5450 recursive competitive equilibrium

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2 5450 recursive compeiive equilibrium recursive compeiive equilibrium The underlying srucure of mos dynamic business-cycle and consumpion-based asse-pricing models is a varian of he neoclassical sochasic growh model. Such models have been analysed by, among ohers, Cass (1965), Brock and Mirman (1972), and Donaldson and Mehra (1983). They focus on how an omniscien cenral planner seeking o maximize he presen value of expeced uiliy of a represenaive agen opimally allocaes resources over he infinie ime horizon. Producion is limied by an aggregae producion funcion echnological (oal facor produciviy) shocks. The soluion o he planning problem is characerized by ime-invarian decision rules, which deermine opimal consumpion and invesmen each period. These decision rules have as argumens he economy s period aggregae capial sock and he shock o echnology. Business cycles, however, are no predicaed on he acions of a cenral planner, bu arise from ineracions among economic agens in compeiive markes. Given he desirable feaures of he sochasic growh paradigm he soluion mehods are well known and he model generaes well-defined proxies for all he major macro aggregaes: consumpion, invesmen, oupu, and so on i is naural o ask if he allocaions arising in ha model can be viewed as compeiive equilibria. Tha is, do price sequences exis such ha economic agens, opimizing a hese prices and ineracing hrough compeiive markes, achieve he allocaions in quesion as compeiive equilibria? This is he essenial quesion of dynamic-decenralizaion heory. Alernaive approaches o dynamic decenralizaion: valuaion equilibrium One way of modelling uncerain dynamic economic phenomena is o use Arrow Debreu general equilibrium srucures and o search for opimal acions condiional on he sequence of realizaions of all pas and presen random variables or shocks. The commodiies raded are coningen claim conracs. These conracs deliver goods (for example, consumpion and capial goods) a a fuure dae, coningen on a paricular sequenial realizaion of uncerainy. Markes are assumed o be complee, so ha, for any possible fuure realizaion of uncerainy (sequence of echnology shocks) up o and including some fuure period, a marke exiss for conracs ha will deliver each good a ha dae coningen on ha realizaion (even). This requires a very rich se of markes. All rading occurs in he firs period: consumers conrac o receive consumpion and invesmen goods and o deliver capial goods in all fuure periods coningen on fuure saes so as o maximize he expeced presen value of heir uiliy of consumpion over heir infinie lifeimes. Firms choose heir producion plans so as o maximize he presen value of discouned profis. Given curren prices, hey conrac o deliver consumpion and invesmen goods o, and o receive capial goods from, he consumer-invesors. Under sandard preference srucures, hese coningen choices never need o be revised. Tha is, if markes reopen, no new rades will occur. In is mos general formulaion, a valuaion equilibrium is characerized simply as a coninuous linear funcional ha assigns a value o each bundle of coningen commodiies. Only under more resricive assumpions can his funcion be represened as a price sequence (Bewley, 1972; Presco and Lucas, 1972; Mehra, 1988). The basic resul is ha for any soluion o he planner s problem ha is, sequences of consumpion, invesmen and capial goods a se of sae-coningen prices exiss such ha hese sequences coincide wih he conraced quaniies in he valuaion equilibrium. This decenralizaion concep is quie broad and applies o cenral-planning formulaions much more general han he neoclassical growh paradigm. I reminds us ha he financial srucure underlying he sochasic growh paradigm is fundamenally one of complee coningen commodiy markes. Neverheless, i is a somewha unnaural perspecive for macroeconomiss (all macro policies mus be announced a ime zero), and i presumes a se of markes much richer han any observed. These shorcomings led o he developmen of he concep of a recursive compeiive equilibrium. Recursive compeiive heory An alernaive approach ha has proved very useful in developing esable heories is o replace he aemp o locae equilibrium sequences of coningen funcions wih he search for ime-invarian equilibrium decision rules. These decision rules specify curren acions as a funcion of a limied number of sae variables which fully summarize he effecs of pas decisions and curren informaion. Knowledge of hese sae variables provides he economic agens wih a full descripion of he economy s curren sae. Their acions, ogeher wih he realizaion of he exogenous uncerainy, deermines he values of he sae variables in he nex sequenial ime period. This is wha is mean by a recursive srucure. In order o apply sandard ime-series mehods o any esable implicaions, hese equilibrium decision rules mus be ime-invarian. Recursive compeiive heory was firs developed by Mehra and Presco (1977) and furher refined in Presco and Mehra (1980). These papers also esablish he exisence of a recursive compeiive equilibrium and he supporabiliy of he Pareo opimal hrough he recursive price funcions. Excellen exbook reamens are conained in Harris (1987), Sokey, Lucas and Presco (1989) and Ljungqvis and Sargen (2004). Since is inroducion, i has been widely used in exploring a vide variey of economic issues including business-cycle flucuaions, moneary and fiscal policy, rade-relaed phenomena, and

3 recursive compeiive equilibrium 5451 regulariies in asse price co-movemens. (See, for example. Kydland and Presco, 1982; Long and Plosser, 1983; Mehra and Presco, 1985.) The recursive equilibrium absracion posulaes a coninuum of idenical economic agens indexed on he uni inerval (again wih preferences idenical o hose of he represenaive agen in he planning formulaion), and a finie number of firms. As in he valuaion equilibrium approach, consumers underake all consumpion and saving decisions. Firms, which have equal access o a single consan-reurns-o-scale echnology, maximize heir profis each period, and are assumed o produce wo goods: a consumpion good and a capial good. Unlike in he valuaion equilibrium approach, rading beween agens and firms occurs every period. (This is in conras o markes in an Arrow Debreu seing where, as menioned earlier, no rade would occur if markes were o reopen.) A he sar of each period, firms observe he echnological shock o produciviy and purchase capial and labor services, which are supplied inelasically a compeiive prices. The capial and labour are used o produce he capial and consumpion goods. A he close of he period, individuals, acing compeiively, use heir wages and he proceeds from he sale of capial o buy he consumpion and capial goods produced by he firms. Consumers hen reain he capial good ino he nex period, when i again becomes available o firms and he process repeas iself. Noe ha firms are liquidaed a he end of each period (reaining no capial asses while echnology is freely available), and ha no rades beween firms and consumer-invesors exend over more han one ime period. Capial goods carried over from one period o he nex are he only link beween periods, and period prices depend only on he sae variables in ha period. Formally, a recursive compeiive equilibrium (RCE) is characerized by ime invarian funcions of a limied number of sae variables, which summarize he effecs of pas decisions and curren informaion. These funcions (decision rules) include (a) a pricing funcion, (b) a value funcion, (c) a period allocaion policy specifying he individual s decision, (d) a period allocaion policy specifying he decision of each firm and (e) a funcion specifying he law of moion of he capial sock. While he resricive srucure of markes and rades makes his concep less general han he valuaion equilibrium approach, i provides an inerpreaion of decenralizaion ha is beer suied o macro-analysis. More recenly, he recursive equilibrium concep has been generalized o admi an infiniely lived firm which maximizes is value. When an RCE is Pareo opimal, is allocaion coincides wih ha of he associaed planning problem. The soluion o he cenral-planning sochasic-growh problem may hen be regarded as he aggregae invesmen and consumpion funcions ha would arise from a decenralized, recursive homogeneous consumer economy. We illusrae his wih he help of an example below, which considers an economy wih a single capial good. The reader is referred o Presco and Mehra (1980) for he more general case wih muliple capial ypes. An example Consider he simples cenral planning sochasic growh paradigm ( ) wðk 0 ; l 0 Þ¼max E XN b uðc Þ (P1) ¼0 c þ k þ1 l f ðk ; l Þ; l 0 ; k 0 given; l ¼ 1 8: In his formulaion, u( ) is he period uiliy funcion of a represenaive consumer defined over his period consumpion c ; k denoes capial available for producion in period and l denoes period labour supply which is inelasically supplied by he consumer-invesor a l ¼ 1, for all. The expression f(k, l ) represens he period echnology (producion funcion) which is shocked by he bounded saionary sochasic facor l. (I is assumed ha l is a saionary Markov process wih a bounded ergodic se.) E denoes he expecaions operaor and he cenral planner is assumed o have raional expecaions; ha is, he uses all available informaion o raionally anicipae fuure variables. In paricular he knows he condiional disribuion of fuure echnology shocks Fðl þ1 ; l Þ. For he purposes of his example we resric preferences o be logarihmic and assume a Cobb Douglas echnology (o he bes of my knowledge, his parameerizaion is he simples example known o resul in closed form soluions): uðc Þ¼ln c and f ðk ; l Þ¼k a l1 a. We also assume ha a, bo1 and ha capial fully depreciaes each period. These condiions are sufficien o guaranee a closed form soluion o he planning problem: c ¼ð1 abþk a l ; and k þ1 ¼ i ¼ abk a l where we idenify as invesmen, i, he capial sock held over for producion in period þ 1. These allocaions are Pareo opimal. We will show ha he invesmen and consumpion policy funcions arising as a soluion o his problem may be regarded as he aggregae invesmen and consumpion funcions arising from a decenralized homogenous consumer economy. We firs qualiaively describe he RCE underlying his model, and hen demonsrae he relevan equilibrium price and quaniy funcions explicily. The one capial good is assumed o produce wo goods a consumer good and an invesmen (capial) good. A he beginning of each period, firms observe he shock o produciviy (l ) and purchase capial and labour from individuals a compeiively deermined raes. Boh capial and labour

4 5452 recursive compeiive equilibrium are used o produce he wo oupu goods. Individuals use heir proceeds from he sale of capial and labour services o buy he consumpion good (c ) and he invesmen good (i ) a he end of he period. This invesmen good is used as capial (k þ1 ) available for sale o he firm nex period and he process coninues recursively. To cas his problem formally as a recursive compeiive equilibrium, we inroduce some addiional noaion. Le k denoe he capial holdings of a paricular (measure zero) individual a ime, and k he disribuion of capial amongs oher individuals in he economy. This laer disincion allows us o make formal he compeiive assumpion: all he economic paricipans will assume ha k is exogenous o hem and ha he price funcions depend solely on his aggregae (in addiion o he echnology shock). Clearly, in equilibrium, k ¼ k for our homogeneous consumer economy. In addiion, le p i, p c and p k be he price of he invesmen, consumpion and capial goods respecively and p l be he wage rae. These prices are presumed o be funcions of he economy-wide sae variables exclusively and all paricipans ake hese prices as given for heir own decision making purposes. The sae variables characerizing he economy are ðk; lþ and he individual are ðk; k; lþ. We use he symbols (c, i, k, l) o denoe poins in he commodiy space for he firm and he consumer. The c in he commodiy poin of he firm is a funcion specifying he consumpion good supplied by he firm and is wrien as c s ðk ; l Þ. Similarly, he c in he commodiy poin of he individual is he amoun of he consumpion good demanded by he individual and is wrien as c d ðk ; k ; l Þ. In equilibrium (as menioned earlier, in equilibrium k ¼ k ), since he marke clears, of course c s ¼ c d. The same commens apply o he oher elemens of he commodiy poin. In he decenralized version of his economy, he problem facing a ypical household is vðk 0 ; k 0 ; l 0 Þ¼max E ( ) XN ¼0 b ln c d ðk ; k ; l Þ (P2) p c ðk ; l Þ c d ðk ; k ; l Þþp i ðk ; l Þ i d ðk ; k ; l Þ p k ðk ; l Þ k s ðk ; k ; l Þþp l ðk ; l Þ l s ðk ; k ; l Þ k þ1 k s ðk þ1 ; k þ1 ; l þ1 Þ¼i d ðk ; k ; l Þ, l s ðk ; k ; l Þ1 and k þ1 ¼ cðk ; l Þ is he law of moion of he aggregae capial sock. Wih capial and labour priced compeiively each period, he firm s objecive funcion is especially simple maximize period profis. The firm s problem hen is max fp c ðk ; l Þ c s ðk ; l Þþp i ðk ; l Þ i s ðk ; l Þ p k ðk ; l Þ k d ðk ; l Þ p l ðk ; l Þ l d ðk ; l Þg c s þ is l ðk d Þa ðl d Þ1 a. Via Bellman s principle of opimaliy, he recursive represenaion of he individual s problem P2 is vðk ; k ; l Þ¼max fcd ;i d ;l s ;k d g ln ðcd ðk ; k ; l ÞÞ Z þ b vði d ðk ; k ; l Þ; cðk ; l Þ; l þ1 ÞdFðl þ1 jl Þg p c ðk ; l Þ c d ðk ; k ; l Þþp i ðk ; l Þ i d ðk ; k ; l Þ p k ðk ; l Þ k s ðk ; k ; l Þþp l ðk ; l Þ l s ðk ; k ; l Þ k þ1 k s ðk þ1 ; k þ1 ; l þ1 Þ¼i d ðk ; k ; l Þ, l s ðk ; k ; l Þ1 and k þ1 ¼ cðk ; l Þ is he law of moion of he aggregae capial sock. The firm of course, simply maximizes is period profis and hence does no have a muliperiod problem. The following funcions ha are a soluion o he individual and firm maximizaion problem above saisfy he definiion of recursive compeiive equilibrium: 1. A value funcion vðk 0 ; k 0 ; l 0 Þ¼E P N ¼0 b ln ½ð1 abþ l k a 1 faðk k Þþk gšg. I can be shown ha vðk 0 ; k 0 ; l 0 Þ¼AþBlnk 0 þ Clnl 0 where A, B and C are consans which are funcions of he preference and echnology parameers. 2. A coninuous pricing funcion pðk ; l Þ¼fp c ðk ; l Þ; p i ðk ; l Þ; p k ðk ; l Þ; p l ðk ; l Þg ha has he same dimensionaliy as he commodiy poin, where p c ðk ; l Þ¼p i ðk ; l Þ¼1 (We have chosen he consumpion good o be he numeraire.) p k ðk ; l Þ¼al k a 1 p l ðk ; l Þ¼ð1 aþl k a Consumpion and invesmen funcions for he individual ha are a funcion of he curren sae of he individual ðk; k; lþ

5 recursive compeiive equilibrium 5453 c d ðk ; k ; l Þ¼ð1 abþl k a 1 faðk k Þþk g l s ðk ; k ; l Þ¼1 i d ðk ; k ; l Þ¼abl k a 1 faðk k Þþk g k s ðk þ1 ; k þ1 ; l þ1 Þ¼i d ðk ; k ; l Þ. 4. Decision rules for he firm ha are coningen on he sae of he economy ðk; lþ c s ðk ; l Þ¼ð1 abþl k a, l d ðk ; l Þ¼1, i s ðk ; l Þ¼abl k a, k d ðk þ1 ; l þ1 Þ¼i s ðk ; l Þ. 5. The law of moion for he capial sock specifying he nex period capial sock as a funcion of he curren sae of he economy ðk ; l Þ k þ1 ¼ cðk ; l Þ¼abl k a. 6. The consumpion and invesmen decisions of he individual c s ðk; k; lþ, l s ðk; k; lþ and i s ðk; k; lþ maximize he expeced uiliy he budge consrain. So ha vðk ; k ; l Þ¼ ln ðð1 abþl k a 1 ðaðk k Þþk ÞÞ Z þ b vðabl k a 1 ðaðk k Þþk Þ; abl k a ÞdFðl þ1jl Þ. 7. The decision rules of he firm c d ðk ; l Þ, l d ðk ; l Þ, i d ðk ; l Þ maximize firm profi. Demand equals supply c d ðk þ1 ; k þ1 ; l þ1 Þ¼c s ðk ; l Þ; l s ðk þ1 ; k þ1 ; l þ1 Þ¼l d ðk ; l Þ and i s ðk þ1 ; k þ1 ; l þ1 Þ¼i d ðk ; l Þ. The law of moion of he represenaive consumers capial sock is consisen wih he maximizing behaviour of agens cðk ; l Þ¼i d ðk ; k ; l Þ. I is readily demonsraed ha since vðk 0 ; k 0 ; l 0 Þ¼wðk 0 ; l 0 Þ, he compeiive allocaion is Pareo opimal. See eqs (P1) and (P2). Having formulaed expressions for he prices of he various asses and heir laws of moion, i is a relaively simple maer o calculae raes of reurn (price raios) and sudy heir dynamics. For an applicaion o risk premia, see Donaldson and Mehra (1984). Some researchers have formulaed models ha can be cas in his same recursive seing, ye whose equilibria are no Pareo-opimal. As a consequence, he model s equilibrium can no longer be obained as he soluion o a cenral-planning-opimum formulaion. These models incorporae various feaures of moneary phenomena, disorionary axes, non-compeiive labour marke arrangemens, exernaliies, or borrowing-lending consrains. Besides increasing general model realism, such feaures enable he models no only o beer replicae he sylized facs of he business cycle, bu also o provide a raionale for inervenionis governmen policies. Moneary models of his class include hose of Lucas and Sokey (1987, a moneary exchange model) and Coleman (1996, a moneary producion model). Bizer and Judd (1989) and Coleman (1991) presen models in which non-opimaliy is induced by ax disorions, while Danhine and Donaldson (1990) presen a model in which non-opimaliy resuls from efficiency-wage consideraions. In hese models, equilibrium is characerized as an aggregae-consumpion and an aggregae-invesmen funcion which joinly solves a sysem of firs-order opimaliy equaions on which marke-clearing condiions have been imposed. Coleman (1991) provides a widely applicable se of condiions under which hese subopimal equilibrium funcions exis. As already noed, however, hese opimaliy condiions canno, in general, characerize he soluion o an opimum problem. RAJNISH MEHRA See also Arrow Debreu model of general equilibrium; decenralizaion; neoclassical growh heory; real business cycles. Bibliography Bewley, T Exisence of equilibria in economies wih infiniely many commodiies. Journal of Economic Theory 4, Bizer, D. and Judd, K Taxaion and uncerainy. American Economic Review Papers and Proceedings 19, Brock, W.A. and Mirman, L.J Opimal economic growh and uncerainy: he discouned case. Journal of Economic Theory 4, Cass, D Opimal growh in an aggregaive model of capial accumulaion. Review of Economic Sudies 32, Coleman, W.J Equilibrium in a producion economy wih an income ax. Economerica 59, Coleman, W.J Money and oupu: a es of reverse causaion. American Economic Review 86, Danhine, J.P. and Donaldson, J.B Efficiency wages and he business cycle puzzle. European Economic Review 34, Donaldson, J.B. and Mehra, R Sochasic growh wih correlaed producion shock. Journal of Economic Theory 29, Donaldson, J.B. and Mehra, R Comparaive dynamics of an equilibrium ineremporal asse pricing model. Review of Economic Sudies 51,

6 5454 recursive conracs Harris, M Dynamic Economic Analysis. New York: Oxford Universiy Press. Kydland, F.E. and Presco, E.C Time o build and aggregae flucuaions. Economerica 50, Ljungqvis, L. and Sargen, T.J Recursive Macroeconomic Theory, 2nd edn. Cambridge, MA: MIT Press. Long, J.B., Jr. and Plosser, C.I Real business cycles. Journal of Poliical Economy 91, Lucas, R.E., Jr. and Sokey, N Money and ineres in a cash advance economy. Economerica 55, Mehra, R On he exisence and represenaion of equilibrium in an economy wih growh and nonsaionary consumpion. Inernaional Economic Review 29, Mehra, R. and Presco, E.C Recursive compeiive equilibria and capial asse pricing. In R. Mehra, Essays in financial economics. Docoral disseraion, Carnegie Mellon Universiy. Mehra, R. and Presco, E.C The equiy premium: a puzzle. Journal of Moneary Economics 15, Presco, E.C. and Lucas, R.E., Jr A noe on price sysems in infinie dimensional space. Inernaional Economic Review 13, Presco, E.C. and Mehra, R Recursive compeiive equilibria: he case of homogeneous households. Economerica 48, Sokey, N., Lucas, R.E. and Presco, E.C Recursive Mehods in Economic Dynamics. Cambridge, MA: Harvard Universiy Press. recursive conracs In conrac heory i is sandard o inroduce a paricipaion consrain (PC) insuring ha he conrac offered o he agen delivers a uiliy higher han he bes ouside opion. In a dynamic se-up agens may abandon he conrac a any poin in ime, even afer he conrac has been in place for a while. For example, workers can leave a labour conrac a almos no cos, or a borrower can sop repaying he loan if he or she declares bankrupcy. The possibiliy ha he agen does no coninue wih he plan of he conrac is usually called defaul. Hence, in a dynamic conex, i is naural o require ha he PC is saisfied in all periods, in order o avoid defaul. I urns ou ha, if a PC in all periods and realizaions is inroduced in he design of he opimal conrac, sandard dynamic programming does no apply, he Bellman equaion does no hold, and he soluion is no guaraneed o be a ime-invarian funcion of he usual sae variables. This complicaes enormously he soluion of hese models. To discuss his in a simple risk-sharing model, consider wo agens i = 1,2 wih uiliy funcion E 0 P N ¼0 b uðc i Þ, where ba(0,1) is he discoun facor and u he insananeous uiliy. Each agen receives a sochasic endowmen w i and he realizaion of endowmens is known boh o he agens and he principal. The principal has full commimen, and will sick o his announced plan. Endowmens provide he only supply of consumpion good so ha he following feasibiliy condiion holds c 1 þ c2 ¼ w 1 þ w2 (1) A Pareo-opimal risk-sharing conrac (implemened by a compeiive equilibrium under complee markes) would se u0 ðc 1 Þ consan for all periods, so ha agens u 0 ðc 2 Þ would share all idiosyncraic risks. This allocaion would be chosen as he opimal conrac if agens would commi o never leave he risk-sharing arrangemen. We refer o his allocaion as he firs bes. The opimum saisfies he usual recursive srucure in dynamic models, namely, ha c = F(w ) where F is a ime-invarian funcion and w ¼ðw 1 ; w2 Þ. Assume now agens canno commi o saying in he conrac for ever. An agen can leave he conrac and consume for ever his individual endowmen, so ha a conrac can only be implemen if i saisfies X N E b j uðc i þj ÞV a i ðw Þ j¼0 a all periods and realizaions, where V a i ðw P N ÞE j¼0 b j uðw i þjþ is he uiliy of consuming in auarchy for ever afer. I is clear ha he above PC is likely o be violaed by he firs bes allocaion. In periods when w i is high, he righ side of he PC is high, bu he agen has o surrender a large par of his endowmen in he firs bes and he lef side of he PC is oo low. Therefore, PC s are ofen binding and hey make he firs bes unfeasible. A Pareo-opimal risk-sharing conrac wih PC s can now be foundpby maximizing he weighed uiliy of he wo agens E N 0 ¼0 b ½luðc 1 cþþð1 lþuðc2 ÞŠ he above PC for all periods and realizaions and for boh agens. The parameer l indexes all such Pareo-opimal allocaions. The resul is an opimal conrac under full commimen by he principal and parial commimen by he agens. The Bellman equaion does no give he soluion o his problem. A key feaure of sandard dynamic programming is ha he se of feasible acions mus depend only on variables ha were deermined las period and he curren shock. Bu i is possible o evaluae if a cerain consumpion level c i saisfies he PC a ime only if fuure plans for consumpion are known. Inuiively, a promise of higher consumpion in he fuure makes a lower consumpion oday compaible wih he PC. Bu in order o implemen his plan he principal has o remember all he promises for higher consumpion ha were made in he pas. Therefore, he opimal soluion is unlikely o be a funcion of only oday s