recursive competitive equilibriumThe underlying structure of most dynamic business-cycleand consumption-based asset-pricing models is a variantof the neoclassical stochastic growth model. Such modelshave been analysed by, among others, Cass (1965), Brockand Mirman (1972), and Donaldson and Mehra (1983).They focus on how an omniscient central planner seekingto maximize the present value of expected utility of arepresentative agent optimally allocates resources overthe infinite time horizon.

Production is limited by an aggregate productionfunction subject to technological (total factor productiv-ity) shocks. The solution to the planning problem ischaracterized by time-invariant decision rules, whichdetermine optimal consumption and investment eachperiod. These decision rules have as arguments the econ-omy’s period aggregate capital stock and the shock totechnology.

Business cycles, however, are not predicated on theactions of a central planner, but arise from interactionsamong economic agents in competitive markets. Giventhe desirable features of the stochastic growth paradigm –the solution methods are well known and the modelgenerates well-defined proxies for all the major macroaggregates: consumption, investment, output, and so on– it is natural to ask if the allocations arising in thatmodel can be viewed as competitive equilibria. That is,do price sequences exist such that economic agents, opt-imizing at these prices and interacting through compet-itive markets, achieve the allocations in question ascompetitive equilibria? This is the essential question ofdynamic-decentralization theory.

Alternative approaches to dynamic decentralization:valuation equilibriumOne way of modelling uncertain dynamic economic phe-nomena is to use Arrow–Debreu general equilibriumstructures and to search for optimal actions conditionalon the sequence of realizations of all past and presentrandom variables or shocks. The commodities traded arecontingent claim contracts. These contracts deliver goods(for example, consumption and capital goods) at a futuredate, contingent on a particular sequential realization ofuncertainty. Markets are assumed to be complete, sothat, for any possible future realization of uncertainty(sequence of technology shocks) up to and includingsome future period, a market exists for contracts that willdeliver each good at that date contingent on that reali-zation (event). This requires a very rich set of markets. Alltrading occurs in the first period: consumers contract toreceive consumption and investment goods and to delivercapital goods in all future periods contingent on futurestates so as to maximize the expected present value oftheir utility of consumption over their infinite lifetimes.Firms choose their production plans so as to maximizethe present value of discounted profits. Given current

prices, they contract to deliver consumption and invest-ment goods to, and to receive capital goods from, theconsumer-investors. Under standard preference struc-tures, these contingent choices never need to be revised.That is, if markets reopen, no new trades will occur.

In its most general formulation, a valuation equilib-rium is characterized simply as a continuous linear func-tional that assigns a value to each bundle of contingentcommodities. Only under more restrictive assumptionscan this function be represented as a price sequence(Bewley, 1972; Prescott and Lucas, 1972; Mehra, 1988).The basic result is that for any solution to the planner’sproblem – that is, sequences of consumption, investmentand capital goods – a set of state-contingent prices existssuch that these sequences coincide with the contractedquantities in the valuation equilibrium.

This decentralization concept is quite broad andapplies to central-planning formulations much more gen-eral than the neoclassical growth paradigm. It reminds usthat the financial structure underlying the stochasticgrowth paradigm is fundamentally one of complete con-tingent commodity markets. Nevertheless, it is a some-what unnatural perspective for macroeconomists (allmacro policies must be announced at time zero), and itpresumes a set of markets much richer than any observed.These shortcomings led to the development of the con-cept of a recursive competitive equilibrium.

Recursive competitive theoryAn alternative approach that has proved very useful indeveloping testable theories is to replace the attempt tolocate equilibrium sequences of contingent functionswith the search for time-invariant equilibrium decisionrules. These decision rules specify current actions as afunction of a limited number of ‘state variables’ whichfully summarize the effects of past decisions and currentinformation. Knowledge of these state variables providesthe economic agents with a full description of the econ-omy’s current state. Their actions, together with therealization of the exogenous uncertainty, determines thevalues of the state variables in the next sequential timeperiod. This is what is meant by a recursive structure. Inorder to apply standard time-series methods to any test-able implications, these equilibrium decision rules mustbe time-invariant.

Recursive competitive theory was first developed byMehra and Prescott (1977) and further refined in Prescottand Mehra (1980). These papers also establish the exist-ence of a recursive competitive equilibrium and the sup-portability of the Pareto optimal through the recursiveprice functions. Excellent textbook treatments are con-tained in Harris (1987), Stokey, Lucas and Prescott (1989)and Ljungqvist and Sargent (2004). Since its introduction,it has been widely used in exploring a vide variety ofeconomic issues including business-cycle fluctuations,monetary and fiscal policy, trade-related phenomena, and

5450 recursive competitive equilibrium

regularities in asset price co-movements. (See, for exam-ple. Kydland and Prescott, 1982; Long and Plosser, 1983;Mehra and Prescott, 1985.)

The recursive equilibrium abstraction postulates acontinuum of identical economic agents indexed on theunit interval (again with preferences identical to those ofthe representative agent in the planning formulation),and a finite number of firms. As in the valuation equi-librium approach, consumers undertake all consumptionand saving decisions. Firms, which have equal access to asingle constant-returns-to-scale technology, maximizetheir profits each period, and are assumed to producetwo goods: a consumption good and a capital good.Unlike in the valuation equilibrium approach, tradingbetween agents and firms occurs every period. (This is incontrast to markets in an Arrow–Debreu setting where, asmentioned earlier, no trade would occur if markets wereto reopen.) At the start of each period, firms observe thetechnological shock to productivity and purchase capitaland labor services, which are supplied inelastically atcompetitive prices. The capital and labour are used toproduce the capital and consumption goods. At the closeof the period, individuals, acting competitively, use theirwages and the proceeds from the sale of capital to buy theconsumption and capital goods produced by the firms.Consumers then retain the capital good into the nextperiod, when it again becomes available to firms and theprocess repeats itself. Note that firms are liquidated at theend of each period (retaining no capital assets whiletechnology is freely available), and that no trades betweenfirms and consumer-investors extend over more than onetime period. Capital goods carried over from one periodto the next are the only link between periods, and periodprices depend only on the state variables in that period.

Formally, a recursive competitive equilibrium (RCE) ischaracterized by time invariant functions of a limitednumber of ‘state variables’, which summarize the effectsof past decisions and current information. These func-tions (decision rules) include (a) a pricing function, (b) avalue function, (c) a period allocation policy specifyingthe individual’s decision, (d) a period allocation policyspecifying the decision of each firm and (e) a functionspecifying the law of motion of the capital stock.

While the restrictive structure of markets and tradesmakes this concept less general than the valuation equi-librium approach, it provides an interpretation of decen-tralization that is better suited to macro-analysis. Morerecently, the recursive equilibrium concept has been gen-eralized to admit an infinitely lived firm which maximizesits value. When an RCE is Pareto optimal, its allocationcoincides with that of the associated planning problem.The solution to the central-planning stochastic-growthproblem may then be regarded as the aggregate invest-ment and consumption functions that would arise from adecentralized, recursive homogeneous consumer econ-omy. We illustrate this with the help of an example below,which considers an economy with a single capital good.

The reader is referred to Prescott and Mehra (1980) forthe more general case with multiple capital types.

An exampleConsider the simplest central planning stochastic growthparadigm

wðk0; l0Þ ¼ max EXN

t¼0

btuðctÞ( )

(P1)

subject to

ct þ ktþ1 � lt f ðkt ; ltÞ; l0; k0 given; lt ¼ 1 8t:

In this formulation, u( � ) is the period utility function ofa representative consumer defined over his period t con-sumption ct; kt denotes capital available for production inperiod t and lt denotes period t labour supply which isinelastically supplied by the consumer-investor at lt ¼ 1,for all t. The expression f(kt, lt) represents the periodtechnology (production function) which is shockedby the bounded stationary stochastic factor lt. (It isassumed that lt is subject to a stationary Markov processwith a bounded ergodic set.) E denotes the expectationsoperator and the central planner is assumed to haverational expectations; that is, he uses all available infor-mation to rationally anticipate future variables. In par-ticular he knows the conditional distribution of futuretechnology shocks Fðltþ1; ltÞ. For the purposes of thisexample we restrict preferences to be logarithmic andassume a Cobb–Douglas technology (to the best of myknowledge, this parameterization is the simplest exampleknown to result in closed form solutions): uðctÞ ¼ ln ct

and f ðkt ; ltÞ ¼ kat l1�a

t . We also assume that a, bo1 andthat capital fully depreciates each period.

These conditions are sufficient to guarantee a closedform solution to the planning problem:

ct ¼ ð1� abÞkat lt ; and

ktþ1 ¼ it ¼ abkat lt

where we identify as investment, it, the capital stock heldover for production in period t þ 1. These allocations arePareto optimal.

We will show that the investment and consumptionpolicy functions arising as a solution to this problem maybe regarded as the aggregate investment and consump-tion functions arising from a decentralized homogenousconsumer economy.

We first qualitatively describe the RCE underlying thismodel, and then demonstrate the relevant equilibriumprice and quantity functions explicitly. The one capitalgood is assumed to produce two goods – a consumergood and an investment (capital) good. At the beginningof each period, firms observe the shock to productivity(lt) and purchase capital and labour from individuals atcompetitively determined rates. Both capital and labour

recursive competitive equilibrium 5451

are used to produce the two output goods. Individuals usetheir proceeds from the sale of capital and labour servicesto buy the consumption good (ct) and the investmentgood (it) at the end of the period. This investment good isused as capital (ktþ1) available for sale to the firm nextperiod and the process continues recursively.

To cast this problem formally as a recursive compet-itive equilibrium, we introduce some additional notation.Let kt denote the capital holdings of a particular (measurezero) individual at time t, and kt the distribution ofcapital amongst other individuals in the economy. Thislatter distinction allows us to make formal the compet-itive assumption: all the economic participants willassume that kt is exogenous to them and that the pricefunctions depend solely on this aggregate (in addition tothe technology shock). Clearly, in equilibrium, kt ¼ ktfor our homogeneous consumer economy. In addition,let pi, pc and pk be the price of the investment, con-sumption and capital goods respectively and pl be thewage rate. These prices are presumed to be functions ofthe economy-wide state variables exclusively and all par-ticipants take these prices as given for their own decisionmaking purposes. The ‘state variables’ characterizing theeconomy are ðk; lÞ and the individual are ðk; k; lÞ.

We use the symbols (c, i, k, l) to denote points in the‘commodity space’ for the firm and the consumer. The cin the commodity point of the firm is a function spec-ifying the consumption good supplied by the firm and iswritten as csðkt ; ltÞ. Similarly, the c in the commoditypoint of the individual is the amount of the consumptiongood demanded by the individual and is written ascdðkt ; kt ; ltÞ. In equilibrium (as mentioned earlier, inequilibrium kt ¼ kt), since the market clears, of coursecs ¼ cd. The same comments apply to the other elementsof the commodity point.

In the decentralized version of this economy, theproblem facing a typical household is

vðk0; k0; l0Þ ¼ max EXN

t¼0

bt ln cdðkt ; kt ; ltÞ( )

(P2)

subject to

pcðkt ; ltÞ cdðkt ; kt ; ltÞ þ piðkt ; ltÞ idðkt ; kt ; ltÞ� pkðkt ; ltÞ ksðkt ; kt ; ltÞ þ plðkt ; ltÞ lsðkt ; kt ; ltÞ

ktþ1 � ksðktþ1; ktþ1; ltþ1Þ ¼ idðkt ; kt ; ltÞ,lsðkt ; kt ; ltÞ � 1

and

ktþ1 ¼ cðkt ; ltÞ

is the law of motion of the aggregate capital stock.

With capital and labour priced competitively eachperiod, the firm’s objective function is especially simple –maximize period profits. The firm’s problem then is

max fpcðkt ; ltÞ csðkt ; ltÞ þ piðkt ; ltÞ isðkt ; ltÞ� pkðkt ; ltÞ kdðkt ; ltÞ � plðkt ; ltÞ ldðkt ; ltÞg

subject to

cst þ is

t � ltðkdt Þ

aðldt Þ1�a.

Via Bellman’s principle of optimality, the recursive rep-resentation of the individual’s problem P2 is

vðkt ; kt ; ltÞ ¼ maxfcd ;id ;ls;kdg ln ðcdðkt ; kt ; ltÞÞ�

þ bZ

vðidðkt ; kt ; ltÞ;cðkt ; ltÞ; ltþ1ÞdFðltþ1jltÞg

subject to

pcðkt ; ltÞ cdðkt ; kt ; ltÞ þ piðkt ; ltÞ idðkt ; kt ; ltÞ� pkðkt ; ltÞ ksðkt ; kt ; ltÞ þ plðkt ; ltÞ lsðkt ; kt ; ltÞ

ktþ1 � ksðktþ1; ktþ1; ltþ1Þ ¼ idðkt ; kt ; ltÞ,lsðkt ; kt ; ltÞ � 1

and

ktþ1 ¼ cðkt ; ltÞ

is the law of motion of the aggregate capital stock.The firm of course, simply maximizes its period profits

and hence does not have a multiperiod problem.The following functions that are a solution to the

individual and firm maximization problem above satisfythe definition of recursive competitive equilibrium:

1. A value function vðk0; k0; l0Þ ¼ EP

N

t¼0bt ln ½ð1� abÞ

�

ltka�1t faðkt � ktÞ þ ktg�g. It can be shown that

vðk0; k0; l0Þ ¼ Aþ Blnk0 þ Clnl0 where A, B and Care constants which are functions of the preference andtechnology parameters.

2. A continuous pricing function pðkt ; ltÞ ¼ fpcðkt ; ltÞ;piðkt ; ltÞ; pkðkt ; ltÞ; plðkt ; ltÞg that has the samedimensionality as the commodity point, where

pcðkt ; ltÞ ¼ piðkt ; ltÞ ¼ 1

(We have chosen the consumption good to be thenumeraire.)

pkðkt ; ltÞ ¼ altka�1t

plðkt ; ltÞ ¼ ð1� aÞltka�1t .

3. Consumption and investment functions for the indi-vidual that are a function of the current state of theindividual ðk; k; lÞ

5452 recursive competitive equilibrium

cdðkt ; kt ; ltÞ ¼ ð1� abÞltka�1t faðkt � ktÞ þ ktg

lsðkt ; kt ; ltÞ ¼ 1

idðkt ; kt ; ltÞ ¼ abltka�1t faðkt � ktÞ þ ktg

ksðktþ1; ktþ1; ltþ1Þ ¼ idðkt ; kt ; ltÞ.4. Decision rules for the firm that are contingent on the

state of the economy ðk; lÞ

csðkt ; ltÞ ¼ ð1� abÞltkat ,

ldðkt ; ltÞ ¼ 1,

isðkt ; ltÞ ¼ abltkat ,

kdðktþ1; ltþ1Þ ¼ isðkt ; ltÞ.

5. The law of motion for the capital stock specifying thenext period capital stock as a function of the currentstate of the economy ðkt ; ltÞ

ktþ1 ¼ cðkt ; ltÞ ¼ abltkat .

6. The consumption and investment decisions of the in-dividual csðk; k; lÞ, lsðk; k; lÞ and isðk; k; lÞ maximizethe expected utility subject to the budget constraint.So that

vðkt ; kt ; ltÞ ¼ ln ðð1� abÞltka�1t

� ðaðkt � ktÞ þ ktÞÞ

þ bZ

vðabltka�1t ðaðkt � ktÞ þ ktÞ;

� abltkat ÞdFðltþ1jltÞ.

7. The decision rules of the firm cdðkt ; ltÞ, ldðkt ; ltÞ,idðkt ; ltÞ maximize firm profit.

Demand equals supply

cdðktþ1; ktþ1; ltþ1Þ ¼ csðkt ; ltÞ;lsðktþ1; ktþ1; ltþ1Þ ¼ ldðkt ; ltÞ

and isðktþ1; ktþ1; ltþ1Þ ¼ idðkt ; ltÞ.

The law of motion of the representative consumers cap-ital stock is consistent with the maximizing behaviour ofagents cðkt ; ltÞ ¼ idðkt ; kt ; ltÞ. It is readily demonstratedthat since vðk0; k0; l0Þ ¼ wðk0; l0Þ, the competitiveallocation is Pareto optimal. See eqs (P1) and (P2).

Having formulated expressions for the prices of thevarious assets and their laws of motion, it is a relativelysimple matter to calculate rates of return (price ratios)and study their dynamics. For an application to riskpremia, see Donaldson and Mehra (1984).

Some researchers have formulated models that can becast in this same recursive setting, yet whose equilibriaare not Pareto-optimal. As a consequence, the model’s

equilibrium can no longer be obtained as the solution toa central-planning-optimum formulation. These modelsincorporate various features of monetary phenomena,distortionary taxes, non-competitive labour marketarrangements, externalities, or borrowing-lending con-straints. Besides increasing general model realism, suchfeatures enable the models not only to better replicatethe stylized facts of the business cycle, but also to providea rationale for interventionist government policies.Monetary models of this class include those of Lucasand Stokey (1987, a monetary exchange model) andColeman (1996, a monetary production model). Bizerand Judd (1989) and Coleman (1991) present models inwhich non-optimality is induced by tax distortions, whileDanthine and Donaldson (1990) present a model inwhich non-optimality results from efficiency-wage con-siderations. In these models, equilibrium is characterizedas an aggregate-consumption and an aggregate-invest-ment function which jointly solves a system of first-orderoptimality equations on which market-clearing condi-tions have been imposed. Coleman (1991) provides awidely applicable set of conditions under which thesesuboptimal equilibrium functions exist. As already noted,however, these optimality conditions cannot, in general,characterize the solution to an optimum problem.

RAJNISH MEHRA

See also Arrow–Debreu model of general equilibrium; decen-

tralization; neoclassical growth theory; real business cycles.

Bibliography

Bewley, T. 1972. Existence of equilibria in economies withinfinitely many commodities. Journal of Economic Theory4, 514–40.

Bizer, D. and Judd, K. 1989. Taxation and uncertainty.American Economic Review Papers and Proceedings 19,331–6.

Brock, W.A. and Mirman, L.J. 1972. Optimal economicgrowth and uncertainty: the discounted case. Journal ofEconomic Theory 4, 497–513.

Cass, D. 1965. Optimal growth in an aggregative model ofcapital accumulation. Review of Economic Studies 32,233–40.

Coleman, W.J. 1991. Equilibrium in a production economywith an income tax. Econometrica 59, 1091–104.

Coleman, W.J. 1996. Money and output: a test of reversecausation. American Economic Review 86, 90–111.

Danthine, J.P. and Donaldson, J.B. 1990. Efficiency wagesand the business cycle puzzle. European Economic Review34, 1275–301.

Donaldson, J.B. and Mehra, R. 1983. Stochastic growth withcorrelated production shock. Journal of Economic Theory29, 282–312.

Donaldson, J.B. and Mehra, R. 1984. Comparative dynamicsof an equilibrium intertemporal asset pricing model.Review of Economic Studies 51, 491–508.

recursive competitive equilibrium 5453

Harris, M. 1987. Dynamic Economic Analysis. New York:Oxford University Press.

Kydland, F.E. and Prescott, E.C. 1982. Time to build andaggregate fluctuations. Econometrica 50, 1345–71.

Ljungqvist, L. and Sargent, T.J. 2004. RecursiveMacroeconomic Theory, 2nd edn. Cambridge, MA: MITPress.

Long, J.B., Jr. and Plosser, C.I. 1983. Real business cycles.Journal of Political Economy 91, 39–69.

Lucas, R.E., Jr. and Stokey, N. 1987. Money and interest in acash advance economy. Econometrica 55, 491–513.

Mehra, R. 1988. On the existence and representation ofequilibrium in an economy with growth and non-stationary consumption. International Economic Review29, 131–5.

Mehra, R. and Prescott, E.C. 1977. Recursive competitiveequilibria and capital asset pricing. In R. Mehra, Essays infinancial economics. Doctoral dissertation, CarnegieMellon University.

Mehra, R. and Prescott, E.C. 1985. The equity premium: apuzzle. Journal of Monetary Economics 15, 145–62.

Prescott, E.C. and Lucas, R.E., Jr. 1972. A note on pricesystems in infinite dimensional space. InternationalEconomic Review 13, 416–22.

Prescott, E.C. and Mehra, R. 1980. Recursive competitiveequilibria: the case of homogeneous households.Econometrica 48, 1365–79.

Stokey, N., Lucas, R.E. and Prescott, E.C. 1989. RecursiveMethods in Economic Dynamics. Cambridge, MA:Harvard University Press.

recursive contractsIn contract theory it is standard to introduce a partic-ipation constraint (PC) insuring that the contract offeredto the agent delivers a utility higher than the best outsideoption. In a dynamic set-up agents may abandon thecontract at any point in time, even after the contract hasbeen in place for a while. For example, workers can leavea labour contract at almost no cost, or a borrower canstop repaying the loan if he or she declares bankruptcy.The possibility that the agent does not continue with theplan of the contract is usually called ‘default’. Hence, in adynamic context, it is natural to require that the PC issatisfied in all periods, in order to avoid default.

It turns out that, if a PC in all periods and realizationsis introduced in the design of the optimal contract,standard dynamic programming does not apply, theBellman equation does not hold, and the solution is notguaranteed to be a time-invariant function of the usualstate variables. This complicates enormously the solutionof these models.

To discuss this in a simple risk-sharing model, considertwo agents i = 1,2 with utility function E0

PN

t¼0btuðci

tÞ,where bA(0,1) is the discount factor and u the instan-taneous utility. Each agent receives a stochastic

endowment wit and the realization of endowments is

known both to the agents and the principal. The principalhas full commitment, and will stick to his announcedplan. Endowments provide the only supply of consump-tion good so that the following feasibility condition holds

c1t þ c2

t ¼ w1t þ w2

t (1)

A Pareto-optimal risk-sharing contract (implementedby a competitive equilibrium under complete markets)

would setu0ðc1

t Þu0ðc2

t Þconstant for all periods, so that agents

would share all idiosyncratic risks. This allocation wouldbe chosen as the optimal contract if agents would com-mit to never leave the risk-sharing arrangement. We referto this allocation as the first best. The optimum satisfiesthe usual recursive structure in dynamic models, namely,that ct = F(wt) where F is a time-invariant function andwt ¼ ðw1

t ;w2t Þ.

Assume now agents cannot commit to staying in thecontract for ever. An agent can leave the contract andconsume for ever his individual endowment, so that acontract can only be implement if it satisfies

Et

XN

j¼0

bjuðcitþjÞ Va

i ðwtÞ

at all periods and realizations, where Vai ðwtÞ � Et

PN

j¼0bjuðwi

tþjÞ is the utility of consuming in autarchy for everafter t.

It is clear that the above PC is likely to be violated bythe first best allocation. In periods when wi

t is high, theright side of the PC is high, but the agent has to sur-render a large part of his endowment in the first best andthe left side of the PC is too low. Therefore, PC’s are oftenbinding and they make the first best unfeasible.

A Pareto-optimal risk-sharing contract with PC’s cannow be found by maximizing the weighted utility of thetwo agents E0

PN

t¼0bt ½luðc1

t cÞ þ ð1� lÞuðc2t Þ� subject to

the above PC for all periods and realizations and for bothagents. The parameter l indexes all such Pareto-optimalallocations. The result is an optimal contract under fullcommitment by the principal and partial commitment bythe agents.

The Bellman equation does not give the solution tothis problem. A key feature of standard dynamic pro-gramming is that the set of feasible actions must dependonly on variables that were determined last period andthe current shock. But it is possible to evaluate if a certainconsumption level ci

t satisfies the PC at time t only iffuture plans for consumption are known.

Intuitively, a promise of higher consumption in thefuture makes a lower consumption today compatible withthe PC. But in order to implement this plan the principalhas to ‘remember’ all the promises for higher consump-tion that were made in the past. Therefore, the optimalsolution is unlikely to be a function of only today’s

5454 recursive contracts