Money, Growth and Sta bi I ity Frank Hahn The MIT Press Cambridge Massachusetts First MIT Press edition, 1985 © Frank Hahn 1985 First published 1985 Basil Blackwell Ltd 108 Cowley Road, Oxford OX4 IJF, UK All rights reserved Except for the quotation of short passages for the purposes of criticism and review, no part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any forrn or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher Library of C::ongress Cataloging-in-Publication Data Hahn, Frank Money, growth and stability Bibliography: p Includes index Money Equilibrium (Economics) I Title HG221.3.H28 1985 339.5 85-15132 ISBN 0-262-08156-3 Typeset by Unicus Graphics Ltd, Horsham Prin tcd in ,reat Bri lain by Bell and Bain Ltd, Glasgow Contents Acknowledgements Introduction In Praise of Economic Theory PART I vu 10 MONEY Money and General Equilibrium The General Equilibrium Theory of Money: A Comment The Rate of Interest and General Equilibrium Analysis Equilibrium with Transaction Costs On Transaction Costs, Inessential Sequence Economies and Money PART II NON-WALRASIAN EQUILIBRIA On Non-Walrasian Equilibria Exercises in Conjectural Equilibria PART III STABILITY On Some Propositions of General Equilibrium Analysis On the Stability of Pure Exchange Equilibrium 10 A Theorem on Non-tatonnement Stability With Takashi Negishi 11 Uncertainty and the Cobweb 31 46 56 75 105 131 159 183 192 202 21O Contents V1 PART IV GROWTH 231 On Two-sector Growth Models I Equilibrium Dynamic with Heterogeneous Capital Goods 14 On Warranted Growth Paths 15 The Stability of Growth Equilibrium 16 On the Disequilibrium Behaviour of a Multi-sectoral Growth Model 17 On Some Equilibrium Paths 301 322 18 19 20 21 341 348 364 377 PART V MISCELLANEOUS Savings and Uncertainty Exe ss Capacity and Imperfect Competition On Optimum Taxation On Equilibrium with Market-dep nd nt Information Index 243 261 278 388 Acknowledgements The publishers acknowledge with gratitude permission to repro­ duce the fallowing texts: 'Money and General Equilibrium' from the Indian Economic Journal, 23, 109-22 (Special Number in Monetary Economics, Oct.-Dec 1975) 'The General Equilibrium Theory of Money: a Comment' from The Review of Economic Studies (1952-3), 19, 179-85 ·The Rate of Interest and General Equilibrium Analysis' from The Economic Journal ( 1955), 65, 52-66 (Cambridge University Press) 'Equilibrium with Transaction Costs' from Econometrica (1971), 39, 417-39 'On Transaction Costs, Inessen­ tial Sequence Economies and Money' from The Review of Eco­ nomic Studies (1973), 40, 449-61 'On Non-Walrasian Equilibria' from The Review of Economic Studies ( 1978), 45, 1-17 'Exer­ cises in Conjectural Equilibria' from the Scandinavian Journal of Economics ( 1977), 79, 210-26 'On Some Propositions of General Equilibrium Analysis' from Economica ( 1968), 35, 424-30 'On the Stability of a Pure Exchange Equilibrium' from the Inter­ national Economic Revieiv ( 1962), 3, 206-13 'A Theore1n of Non-tatonnement Stability' with Takashi Negishi frotn Econo­ n1etrica ( 1962), 30, 463-9 'Uncertainty and the Cobweb' from The Review of Economic Studies ( 1955-6), 35, 65-75 'On Two Sector Growth Models' from The Review of Economic Studies (1965), 32, 339- 46 'Equilibriun1 Dynamics with Heterogeneous Capital Goods' fro111 The Quarterly Journal of Econonzics ( 1966), 80, 633-45 © The President and Fellows of Harvard College 'On Warranted Growth Paths' from The Revie,v of Econo1nic Studies (1968), 35, 175-84 'The Stability of Growth Equilibriu111' from The Quarterly Journal of Econo,nics ( 1960), 74, 206-26 © The President and Fellows of Harvard College 'On the Disequilibriu111 Behaviour of a Multi-sectoral Growth Model' frotn Tlze Quarterly 111 Acknowledgements Journal of the Royal Economic Society ( 1963), 73, 442-57 (Cambridg University Press) 'On Some Equilibrium Paths' from Jod l of Economic Growth, ed J A Mirrlees and N H Stern (Proceedings of a Conference held by IEA at Jerusalem), pp 193_06 Sa ings and Uncertainty from The Review of Economic Studies ( 19 70) 37, 21-4 'Excess Capacity and Imperfect Compe­ tition from Oxford Economic Papers, new series volume 7, no 3, October 1955 (Oxford, Clarendon Press), pp 229-40 'On Opti1num Taxation' from The Journal of Economic Theory ( 19 73), 6, 96-106 © Academic Press, New York and London 'On Equi­ librium with Market-Dep ndent Information' from Quantitative Wirtschaftforschung, pp 245-54 Introduction The papers which follow are more technical than those of the pre­ vious volume, Equilibrium and Macroecono,nics (Basil Blackwell; Oxford, 1984) But they should be accessible to anyone familiar with a n1odern text By way of introduction I have included my Jevons Lecture, 'In Praise of Economic Theory' In it I discuss what I think we are doing when we theorize and I argue that this is a worthwhile activity In the remainder of this introduction I briefly comment on the papers which follow I MONEY My interest in monetary theory derives largely fro1n the difficulty one has in incorporating it into classic theories of the econo1ny which in other respects seem satisfactory It is, of course, widely known that Arrow-Debreu theory will not allow such incorpora­ tion at all But recent work on models of overlapping generations in which money appears are equally unsuitable as long as it is the only asset, and when it is not it raises all sorts of difficulties Some of these are quite old in the literature Someti1nes it seen1s as if the world is coming to the aid of the theorist Thus, interest-paying current accounts are beco1ning 1nore \Videspread and thereby one old puzzle is sidestepped But this pheno1nenon also shows that the theoretical puzzle has real substance: there is a tendency for other assets to drive out 'sterile' money In paper I give an account, of a rather non-technical kind, of the way general equi­ librium theorists have tried to incorporate non-interest-bearing, intrinsically worthless, money into their theorizing In the 1950s the most notable atten1pt in this direction was that of Patinkin (2nd edn, 1965) In son1e of his early papers he failed Introduction to n1odel adequately the essential intertemporal aspect of money In paper I pointed out why that was wrong Of course, it was only a slip on Patinkin's part, and I reprint it only because it 111akes the point concerning the central importance of the inter­ ten1poral aspect rather sin1ply In the 1950s also the battle still raged between 'loanable funds' and 'liquidity preference' theories of the rate of interest This episode is now, deservedly, almost forgotten But the bad habits of the debate have persisted in other contexts In particular, there is the habit of writing that some price is 'determined' in one particular market without giving this clai111 the precision it needs to be meaningful; examples are 'Inflation is a Monetary Phenome­ non' and 'Investment determines Savings' This means that paper may still be of interest since it is an attempt, in a particular context, to sort out this sort of muddle The next two papers represent my most serious attempt to fashion a general equilibrium theory in which money can find a place The first of these is not easy or pleasant reading because the con1plications of an econo1ny with transaction technologies are rather large Foley (1970) had pursued the same line at the same time and independently However, he did not examine the inter­ temporal problems which gave rise to the possibility of inefficient equilibria This came as a surprise to n1e, since the inefficiency in question is relative to the transaction technology Starrett' s example ( 1973) is a beautifully clear one, and readers interested in these matters should read it as a companion piece to paper Kurz ( 1972) made further clarifications possible, and the \Vhole rnatter has recently been rnagisterially discussed by Gale ( 1982) I think that this whole episode has i111proved our understanding of monetary theory But there are still large gaps The 'transaction technology' is ad hoc and not properly grounded, as it should be, in inforn1ation theory Moreover, although Heller and Starr ( 1976) huve relaxed the assurnption sornewhat, it is still the case that we cannot deal at present with the likely significant non­ convexity of transaction technologies Clearly, one needs to dig more deep!� to the level at which the infonnation of exchanging agents can be studied II NON-WALRASIAN EQUILIBRIA Perfectly cornpetitive econornies arc now understood rather well I have, however, been puzzled by the many econo1nists who seern Introduction to consider that this abstraction is satisfactory both intellectually and for the study of actual economies To take just one exa1nple, it seems that the fraction of UK GNP spent on advertising was for long comparable to that spent on education Such a dissonance between theory and gross observation would certainly give pause to, say, physicists, but not to n1any economists, who are apt to mun1ble something about looking for lost keys under a street light or appealing to the metaphysics of 'as if ' Even if somehow this odd stance could be defended, it would still be highly peculiar not to be interested in the challenge of studying economies in which agents must act strategically In any case, for the last ten years or so I have been much interested in departures from Walrasian economies, although I cannot claim to have advanced far along this difficult road The Belgians led by Dreze (197 5) and the French led by Malinvaud (1977) went, as it \Vere, to the other extreme and studied econo1nies with exogenously given prices and consequent rationing They considered what in paper I called 'orderly equilibria', that is, equilibria where only the 'short side' in any market is rationed This was the basis of the dynamic process studied by Negishi and myself (see paper IO below) In the first part of paper I was interested in an existence proof for a Dreze equilibrium I found it useful to study an associated economy with coupon prices as rationing devices for this purpose But the aim of that paper was to move beyond exogenously given prices For that purpose I introduced the term 'conjectures' to describe theories of firms of the reaction of the economy to their own actions Of course, this was related to Negishi's (1961) perceived de1nand curve but was rather more general Paper considers so1ne examples of conjectural equilibria In the theory which I develop in paper I introduced the notions of rational and of reasonable conjectures The first of these was badly christened since it triggers a \Vrong reaction in game theorists (see Makowski, 1983) I should have named the1n correct conjectures, and even that tenninology needs further elaboration The idea is this Agents have theories of what will happen to, say, their profits if they deviate fron1 so1ne given status quo I considered only small deviations and implicitly supposed that the direct consequences of such a deviation would be spread over many agents I did not have a duopoly or oligopoly ga1ne in mind I therefore took the consequences of a sn1all deviation by a single agent to be calculable from the equilibriu111 of the rest of the Introduction econo1ny after the deviation and when all other agents keep to their conjectures This becomes clear in the examples of paper I certainly \vas not thinking of dynamics, nor of games in extensive form No agent is playing against another who reacts to his devia­ tion by calculating the best responses Rather, any one agent has an anonymous opponent - the rest of the economy - and for 1ny purposes the agents who constitute that economy need not know \vho has deviated and may indeed regard the ne\v signal as being due to some change in the state of the economy I hope it is clear fro1n this that n1y intention was not to take a shortcut through the intricacies of rationality in an economy­ \vide ga1ne As far as I kno\v, game theory and general equilibrium theory have met only in circumstances in which a perfect competi­ tive equilibrium is the outcorne Moreover, those studies seem to have all been concentrated on pure exchange The main exception to this is the book by Marschak and Selten ( 1979), which proceeds in a manner not markedly dissimilar from the one \vhich I fol­ lo\ved (It preceded my own work, but I only became aware of it later.) In any case, I have to confess that I have some doubts that it \vill prove possible to bring about a marriage of recent rather beautiful advances in game theory and economy-\vide analysis Certainly when I wrote my papers I knew that it was beyond me I accordingly opted for what is essentially a Chamberlinian approach (see Hart, 1983) I will not be surprised if it transpires that that was a wrong move III STABILITY I can be brief in m y co1nments on papers , and In the first of these I am concerned rather less with stability than with the existence of equilibriu111 I showed how, in an econorny with the gross substitutability property, one could develop an existence proof by induction (and so without a fixed point theore1n) I needed gross substitutes because I needed equilibriu111 to be unique for every way of fonning a certain co1nposite con1n1odity out of two con1n1odities I now believe that it n1ay be possible to carry out the same proof with only the postulate that the economy is always regular (Debreu, 1970) But I have not tried to this since there arc more interesting things to instead Both the ren1aining papers concern a price adjustn1ent process for a pure exchange econon1y in which trading takes place at all 1\J iscella, 1eo 11s 78 states of th e world in w h ic h a h ouse h old of type i (i e i == H or i == L ) h as an accide n t and SiO the se t of sta tes of t he world i n w h ic h a house hold of type i does not h ave an accid ent An insur­ a n c e co1n p any is a finn trading units of one pound contingen t on an accid ent to th e buyer I f th e insurance company is able to p a rt ition th e sta tes of the world in to the fou r se ts Si l S;0 • i == H, L th en we say th a t it h as th e fin e partition, and if it can only par­ tition th e sta tes in to th e two se ts S11 U SL and SH o U SL we say it h as the coarse p a rtition Th e inte rest of this e x am p le , as we sha ll see derives fro1n th e fac t th a t we cannot speci fy in advance w h e th er insurance co1n p anics h ave the fine or the coarse p a rtition Le t C; == (Yi l · YiO ) w h ere J 'i < is ne t pay 1ne n t to agent type i i f h e h as an accid ent and J ';o > is t h e payn1cnt by agent type i i f h e docs no t h av e an accident \Ve write iv(c;, ex ;) a s the ex pe c ted u tility of agent i fro1n th e ·contrac t' c; Notice th a t this assun1es th a t a ll age n ts h ave th e sa 1ne u tili ty function a nd endow1ncnt (Th e u tility func tions a rc taken as stri ctl y concave ) Eve ry insu r­ ance con1pany knows the con1n1on u tility fu n c tion and cncl ow­ n1cnt I nsu rance con1panics, because th ey h ave 1nany c usto1n e rs a rc risk-ne u tral and seek to 1n a xi1ni;,c c xpcc tc cl p rofi t Le t a ; == ( ex; - CX.;) Th en an insu n ce co1n p any h aving th e fine partition n1 a xin1izcs a;c; If insu ran ce con1 p a nies have the coa rse p a rtition th e y n1a xi1nizc 2' /\ ;G ; C; The fol lo,ving in1 portant assu 1n p tion is 1nade by Rothsc hild and S tiglitz ( R-S ) · l ssun1p tio1 ( Free 1:·n t ry ) ( ) I f the e x pected p rofit on �,ny ­ tract is positive a l a rge nun1be r of co1n p a nics wil l ente r th e 1n a rkc t fo r t h a t t r �, c t ( ) I r {' == ( C f l , C L ) is i.l n y gi Ven C O n tr a C t pair On O ffe r t he n , i f th e re exists a n o ffe r c' w hic h wil l h ave positive e x pe c t e d p rofit gil'e11 tha t the cu 11 tracts c co11 ti11uc 1 offer , so1n c i nsu rance co1n­ p an y W i I I O f fc r {1 • \Ve wil l n ow show t h a t e a c h of the two 1nost na tu l concL' p ts of cq u i lih riu 1n ru n in to diffic u l ties wit h t h is e x a 1n p l c Ill COM P ET I T I V E E Q U I L I B R I U M L c t / J ; == ( / J ; , /J , ) (i - I I , I, ) a p r icc v L' t: t o r, w h c r e /J ; i s t h c p ric c p l: r pou n d co n tinge n t o n a n a c t: i d c n t t o a n �1gL' n t o f t y pe i c t t: Equilibriunz lVith AJarket-dependen t lnfonna tion 79 Suppose (pt , p f ) is a con1 petitive equilibriun1 a t which agents of each type transact Then p art (1 ) of the Free Entry assumption requires p?c? = i = H L or , on a suitable norm alization P l9 = a l• i = H L (2 I ) But in a con1p etitive econo1ny all households will wish to transact at prices pf , since they give n1ore favourable tenns That is, no household will de1nand cR if it can get insurance at pf But an insurance con1p any trading at p f can only recognize type h after it has observed its de1nand (it de1nands more than does type l ) Since by definition the com p any stands ready to supply whatever is de111anded at the given price, the observation does it no good Hence there cannot be a con1petitive equilibriun1 where both types trade and insurance co111p anies have the fine p artition Let p = (p , p 0) where p is the price of a pound contingent on an accident to a household (regardless of type) and P o is the price of a pound contingent on no accident to a household If p is a con1petitive equilibriun1 relatively to the coarse p artition, it 1nu st be consistent with zero expected profits of the insurance con1p any If p = a L then, since both types buy at p , expected profit is negative If p = a H it is positive Since one easily shows expected profit to be continuous in p , there will be p such that CXL CX.H < p ? /p g < ­ I - cx.L I - cx.H and expected profits are zero One therefore concludes that a co111petitive equilibriun1 relatively to the coarse partition exists Sup pose now that c = (ci c2) are the contracts of the typ ical agents of each type in such an equilibriun1 By definition Since every co111ponent of ci exceeds in absolute value the corn> sponding co111 ponent of c2 , there is a set of contracts c such that, if c i = c , i = H l , (2 ) Let c satisfy ( ) with equality and write j) = � /\ i CX i µ = 0 ere /r L A l• ( l - cx.l- ) · But p 01 = L CX.l·/\ l· �1 ' l91 /1· � • p = L ( l - cx l ) A l- · 10 wh Miscellaneous o - ;P o; Po > P Po · (2 3) o e rt a i n l for 111 k > t h re i a t ract kc w h ich if offe red to ry hou hold wou ld b accepted since it makes th em b etter ' off irre p c t i of typ and wou l d leave t h e i n u nce comp any \vi t h zero ; pect d p rofi t By t i n u i ty , t h e n , t h r is a t ract c \ h ich w i l l b a p t d by every hous hol d a t p r n t b uyi n g c? a nd \ ill i Id po i t i pect d p rofit to t h com p a ny H e nc Proposition ( P ) Th Com p t i t ive Compet i t ion E q u i l i b ri u m A l l ocat ion is u nstable u nder p a r t ( ) of t h e F ree E ntry assump tion I f on t h ink of a l l house hol d e n teri ng a coa l i t ion w i t h a n i nsu r­ an on1 p a ny a nd t radi ng c w i t h i t w m ay ay t h at t h is coali tion b lock th Com p t i tiv A l locat ion So if th Cor i any allocat ion of t ract wh ich can not b blocked w t h a t t h omp e t itiv q u i l i b ri u m i n ot i n th Cor Thi will b o h ow v r l a rg t h e onomy H e n ce com p t i t jv equil i b ri u m i not rob u t i n t h i a m p l F igu re l r pea t t h argu me n t lead i ng t o ( P l ) a b A Yo f JG R � = P� pg = P i , Po Equilibrizun with Marke t-depen den t lnfonna tio n 38 In figure 21.1, H and L are indifference curves for typical households o f these types One verifies that any contract which is bought by both types and is south-west o f the line B yields positive expected profits What is the 'explanation' o f (P l)? If insurance con1p anies with a coarse partition quote a price and supply whatever is de1n anded, then the probability o f, say, paying a pound depends not only on the given type frequencies A H and A £ but also on the differences in the amount of insurance bought by each type But the latter is not independent of price, and hence the possibility o f paying a pound is not independent o f price The blocking probability arises fron1 the fact that insurance cotnpanies acting as perfect con1petitors not use this dependence o f probabilities on prices All this in tun1, o f course, is a consequence o f only t,1e coarse parti­ tion being possible in a con1petitive equilibriu 111 IV AN R-S EQUILIBRIUM I not know whether (P1) or son1ething like it led R-S to con­ sider an alternative equilibriu1n concept or rather an alternative institutional arrange1nent Certainly (P I ) justifies such an atte1npt It is now assu1ned that co1npanies not quote prices at which they are willing to transact whatever is desired, but rather that they o ffer particular contracts Once again let c = (ci , c2) If in an equilibriun1 o f this new econo111y c.Pt, =I=- cf , R-S say that the equilibriun1 is separating (co1npanies have the fine partition), and if c.Pt, = cf = c , say it is pooling (i.e , the partition is coarse) An equilibriun1 is an offer c which yields zero expected pro fit, is accepted by households and cannot be blocked in the sense o f part (2) of the Free Entry assun1ption One now proves Proposition (P2) There are econon11es for which no R-S equi­ librium exists Proof ( ) Let a = La ;A.; and suppose a pooling R-S equili­ briun1 c = (c , c 0) exists Then certainly ac = whence a L c > (since o:.L < o:.H) · But at c the indifference curve o f agent type H cuts that o f agent type L fron1 below (once again, since o:.1, < o:.11 and both types have the saine utility function) Hence the n' is c ' such that c' > c0 L and c' � c0 H and 382 Miscellaneo us i n a s111 a l l n e igh bou rh ood of c B u t t h e n a L c ' > , a n d si n ce t h e co1n p a n y offe ri n g c ', given c , h as now t h e fin e p a rt i t ion ( no agen t t y pe JI acce p ts t h e offer) , i t fol lows for p art ( ) of t h e Free E n try assu111 p t ion t h a t c ' w i l l ' b loc k ' c H e nce n o R-S equ i l i b ri u 1n rel a­ t ively to t h e coa rse p a rti t i on e x ists ( ) Le t us w r i te c ,- ( p ,-) as t h e choi ce of age n t type i i n t h e con1p e t i t ive e cono111 y of t h e p revious sec t ion , i f t h a t agent faces t h e b udge t l i ne p ,-c ,- = Su p pose ( 4) \vh e n c = (cJ1 , cf) i s a n R-S separa t i ng e q u i l i b ri u n1 Th e n for eve ry c ' i n a s1n al l n e igh bourhood of c L (a ) , age n t type L p re fe rs c ' to cf a nd wi l l acce p t t h e fonn er Si n ce ac ,- (a ) = , t h e re w i l l t h us be c i n t h e n eigh bou rhood of c L (a ) \vh ich L accep ts and for w h ich ac > ( 5) B u t c wi l l a lso be acce p te d by type H For i f not , t h e re wou ld h ave bee n a se p a t i ng tra c t ( c , cf1 ) at \vh ich posi t ive e x pe cted profi t s can be earned So i f ( ), t h e re e x ists a pool i ng t c t wh i c h bre a ks t h e R-S se p ara t i ng e q u i l i briu111 Bu t by part ( ) of t h is p roof n o pooli ng t ract can be a n R-S equ ilibriu 1n So th en no R-S equ i l i b ri u 111 e x ists ( ) I t re1n a i ns t o sh o\v t h a t ( ) i s poss i b l e Ce rt n ly i f c is a se p a t i n g equ i l i b ri u 111 one h as ( 6) I f t he fi rst eq u a l i ty d id not hold , t h e n iv [ c 1 (a 1 ) , 0'.1 ] > �V(cf1 , 0'.11 ) (si n ce a 1 c,P1 = ) B u t t h e n , every c; i n a s1n a l l n e igh bou rh ood of c 11 (a 1 ) wo ul d , by // , be p re ferred to cf1 Also, t h e re is c 11 i n t h a t n e igh bo u rh ood w i t h a 1 c 1 > a n d c 11 acce p t a b le t o H w h i c h t rad i c ts t h e d e fi n i t i o n o f c f1 as a n cqu i l i b ri u 111 t rac t Th e last i n equ a l i ty o f ( ) is requ ired i f t h e i n sura n ce con1 p a n ie s a rc t o h a ve th e fi n e pa rt i t i o n Now let and 383 Eq uilibriun1 \Vitlz Marke t-depen den t fnforn za tio n Then it is easy to see that there is &.17 such that (2 7) For as cxH ➔ , R (cxH) converges to the singleton c1., = But now one also has (2 8) For by the usual argu111ent if not, there would be cL elose to cf which is preferred by L and for which a L c L > while cL is inferior to c2- for H But then (ct, cf) is not an equilibri1in1 So for cx.H � &.H, (2 8) defines cf as a function of cx.H so the right-hand side of (2 4) is given once cx.H is given Now let /\H ➔ so that a ➔ aL , so Hence there is so111e "- H such that (2 4) holds for all /\ H � Xu The 'explanation' of this 111elancholy result at a technical level is to be found in an essential discontinuity in the 'reaction functions ' of insurance eon1panies, and the econo111ic reason for that is an essential infonnational externality Let c be a given contract pair which is accepted by households Let and consider the action of a potential co111pany We interpret it as choosing in F(c) n I a pair of contracts which will be accepted by households In particular, if it chooses c; = C ; this is interpreted as the potential finn 111aking no new offer to agent of type i The - potential finn then chooses new offers in J, (c) n / which 111axi111ize its expected p rofits Let '¥ (c) be the finn 's choice correspondcnee Then c E '¥ (c) if and only if (a) the expeeted profits of c arc zero and (b) there is no offer in F(c) J \vith positive expected profits Hence c E '¥ (c) is an R-S equilibriun1 Now '¥ (c) is a correspondence of the space of contracts into itself The space of contracts 111ay be taken as a co111pact convex fvlis c cllan eo us su bsp a L'L' o f R H o \vever t h e co r respo nden ce 1nay lack t h e fixed­ p o i n t p ro p e rty i e i t 1n ay n o t be co nvex-valued S u p pose c i s se p a ti ng \vh e re c f-I s a t i s fies a n d c 1, s a t i s fies Th e n ce r t a i nly t h e re will be a se p ara t i ng co n t rac t c' yieldi n g posi­ t i\'c e x pe c t e d p ro fi t s so c $ '1J (c ) Assu 1ne t h a t '¥(c) yields t h e si n gle e k 1n c n t c' N o w t a k e t he R-S case alre a d y discussed T h a t i s le t cL satisfy h'( c 11 , 0:11 ) i i' [ c L = Jv ( c1., cx.1 ) , (a ) o:.L l > ii1 ( cL ex1J Co n s i d e r a seq u e n ce e r wi t h c = c cH = c 11 all r a nd cL a L cL = a l l r T h e n t h e re \vi i i be F s u c h t h a t '¥ ( e r ) i s sep a t i ng r '¥ (c1 is p o o l in g r ➔ cL , F Also '¥ ( e r ) will be a t wo-ele nc n t se t c o n s is t i ng o f a pooli ng a nd a se p a t i ng c o n t c t p a i r H e n ce '¥ ( · ) is n o t co n ve x -v a l u e d a t /i-_ T h e se t e c h n i cali t ie s a rc si 1n p k b u t o ne w a n t s to u nders t a n d t h e d i ffi c u l t i es i n a kss fo nn al 111 a n n c r I f eve ry o n e i n t h e e c o n o n1y w e re a l ways full y i n fo nn e cl t h e n t he cco11 111 y \VO Li I d h a ve fo u r c o n t i nge n t goo d s ; i f all age n t s h a ve t h e: coa rse p a rt i t i o n t h e n t h e re wo u l d be o n ly t wo N o w t h e C o u rn o t -l i k c ass u n1 p t io n ( p a rt ( ) o f t h e Free E n t ry assu 1n p t io 11 ) h a� t h L' c o n se q u e n ce t h a t t h e d i 111 c n s i o n o f t h e releva n t co n1 n1 o d i t y space d e pe n d s o n { , i e o n t h e se t o f co n t c ts a gi ve n gro u p o f fi rn1 s i \ a c t u all y se l l i ng a n d w h i c h p o te n t i a l c o n1 p e t i t o rs �1 ssu n1 e W i I I C O 11 t i Il LI C t O be O f re red J l l p a r t i C LI l a r , i f e V C ry O n C i S fLI 1 y i n fo nne d a n cl e a rn i ng 1.e ro-e x pec t e d p ro fi t , t h e n i n t h e eco n o 11 y o f ( P ) t h e i n fo rn1 a t i o n i s wo rt h less, s i n L· c �1ge n ts Ci.I ll d o as well ( �t ri c t l y , be t te r) w i t h a coa rse p a r t i t i o n On t h e o t h e r h a n d , w h e n a l l age n t � h a ve t h e coa rse p a rt i t i o n , t h e va l ue o f t h e i n fo n n a t io n i n d u cc:d b y o bse rv i ng a c t u a l t r�1 c t i s posi t i ve Th us t h e v a l u e o f i n fo nn a t i o n c.k p c: n d s d i �co 11 t i n u o u s l y o n t h e o bse rve d co n t c t s c Equilibrizan with Market-dependen t Informa tio n 385 The reason why th is exainple is of such interest is because it has illu111inated a kind of externality which h ad previousl y not been noticed Th is consists of the actions of so1ne agents affecting the nun1ber of goods other agents can distinguish , i.e., the dimension of the con1n1odity space As R-S notice, there are a good nun1ber of other exan1ples of th is kind of externality The fact th at in this sort of situation an R-S equilibriun1 m ay not exist (P2) is of interest But even when an equilibriu1n exists it h as so1ne new fea tures In p articular, as the above discussion n1akes clear, the cost of sustaining the fine p artition will be borne entirely by those h ouseholds th at h ave so111eth ing to gain from a fine p artition That is , the high-risk h ouseholds would as well in an R-S equilibrium as they could in a perfectly cotnpetitive equilibriu111 relatively to the fine partition (i.e , cH (a H) = c2,) It is the low-risk h ouseh olds wh o bear the utility cost R-S call th is a dissipative externality If one thinks about it, one sees th at one h as inforn1ally known of this pheno111enon for a long ti111e: for instance, the h onest bear the cost distinguishing between then1 and the dishonest V AN ALTERNATIVE EQUILIBRIUM The R-S equilibriutn 111 ay be understood in the context of the fallowing game A firm proposes an offer c and announces this to other firms An objection to this p roposal by another firm is a proposal c' such that, if c is persisted in, it will either attract no custo1ners or bankrupt the proposer A solution c is an offer which h as no objection If every offer in a compact set of offers h as an objection, no solution exists In a sense, one 1nay think of an R-S equilibriun1 as the stationary point in a tatonnement of offers The game h as the fallowing curious feature An obj ection to c 1nakes c unviable for its proposer Yet the objector bases this objection on information wh ich is contingent on c being offered This certainly does not 1nake the story very convincing When a somewh at dubious assumption leads to an unsatisfactory theory , the possible non-existence of equilibrium here , one is encouraged to look again 86 1vliscella11eo us One idea \Vhich I brie fly explore in this section is the follo\ving If a finn observes that son1e households have entered into a bind­ ing contract \Vith another finn , then it treats the resulting infor1nation as non-contingent For instance, i f all high-risk households have entered into a binding contract, then the fin11 treats the infonnation that al l its custo1ners will be low-risk as certain Ho\V­ ever the finn kno\vs that its own offer will destroy any infonna­ tion it 1night receive fro1n another offer which has not been the subject of bind ing contracts So let us think of coalitions of 1nore than one finn A coalition is a binding agrce1nent between household 1ne 1nbers and the finn on a contract c A finn wil l join such a coalition only i f it yields non-negative profits A household \Vill join such a coalition only if there is no other \Vhich it pre fers An cqu ilibiu1n is c su ch that all households arc 1ne 1nbers of a coalition No\v if c is an R-S equilibriu1n , it is also an equilibriun1 in the above sense But consider e x a pooling equilibriu1 n \Vith Then c* \vill also be an equilibriu1n for the p resent p roposal For consider a household type I I f it is not prepared to join the coali­ tion \Vith c * , then no p re ferred contract will be available to it in another coalition For by its · re fusal to join a c * coal ition it destroys the latter, and so with it the inforrnation \vhich allo\vs it to be distinguished as a low-risk household Hence there will be no finn with \vhich it can forn1 a pre ferred coal ition But if c* is an equil ibriu1n, it now follo\vs that there ahvays exists an eq ui­ l ibriu111 in the sense here proposed There arc alinost certainly other e q u ilibriu 1n concepts, and they n1ay be n1ore ap pealing than the prcsen t suggestion The latter \Vas 1nadc to illustrate the ad vantage there 1n ight be in not follo\ving R-S in throwing away hal f of their story This they d o \vhcn they consider only the infonnational cxtcrnality of a given offer and not the possible d estruction of this bene fit when son1eone tries to tak e ad van tagc of it A pooling offer, for instance , can re rnain viable and so persist only if it yields infonnation which is of no value The 11101nent the cxternality is valuable ( i e , gives rise t o an offer to the low-ris k households), it 1n u st be d es t roye d by the act T h e rea d e r w ill n o t ice t h e fu dge : a l l h o u se h o l d s o f a g ive n t y p e su p p o se t h a t t h e y w i l l a c t i n t h e �a m c m a n n e r Th i -; m ay h e a se r io u s o b j e c t i o n Equilibriznn with Market-dependen t Infonna tio n 87 of appropriating the value, This is an essential and interesting feature of the situation, and in the present context it seen1s q uite unconvincing to suppose that agents are not aware of it VI CONCLUDING REMARKS The discussion of the previous section leads one to a related but different class of problen1s Suppose there are two states of nature which can only be distin­ guished directly by a subset of agents, say M C N, there being N agents altogether However, when that is the case , the equilibriu1n prices of the econo1ny are different for each of the two states whence we can assun1e that the remaining agents be con1e in­ farmed, i.e., can distinguish between states by virtue of the information carried by prices On the other hand, it 1n ay be that, in an economy in which all agents can distinguish between states, there is one where equilibriu1n prices are identical for the two states But then there cannot exist an equilibriun1 in wh i ch actions based on expected prices yield these as equilibria This proble1n also arises from inforn1a tional externalities How­ ever, it will be harder to resolve than is the R-S proble1n I t is an essential feature of the latter that an agent can calculate the infor­ n1ation externality of this action This is not the case in the above situation REF E RENCES Radner, R ( 968), 'General Equilibrium with Uncertainty', Econometrica, 36 Rot h schild, M and Stiglitz, J ( I 976), ' Equilibrium i n Competitive Insurance Markets : An Essay on the Economics of hnperfect Infonnation ' , Quarter(v Journal of Economics, 90 These ideas will be formed in recent short m imeo notes of Roy Radner I n dex accelera tor, , , accidents (probability ) , 7-9 accumulation, , 264 , 27 2-5 , 34 -7 advertising, , , Arrow, K J , , , 84, 87 , 20 -Debreu model, , 9-2 , - , 9-40 , 75 , 7 , , 94, 2-4 , -Hah n model, 35 , , 95 , , 2-3 assets (portfolio selection), 345-7 auctioneer, 40, , 34 , 60 axioms/assu mptions, 2- , 6-22, 24-5 Azariades, C , 24 balanced growth path , 249-5 0, 25 8-60 bankru ptcy, , 3-4 barter, 2, 3 , 00 Bau mol, 1 Benassy , J P , , 34 , bond s, , , 8-5 , - , 6-73 borrowing, , 2-3 , 86-7 , 05 , 26 Bray , M , budget constrain ts, , 8-9 0, 6-9, 05 capaci ty , excess, , 8-6 capital accu mulation, , 264, 272-5 , 34 -7 -labour ratio , 2- , 244-8 , 25 0, 280-8 , 28 7-8 , 297-8 -ou tput tio, -6 , 289-90, 298-300 satiation , stock (errors) , 2- , , 20 capi tal goods, 3- , 7-8 , 70, equilibrium pa ths, 8-9 , 3 2-7 heterogeneous, 24 3-4 , 25 8-9 two-goods model, 23 - two-sector mo del, 23 - Cass, D , , 24 , 43 ca tcnary propert y , 270, , 276 causal theorizing ( J evons), - CI owe r, R W., 3 , , I coali tio ns, 86 C obweb th eore m , 5, 0-28 , 28 commodity prices, 7-9 , 8-9 , 282-8 , 290, space, 7-9 , , 7, 84-5 com pe titio n imperfect, , 348-6 monopolistic, -2 , perfect, 20, 40- , competitive conjectures, see conjectures competi tive equilibri u m , 27 5-6, 8-8 conjectu ral equ ilibrium, 2- , 2-5 existence problems, , 7-74 model (simple), 2-7 Nash rat ional, 6-7 , 4-5 ortho dox theory and, 9-6 conjectures com petitive , 3- , 2-3 , 5 , , 64 , linear/constant elasticity , 76-80 tional , 3- 4, 34 , 9-5 , -6 , 70, 2-6 consumer behaviour, 5- , 7-9 consumption, , , goods, 23 -2 , 3 4-5 contracts, 8-80, -5 , 86 Dasgup ta, P , De bassy , , Debreu , G , , , , , eco nomy , 05 , 06- 1 , 1 7-26 , property , , , , 9 , 1 3-1 see also Arrow, K J dece n traliza tion, , 2 demand , excess, see excess d emand d escriptive model, 26 -6 , 2-5 D iamond , P , , , 64-5 , 74-6 discou n t factors, , diseq u ilibriu m , , , , -2 d ivision of lab our, 3 , dom inance notio n , , l 97 , 200 Dosso , 26 Drezc, J , , , , 3 , , d u al i ty th eory , , 22-9 Ducscn bcrry's hy pothesis, 2 , 225 Index economic theory, , 0-28 efficiency, 6-7 production, 6-7, 69-74, 76 sequence economies, 1 -1 , , 23-5 transactions and, 3-9, 02, 04 endowments, , h ousehold , 76-7 , 80, 82, 8 , 93-4, 99 entrepreneur (roles), 34 8-6 equilibrium competitive, 27 5-6 , 78-8 dynamics, 24 3-60 existence proof, 4, 83-9 fixed-price, , 3-42, 56-7, 302-7 interest rate and, 6-7 market-dependent information, 77-8 non-existence, , 85-7 non-Walrasian , 2- , -5 orderly, , 3 , 5-7, , pure exchange, 4-5 , 2-20 R-S , , 7 , , -7 transaction costs, 74, -9 uniqueness, 4, 7-90 see also conjectural equilibrium equilibrium , momen tary , , 35 -40, 23 equ ilibrium dynam ics and, 25 0-5 4, 25 existence proof, 25 -8 u niqueness , 234-4 , 28-9 , 3 2-6 equilibrium paths duality theory and , , 22-9 with heterogeneous goods , 24 3-6 inelastic expectations and , , 29-3 in momen tary equilibrium, 7-4 warranted growth , 260-7 equilib rium theory , , 9-23 growth and, 5, 23 - m oney m, see money Euler equations, 26 , 27 excess capacity , , 34 8-6 excess demand, , 200, 27 8-9 non-tatonnement stability , , 20 2-9 see also liquidity preference exchange, m ediums of, 2- exch