (a,p ) and (x(a), -y(a)) e closure {(x n(a), -y n(a)} implies by the upper semi-con-k k t iniuty of 6 in p that (x(a), -y(a)) e 6 (a,p). Therefore, (x,-y) e G . , i .e . the graph of i|>k is closed. ty It is again possible to show that ^ k(a,p) f 0 for almost every a e A and that the correspondence ty (*,p) is v-measurable. Then since the integral of a correspondence with respect to an atomless measure is convex, ty .(p.) is convex for every p e A. 1 20 Part b. The supply correspondence for the coalit ion of traders of the entire economy is defined for p e A by n (p) - | (x , -y ) e T (A) | p b • x - p g • y = n(p,A) It is easy to see that n has the following properties: i) for every p e A, n(p) is closed and since i t is contained in the compact set T(A), i t is also compact, i i ) - for every p e A, n(p) is nonempty and convex, and i i i ) the graph of the correspondence n is closed By def in i t ion, n(p,A) = max (x.-y)eT(A) P b • x - p s • y The f i r s t two properties fellow, from the continuity and l ine-arity of the function Pb * x " P s * y l n (x>"y) a n c ' f r o m t n e compactness of T(A). Property i i i ) holds since the function b^ * x " Ps * y 1 S a l s o continuous in ( p b , p s ) . 121 P a r t c. Now d e f i n e t h e e x c e s s q u a s i - d e m a n d c o r r e s p o n d e n c e f o r t h e k - t h t r u n c a t e d economy by ? k ( p ) = 4> k(p) - n ( p ) . k £ £ The c o r r e s p o n d e n c e E, : A R x R has t h e f o l l o w i n g p r o p e r t i es . 2 £ i ) t h e r e i s a c o m p act s e t N c R s u c h t h a t £ k ( p ) c N f o r e v e r y p e A, i i ) t h e g r a p h o f £ i s c l o s e d , i i i ) f o r e v e r y p e A, £ ( p ) i s nonempty and c o n v e x , and i v ) f o r e v e r y p e A , p • 5 (p) < 0. P r o p e r t i e s i ) - i i i ) a r e i m m e d i a t e c o n s e q u e n c e s o f t h e p r o p e r t i e s o f t h e t o t a l q u a s i - d e m a n d and s u p p l y c o r r e s p o n -d e n c e s . To e s t a b l i s h i v ) l e t z e £(p) =* z = ( x ' - x , -( y ' - y ) ) s u c h t h a t ( x ' ,-y') e / ( p ) , (x,-y) e n ( p ) . B u t ( x ' , - y ' ) e ^ ( p ) i m p l i e s t h a t t h e r e e x i s t v - i n t e g r a b l e f u n c t i o n s x,y : A -> R +, s u c h t h a t ( x ( a ) , - y ( a ) ) e 6 ( a , p ) and x' x ( * ) d v , y y ( • ) d v , But ( x ( a ) , - y ( a ) ) e 6 k ( a , p ) (x(a) , - y ( a ) ) e 3 k ( a , p ) and t h e r e f o r e p b • x ( a ) - p s • y ( a ) < 7 t ( p , a ) , a.e. i n A I f we i n t e g r a t e t h e l a s t i n e q u a l i t y we g e t x ( * ) d v - p, y(«)dv < Tr (p,a) A JA o r P b • x' - p s • y' < n(p,A) However, ( x , - y ) e n(p) =* P b •• x - p g • y = n(p,A) Thus o r P b • x 1 - p s • y 1 < p b • x - p s • y , P b • [ x ' - x ] - p s • [ y ' - y ] < 0 A-T h a t i s , , p-is z = ( P b > P s ) • (x 1 - x , - ( y ' - y ) ) < 0 . 1 23 P a r t d By D e b r e u ' s lemma [ 1 3 , p 8 2 ] , t h e r e i s a p e A s u c h t h a t 5 k ( p ) n R* £ f $. T h a t i s , t h e r e e x i s t v - i n t e g r a b l e 0 f u n c t i o n s x,y : A -> R + s u c h t h a t x ( a ) , - y ( a ) e <S ( a , p ) , a.e. a e A , and (3) i f x = x ( • ) d v and y = A y ( • ) d v , t h e n ( x , - y ) e T ( A ) , (4) and P b • x - p s • y = n ( p , A ) (5) E q u a t i o n ( 4 ) f o l l o w s f r o m p r o p e r t y i i ) o f t h e t r a n s a c t i o n s c o r r e s p o n d e n c e T. by I f we d e f i n e t h e v - i n t e g r a b l e f u n c t i o n f : A R + f = to + x-y M M t h e n f i s a ts/.t.a^t © fo f. hfeh e ce&&mm$ 2 a na1n d- ht-h e>ap g'i r ^ tp^,'f) : i s a q u a s i - c o m p e t i t i v e p r i c e e q u i l i b r i u m f o r t h e " k - t h t r u n c a t e d economy." Thus f o r e v e r y p o s i t i v e i n t e g e r k, t h e r e i s a p r i c e k k k I v e c t o r p e A and v - i n t e g r a b l e f u n c t i o n s x ,y : A R + s u c h t h a t i f we l e t 1 24 f k = (o + x k - y k , k k then the pair (p ,f ) form a quasi-competitive price equi l -ibrium for the "k-th truncated economy." Since A is compact, w . l .o .g . lim p k = p* = (p£,p*) Furthermore, since for every positive integer k, x k ( - )dv, - y k ( - )dv £ T(A) A J A J and since T(A) is compact, each of the sequences 4 x («)dv ff k ) A J and y (-)dv.- has a convergent subsequencess Without loss A • s 6f:-,genenail;i;ty.,ythere; ex.tsttx?#, e R* such that 1 im k+co i A x k(-)dv = x, lim f y k ( - )dv = y k-»-» -"A Since no agent can sell more than his in i t i a l endowment we also have, for al l k, that 0 < y k (a) < coi(>a), for every a e A . By applying Schmeidler's [56] version of Fatou's lemma to the f i r s t sequence and Schmeidler's corollary to the second sequence, we get that there exist v-integrable function x*,y* : A R£ such that 125 x*(a),-y*(a)j is a cluster point of the (6) sequence x k (a ) , -y k (a) for a.e. a e A, and A X * ( r )dv < X, y*(- )dv = y, (7) But T(A) compact implies that (x,-y) e T(A). From property i i ) on the transactions correspondence T and (7) above we also get x*(-)dv, - | y*(-.)dvjee T (A) ( 8 ) Since (x*(a),-y*(a)) is a cluster point of the sequence {(x k(a), -y k (a)}, there is a subsequence {k'} of the positive integers such that x*(a) = lim x k ' (a) , y*(a) = lim y k ' ( a ) k '•*•» k'->~ Furthermore, x k ' ( a ) , - y k ' ( a ) l e 6 k , ( a , p k ' ) c B ^ U . p ^ ) implies that P K ' ' x k ' (a) - p k ' • y k ' (a ) < 7r(p k ' ,a) Since T(A) is compact, the function 7r(»,a) is continuous A. Taking l imi ts , we get that p£ • x*(a) - p* • y*(a) < ir(p*,a) Since X(a) is closed, 1 im k 1 co(a) + x k ' (a) - y k ' (a) e X(a) Thus , x*(a),-y*(a) e 3(a,p*) Suppose for a e A, we have inf - p. • [r-w(a)] - p. • [r-w(a)]~f < ir(p ,a) reX(a)1- D s i and there exists (x, =y) e $(a,p*) such that p£ • x - p* • y < 7T(p*,a) But since lim p k = p*, for k e {k'} large enough k->oo 127 Pb * x - p k • y < 7 r ( p k , a ) , and oj(a) + x-y e X k(a) But (x k (a), -y k (a)) e S k (a ,p k ) implies that 03(a) + x-y 2 a u ( a ) + x k(a) - y k (a) Since ~ is continuous, after taking limits we get oj(a) + x-y w(a) + x*(a) - y*(a) Thus for r e X(a) satisfying P b * [r-o)(a)] - p* • [r-oj(a)]- < 7T(p*,a) (11) we have r ~ a f* (a) . a Following Hildenbrand [33,p. 620] when (10) holds for every s e X(a) with Pb .• [s-co(a)]+ - p* • [s-(&,(.? i r = = TT#?§a) is the l imit of vectors f e X(a) with n 1 28 Thus x*(a), - y * ( a ) j e S(a,p*) ( 1 2 ) Final ly , I claim that Ph ' xt(*)dv - p' A y * ( r ) d v = n(p*,A) ( 1 3 ) We know for k e {k'} that x k(-)dv - p k y k ( . )dv = n (p k ,A ) , and that pb * x k ( a ) " p s * y l < ( a ) = ^ P ^ 9 ) . f o r a l l a e A The last two equations imply there exists A x e Q, such that v ( A i ) '= v ( A ) and p b * x k ( a ) " p s ' y l < ( a ) = ^ P ^ 9 ) ' f o r a 1 1 a 8 A i -Taking limits on the last equation we get p* • x*(a) - p* • y*(a) = T r (p * ,a ) , f o r a l l a e A i 129 Integrating the last equation gives (13). Equations (13), (12) and (8) imply (p* , f * ) , where f* = w + x* -y* , is a quasi-equi1ibrium of the "monetary" economy ^ . Q.E.D. Corollary: If the measure space of agents for the "monetary" M economy E is non-atomic, or i f for every a e A with v({a}) > 0, the preference-or-nVridif f erenee:''reiI.atdion~2 ' :i scconvex , then there '-. a exists a quasi-competitive price equilibrium. Proof: Same as in the corollary to Theorem 1. £. The assumptions made in Chapters 2 and 3 were suff icient to prove the existence of a quasi-competitive price equilibrium in both the "barter" and "monetary" economies. To ensure that these equil ibria are also competitive price equi l ibr ia , additional assumptions must be made. Suppose for both the "barter" and "monetary" economies: 1) X(a) = \\\ for a l l a e A, i .e . each agent's consumption set is the non-negative orthant of the Euclidean space of dimension £, 1 30 2) The preference relation <* is monotonic a 0 for al l a e A, i .e . x,y e R+ with x < y implies x <* y, a 3) co(a) >> 0 for a l l a e A , i .e . each agent possesses positive quantities of every good. I conjecture that these three additional assumptions are suff icient to ensure that the quasi-competitive price equilibrium in the monetary economy is .a lso a competitive price equilibrium (see Aumann [6], Hildenbrand [33], Kurz [41] and Schmeidler [55]). To ensure that the quasi-competitive price equilibrium of the .^barter" economy is also a competitive price equil ibrium.. I conjecture that the assumption: c 4) S(a) is convex for al l a e A , j plus the f i r s t three assumptions are suf f ic ient . Assumption 4) is necessary, tbeensure that an agent will be able to buy positive quantities of al l commodities. Otherwise, the transaction cost of s'tar'ti.h'g ah' exchange": pfl.tis the'amouht given up in the exchange may exceed an agent's in i t i a l endowment (see Kurz [43] ) . 1 0 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            data-media="{[{embed.selectedMedia}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.831.1-0100098/manifest

Comment

Related Items