Increasing Returns and Optimal Oscillating Labor Supply

Increasing Returns and Optimal Oscillating Labor Supply (cid:3) William D. Dupor Andreas Lehnert The WhartonSchool Board ofGovernors ofthe UniversityofPennsylvania Federal Reserve System 2300Steinberg Hall-DietrichHall MailStop 93 Philadelphia, PA19104 Washington,DC 20551 (215)8987634 (202)452-3325 Dupor@wharton.upenn.edu Andreas.Lehnert@FRB.GOV This Version: April2002 LastRevised: April17,2002 (cid:3) DuporthankstheHooverInstitutionforitssupportandhospitality.Theviewsexpressedinthe paperareoursaloneanddonotnecessarilyreflectthoseoftheBoardofGovernorsoftheFederal ReserveSystemoritsstaff.TheauthorswouldliketothankLarryChristiano,JoaoGomes,Robert Hall, Ken Judd, Kevin Lansing, Per Krusell and Robert Shimer for helpfulconversations. This paper has additionallybenefitted from the commentsof seminar participantsat the 2001NBER summerinstituteandtheFederalReserveBoard.

Increasing Returns and Optimal Oscillating Labor Supply Abstract Models featuring increasing returns to scale in at least one factor of production havebeenusedtostudytwoseparatephenomena: (1)multiplicityofself-fulfilling rational expectations equilibria (i.e. sunspots), and (2) production schedules that optimallyfeaturebunching. Weshowinacontinuous-timemodelwithincreasing returnstolabor(IRL)thatiftheeconomyfeaturesmultiplecompetitiveequilibria, theoptimalpathofinvestment,employmentandconsumptioncannotbeconstant, or even smoothly-varying. Any macroeconomic policies that shielded the economyfromsunspotfluctuationswouldnecessarilynotbeoptimal. Wethencharacterize the optimal allocation (the solution to the planner’s problem) in a discrete time version of the model. We find that the optimal investment, employment and consumption policies under increasing returns can feature (1) discontinuous jumps, (2) endogenous cycles (with time-varying cycle limits) and (3) stochastic controls(lotteries). Ourdiscrete-timemodelisveryclosetothatstudiedbyChristianandHarrison(1999);they,however,findthatfluctuationsarenotoptimal. We showthat thisdifference is drivenby theirassumptionthat productionis linearin capital. JournalofEconomicLiteratureclassificationnumbers: E32,E33,C61,C62,D62 Keywords: Increasing returns,externalities,fluctuations,lotteries

1 Introduction Economistsroutinely use dynamicmodels which feature many competitiveequilibria to study macroeconomic cycles. In these models, severe swings in output and employment can follow phenomena that have no fundamental economic significance(usuallyreferred toassunspots). Research hasnaturallyfocusedtheexistenceandnatureofendogenousequilibriumcycles,aswellasondesigningfiscal and monetary policies to rule them out. However, despite their ultimate goal of formulating macroeconomic policies, very few papers have explicitly studied the optimalpathsofemploymentandinvestment. Inthecontextofapopularmultipleequilibriummodel, we show that thenecessary conditionsfor multipleequilibria are sufficient to preclude a smooth optimal path for investment and labor. Any policy that successfully eliminated equilibriumcycles would thus necessarily not beoptimal. The model we study involves increasing returns to labor to generate multiple equilibria. This is a popular choice in the literature, and leads to a natural interpretation of multiple equilibria. Intuitively, consider the simple case of a static model in which many identical households must decide how hard to work. The production function features increasing social returns but constant privatereturns tolabor. Ifhouseholdsexpectahighwage,theysupplyhighlabortothemarket. If theyexpectalowwage,theysupplylesslabortothemarket. Becausethemarginal product of labor is increasing in aggregate labor supply, in a symmetric equilibriumhouseholds’expectationsare confirmed. Thus eitherset of expectationsand 1

correspondingactionssatisfytheconditionsforan equilibrium. Moresophisticateddynamicmodelsexploringtheconnectionsamongincreasingreturns,expectationsandequilibriumindeterminacyincludeKiyotaki’s(1988) study of equilibrium investment and government policy; Murphy, Vishny and Shleifer’s (1989a) argument that externalities must play a key role in business cycles; Baxter and King’s(1991) study of externalities and business cycles; Benhabib and Farmer’s (1994, 1996) and Farmer and Guo’s (1994, 1995) studies of “animalspirits”models;BoldrinandRustichini’s(1994)studyoftwo-sectormodelswithexternalitiesandSubrahmanyamandTitman’s(1999)studyoftheroleof equityprices asan equilibriumselectionmechanism. A different literature has pursued the implications of increasing returns on optimal production policies. In a spatial context, for example, economists have studied how small initial differences across locations can powerfully affect the long-run distribution of industries.1 In the same way that production bunching across space might be optimal under increasing returns, in a dynamic setting, it may be optimal to bunch production across time. That is, a social planner, faced with increasing returns to labor, might optimally direct households to work hard forbriefperiodsoftimeandthenrelax. Duringthespurtsoflabor,householdsaccumulatealargecapitalstockwhichtheythenconsumeduringtherelativelyquiet time. If households are sufficiently willing to substitute leisure between periods, andhavesomemeansofstoringoutputfromoneperiodtothenext,theywilltake 1We can at best merely scratch the surface of the large literature on increasing returns, spillovers and the development of countries and cities; see, e.g. Murphy, Vishny, and Shleifer (1989a),Lucas(1993)orKrugman(1999). 2

advantage of increasing returns to labor and work hard when labor effort is already high,becausethemarginalproductoflaborisincreasing. Murphy,Shleifer and Vishny (1989a, p. 250) recognize this point in a durable consumption goods model with increasing returns to labor, but without capital: “[i]t is efficient for thisindustrytoproduceatcapacitysomeofthetimeandtorestothertimes,rather thantoalways produceataconstantoutputlevel.” A complementary,data-driven view is that business cycles are largely driven by inventory cycles. For example, inventoryadjustmentsaccountfor87percent ofthedropinGNPduringtheaveragepost-warU.S. recession (Blinderand Maccini, 1991). Further, therehas been increasinginterestrecentlyinspillovers,orexternalitiesinlaborproductivity,asa partofthe“new”economy. Thusthetypeofcycledrivenby laborexternalitiesis notutterlyunlikethekindofbusinesscyclethatcharacterizes theU.S. economy. Despite this natural intuition and a decade’s worth of theoretical and applied interest, there have been few efforts to characterize optimal resource allocations underincreasingreturnstolaborintheneoclassicalgrowthmodel.2 Instead,most work has focused on designing policies to eliminate equilibrium cycles in the model. Putanotherway,researchhasconcentratedontheconditionsunderwhich competitive equilibria are unique, generally ignoring the optimal plan. For example, Guo and Lansing (1998)show that a progressivetax can render otherwise indeterminateequilibrialocallyunique,shieldingtheeconomyfromsunspotfluctuations. In this paper, we characterize the connection between equilibrium indetermi- 2Wewillconsideroneexception,ChristianoandHarrison(1999),ingreatdetailbelow. 3

nacy and optimalcycles in the neoclassical growthmodel with increasing returns to labor. We begin by finding the equilibria of a continuous time neoclassical growth model featuring increasing social returns to labor but diminishing social returns to capital. FollowingBenhabib and Farmer (1994), we show that increasing returns to labor in production is a necessary condition for local equilibrium indeterminacy of the steady-state equilibrium. We establish that this necessary condition for indeterminacy is sufficient to ensure that the optimal paths of labor and investment cannot be constant. In fact, the optimal path cannot even be smoothly varying because we show that, if it exists, the path is not a piecewise continuousfunctionoftime. Insteadofattemptingtocharacterizetheoptimalallocationinthiscase(forexample,byimposingadjustmentcostsorrecasting theproblemas oneofchoosing aprobabilitydistributionovereffortlevels)wemoveontostudythediscrete-time versionofthemodel. Indiscrete–asopposedtocontinuous–time,theplannermust chooseonelevelforeffort andmaintainitthroughouttheperiod. Wefind thatthe planner’s optimal labor policy (as a function of beginning of period capital) generally features a sharp drop at a critical capital level, so that when the economy is richer than the critical point, optimal labor effort drops to zero. At this critical point,householdsconsumesomeoralloftheexistingcapitalstock. Ifthiscritical pointliesalongtheoptimaldynamiccapitalaccumulationpath,theeconomywill feature oscillating labor supply and the capital stock will cycle over time. Moreover, we find that, although the planner could completely smooth consumption over time, he chooses not to. Intuitively,the shadow price of consumption varies 4

witheffortand investment,leadingtheplannerto tradeofftheutilitybenefits and thetechnologicalcostsofsmoothing. Even aside from the single discontinuous drop down in effort at the critical levelof capital, theoptimallabor effort schedulecan feature several separate discontinuous jumps up at lower levels of capital. In essence, the increasing returns tolaborcausethehousehold’sutilityasafunctionofinvestmenttofeatureseveral local maxima. As beginningofperiod capitalincreases, theglobalmaximumcan hop among these various local maxima, and their associated investment policies. Thus not only is it possible for the optimal path of labor to cycle over time, the limitsofthecyclecan, as aresultofthesesharp jumpsup,also cycleovertime. Finally, we show that, under certain circumstances, the planner prefers using stochastic policies or capital gambles. This makes sense; our results are driven by the fact that the planner’s static return function features a sharp kink at the point where optimal labor effort drops from a positive level to zero. Faced with an initialcapitalstock withinthekinkedregion,theplannerwouldprefer togamble over a high capital stock and a low capital stock rather than use the inherited capital stock. Note however that the planner does not always want to use lotteries; by cleverly choosing non-stochastic policies, the planner will, under certain parameters, beableto comearbitrarilyclosetothesolutionwithlotteries. The discrete-time version of our model is close to that studied by Christiano andHarrison(CH, 1999),whoalso characterizetheoptimalplan. They,however, find that labor supply is smooth in the capital stock, that production cycles are not optimal and, as a result, that the planner would not use stochastic controls if 5

offeredthem. Thekeydifferencebetweenourmodelandtheirsisthesocialreturn to capital. We assume that social returns to capital are decreasing; in contrast, they assume that social returns to capital are constant. In essence, in our model, with decreasing returns to capital, the planner finds it optimal to work very hard today in order to build up a large capital stock. In the next period, although the marginal product of labor will be higher (because the capital stock is higher) it will be less than linearly higher, as in CH’s model. If the planner in CH’s model attempted to replicate the optimal plan from our model by working hard today, he would find that, tomorrow, labor’s productivity would be linearly higher than today’s. But labor’s productivity is also the opportunity cost of taking vacations, andtheplannerwouldfindavacationtooexpensivefollowingaburstofeffort. In ourmodel,labor’sproductdoesnotincreaseone-for-onewiththecapitalstockas itdoesintheirmodel. Beyondsomecriticalpoint,labor’sproductdipsenoughso thattheplannerprefers totakeavacationand consumeleisure. The plan of our paper is as follows: in section 2, we develop the implications of equilibrium indeterminacy for the nature of optimal labor supply paths in the continuoustimeBenhabib-Farmermodel. Insection3,westudyoptimalresource allocations in a discrete-time model. We characterize the planner’s static return function and develop a simplified finite-horizon model to provide some intuition forourresults. Wealsoprovideresultslinkingincreasingreturnstoscaleinlabor to stochastic controls. In section 4 we numerically characterize the discrete-time planner’s problem. We choose a benchmark set of parameters and then vary all the parameters of interest around the benchmark. Cycles are more likely to be 6

optimalwithlargeincreasingreturnstolabor,slowcapitaldepreciationandalow utility time discount rate. We also compute the optimal policies with lotteries. Section 5briefly concludes. All proofsofourtheorems andlemmasare relegated toappendixA.InappendixBweprovideacomparisonwiththeexactChristiano- Harrisonspecification. 2 Increasing Returns in Continuous Time 2.1 IncreasingReturnstoLaborAreNecessaryforLocalEquilibrium Indeterminacy Consider the neoclassical growth model with external returns to scale in production. A representative household maximizes the present discounted valued of a log-linearfelicityfunctiondefinedintermsofflowsofconsumption( c )andlabor effort( n ). Thediscountrateis (cid:26) > 0 , sothemaximizationproblembecomes: Z 0 1 e (cid:0) (cid:26) t [ l o g ( c ) (cid:0) (cid:30) n ] d t ; subjectto: _k = ( r (cid:0) Æ ) k + w n + (cid:25) (cid:0) c ; (2.1) andaninitialcondition k ( 0 ) = k 0 > 0 . Asusual, k denotesthelevelofaggregate capital. Here, the scalar parameters (cid:26) ; Æ ; and (cid:30) are all strictly positive and the 7

householdtakes thereal interestrate r , wage w and profits (cid:25) as given. Firm-levelproductionis givenby: y = k a n g h k (cid:11) (cid:0) a n 1 + (cid:13) (cid:0) g i wherethe scalarparameters a ; g ; (cid:11) ,and (cid:13) satisfy a ; g > 0 , 1 > (cid:11) > a , 1 + (cid:13) > g and a + g = 1 . Here, k ; n denote firm-level capital and labor usage, while k ; n aretheeconomy-wideaverages. Theparameters a ; g governtheprivatereturnsto capital and labor (as opposed to the social returns). The social (total) production functionisgivenby: y = k (cid:11) n 1 + (cid:13) : (2.2) Firmsmaximizeprofits,takingasgivenfactorpricesandeconomy-wideaverageinputuse. Profit-maximizationimplies: y = k = r = a ; (2.3) and: y = n = w = g : (2.4) By assumingconstant privatereturns to scale ( a + g = 1 ), firm profits equal zero inacompetitiveequilibrium. 8

Thenecessary conditionsforconsumeroptimizationare: c = w (cid:30) ; (2.5) _c c = r (cid:0) (cid:26) (cid:0) Æ : (2.6) Also,thepathsofcapitalandconsumptionmustsatisfyatransversalitycondition: T l i ! m 1 e (cid:0) (cid:26) T k c ( ( T T ) ) = 0 : (2.7) For the remainder of the analysis, we study symmetric equilibria, in which the aggregate quantities of capital and labor, k and n , are given by the representative household’soptimalchoicewhen ittakes k and n asgiven. Usetheprofit-maximizationconditions,equations(2.3)and(2.4),toeliminate factorprices r and w fromthenecessaryconditionsfromtheconsumer’sproblem, equations(2.5)and (2.6): c = g (cid:30) y n ; (2.8) _c c = a y k (cid:0) (cid:26) (cid:0) Æ : (2.9) After substituting out factor prices, the law of motion for capital, equation (2.1), becomes: _k = y (cid:0) Æ k (cid:0) c : (2.10) 9

Equations (2.2), (2.8), (2.9), (2.10), along with the transversality condition (2.7), describetheequilibriumdynamicsoftheeconomy. We can use the production function, equation (2.2), and the relationship of consumption to the output-labor ratio, equation (2.8), to substitute output y and labor n out of the differential equations governing the evolution of consumption (2.9)andcapital (2.10). Begin byeliminatinglaboreffortfrom output,so that: y = k (cid:0) (cid:11) = (cid:13) (cid:18) (cid:30) g c (cid:19) ( 1 + (cid:13) ) = (cid:13) : Thus equilibriumconsumptionand capital jointlyevolveaccording to the system ofequations: _c c = a (cid:18) (cid:30) g (cid:19) 1 + (cid:13) (cid:13) k (cid:0) (cid:11) + (cid:13) (cid:13) c 1 + (cid:13) (cid:13) (cid:0) (cid:26) (cid:0) Æ ; (2.11) _k = (cid:18) (cid:30) g (cid:19) 1 + (cid:13) (cid:13) k (cid:0) (cid:11) (cid:13) c 1 + (cid:13) (cid:13) (cid:0) Æ k (cid:0) c : (2.12) Wenextturnto determiningthenatureofequilibriagenerated bythissystem. We begin by log-linearizing the system. First, define C = l o g ( c ) and l o g ( k ) K = . The dynamicsystem, equations (2.11) and (2.12), can be rewritten as the autonomousdifferentialequations: C _ = a e x p ( D 0 (cid:0) D 1 K + D 2 C ) (cid:0) (cid:26) (cid:0) Æ (2.13) K _ = e x p ( D 0 (cid:0) D 1 K + D 2 C ) (cid:0) Æ (cid:0) e x p ( C (cid:0) K ) : (2.14) 10

Forconvenience,wedefined theconstants D 0 ; D 1 and D 2 as: D 0 = 1 + (cid:13) (cid:13) l o g (cid:18) (cid:30) g (cid:19) ; D 1 = (cid:11) + (cid:13) (cid:13) ; and D 2 = 1 + (cid:13) (cid:13) : Every path for capital and consumption that satisfies the transversality condition, equation(2.7),thesystemofequations(2.13)and (2.14),and theinitialcondition for capital is a perfect foresight equilibrium. It is straightforward to show that there is a unique steady-state equilibrium. Define K s s and C s s as the equilibriumsteady-statelevelsofcapitalandconsumption. Weexploitthefollowingtwo steady-stateimplicationsofequations(2.13)and (2.14): e x p ( D 0 (cid:0) D e 1 x K p s ( s C + s s D (cid:0) 2 C K s s s s ) ) = = (cid:26) (cid:26) + a + a Æ Æ ; (cid:0) Æ : We next wish to study the local determinacy properties of the steady-state equilibrium. Linearizingthesystemequilibriumequationswrittenintermsoflog capitaland logconsumption,equations(2.13)and (2.14), wehave: 2 6 4 C K _ _ 3 7 5 = 2 6 4 ( (cid:26) + Æ Æ + ) 1 (cid:26) + (cid:16) a (cid:13) + (cid:13) Æ (cid:13) ; (cid:17) ; (cid:0) ( (cid:0) (cid:26) h + Æ Æ + ) (cid:11)(cid:16) 1 ( (cid:26) (cid:13) + + a Æ ) (cid:11) (cid:13) i (cid:17) 3 7 5 2 6 4 C K (cid:0) (cid:0) C K s s s s 3 7 5 (2.15) Let A denotethesquarematrixin(2.15). Wearereadytopresentaresultconcerningindeterminacywhich was first workedoutby Benhabib and Farmer(1994). 11

Theorem 1 The ( C s s ; K s s ) equilibriumis locallyindeterminateif andonlyif: 1. Returnstolaborareincreasing: (cid:13) > 0 , 2. Thetraceof thetransitionmatrix A is negative,or: Æ > ( (cid:26) + Æ ) [ ( 1 + (cid:13) ) a (cid:0) (cid:11) ] = ( (cid:13) (cid:11) ) : Theorem 1 holds that a necessary condition for equilibrium indeterminacy is social returns to labor that are increasing in scale (in other words, that (cid:13) > 0 ). This is intuitive; as Benhabib and Farmer (1994) explain, when (cid:13) > 0 , aggregate labor demand is upward-sloping and, in fact, steeper than aggregate labor supply athighlevelsoflabor. Further,when (cid:13) > 0 itisalsopossiblethattheequilibrium is a source, so that any small deviation from the steady-state will result in an infiniteexpansion. Forexample,weassumedthat theprivatereturnsto capital, a , arelessthanthesocialreturns, (cid:11) . Ifinsteadweassumethat a = (cid:11) ,thetraceofthe transition matrix A becomes tr ( A ) = (cid:26) > 0 (violating condition 2 in theorem 1). As a result, without a wedge between social and private returns to capital (even thoughbothare decreasing inscale), theequilibriumwillbelocallyexplosive. Requiringawedgebetweenthesocialandprivatereturnstocapitalisthesame as assuming that at least some of capital’s product is external to the firm. These capital externalities militate against explosiveness of the equilibrium dynamics; theybringthesystembacktothesteadystateafteranexpectationsshock. Imagine 12

theeconomybeginswithcapitalatthesteady-statelevel. Attimezero,households develop expectations that their neighbors will work harder than the steady-state effort level, both in the instant t = 0 , and for at least a short whilethereafter. Associatedwiththisexpectationofthepathofaveragelabor,householdsalsoexpect agradualriseandthendeclineinaveragecapitalholding. Expectinghighershortrun(laborandcapital)productivity,householdsincreaseownlaborsupply,which validatestheir expectationsof higheraverage labor input, and increase savings to smooth consumption. The increased savings, in addition, augments the marginal productivity of labor in the future. If the household fully internalized the returns tocapital ( a = (cid:11) ), itsfuturepathofcapital wouldcontinueto increase. However, because of the wedge in capital’s product, the average household persistently undersaves,allowingcapital todecline. 2.2 Increasing Returns to Labor Are Sufficient to Preclude a Smooth Optimal Path for Labor Inadditiontobeinganecessary conditionforequilibriumindeterminacy,increasing returns also has implicationsfor the Pareto optimal resource allocation in the Benhabib-Farmermodel. Tostudytheoptimalallocation,weusesomeofthetheoretical apparatus of Pontryagin’s Maximum Principle (PMP). We briefly review therequired resultsbefore turningourattentiontotheplanner’sproblem. 13

Considerthefollowinginfinitehorizonproblem(orIHP): u m ( t) a ; x 0 (cid:20) t Z 0 1 e (cid:0) (cid:26) t f (cid:0) x ( t ) ; u ( t ) (cid:1) d t ; (2.16) where x ( t ) is a scalar function which is free as t ! 1 and has an initial value given by x ( 0 ) = x 0 . The controls u ( t ) 2 U (cid:26) R m influence the evolution of the state x as: _x ( t ) = g (cid:0) x ( t ) ; u ( t ) (cid:1) : Wewillrequirethatunderalladmissibletrajectoriesofthecontrols u ( t ) 2 U , the integralintheobjective(2.16)converges.3 Thecurrent valueHamiltonianfortheinfinitehorizonproblemis: H c ( x ; u ; (cid:21) ) = (cid:21) 0 f ( x ; u ) + (cid:21) g ( x ; u ) (2.17) Wenowappeal totheresultsinPontryagin etal (1964,theorems1 and 2). Definition1 (The PontryaginMaximum Principle) If x ? ( t ) and u ? ( t ) are admissible paths for the IHP and are both piecewise optimal, then there exists a constant (cid:21) 0 anda continuousscalarfunction (cid:21) ( t ) suchthatforall t (cid:21) 0 : 1. (cid:21) 0 = 0 or 1 , and ( (cid:21) 0 ; (cid:21) ( t ) ) isnever ( 0 ; 0 ) . 3Thegimlet-eyedreaderwillhavenoticedthatwehavealsoassumedthattheobjective f and thecontrol g areautonomousfunctionsofthecontrolsandthestate. Thisismerelytotiethismore generaldiscussiontoourspecificproblem.Itisotherwiseinnocuous. 14

2. Forallcontrols u in U : H c (cid:0) x ? ( t ) ; u ; (cid:21) ( t ) (cid:1) (cid:20) H c (cid:0) x ? ( t ) ; u ? ( t ) ; (cid:21) ( t ) (cid:1) : 3. Atpointswhere u ? ( t ) is continuous: _(cid:21) (cid:0) (cid:26) (cid:21) = (cid:0) @ @ x H c ( x ? ; u ? ; (cid:21) ) : We next turn to theapplication of thePMP forthe particular IHP first studied byBenhabib and Farmer. Definition2 (Social Planner’s Problem) Theplannermaximizes: Z 0 1 e (cid:0) (cid:26) t (cid:20) l o g (cid:0) c ( t ) (cid:1) (cid:0) (cid:30) n ( t ) (cid:21) d t ; (2.18) where the control vector [ c ( t ) ; n ( t ) ] 0 (cid:21) 0 for all t . The capital stock k ( t ) (the single state variable) is free as t ! 1 and the initial capital stock is given by k ( 0 ) = k 0 > 0 . The dynamics of the capital stock are governed by the law of motion: _ k ( t ) = k ( t ) (cid:11) n ( t ) 1 + (cid:13) (cid:0) c ( t ) (cid:0) Æ k ( t ) : We assume, as in the decentralized model, that 0 < (cid:11) < 1 , Æ ; (cid:30) > 0 , and (cid:11) (cid:0) 1 (cid:13) (cid:21) . 15

Theorem 2 If (cid:13) > 0 ,theredoesnotexistapiecewisecontinuousoptimalpathfor consumptionandlaborforthesocialplanner’sproblemfromdefinition2 above. Theintuitionfortheorem2isstraightforward: condition(2)ofthePMPholds that for a given level and shadow value of the state variable, k ? ( t ) and (cid:21) ( t ) , the controlsmust be chosen to maximizethecurrent valueHamiltonian. However, at eachinstant,thesocialplannerfacesincreasingreturnstolaborbutlineardisutility of work. For a given productivity of labor ( k ? ) (cid:11) and shadow value of consumption (cid:21) , increasing returns to labor implies that there is no interior solution to the social planning problem at each instant. Indeed, every piecewise continuouspath for labor and investment is dominated by some other piecewise continuous path. Weconjecturethattheoptimalplanmayinvolveachatteringsolution,orarbitrarily frequent discontinuous jumps in the labor and investment time paths. Romer (1986) points out that such solutions have little economic meaning, and advises puttingmorestructureon problemswiththisfeature. Theparticularstructurethatweadoptistomoveouranalysistodiscretetime. This strategy has two benefits. First, moving to discrete time provides an elegant circumventionofthechatteringproblem. Inessence,wearerequiringtheplanner tocommittoaparticulareffortlevelforthelengthofperiod,inthespiritofStokey (1981). Thisforcestheoptimalplantobeapiecewisecontinuousfunctionoftime. Second,giventhatourresultsdifferfromthoseofChristianoandHarrison(1999), movingtodiscretetimewillpermitamoredirectcomparison. 16

3 Increasing Returns in Discrete Time In this section we describe the discrete time analog to the continuous time social planner’s problem from definition 2 above. We show that the static part of the planner’s objective function features a distinct kink, that the planner’s choices of investment and labor policies jump down at this kink and, further, that they can jump up to the left of this kink. We also show that the planner would use certain typesofstochasticcontrols(thatis,lotteries)ifpermitted. One reason that we study a discrete-time model is to make our results comparable with Christiano and Harrison, who study a similar model but find that the Pareto optimal policy involves a constant labor supply. Our model differs from theirs in three ways. The most important difference is that we assume diminishing–as opposed to constant–returns to scale in capital. The other differences are thefunctionalforms ofpreferences andtechnology;inparticular,we assumelinear disutilityof labor and returns to labor of n 1 + (cid:13) as opposed to n 2 . In appendix B we solve the exact Christiano-Harrison model with diminishing returns to capital and find that it produces endogenous optimal cycles. Finally, it is worth noting that Christiano and Harrison explicitly recognized the important role that constant returns to capital might have in their model: “We have shown that this [optimalityof constant labor input]is so under a particular homogeneity assumption on the resource constraint. But, standard models do not satisfy this condition(1999,p. 24).” 17

3.1 Model At the beginning of period t , the social planner faces the problem of choosing streamsoflaborandconsumptiontomaximizethehousehold’sutility: v t (cid:0) k t (cid:1) = f c +t m ;n i a x +t gi 1 =i 0 1 i= X 0 (cid:12) i (cid:2) l o g (cid:0) c t+ i (cid:1) (cid:0) (cid:30) n t+ i (cid:3) ; 0 < (cid:12) < 1 : (3.1) The economy-wide technology transforms periodt labor n t and capital k t into outputas: Y ( k t ; n t ) = k (cid:11) t n 1 t + (cid:13) : (3.2) Wewillgenerallytake 0 (cid:20) (cid:11) < 1 and (cid:13) (cid:21) 0 . However,notethatif (cid:0) 1 < (cid:13) (cid:20) (cid:0) (cid:11) the production function is Cobb-Douglas with decreasing (or at best constant) returnsto scaleincapital and labor. Thecapitalstock evolvesas: k t+ 1 = k (cid:11) t n 1 t + (cid:13) + ( 1 (cid:0) Æ ) k t (cid:0) c t : (3.3) In each period t , the planner maximizes the continuation utility (3.1) subject to thecapital evolutionequation(3.3). Wewillfinditmoreconvenientnottoworkwiththeplanner’ssequenceproblem, equation (3.1) above, but rather with the recursive version of the planner’s 18

problem. Writtenrecursively,theplanner’sproblembecomes: V ( k ) = m k a 0 ;c x ;n (cid:26) l o g ( c ) (cid:0) (cid:30) n + (cid:12) V (cid:0) k 0 (cid:1) (cid:27) ; (3.4) subject to: k 0 = k (cid:11) n 1 + (cid:13) + ( 1 (cid:0) Æ ) k (cid:0) c ; and k ; k 0 2 K = [ k m in ; k m a x ] ; k m in > 0 ; k m a x < 1 : Define B ( K ) as a space of bounded continuous functions f : K ! R (where R (cid:26) R ). Define thefunctionaloperator T : B ( K ) ! B ( K ) as: (cid:0) T f (cid:1) ( k ) = m n a ;k x0 l o g (cid:0) k (cid:11) n 1 + (cid:13) + ( 1 (cid:0) Æ ) k (cid:0) k 0 (cid:1) (cid:0) (cid:30) n + (cid:12) f (cid:0) k 0 (cid:1) : (3.5) In section A.5 below we establish that the solution to the recursive problem, the value function V ( k ) , is the same as the solution to the sequence problem, v t ( k ) for all periods t . Moreover, we show that the operator T satisfies the conditions foracontractionmapping. Thusunlikethecontinuoustimeversionofourmodel, the discrete time version will at least have some kind of solution that we have a hopeoffinding withstandardtechniques,evenunderincreasingreturns toscale. 3.2 The Static Return Function We now show that we can decomposethe recursiveproblem, equation 3.4 above, into a static piece and a dynamic piece; moreover, we show that the static piece is purely a function of initial capital and the choice of continuation capital. We analyse this static return function and show that it features a distinct set of kinks 19

at critical valuesof thecurrent capital stock. In oneregion, theplannerwill work hard;beyondthekinks,though,theplannerwillset laboreffortto zero. For a given level of capital today, k , imagine first fixing the choice of capital tomorrow, k 0 , and thendeciding howmuch towork today, n . This willgeneratea one-periodreturn function X (cid:0) k ; k 0 ; n (cid:1) defined as: X (cid:0) k ; k 0 ; n (cid:1) (cid:17) l o g (cid:18) k (cid:11) n 1 + (cid:13) + ( 1 (cid:0) Æ ) k (cid:0) k 0 (cid:19) (cid:0) (cid:30) n : (3.6) By bringingout the k (cid:11) term,thismay berewrittenas: X (cid:0) k ; k 0 ; n (cid:1) = (cid:11) l o g ( k ) + l o g (cid:18) n 1 + (cid:13) + ( 1 (cid:0) Æ k ) k (cid:11) (cid:0) k 0 (cid:19) (cid:0) (cid:30) n : Define: F (cid:0) k ; k 0 ; n (cid:1) = l o g (cid:0) n 1 + (cid:13) + h (cid:1) (cid:0) (cid:30) n ; where h (cid:17) ( 1 (cid:0) Æ k ) k (cid:11) (cid:0) k 0 : (3.7) Note that h can take on large negative or positive values. The planner’s goal, given k ; k 0 is to choose effort n to maximize F (cid:0) k ; k 0 ; n (cid:1) and with it the static return function X . Thus F (cid:0) k ; k 0 (cid:1) is: F (cid:0) k ; k 0 (cid:1) (cid:17) m a n x (cid:8) l o g (cid:0) n 1 + (cid:13) + h (cid:1) (cid:0) (cid:30) n (cid:9) : (3.8) Define n ? (cid:0) k ; k 0 (cid:1) as the optimal effort policy from this problem. The recursive 20

problem(3.4)nowbecomes: V ( k ) = m a0 k x (cid:8) (cid:11) l o g ( k ) + F (cid:0) k ; k 0 (cid:1) + (cid:12) V ( k 0 ) (cid:9) ; Wewillnowcharacterize F ,withthegoalofcharacterizing,totheextentpossible, V . Webeginwithatechnical result: Lemma 1 The functions F (cid:0) k ; k 0 (cid:1) and n ? (cid:0) k ; k 0 (cid:1) defined in equation (3.8) above satisfy: 1. Forall k ; k 0 in K (cid:2) K , F isboundedand continuous. 2. Forall k ; k 0 in K (cid:2) K , n ? isbounded. Wenextturn tocharacterizing theoptimaleffort policy: Lemma 2 Theoptimaleffortpolicy, n ? (cid:0) k ; k 0 (cid:1) satisfies: 1. If (cid:13) < 0 , n ? (cid:0) k ; k 0 (cid:1) > 0 all k ; k 0 in K (cid:2) K . 2. If (cid:13) > 0 there exists a scalar h m a x > 0 such that for all k ; k 0 in K (cid:2) K satisfying k 0 < (cid:0) h m a x k (cid:11) + ( 1 (cid:0) Æ ) k ,theoptimaleffortiszero, n ? ( k ; k 0 ) = 0 . 3. If (cid:13) > 0 andlaboreffortisgreaterthanzero, n ? > 0 ,thenlaboreffortmust beabovea positivethresholdlevel, n ? (cid:21) (cid:13) = (cid:30) > 0 . Lemma2essentiallystatesthatincreasingreturnstoscaleinlabor(thatis, 0 (cid:13) > )isanecessaryconditionfortheoptimalityoflaborcycles. Iftheoptimalchoice of labor is smoothly varying the problem collapses to the standard neoclassical 21

growthmodel. In the presence of increasing returns, however, theoptimal choice oflaborfeaturesadiscretejumpdown. Thiswillinduceasetofkinksinthestatic return function F (cid:0) k ; k 0 (cid:1) ; everywhere that n > 0 , F willhave arelativelyshallow slopein k (positive)and k 0 (negative),becausetheplannerisalsoadjustinglabor. Intheregionwhere n = 0 ,bycontrast,theslopewillberelativelysteeper,because consumptionwillbe ( 1 (cid:0) Æ ) k (cid:0) k 0 . In figures 1 and 2 below we plot the region in which the planner works hard and that in which he does not, and we also plot the static return function F for several values of (cid:13) . Notice the distinct kink in F for even small values of (cid:13) > 0 ; this kink disappears in the case where (cid:13) < 0 , that is, diminishingreturns to scale inlabor. Lemma 2 also shows that if (cid:13) > 0 there will some region of low, but nonzero,laboreffortthattheplannerwillsimplyneverchoose. Asheapproachesthis region,theplannerjumpsdownto zero. Thisalsounderliesthekinksinfigure 2. 3.3 Finite Horizon Inordertounderstandhowtheinfinite-horizonplanner’sproblembehaves, V ( k ) , it is useful to consider the solution to the finite horizon case. In particular, using a finite horizon version of the planner’s problem, we show that this model can exhibit other unusual features, even beyond oscillating labor supply. As we saw in lemma 2 above, the planner never chooses a non-zero labor level that is also lessthan (cid:13) = (cid:30) . Thuswhen (cid:13) > 0 ,theplanner’slaborpolicyfunctioncansuddenly jump down to zero. This behavior underlies the oscillations in labor and capital 22

g=0.01 4 2 0 0 2 4 Capital Today: k ’k :worromoT latipaC Figure 1: Planner’s work decision for k ; k 0 ; the circles give points at which the planner works hard, the dots where hedoesnot. Thedashedlinegives Æ ) k ( 1 (cid:0) 0 g=1 −2 g=0.01 g=−0.5 −4 −2 0 2 h . )h(F Figure 2: The static return function F writtenasafunctionof k 0 (cid:3) k (cid:0) (cid:11) h = (cid:2) ( 1 (cid:0) Æ ) k (cid:0) . Note the distinct kinks when (cid:13) > 0 : here the planner switches from workinghard to notworkingat all. in our model. However, as we show in this section, the planner’s labor supply function can also feature jumps up. Thus the limits of the cycle can themselves shiftovertime. Intuitively,astheplannergetsricher(beginswithmorecapital),thewealtheffectleadshimtoworklessbuttheproductivityeffectleadshimtoworkmore. This lattereffectemergesbecauselabor’sproductisproportionalto k (cid:11) . Intheneoclassical growth model capital provides two services: it is a productivity-enhancing input to production and it is a risk-free storage technology. When (cid:13) < 0 , there is normallynobenefittousingcapitalforstorage. When (cid:13) > 0 ,though,theoptimal productionplan may feature periods ofhard work interspersed with vacations. In thefaceofsuchaproductionplan,societymusthaveastoragetechnologyinorder 23

to smooth consumptionover time. As we have seen, the only storage technology available is one that also augments labor productivity, giving rise to even more complicateddynamicsthan acyclewithconstantlimits. With a finite horizon, there is some terminal period T , beyond which the household (and hence the social planner) does not survive. In this ultimate period, t = T , we know that the optimal investmentpolicy satisfies k T + 1 = 0 . As a result,wecan writedowntheperiod- T valuefunction: V T ( k T ) = X (cid:0) k T ; 0 (cid:1) = (cid:11) l o g (cid:0) k T (cid:1) + F (cid:0) k T ; 0 (cid:1) : We know from lemma 2 above that if (cid:13) > 0 , F (cid:0) k T ; 0 (cid:1) will feature a kink at a criticalcapitallevel, ~ k T > 0 ,wheretheoptimallaborsupplyjumpsdowntozero. Working backwards from the terminal period T , the planner’s value function inthepenultimateperiod, T (cid:0) 1 , is: V T (cid:0) 1 ( k T (cid:0) 1 ) = m k a T x X T (cid:0) 1 (cid:0) k T (cid:0) 1 ; k T (cid:1) + (cid:12) X (cid:0) k T ; 0 (cid:1) : This function has, potentially, two or more local maxima. The discontinuous jumps in the planner’s policy functions will be associated with jumps from one localmaximumto another. Imagine fixing the penultimate period’s capital stock, k T (cid:0) 1 , and varying terminal capital, k T , beginning from zero and moving up. At very low choices of k T , the planner will not work today (because investment is low), but will work tomorrrow (because k T is low). Thus the penultimate period’s static return func- 24

tion, X T (cid:0) 1 ,willhavearelativelyhigh(negative)slopein k T : becausetheplanner is not working in period T (cid:0) 1 , changes in k T affect consumption one-for-one. Bycontrast,theultimateperiod’sstaticreturnfunction, X T ,willhavearelatively small(positive)slopein k T : because theplanner is working in period T , changes in k T havea lessthanone-for-oneeffect onconsumption. At higher choices of k T , though, the situation is reversed. Now the planner is working in period T (cid:0) 1 (because investment is high) and not working in period T (because the planner has inherited a relatively high capital stock). Thus the relativemagnitudesoftheslopesofthestaticreturnfunctions, X T (cid:0) 1 and X T ,will bereversed. Now X T (cid:0) 1 slopesdownlessand X T slopesupmore. As a result, there will be (at least) two local solutions to the planner’s problem; in one, he is not working today but is working tomorrow, in the other, he is working today but is not working tomorrow. In both cases there will be an optimal choice of investment. The planner’s global problem is to choose which of theselocalsolutionsisbest. Onecanalsoimagineothersolutionstotheplanner’s problem,associatedwithotherpolicycombinations,e.g. workinginbothperiods. Theplanner’sdilemmaisdisplayedinfigure3below,whereweplottheplanner’speriod T (cid:0) 1 valuefunctionasafunctionofinitialcapital(whichtheplanner inherits)and terminalcapital (which theplannermustchoose): V T (cid:0) 1 (cid:0) k T (cid:0) 1 ; k T (cid:1) = X T (cid:0) 1 (cid:0) k T (cid:0) 1 ; k T (cid:1) + (cid:12) X T (cid:0) k T ; 0 (cid:1) : We mark the global maximaoverchoices of terminal capital. Notice that the two 25

local maxima occur at widely separated choices of terminal capital. As initial capital increases, the right hand maximum, associated with high labor today and lowlabortomorrow,increases relativetotheleft handmaximum. More generally, notice that the relative magnitudes of the local maxima will depend on the planner’s penultimate wealth, k T (cid:0) 1 ; however, the planner will be trading off the wealth effect and the productivity effect of capital. At low values of k T (cid:0) 1 , the marginal utility of consumption will be high, prompting the planner to work, but the marginal product of labor, which depends on k (cid:11) T (cid:0) 1 , will be low, promptingtheplannernottowork. Athighvaluesof k T (cid:0) 1 theoppositeistrue. We knowfromlemma2 thatundernocircumstanceswillbetheplannerbewillingto work anon-zero amountless than (cid:13) = (cid:30) . Thuseven iftheplannercould equatethe marginalvalueoftheconsumptionproducedbylaborwiththe(constant)marginal costoflaborat alowleveloflabor, theplannerwouldspurnsuch asolution. In the special case that (cid:11) = 0 the productivity effect vanishes. The planner’s policyfunctionswillstillhavedownwarddiscontinuities,butthejumpsupvanish. When (cid:11) = 0 capitalisonlyusefulasalow-returnstoragetechnologyInthiscase, the marginal product of labor will be constant no matter what the capital stock. Thuswecanthinkabouttheplanner’sproblemasoneofminimizingtheeffortcost necessary to finance an optimal stream of consumption. We explore this special casein somedetailbelow. The planner’s optimal choices of investment, k T (cid:0) k T (cid:0) 1 (cid:1) and effort n T (cid:0) 1 ; n T as a function of the penultimate period’s capital stock are displayed in figure 4 below. Noticethatlaborinperiod T (cid:0) 1 andinvestmentbothjumpupatprecisely 26

the values of initial capital, k T (cid:0) 1 , where the global maximum switched from one local maximum to the other. Also notice that at this critical level of capital the planner’s optimalchoiceof laboreffort in the nextperiod, period T , falls to zero, even as the planner’s optimal choice of labor effort in the current period, period T (cid:0) 1 , jumpsup. The policies displayed in figure 4 have other notable features. First, effort in period T (cid:0) 1 is increasing in the inherited capital stock, k T (cid:0) 1 . Here the productivityeffect is dominating(notice that these figures use a large value for capital’s socialproduct, (cid:11) = 1 );atlowlevelsofinitialcapital,labor’sproductistoolowto botherworking(consumptioncanbereadasthedifferencebetweenthe45degree lineand theoptimalchoiceof k T ). Second, effort inperiod T is decreasingin the initial capital stock, k T (cid:0) 1 . Because in the second period the capital stock will be completely consumed, the wealth effect dominates. Third, effort, n T (cid:0) 1 , and investment, k T , display several (small) jumps at relatively high values of the initial capitalstock, k T (cid:0) 1 . Thesearetheresultofthecompetingwealthandproductivity effects. Notice that effort tends to gradually rise, the result of the productivity effect augmenting labor’s product, and then suddely fall, the result of the wealth effect. More generally, with multiple periods, the relevant objective functions can featuremultiplelocal maximaand henceseveraljumps. 27

a =1 b =0.9 g =0.17 d =0 f =2 k =0.458 T−1 −4 k =0.456 T−1 k =0.454 T−1 −4.05 k =0.452 T−1 0.25 0.5 k T ) k; k( V T 1−T 1−T Figure 3: The planner’s penultimateperiod value function, V T (cid:0) 1 , over choices of investment, k T , for several different levels of initial capital, k T (cid:0) 1 . The stars give theglobalmaxima. 1.5 1 0.5 0 0 0.5 1 1.5 k T−1 k T a =1 b =0.9 g=0.17 d =0 f =2 Optimal investment policy in period T−1 0.5 n T n T−1 0 0 0.5 1 1.5 k T−1 a. Investment n , n T 1−T a =1 b =0.9 g=0.17 d =0 f =2 Optimal effort policy in periods T−1 and T b. Labor Figure 4: Optimalinvestmentpolicies, k T (cid:0) k T (cid:0) 1 (cid:1) , and laborpolicies, n T (cid:0) 1 (cid:0) k t(cid:0) 1 (cid:1) and n T (cid:0) k T ( k t(cid:0) 1 ) (cid:1) in thefinite-horizon case. 28

3.4 Stochastic Control Until we now we have forced the planner to use policies that are non-stochastic; the planner simply chooses effort and investment levels given an initial capital level. The non-convexity in the period return function, F , identified in lemma 2, suggests that the planner would use lotteries if permitted. In this section we liberalize the planner’s control set to permit stochasticpolicies. We show that, in certain cases, theplannerwouldinfact uselotteriesto increaseexpected utilities; however,theselotteriesultimatelytaketheform ofgamblesoverthecurrentcapital stock. In our setting, such lotteries would impose aggregate risk, and would require a counterparty willing to take the other side of a gamble involving a significantpartofanationalcapitalstock. Asaresult,suchgamblesareprobablynot a feature of the U.S. economy as whole. On the other hand, they may be feasible forparticular sectorsoftheU.S. economyand forsmaller,developingcountries. As the horizon faced by the planner grows to infinity, the region of capital stocks within which the planner would use lotteries may shrink to nothing. By sharply varying labor and investment over time, the planner may be able to get arbitrarily close to the value function with lotteries. We show that in the case where capital is purely a storage technology, so that (cid:11) = 0 , even at the infinitehorizonlimittheplannerbenefitsfrom usinglotteries. Lotteries over inputsto a production function have appeared in other applications. Prescott and Townsend (1984a,b) laid the theoretical framework for their usewhencontracts mustsatisfyan incentivecompatibilityconstraint. Phelan and Townsend(1991)demonstratedhowtocomputemulti-periodandinfinite-horizon 29

planner’s problems using lottery contracts. Lehnert (1998) used this framework to characterize the effect of stochastic capital inputs on growth models. Most recently,PaulsonandTownsend(2000)havefoundsomeempiricalevidenceforthe presenceoflottery-basedcontracts inmicro-leveldatafrom Thailand. GeneralizedPlanner’s Problem WithLotteries We follow Phelan and Townsend (1991) in generalizing our model to allow the planner to use lotteries. As before, we require capital and effort to live in the closed and bounded intervals of the real line, K = [ k m in ; k m a x ] , k m in > 0 and N = [ 0 ; n m a x ] . We now generalize the planner’s choice set each period to be a probability measure over joint events s in B ( S ) , where s is a triplet ( n ; k 0 ; k ) , S is the cross-product N (cid:2) K (cid:2) K and B ( S ) are all the Borel subsets of S . The planner’sproblemnowbecomes: (3.9) W ( k ) = m a (cid:24) x Z ( n ;k 0 ;z ) 2 N (cid:2) K (cid:2) K (cid:26) l o g (cid:0) z (cid:11) + n 1 (cid:12) + W (cid:13) + ( k ( 1 0 ) (cid:27) (cid:0) (cid:24) Æ ( ) n z ; (cid:0) k 0 ; k z 0 (cid:1) ) (cid:0) d ( n (cid:30) ; n k 0 ; z ) : Thechoiceobject (cid:24) ( n ; k 0 ; k ) isbestthoughtofasthejointprobabilitydensityover today’seffort n ,tomorrow’scapitalstock k 0 andtoday’scapitalstock k (whichwe denotedwiththedummyvariable z in theplanner’sproblem,equation3.9). 30

Thisoptimizationproceeds subjecttoan economy-wideresource constraint: Z ( n ;k 0 ;z ) 2 N (cid:2) K (cid:2) K z (cid:24) ( n ; k 0 ; z ) d ( n ; k 0 ; z ) = k : (C1) This constraint requires that the expected value of the capital input actually used in production be equal to the total outstanding stock of capital. Thus the planner canneithergain(nor,ofcourse,lose)capitalinexpectedvalue. Analternateview wouldbetoruleoutalllotteriesthatvarythecapitalinputawayfromtheinherited, beginning-of-periodcapitalstock. Such aconstraintwouldbe: (cid:24) ( n ; k 0 ; z ) = 0 ; all z = 6 k : (C2) Thisconstraintforces theplannerto putmeasurezero on any outcomethatvaries today’scapitalinput z fromitsinitiallevel k ,buttheplannermaystillusestochasticchoices ofinvestmentandlaboreffort. Both constraints (C1) and (C2) are linear in the choice object (cid:24) and define closed and convexsets of permitted choices (cid:24) . We can therefore define two additionaloperatorsaslottery-basedcounterpartstotheoperator T definedinequation (3.5) above. Each new operator will be associated with a different restriction on lotteries. Asbefore, theoperators map B ( K ) intoitself: T C1 f = m a (cid:24) x Z S (cid:26) l o g ( c ) (cid:0) (cid:30) n + (cid:12) f (cid:0) k 0 (cid:1) (cid:27) (cid:24) (cid:0) s ) d (cid:0) s ) ; subjectto (C1) T C2 f = m a (cid:24) x Z S (cid:26) l o g ( c ) (cid:0) (cid:30) n + (cid:12) f (cid:0) k 0 (cid:1) (cid:27) (cid:24) (cid:0) s ) d (cid:0) s ) ; subjectto (C2) 31

Herewehavesuppressedfornotationalconveniencetherelationships: c = k (cid:11) n 1 + (cid:13) + ( 1 (cid:0) Æ ) k (cid:0) k 0 ; and: s 2 S = N (cid:2) K (cid:2) K : From Phelan and Townsend (1991) we know that the operators T and C1 T are C2 contractionmappings. Thuswe definetwo additionalvaluefunctions as thefixed pointsoftheappropriateoperators: W ( C 1 ) ( k ) = (cid:0) T C1 W ( C 1 ) (cid:1) ( k ) and W ( C 2 ) ( k ) = (cid:0) T C2 W ( C 2 ) (cid:1) ( k ) : Results WithoutCapitalGambles Define a capital gamble as any policy (cid:24) that satisfies constraint (C1) but violates (C2). Such policies involve variations of the capital input away from the beginning-of-periodinheritedcapitalstock. As wediscussedearlier, capitalgambles are probably not feasible for large economies taken as a whole. Our first resultshowstheeffect ofrulingcapital gamblesout: Lemma 3 (cid:0) T C2 f (cid:1) ( k ) = (cid:0) T f ) ( k ) allboundedand continuousfunctions f . Notice as an immediate consequence of lemma 3 we know that the the fixed points of the two operators, T and T , must be equal, hence the value functions C2 W ( C 2 ) ( k ) and V ( k ) mustbeequal. 32

Results WithCapitalGambles We now explore the consequences for our model of allowing the planner to use fairgamblesovertheinheritedcapital stock. Inadditiontoarbitrary boundedand continuous functions f : K ! R (cid:26) R , we will find it convenient to define the zerofunctionas f ( 0 ) = 0 all k 2 K . Lemma 4 Theoperator T satisfies: C1 1. (cid:0) T C1 f (cid:1) ( k ) (cid:21) (cid:0) T f (cid:1) ( k ) all k 2 K , 2. (cid:13) < 0 = ) (cid:0) T C1 f ( 0 ) (cid:1) ( k ) = (cid:0) T f ( 0 ) (cid:1) ( k ) all k 2 K , 3. (cid:13) (cid:21) 0 = ) (cid:0) T C1 f ( 0 ) (cid:1) ( k ) > (cid:0) T f ( 0 ) (cid:1) ( k ) atleastone k 2 K . Notice what lemma 4.3 does not say, namely, that an economy with gambles would be strictly better offthan an economy without gambles at the infinitehorizonlimit. Instead,thelemmaappliesonlytothefinitehorizoncase, whenthe continuation utility is given by the zero function f ( 0 ) . Iterating backwards, however, theplanner withoutcapital gamblescan choosepolicies that cleverlymimic lotteries, and, indeed, can often (but not always) achieve the same utility as in the world with capital gambles. That is, for certain parameters, it is the case that V ( k ) = W ( C 1 ) ( k ) . However, we can identify one functional form where this is notthecase: Theorem 3 If (cid:11) = 0 then: 1. If (cid:13) < 0 then W ( C 1 ) ( k ) = V ( k ) all k 2 K . 33

2. If (cid:13) > (cid:13) ? where: (cid:13) (cid:21) (cid:13) ? (cid:17) 1 (cid:12) + (cid:12) l o g (cid:2) l o (cid:0) ( 1 g ( (cid:0) 1 + Æ ) (cid:12) (cid:12) ) (cid:1) (cid:0) 1 (cid:3) > 0 ; then W ( C 1 ) ( k ) (cid:21) V ( k ) all k 2 K and W ( C 1 ) ( ^k ) > V ( ^k ) atleastone ^k 2 K . Intuitively, when (cid:11) = 0 , we saw in section 3.3 that the productivity effect vanishesandcapitalisonlyusedasalow-returnstoragetechnology. Thuswecan imaginebreakingtheplanner’sproblemintotwopieces. Firsttheplannerchooses a time path for consumption that does not depend on the capital stock and then decides how much to work in order to finance this consumption stream. When (cid:13) is large enough to overcome the effects of capital’s depreciation rate, Æ , and the discount factor, (cid:12) , it is optimal for the planner to bunch production; moreover, the planner decides how hard to work (in those periods that he does work) by choosing the number of vacation periods that a particular level of effort will net him.4 Imagine beginning with initial capital of zero: the planner works hard enough to finance, say, a three-period vacation. If the planner were instead endowed with a small positive amount of capital, he responds by working a little less, but maintaining the same target level of capital next period (enough to finance a three-period vacation). As the stock of initial capital grows, though, the planner’schoiceofhowmuchtoinvestcanjumpdown. Heretheplannerswitches to a policy of working hard enough to finance a two period vacation. Before the jump, the planner’s value function will have a relatively shallow slope in initial 4Note that when capital does not depreciate, so Æ = 0 , and the discount factor approaches unity, (cid:13) ? approacheszero. 34

capital; the planner is only adjusting one margin, how much to work. After the jump,theplanner’svaluefunctionwillhavearelativelysteeperslopebecausethe planner has adjusted the amount of target capital next period. A capital-gamble style lottery would allow the planner to convexify the kink, increasing the value functioninexpected value. 4 Numerical Results Although we were able to make some progress in characterizing the planner’s problemindiscretetime,acompleteanalyticcharacterization isimpossiblewithout assuming constant returns to scale in capital, which we have seen rules out optimal cycles. In this section we characterize the solution to the planner’s problem using numerical techniques. We begin by (briefly) justifying the technique we use, we then establish a benchmark specification of parameters and vary parameters of interest around the benchmark. Next, we explore the implications of theorem 3 by solving the planner’s problem when (cid:11) = 0 with and without lotteries. One final question of interest is whether or not the solution at a particular parameter combination will feature endogenous cycles; we explore this issue by solvingthemodel at many combinationsofparameters and finding the regionsin parameterspacewherecycles arise. 35

4.1 Numerical Technique Because our basicproblem is, from thestandpointofnumerical techniques,quite straightforward, we can afford to use one of the simplest numerical techniques available,namelydiscrete-statedynamicprogramming. Wespecifyagridofcapitalpoints K: K = (cid:8) k m in ; k 2 ; : : : ; k m (cid:0) 2 ; k m (cid:0) 1 ; k m a x (cid:9) ; k m in > 0 : Here the integer m is the number of grid points in the capital state space approximation. Associated with every combination of capital today k and tomorrow k 0 , where k ; k 0 are in K, we compute the static return F, defined in equation (3.8) above: F (cid:0) k i ; k j (cid:1) = m a n x (cid:26) l o g (cid:0) k (cid:11) i n 1 + (cid:13) + ( 1 (cid:0) Æ ) k i (cid:0) k j (cid:1) (cid:0) (cid:30) n (cid:27) ; all k i ; k j 2 K : (Noticethatbyworkinghardenough,theplannercanrealizepositiveconsumption for any combination k i ; k j , as long as k i > 0 .) Associated with each k i ; k j there willbean optimalchoiceoflaboreffort n ij . Forvaluesof k i ; k j such that: h ij (cid:17) ( 1 (cid:0) Æ ) k k (cid:11) i i (cid:0) k j > h m a x ; we know that the optimal solution cannot be to work hard, so that n ij = 0 . If h ij (cid:20) h m a x , we solve the planner’s first order condition (given in equation A.2) to find a candidate solution involving high labor effort. If h ij > 0 , we also have 36

to check the corner, n ij = 0 , because the planner might prefer not working to working. With the static return function F computed, we next solve a recursive versionoftheplanner’sproblem: V ( ‘ + 1 ) ( k i ) = m k a j x (cid:26) F (cid:0) k i ; k j (cid:1) + (cid:12) V ( ‘ ) ( k j ) (cid:27) ; where k j 2 K; all k i 2 (4.1) K. We begin with an initial guess at the value function V ( 0 ) and implement the discrete operator defined by equation (4.1) until successive iterations produce value functionsthatarenumericallyindistinguishablefromoneanother. SeethediscussioninJudd(1999)foracompletediscussionofthistechnique. 4.2 Benchmark Parameterization Because the case that we are interested in (when (cid:11) < 1 and (cid:13) is relatively low) is precisely the case without an analytic solution, we pick a central benchmark calibration for our parameters and then vary parameters to study their effect on ourmodel. Thebenchmarksand variationranges aregivenintable1below. The parameter of primary interest in our model is (cid:13) , which governs the total returns to labor, 1 + (cid:13) . We first vary this parameter, then the returns to capital (cid:11) (when (cid:11) > 0 ),then thedepreciation rate Æ and finallythediscountfactor (cid:12) . Our chosen benchmark is (cid:13) = 1 = 6 ; (cid:11) = 1 = 3 ; (cid:12) = 0 : 9 8 ; Æ = 0 : 1 and 2 (cid:30) = . We specify a relatively low value of returns to labor, 1 + (cid:13) , to demonstrate that optimal labor cycles can occur under mild increasing returns to labor. The return to capital parameter, (cid:11) , is chosen to match the social returns to capital 37

used in many business cycle models. The total returns to scale in the benchmark specification equal 1.5. Next, we select a discount factor of (cid:12) = 0 : 9 8 and a depreciation rate of Æ = 0 : 1 , which are reasonable calibrations for an annual model. Inthesensitivityanalysis,weshowthatlaborcyclesappearunderalarger discountfactor ora smallerdepreciationrate. VaryingReturns to Labor: (cid:13) Figure5plotstheoptimalinvestment(thatis,nextperiod’scapital)andlaborpolicy functions for the benchmark parameterization and two alternate levels of (cid:13) . Figure 6 shows the dynamics of labor effort, capital and consumption under the benchmark andalternate levelsof (cid:13) . Asexpected, thepronounced,discontinuous drop in the policy functions, when they occur along the optimal capital accumulation path, generate cycles. (Interestingly, although consumption cycles can be eliminated, the planner chooses not to.) Notice also that the limits of the cycles shift over time; this is the result of the discrete jumps up in the policy functions, discussedinsection3.3 above. The labor and capital policy functions for all three parameterizations feature downward discontinuities, where the social planner chooses to stop working and consume only out of the capital stock. Examining the labor policy functions, a larger (cid:13) implies a higher level of the labor input for any level of current capital. Likewise, for the three values of (cid:13) considered here, a larger (cid:13) is associated with a higher cut-off capital stock for supplying labor. As predicted by lemma 2.3 above, the minimum positive labor value (the smallest non-zero amount of labor 38

10 5 5 10 Capital Today worromoT latipaC Optimal Capital Policy Optimal Labor Supply 3 g=0.20 g=0.20 2 g=1/6 g=1/6 1 g=0.05 g=0.05 0 0 5 10 Capital a. Investment robaL b. Labor Figure 5: Optimal investment and labor policies (as a function of capital) for varyinglevelsof (cid:13) : thattheplanneriswillingtosupply)alsoincreasesin (cid:13) . Sincethereturnstolabor are higher, the optimalallocation involves greater labor effort for any level of the capital stock. Examining the capital policy function, a larger (cid:13) implies a higher levelof(nextperiod)capital foranycurrent capitalstock. In the cases of (cid:13) = 1/6 or 0.2, the optimal capital policy function does not cross the 45 degree line at a continuous point; instead, at a point–call it ~k –the optimalinvestmentpolicy features a discontinuousjump down. Forboth of these parameterizations,theoptimalcapitalsequencedoesnotconvergeasymptotically, but instead cycles between periods of positive labor input and savings to periods of zero labor input and dissaving. Intuively, the social planner would like to find an intermediate average level of the capital stock at which to spend his time saving, consuming and working. Such an ideal capital stock may not exist under increasing returns. Low capital stocks do not exploit capital’s ability to augment 39

g =0.20 2 1 g =0.05 0 g =1/6 troffE robaL 9 0 latipaC 1 0 0 5 10 15 20 25 30 Time noitpmusnoC Figure6: Dynamicpathsoflaboreffort,capital stockand consumptionunderdifferent valuesof (cid:13) . 40

labor productivity, while high capital stocks are expensive to maintain because of capital’s diminishing marginal product. By oscillating labor supply, the social planner takes advantage of increasing returns to labor by jumping between positive (and high) labor and zero labor. Notice than in our example, the planner workshardinburstsofoneperiodonly;however,theplanner’svacationsaretypicallylongerthanasingleperiod. Inordertosmoothconsumptionandkeepcapital nearitstargetlevel,whentheplannerisfollowinganoscillatinglaborsupplypolicy,capital mustbeabovethesteady-statewhenlaboriszero becausehouseholds consume out of the capital stock. Positive consumption and capital depreciation imply capital falls during zero labor periods. On the other hand, capital is below thesteady-statewhen laboris positive. To finance consumptioninthefuturezero labor phase of the cycle, the social planner increases the capital stock during the positivelaborphaseofthecycle. Forboththe (cid:13) = 1 = 6 and (cid:13) = 0 : 2 cases,thecapitalandlaborpolicyfunctions display several upward discontinuitieswhen k < ~k . There are the discontinuities discussedinsection3.3 above. Anoscillatinglaborplanhasonebuilt-ininefficiency: theperiodofhighlabor effort occurs when the capital stock is lowest, but a low capital stock implies a relatively low marginal product of labor. If the degree of increasing returns is sufficiently low, oscillating labor supply will not be optimal. This occurs when (cid:13) = 0 : 0 5 : although the policy functions continue to display the characteristic downward discontinuity, it occurs at capital levels above the steady-state. The capital policy function in this case crosses the 45 degree line continuously and 41

from above, so there exists a steady-state optimal capital stock. Labor cycling is not optimal because the benefit of bunching production in the face of small increasing returns is outweighed by the cost of having to work when the capital stockislow. Varyingthe Depreciation Rate: Æ In our model, capital provides two services to households: (i) it is a risk-free constant returns storage technology, with a rate of return decreasing in Æ , and (ii) an investmentwhichaugmentstheproductivityofnextperiodlabor. Wecan vary (i)bychangingthedepreciationrate. Figure 7 plot the optimal investment and labor policy functions for three different depreciation rates Æ = 0 : 1 , the benchmark case, Æ = 0 : 0 7 5 . and Æ = 0 : 2 5 . There is a general downward shift in both policy functions as Æ rises. As capital depreciates more quickly, the planner reduces gross physical investment and labor. Also, the point at which labor input goes to zero as a function of the current capital stock decreases as Æ rises. For the largest depreciation rate, Æ = 0 : 2 5 , the optimalallocationconvergesto aconstantlaborandcapital steady-state. Varyingthe Returns to Capital: (cid:11) > 0 One parameter of particular interest is the return to capital, (cid:11) . Theorem 3 characterizes the planner’s problem in the special case that (cid:11) = 0 ; further, in section 4.3 below we compute numerical solutions using lotteries in that case. However, itisalsointerestingtostudytheeffectofincreasing (cid:11) outsideofthisspecialcase. 42

10 d=0.075 8 d=0.1 6 4 2 d=0.25 0 0 2 4 6 8 10 Capital Today worromoT latipaC b =0.98 a =1/3 g=1/6 f =2 3 d=0.075 2.5 d=0.1 2 1.5 1 d=0.25 0.5 0 0 2 4 6 8 10 12 Capital Today a. Investmentpolicy yadoT troffE b =0.98 a =1/3 g=1/6 f =2 b. Laborpolicy Figure7: Effectofvaryingthedepreciationrate, Æ . Figure8 displaystheoptimalpoliciesforthreelevelsof (cid:11) , f 1 = 6 ; 1 = 3 ; 1 = 2 g . As (cid:11) increases, the critical capital level at which the investmentpolicy jumps down, ~k , shifts out. In all cases (as, indeed, in the case when (cid:11) = 0 ) endogenous cycles arise. Thus the average capital level during oscillations is increasing in (cid:11) . Finally, notice that the sharp jumps up in the policy functions appear at all levels of (cid:11) ,so thelimitsofthecycleswillalsoshift. Varyingthe DiscountFactor: (cid:12) The discount factor (cid:12) defines the length of the period. As (cid:12) approaches 1, the discrete time model becomes more like the continuous model. Low values of (cid:12) correspondto alongerhorizon duringwhichtheplannermustfix inputs. We consider three levels for (cid:12) , f 0 : 8 ; 0 : 9 8 ; 0 : 9 9 g . In figure 9, we plot the investment and labor policy functions for the benchmark model at these three discountfactors. As (cid:12) rises,thepolicyfunctionsallshiftup,implyinghigheraverage 43

30 a =1/2 25 20 15 10 a =1/3 5 a =1/6 0 0 5 10 15 20 25 30 Capital Today worromoT latipaC d=0.1 b =0.98 g=1/6 f =2 3 a =1/2 2.5 2 a =1/3 1.5 1 0.5 a =1/6 0 0 5 10 15 20 25 30 Capital Today a. Investmentpolicy yadoT troffE d=0.1 b =0.98 g=1/6 f =2 b. Laborpolicy Figure8: Effect ofvaryingreturns tocapital, (cid:11) . capitaland effort levels. Also,thecriticalcapital level ~ k shiftstotheright. For the lowest value of (cid:12) , 0.8, cycles do not arise endogenously. This arises from the discrete-time formulation; from our analysis of the benchmark model, theoptimalallocationinvolveskeepingcapitalclosetoatargetsteady-state. Such apolicybecomesmoredifficulttoimplementusingcyclesiftheplannerisforced tomaintainalaborchoiceovera longerand longerhorizon. 4.3 Capital Gambles When (cid:11) = 0 Theorem 3 assures us that capital gambles will strictly improve welfare (in expected value) when (cid:11) = 0 and (cid:13) is large enough to stimulate cycles. Under our benchmarkparameterization,thecriticalvalueof (cid:13) isabout0.0910;thusourcentral level of (cid:13) , 1/6, is large enough to provokecycles and hence for lotteriesto be welfare-improving. We now display some numerical results for our benchmark model(with (cid:11) = 0 )withand withoutcapitalgambles. 44

10 8 b =0.99 b =0.98 6 4 b =0.8 2 0 0 2 4 6 8 10 Capital Today worromoT latipaC d=0.1 a =1/3 g=1/6 f =2 3 2.5 b =0.99 2 b =0.98 1.5 1 0.5 b =0.8 0 0 2 4 6 8 10 12 Capital Today a. Investmentpolicy yadoT troffE d=0.1 a =1/3 g=1/6 f =2 b. Laborpolicy Figure9: Effect ofvaryingthediscountfactor, (cid:12) . We calculated the optimal stochastic policy using the linear programming techniquespioneeredby Phelan and Townsend(1991). Ourparticularapplication here of these techniques is quite straightforward; unlike Phelan and Townsend’s problem, our model does not feature moral hazard or exogenous shocks, so our lotteries do not have to satisfy incentive-compatibilityor Bayes-consistency constraints. In the same way that we computenon-stochasticpolicies using discretespace approximations to the operator T , we compute optimal stochastic policies using a discrete-space approximation to the operator T . Instead of choosing C1 a general measure over the Borel subsets of effort, investment and capital, we chooseaprobabilitymassfunctionoverinvestmentand capital gridpoints. Figure10displaysthelevelsandslopesof W ( C 1 ) ( k ) and V ( k ) ,thevaluefunctions with and without capital-gamble style lotteries. Without lotteries, the value function V displaysseveraldistinctkinks,whiletheplanneruseslotteriesto convexify around these kinks, so that W is linear in these regions. The effect of 45

lotteries on the slopes of the value functions is quite striking: without lotteries, the slope of the value function swings wildly, while with lotteries it is smoothly decreasing, and flat at the kinks in V . Notice that the planner does not always use lotteries; in regions where V is not kinked, the planner can do no better with lotteriesthanwithoutthem. Figure11displaystheassociatedinvestmentandeffortpolicies. Noticethat,in the case of lotteries, we display the expected values of investment and labor. The planner is using lotteries to bounce between two extreme points–high labor and highinvestmentagainstlowlaborandlowinvestment. Asaresult,eventhoughthe expected value of next period’s capital level crosses the 45-degree line smoothly and from above, there is no steady-state capital in the usual sense. Instead, the plannerisundertakingex antecapitalgamblesbetween a higherand a lowercapitalstock. −84 −85 W(k) V(k) −86 −87 0 1 2 Capital Today snoitcnuF eulaV f =2 a =0 g=1/6 d=0.1 b =0.98 No Lotteries Lotteries V’(k) 2 W’(k) 0 1 2 Capital Today a. Levels snoitcnuF eulaV fo sepolS f =2 a =0 g=1/6 d =0.1 b =0.98 No Lotteries Lotteries b. Slopes Figure10: Valuefunctionswith andwithoutcapitalgambles. 46

2 1 0 1 2 Capital Today worromoT latipaC f =2 a =0 g=1/6 d =0.1 b =0.98 No Lotteries Lotteries 1 0 0 1 2 Capital Today a. Investment yadoT troffE f =2 a =0 g=1/6 d =0.1 b =0.98 No Lotteries Lotteries b. Labor Figure11: Optimalpolicieswithand withoutcapital gambles. 4.4 Optimality of Cycles We now present results from jointly varying the parameters (cid:11) ; (cid:12) ; (cid:13) over a wide range. Instead of presenting the policy functions in each case, we determine whether, under each parameter combination, the policies give rise to endogenous cycles. Figure12displaystheresultsin ( (cid:11) ; (cid:13) ) space,andfigure13displaystheresultsin ( (cid:12) ; (cid:13) ) space. Theresultsshowthatcyclesarelesslikelyatlowervaluesof (cid:12) ;ahighervalueof (cid:13) is requiredto overcomethenaturaltendencyoftheplanner to converge to a fixed capital stock as the horizon grows. Also, as (cid:11) grows, cycles become less likely. Thus as the model approaches Christiano and Harrison’s (cid:11) = 1 case, cyclesgraduallyvanishfromthemodel. 47

Table1: Benchmark parameterizationand variations Parameter min benchmark max (cid:13) Labor’sexternality -1/3 1/6 2 (cid:11) Capital’sproduct 0 1/3 0.8 (cid:12) Discountfactor 0 0.98 0.999 Æ Capital survival 0.10 (cid:30) Disutilityoflabor 2 Optimal Cycles: b =0.98 d =0.1 f =2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −0.5 0 0.5 1 1.5 2 g a Figure12: Cycles in ( (cid:11) ; (cid:13) ) space. 48

Optimal Cycles: a =0.33333 d =0.1 f =2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 −0.5 0 0.5 1 1.5 2 g b Figure13: Cycles in ( (cid:12) ; (cid:13) ) space. 49

5 Conclusion Consider two economies, called “A” and “B”. In both economies technological progress allows workers to communicate their ideas, needs and resources much more effectively than before; so much so, in fact, that labor productivity noticeably increases even as the aggregate supply of labor also increases. In economy A, thanks to strenuousinterventionsby the government,investment,employment and output fluctuate little. In economy B, by contrast, investment, employment and output fluctuate, often apparently at random. In economy B, inventories of goods accumulate towards the ends of booms, and then run down, signaling the beginningofrecession. Employmentpeaks, paradoxically, exactly when thecapital stock has run down significantly and thus augments labor’s productivity the least. In whicheconomy arehouseholdsbetteroff? Thispaperis,atleastinpart,apleanottodismissthepossibilitythateconomy B is actually following the optimal path. We showed that, within an increasingreturnsmodel,multiplecompetitiveequilibriaexistpreciselyinthosecases when itisat least possiblethat theoptimalpathofoutputisnotconstant. Indeed,weshowedthattheoptimalpolicyunderincreasingreturnscanfeature (1) endogenous cycles, (2) shifting endpoints for these cycles and (3) stochastic capital gambles. None of these phenomena are normally taken as the hallmarks ofwiseeconomicpolicy;andyet,incertaincasesinourmodel,theyareprecisely that. Our results abstracted from many features of the real world, and thus should 50

not be taken as a positive prescription to allow (or even encourage) cycles. In particular,thesolewelfarecostofourcyclescamefromthefluctuationsinaggregate consumption. In reality, because households retain a significant amount of idiosyncraticrisk,itislikelythatsomefractionofhouseholdswouldbearmostof theconsumptiondeclinesassociated withthecycles. 51

A Proofs A.1 Proof of Theorem 1 We can easily compute the eigenvalues (cid:23) of the linear system given by equation (2.15)by solvingthecharacteristicequation: d e t ( A (cid:0) (cid:23) I ) = 0 : Theeigenvalues (cid:23) arethesolutiontothequadraticequation: (cid:23) 2 (cid:0) tr ( A ) (cid:23) + d e t ( A ) = 0 : Applyingthequadraticformula: (cid:23) = 1 2 (cid:20) tr ( A ) (cid:6) q ( tr ( A ) ) 2 (cid:0) 4 d e t ( A ) (cid:21) : The eigenvalues will have opposite signs if and only if d e t ( A ) < 0 . Thus, for any given starting value of capital, there will a unique path to the steady-state if and only if d e t ( A ) < 0 . In the same way, for any given starting value of capital closetothesteady-state,therewillbemanypathstothesteady-state(thatis,both valuesof (cid:23) willbenegative)ifand only d e t ( A ) > 0 andtr ( A ) < 0 . Thedeterminantisgivenby: d e t ( A ) = ( 1 (cid:0) (cid:11) ) (cid:13) ( (cid:26) + Æ ) (cid:20) (cid:26) + a Æ (cid:0) Æ (cid:21) : Givenour previous assumptionson the parameters, 0 < a < (cid:11) < 1 and (cid:26) ; Æ > 0 , itmustbetruethat: sign ( d e t ( A ) ) = sign ( (cid:13) ) : Notice that (cid:13) > 0 is sufficient to rule out the usual local saddle-path stability. Condition2 ofthetheoremgivesthenegativetrace condition. A.2 Proof of Theorem 2 Proof by contradiction. Assume first that there exist paths for capital, consumption and labor k ? ( t ) ; c ? ( t ) ; n ? ( t ) where the control function is piecewise contin- 52

uous and the state is piecewise differentiable, that maximizes the social welfare function, equation (2.18). According to the maximum principle, at each instant thecurrent valueHamiltonianismaximizedat c ? ( t ) ; n ? ( t ) . Then: (cid:21) 0 [ l o g (cid:20) ( c (cid:21) ) 0 (cid:0) [ l (cid:30) o g n ( ] c + ? ) (cid:21) (cid:0) ( (cid:2) (cid:30) k n ? ? ) ] (cid:11) + n 1 (cid:21) + (cid:2) (cid:13) ( (cid:0) k ? c (cid:11) ) (cid:0) ( n Æ ? k ) ? 1 +(cid:3) (cid:13) (cid:0) c ? (cid:0) Æ k ? (cid:3) forall ( c ; n ) 2 U Fortheparticularproblemthatwearestudying,householdscanneverbesatiated; thus the objective can never be disregarded. Thus we can dispense with the case of (cid:21) 0 = 0 , where the Hamiltonian would be strictly decreasing in consumption, and increasinginlaboreffort. We wish to show that an alternate choice of policies, ( c ? ; (cid:18) n ? ) , increases the valueof H c above ( c ? ; n ? ) forsome (cid:18) > 0 . Plugginginourconjecturedimproved controlpair, theaboveinequalitybecomes: (cid:21) ( k ? ) (cid:11) ( n ? ) 1 + (cid:13) (cid:0) (cid:18) 1 + (cid:13) (cid:0) 1 (cid:1) (cid:20) (cid:30) ( (cid:18) (cid:0) 1 ) n ? : (A.1) Let v ( (cid:18) ) and w ( (cid:18) ) denote the left and right-hand side of (A.1). The functions coincidewhen (cid:18) = 1 ,andbotharedifferentiablein (cid:18) . If v 0 = 6 w 0 at (cid:18) = 1 ,thenthe functionscrossatthispoint,andcondition(A.1)mustbeviolatedforsome (cid:18) . The more interesting case, when v 0 = w 0 is illustrated in figure 14 below. If (cid:13) > 0 , then v isconvexand increasingin (cid:18) and v > w for (cid:18) = 6 1 . 53

v(q ), g <0 0 v(q ), g >0 w(q ) 1 q ) (q v ,) (q w Figure14: Relation of v and w . 54

A.3 Proof of Lemma 1 We first show that n ? (cid:0) k ; k 0 (cid:1) is bounded, and then it a natural result that F is also bounded. Firstnoticethatthevariable h = (cid:2) ( 1 (cid:0) Æ ) k (cid:0) k 0 (cid:3) k (cid:0) (cid:11) definedinequation(3.7) above is sufficient for F (cid:0) k ; k 0 (cid:1) , so that F is a function of k and k 0 only as they affect h . Nowwecan writethederivativeof F as: F n (cid:0) h ( k ; k 0 ) ; n (cid:1) = ( 1 n 1 + + (cid:13) (cid:13) + ) n h (cid:13) (cid:0) (cid:30) : (A.2) Thus if we were to imagine the planner choosing an unboundedly large level of effort,wecan find thelimitofthisderivative: l i n ! m 1 F n (cid:0) h ; n (cid:1) = (cid:0) (cid:30) < 0 : If (cid:13) > 0 ,thefunction F isnon-monotonein n andalsonon-concave,thusopening thepossibilityofa cornersolution. However,if h > 0 and (cid:13) > 0 , then 0 ) = (cid:0) (cid:30) F n ( h ; n = also. The planner might like to force labor effort even lower than zero, but,ofcourse, cannot. Thusit istheleftconstraintwehavetocheck. So now we haveestablished that F is the upper envelopeof two functions: F and F . Thesefunctionsaredefined as: F (cid:0) h ( k ; k 0 ) (cid:1) = l o g (cid:0) n 1 + (cid:13) + h (cid:1) (cid:0) (cid:30) n ; (A.3) where: n : n 1 + (cid:13) (cid:0) [ ( 1 + (cid:13) ) = (cid:30) ] n (cid:13) + h = 0 ; (A.4) and F (cid:0) h ( k ; k 0 ) (cid:1) = l o g ( h ) : (A.5) If there is no interior solutionto the planner’s first order condition (A.4), we take itas undefined. Figure15belowplots F ; F and theirupperenvelope. We are guaranteed that h is finite by our assumptionson K . This then in turn guarantees us that n (if it exists) is also finite. Optimal labor n ? is either 0 or n , bothofwhichare finite. But if n and h arefinite, F mustalso bebounded. A.4 Proof of Lemma 2 To show lemma 2.1, we show that F ( k ; k 0 ; n ) is strictly concave in n if (cid:13) < 0 , so that the planner is always picking an interior choice of n , n > 0 . The second 55

0 h* h )h(F n>0 n=0 * n=n Figure15: Staticreturn F . derivativeof F isgivenby: d d n 2 2 F (cid:0) k ; k 0 ; n (cid:1) = (cid:13) n M ( n ) (cid:0) (cid:2) M ( n ) (cid:3) 2 ; where: M ( n ) (cid:17) ( 1 n 1 + + (cid:13) (cid:13) + ) n h (cid:13) : Here h istheconvenientfunctionof k and k 0 , h = (cid:2) ( 1 (cid:0) Æ ) k (cid:0) k 0 (cid:3) k (cid:0) (cid:11) firstdefined in equation (3.7) above. When (cid:13) < 0 , M ( n ) > 0 if n > 0 , so the sign of the secondderivativeof F in n iscertainly negativeif (cid:13) isnegative. To show lemma 2.2, notice first that an interior solution to the static problem requiresthatlaborbechosensothat F n (cid:0) h ( k ; k 0 ) ; n (cid:1) = 0 ,or H ( n ) + h = 0 ,where: H ( n ) (cid:17) n 1 + (cid:13) (cid:0) 1 + (cid:30) (cid:13) n (cid:13) : Positive values of h are associated with negative investment and increased consumptiontoday relative to tomorrow. The larger h is, the better off the planner is today. We show that H ( n ) + h = 0 will always have a unique solution in n if (cid:13) < 0 butmayhavenosolutionsif (cid:13) > 0 and h islargeand positive. Notice that if (cid:13) < 0 , H ( n ) becomes unboundedlynegativeas n approaches 0 from the left; further, H 0 ( n ) is positive everywhere n (cid:21) 0 so H ( n ) is monotone 56

increasing. Thusforall h 2 ( (cid:0) 1 ; 1 ) asinglevalueof n willsatisfytheplanner’s first orderconditionin n : H ( n ) + h = 0 . If instead (cid:13) > 0 , we show H ( n ) has a finite minimum and hence for large positivevaluesof h ,novalueof n (cid:21) 0 satisfies H ( n ) + h = 0 . Forsuchvaluesof h the planner must set effort to zero (although he may choose to do so at smaller valuesas well). Thefirst and secondderivativesof H ( n ) are: H H 0 ( 00 ( n n ) ) = = (cid:13) ( ( 1 1 + + (cid:13) (cid:13) ) ) n n (cid:13) (cid:13) (cid:0) (cid:0) 1 2 [ [ n n (cid:0) (cid:0) ( ( (cid:13) (cid:13) = (cid:0) (cid:30) ) ] 1 ) = (cid:30) ] : Thus H ( n ) forms an upward-pointing parabola in n when (cid:13) > 0 , with a global minimum at n = (cid:13) = (cid:30) . If there are two solutions to H ( n ) + h = 0 we need only check the one to the right of the parabolic minimum, because the planner’s objectivefunctionslopesdowntotheleftofthispoint. Attheparabolicminimum, H takes onthevalue: H (cid:0) (cid:13) = (cid:30) (cid:1) = (cid:0) 1 (cid:13) (cid:18) (cid:13) (cid:30) (cid:19) 1 + (cid:13) : Notethat H (cid:0) (cid:13) = (cid:30) (cid:1) < 0 inthe (cid:13) > 0 case. Thusforvaluesof h greater than h m a x : h m a x = (cid:0) H (cid:0) (cid:13) = (cid:30) (cid:1) ; (A.6) the planner’s first order condition H ( n ) + h = 0 has no non-imaginary solution, and theplannermustset n = 0 . Thefinalelementofthelemma,lemma3,requiresustoshowthatif (cid:13) > 0 ,the planner’s choice of optimal effort jumps down to zero at a certain point. In other words, there is some region of labor effort greater than zero which the planner never chooses. But we have seen that the planner never chooses labor effort (cid:13) = (cid:30) n < as thisputshimonthedown-slopingpart oftheparabola H ( n ) . A.5 Proof that the Planner’s Problem is a Contraction Mapping Theplanner’sproblemisinherentlyasequenceproblem. (HerewefollowStokey and Lucas (1989).) In equation(3.1)abovewedefined theplanner’s continuation 57

utility at period t when beginning with a capital stock of k t as v t (cid:0) k t (cid:1) . This is the solutionto the sequenceproblem. In equation(3.4) abovewe wrotetheplanner’s problemrecursivelyand defined V ( k ) at thevaluefunction. Wenowshowthat: v t ( k ) = V ( k ) ; all k 2 K : We begin by re-writing the sequence problem using the static return function X defined inequation(3.6)above: v t (cid:0) k t (cid:1) = f k m +t a g i x =i =i 1 0 1 i= X 0 (cid:12) i X (cid:0) k t+ i ; k t+ i+ 1 (cid:1) ; k t+ i 2 K ; i = 0 ; 1 ; 2 ; : : : 1 : (A.7) Next, we show that v t (cid:0) k t (cid:1) must be finite for all starting values of capital. Define the sequence (cid:20) t as the optimal plan from the planner’s sequence problem, (A.7) above: (cid:20) t = (cid:8) k ? t+ 1 ; k ? t+ 2 ; : : : (cid:9) : We now show that the present-discounted value of following the capital plan (cid:20) t from period t forward is less than the present discounted value of an impossible plan. We then show that the PDV of this impossible plan is bounded. The particular impossibleplan that we will consider will be this: the planner will follow the optimal path (cid:20) t going forward, except that each period we will allow him to completely consume the current capital stock without running it down; the next period, even after his capital binge, the capital stock called for by (cid:20) t appears (by magic). Thusthevalueoftheoptimalplan isgivenby: 1 i= X 0 (cid:12) i X (cid:0) k ? t+ i ; k ? t+ i+ 1 (cid:1) < (cid:20) 1 i= X 1 i= X 0 0 (cid:12) (cid:12) i i X X (cid:0) (cid:0) k k ? t+ m ; i a x 0 ; (cid:1) 0 ; (cid:1) ; 58

forsimplicity,assumethat k m a x > h m a x , defined in equation(A.6)above: = = 1 i= X 1 0 1 (cid:0) (cid:12) (cid:12) i l l o o g g (cid:0) (cid:0) k k m m a a x x (cid:1) (cid:1) ; < 1 : Next we turn to the recursive problem, equation (3.4) above. Define B ( K ) as a space of bounded continuous functions f : K ! R (where R (cid:26) R ). Consider thefunctionaloperator T : B ( K ) ! B ( K ) defined as: (cid:0) T f (cid:1) ( k ) = m k a 0 2 x K (cid:26) X (cid:0) k ; k 0 (cid:1) + (cid:12) f (cid:0) k 0 (cid:1) (cid:27) ; all k in K . We now show that T satisfies Blackwell’s sufficient conditions for a contraction, monotonicityand discounting. Firstweshowthat T satisfiesmonotonicity. Ifthefunctions f a ; f b arebothin B ( K ) and f a ( k ) (cid:20) f b ( k ) all k in K , we must show that T f a ( k ) (cid:20) T f b ( k ) all k in K : (cid:0) T f b (cid:1) ( k ) = = (cid:21) (cid:21) m k m k (cid:0) (cid:0) a 0 2 a 0 2 T T x K x K f f (cid:26) (cid:26) a (cid:1) a (cid:1) X X ( ( k k (cid:0) (cid:0) ) ) k k + : ; ; k k (cid:12) 0 (cid:1) 0 (cid:1) m + + a0 k x (cid:12) (cid:12) f f (cid:26) b a f k (cid:0) k (cid:0) b (cid:0) 0 (cid:1) 0 (cid:1) k (cid:27) + 0 (cid:1) ; (cid:0) (cid:12) (cid:2) f f a b (cid:0) (cid:0) k 0 k (cid:1) 0 (cid:27) (cid:1) (cid:0) ; f a (cid:0) k 0 (cid:1) (cid:3) (cid:27) ; Next we show that T satisfies discounting. For any f 2 B ( K ) we must show that there is 0 < (cid:17) < 1 such that (cid:2) T ( f + a ) (cid:3) ( k ) (cid:20) (cid:0) T f (cid:1) ( k ) + (cid:17) a for a (cid:21) 0 and 59

k in K : (cid:2) T ( f + a ) (cid:3) ( k ) = = = m k m k (cid:0) a 0 2 a 0 2 T x K x K f (cid:1) (cid:26) (cid:26) ( X X k ) (cid:0) (cid:0) + k k ; ; (cid:12) k k a 0 (cid:1) 0 (cid:1) : + + (cid:12) (cid:12) f f (cid:0) (cid:0) k k 0 (cid:1) 0 (cid:1) + (cid:27) (cid:12) + a (cid:12) (cid:27) a ; ; Therefore, any value for (cid:17) such that (cid:12) (cid:20) (cid:17) < 1 will satisfy the condition. We assumedthatthediscountfactor (cid:12) was strictlybetween0 and 1. A.6 Proof of Lemma 3 Beginby definingthebudgetset B ( k ) as: B ( k ) = (cid:26) (cid:0) n ; k 0 (cid:1) 2 N (cid:2) K : k (cid:11) n 1 + (cid:13) + ( 1 (cid:0) Æ ) k (cid:0) k 0 (cid:21) 0 : (cid:27) Notice parenthetically that if (cid:13) > 0 , B ( k ) is non-convex. Define g k to be the shared objective, g : b 2 B ( k ) ! R : g k = l o g (cid:0) k (cid:11) n 1 + (cid:13) + ( 1 (cid:0) Æ ) k (cid:0) k 0 (cid:1) (cid:0) (cid:30) n + (cid:12) f (cid:0) k 0 (cid:1) : Forall k 2 K , g k willbeafinite,real-valuedfunctionofallpoints b inthebudget set B ( k ) exceptforthosepointsattheedgesofthebudgetset,whereconsumption iszero. Atthesepoints, g k willbenegativeinfinity. Definethesetofpoints b ? k as: b ? k (cid:17) (cid:26) b 2 B ( k ) : g k ( b ) (cid:21) g k ( b 0 ) ; all b 0 2 B ( k ) . (cid:27) Define g ? k as g k ( b ) for b in b ? k . Notice that g ? k will be finite, because for all k > 0 theplannercanfindapolicy n ; k 0 thatdeliversnon-zeroconsumption. Now,notice thatfrom constraint(C2)above,thechoiceobject (cid:24) (cid:0) n ; k 0 ; z (cid:1) mustsatisfy: Z N (cid:2) K (cid:24) (cid:0) n ; k 0 ; z (cid:1) d (cid:0) n ; k 0 (cid:1) = (cid:26) 1 0 z z = = 6 k k ; : 60

We can therefore write the problem as one of choosing lotteries (cid:24) k ( b ) for each capitalstock k and point b in thebudget set B ( k ) . Therefore: Z B ( k ) g k ( b ) (cid:24) k ( b ) d b (cid:20) g ? k all (cid:24) k ( b ) : The best that the planner can do is write lotteries whose support is only b ? k . But these lotteries can never produce outcomes greater than g ? k , which is available to theplannerwithoutlotteries. As aresult,itmustbethecasethat: (cid:0) T C2 f (cid:1) ( k ) = (cid:0) T f (cid:1) ( k ) ; all k 2 K : Thisconcludestheproof. A.7 Proof of Lemma 4 The first item in the lemma is easy: because the planner can always replicate any non-stochasticplanwithdegeneratelotteries,theplannercanneverdoworsewith lotteries;thus T C1 f (cid:21) T f . Beforeaddressingparts2and3ofthelemma,weanalyzethejointdistribution (cid:24) further. Notethat (cid:24) maybewrittenas: (cid:24) (cid:0) n ; k 0 ; k (cid:1) = (cid:25) ( k ) (cid:16) (cid:0) n ; k 0 j k (cid:1) : Here (cid:25) ( k ) is thecapital gambleand (cid:16) istheex post lotteryovereffort and investmentconditionalonoutcomesofthecapitalgamble. Weknowfromlemma3that the conditional distributions (cid:16) (cid:0) n ; k 0 j k (cid:1) are degenerate; thus the general contracts can be written as a stochastic part (the capital gamble) and a non-stochastic part (the optimal choice of effort and investment given an outcome k from the capital gamble). Nowtheoperator T can bewrittenas: C1 T C1 f = m a (cid:25) x Z K (cid:25) ( z ) (cid:26) m n a ;k x0 l o g (cid:0) z (cid:11) n 1 + (cid:13) + ( 1 (cid:0) Æ ) z (cid:0) k 0 (cid:1) (cid:0) (cid:30) n + (cid:12) f (cid:0) k 0 (cid:1) (cid:27) d z : Theconstraint(C1) isnowwrittenas: Z K (cid:25) ( z ) z d z = k : Turn now to item 2 of the lemma. Intuitively, without increasing returns to 61

labor,theplannervarieseffortcontinuouslywiththecapitalstock. Becauseagents are risk-averse, the planner finds no benefit in using stochastic controls (or, of course, sharply varying effort and investment over time). If (cid:13) < 0 then the static return function F defined in equation (3.8) above is concave in n for all n (cid:21) 0 . Theplanner’sproblemisnow: T C1 = m (cid:25) a ( k x ) Z K (cid:25) ( k ) (cid:26) m n a ;k x0 (cid:11) l o g ( k ) + F (cid:0) k ; k 0 ; n (cid:1) + (cid:12) f ( 0 ) (cid:0) k 0 (cid:1) (cid:27) d k : The optimal investment policy is clearly degenerate, k 0 = 0 for all values of k . Further, because F is concave for all n (cid:21) 0 , the optimal effort policy can be found at thecritical pointwhere F n (cid:0) k ; k 0 ; n (cid:1) = 0 , defined in equation A.2 above. Here: (cid:0) n ( k ) (cid:1) 1 + (cid:13) (cid:0) 1 + (cid:30) (cid:13) (cid:0) n ( k ) (cid:1) (cid:13) + ( 1 (cid:0) Æ ) k 1 (cid:0) (cid:11) = 0 ; (A.8) for all k 2 K . As a result, we know that if (cid:13) < 0 , optimal effort n is non-zero and smoothlydecreasing in k for all k > 0 . Define F ( 0 ) ( k ) to be the static return function F ( k ; 0 ; n ) when effort n is replaced by its optimallevel, n . The slopeof F ( 0 ) ( k ) becomes: d d k F ( 0 ) ( k ) = d d k F (cid:0) k ; 0 ; n (cid:1) + d d n F (cid:0) k ; 0 ; n (cid:1) : When (cid:13) < 0 allofthesederivativesexistand, from equation(A.8)above: d d n F (cid:0) k ; 0 ; n (cid:1) = 0 ; all k > 0 . Thusthesecond derivativeis: d d k F ( 0 ) ( k ) = (cid:0) (cid:11) ( 1 (cid:0) (cid:11) ) k (cid:11) (cid:0) 2 n 1 + (cid:13) c + 1 (cid:0) Æ (cid:0) (cid:2) k (cid:11) (cid:0) 2 n 1 + (cid:13) c 2 + 1 (cid:0) Æ (cid:3) 2 < 0 : Thus the payoff function is strictly concave everywhere in k > 0 and the planner hasno motivetouselotteries. Thisgivesusthesecond iteminthelemma. Turningnowtothefinalitem: when (cid:13) (cid:21) 0 ,weshowthattheplanner’soptimal choice of labor effort will, for values of k (cid:21) ~k , put the planner at the corner solution n = 0 , inducing a kink in the payoff function F ( 0 ) ( k ) . Although the household is risk-averse, the sudden change in the slope of the payoff function 62

will create a region where the household prefers capital gambles. To see this, consider again the locus of effort points defined by F n ( k ; 0 ; n ) = 0 , equation (A.8)above. Thismay bere-written as: n 1 + (cid:13) (cid:0) 1 + (cid:30) (cid:13) n (cid:13) = (cid:0) ( 1 (cid:0) Æ ) k 1 (cid:0) (cid:11) : For (cid:13) (cid:21) 0 ,thelefthandside(LHS)ofthisequationisconvexincreasingin n (cid:21) 0 . Theglobalminimumisat thecriticalpoint n = (cid:13) = (cid:30) ,at whichpointtheLHSis: L H S (cid:18) n = (cid:13) (cid:30) (cid:19) = (cid:0) 1 (cid:13) (cid:18) (cid:13) (cid:30) (cid:19) 1 + (cid:13) < 0 : Thusforvaluesofinitialcapital greaterthan thecritical point k : CRIT k CRIT = 1 1 (cid:0) Æ 1 (cid:13) (cid:18) (cid:13) (cid:30) (cid:19) 1 + (cid:13) ; there is no real solution to equation A.8; for values of the capital stock k > k , CRIT the optimal policy must be to set effort to zero. At the other extreme, for very small values of initial capital k ! 0 , we know that the optimal policy is to work hard. At someintermediatecapital point, 0 < ~ k (cid:20) k , thepolicyswitches from CRIT working hard to not working at all. We know that the slope of the value function F ( 0 ) ( k ) is lowerat any capital level when thehousehold works than when it does not(becauseitisconsumingmoreintheformercasethaninthelatter). Although the derivative of the value function is not defined at the kink ~k , we know that in an (cid:15) -neighborhoodof ~ k : l (cid:15) i ! m 0 d d k F ( 0 ) ( k ) (cid:12) (cid:12) (cid:12) (cid:12) k = ~k (cid:0) (cid:15) (cid:0) d d k F ( 0 ) ( k ) (cid:12) (cid:12) (cid:12) (cid:12) k = ~k + (cid:15) > 0 : Marginal utility is briefly increasing in wealth around the kink. In the neighborhood of ~ k , the agent becomes in effect a risk-lover, prompting the social planner toimplementcapital gambles. A.8 Proof of Theorem 3 Webeginwithpart1ofthetheorem. Weknowfromlemma2thatwhen (cid:13) < 0 the planner’s static problem is smooth; his choice of labor effort never drops to zero 63

no matter how rich he is. As a result, the planner’s static return function F does not feature a kink. But the planner’s static objective is X = (cid:11) l o g ( k ) + F ; with theassumptionthat (cid:11) = 0 thiscollapsesto X = F ,sowearealsoassuredthatthe planner’s static objective, X , is smooth and concave everywhere. From the proof of lemma 4 above we see that in such cases, there are no gains even in the finite horizontocapitalgambleswhenthestaticobjectiveissmooth. Withasmoothand concavestaticobjective,theinfinitehorizonvaluefunctioninheritstheproperties ofthestaticobjective,andsoisalsosmoothandconcave. Thustherearenokinks tobeconvexifiedaroundwithlotteries,andsocapitalgamblesprovidenoservice, and areruled outbyrisk aversion. The proof of the second part of the theorem proceeds as follows: (1) We establish that (cid:13) (cid:21) (cid:13) ? is sufficient to establish that cycles are optimal; (2) We show thatifcyclesareoptimaltheplannerworkshardatlowcapitalstocksbutdoesnot work at all at higher capital stocks; (3) We show that in the neighborhood of this critical capital region the planner’s value function V ( k ) is kinked; finally (4) we appeal tolemma4 sothat T C1 V ( k ) > V ( k ) in thisneighborhood. Theplanner’ssequenceproblemis: f c t m ;n a : t x t(cid:21) 0 g 1 t= X 0 (cid:12) t (cid:26) l o g ( c t ) (cid:0) (cid:30) n t (cid:27) ; (SP) subjectto: k t+ 1 = ( 1 (cid:0) Æ ) k t + n 1 t + (cid:13) (cid:0) c t ; and k t (cid:21) 0 ; t = 1 ; 2 ; : : : Ifthenon-negativityconstraintoncapitaldoesnotbind,theEulerequationimplies thatconsumptionfollowsthesequence: c t+ 1 = ( 1 (cid:0) Æ ) (cid:12) c t : (A.9) Butthenon-negativityconstraintwillingeneralbind. Ifitbindseveryperiod,then theplanner follows a constant-effort policyin which k t = 0 always. If it binds in someperiodsbutnotothers,theplannerfollowsacyclicalpolicyinwhich k t > 0 in some period but k t = 0 in others. Define v as the value of the constant-work policy: v = m a c x 1 t= X 0 (cid:12) t (cid:8) l o g ( c ) (cid:0) (cid:30) c 1 = ( 1 + (cid:13) ) (cid:9) : Now define a T (cid:0) period cycle as a policy of working hard in the initial period, 64

t = 0 ,butexertingnoeffortinthesubsequent T (cid:0) 1 periods, t = 1 ; 2 ; : : : ; T (cid:0) 1 . The non-negativityconstraint on capital binds only from period t = (cid:0) 1 to t = 0 and from period T (cid:0) 1 to T and so on. In the other periods, t = 0 ; 1 ; : : : ; T (cid:0) 2 , consumptionfollowstheEuler-equationpath, equation(A.9). Let (cid:23) T ( k 0 ; k T ) bethevaluetoa T (cid:0) periodcyclewhentheplannerbeginswith initialcapital k 0 butmustbequeathcapital k T at theend ofthecycle: (cid:23) T ( k 0 ; k T ) = n ;c m a 0 ;::: x ;c T (cid:0) 1 ( T (cid:0) t= X 1 0 (cid:12) t l o g ( c t ) ) (cid:0) (cid:30) n ; subjectto: k 1 = ( 1 (cid:0) Æ ) k 0 (cid:0) c 0 + n 1 + (cid:13) ; k t+ 1 = ( 1 (cid:0) Æ ) k t (cid:0) c t ; t = 1 ; : : : ; T (cid:0) 1 : Thisproblemcan berewrittenas: (cid:23) T ( k 0 ; k T ) = m a n x m l o g (cid:20) 1 m (cid:18) n 1 + (cid:13) + ( 1 (cid:0) Æ ) k 0 (cid:0) ( 1 (cid:0) k T Æ ) T (cid:0) 1 (cid:19) (cid:21) (cid:0) (cid:30) n + M (A.10) Here: m M = = 1 1 l o (cid:0) (cid:0) g (cid:0) (cid:12) (cid:12) ( 1 T (cid:0) Æ ) (cid:12) (cid:1) (cid:26) ( (cid:12) 1 (cid:0) (cid:0) (cid:12) (cid:12) T ) 2 (cid:0) T 1 (cid:0) (cid:0) 1 (cid:12) (cid:12) T (cid:27) : The planner’s choice of n , given T , pins down initial consumption, c 0 ; the Euler equation(A.9)pinsdownthesubsequenttrajectoryofconsumptionforthebalance ofthecycle. Note that the constant-labor policy is just equal to the value of pursuing a one-period cycle endlessly, v = ( 1 = ( 1 (cid:0) (cid:12) ) ) (cid:23) 1 ( 0 ; 0 ) . With this in mind, we can computethevalueofpursuingatwo-periodcycleendlessly; 1 = ( 1 (cid:0) (cid:12) ) (cid:23) 2 ( 0 ; 0 ) . It turnsoutthat: (cid:23) (cid:23) 1 2 ( ( 0 0 ; ; 0 0 ) ) = = ( 1 + (cid:12) ) ( ( 1 1 + + (cid:13) (cid:13) ) ) (cid:20) (cid:20) l l o o g g (cid:18) (cid:18) 1 1 + (cid:30) + (cid:30) (cid:13) (cid:13) (cid:19) (cid:19) (cid:0) (cid:0) 1 1 (cid:21) (cid:21) ; + M : 65

The associated value functions are (cid:23) 1 = ( 1 (cid:0) (cid:12) ) for the constant-effort policy, and (cid:23) 2 = ( 1 (cid:0) (cid:12) 2 ) for the two-period cycle. For the two period cycle to provide more utilitythantheconstant-effortcase, itmustbethecasethat: (cid:13) (cid:21) 1 (cid:12) + (cid:12) l o g (cid:2) l o (cid:0) ( 1 g ( (cid:0) 1 + Æ ) (cid:12) (cid:12) ) (cid:1) (cid:0) 1 (cid:3) This is exactly our definition of (cid:13) ? from the theorem. Thus we know that when (cid:13) (cid:21) (cid:13) ? someform ofcyclewillbeoptimal. Let V ( k ) be the fixed point of T as before; also let c ( k ) and n ( k ) denote the optimalconsumptionand effortpoliciesassociatedwith V ( k ) . Wenowargue that: V ( k ) = m a T x (cid:26) (cid:23) T ( k ; 0 ) + (cid:12) (cid:23) 1 T ( (cid:0) 0 ; (cid:12) 0 T ) (cid:27) : Consider the case when (cid:13) = (cid:13) ? . As we saw, a two-period cycle provided more utility than the constant-labor policy. Could the planner do better than a twocycle? The only other possibility is that the planner work some positive amount eachperiod. Ifthatisthecase,thenthenon-negativityconstraintsoncapitalnever bind,and fromthesequenceproblem(SP) above: n t+ 1 = (cid:2) ( 1 (cid:0) Æ ) (cid:12) (cid:3) 1 = (cid:13) n t : However, because ( 1 (cid:0) Æ ) (cid:12) < 1 and (cid:13) > 0 , this implies that n t+ 1 < n t , or that consumption falls smoothly over time. This in turn implies that at some period (cid:28) in the future, no matter how large n 0 is, n (cid:28) < (cid:13) = (cid:30) . This is contradiction of lemma 2.3, which holds that effort is either zero or greater than (cid:13) = (cid:30) . Thus the non-negativityconstraints must bind sometime. If they do not bind every period, the planner works someamount and consumptionis guided by the unconstrained Eulerequationthereafter, untilthenon-negativityconstraintbindsagain. Consider the policy associated with a T (cid:0) cycle. When initial capital is zero, theplannerworks andconsumes n 0 and c 0 givenby: n c 0 0 = = m m ( T 1 ( T ) ) 1 n + (cid:30) 1 + 0 (cid:13) (cid:13) : ; 66

Wecanthenderivetheend-of-periodcapitalstock, k 1 ,associatedwiththispolicy. Aftersubstitutingback infortheconstantterm m ( T ) wefind that: k 1 = (cid:18) (cid:12) 1 (cid:0) (cid:0) (cid:12) (cid:12) T T (cid:19) (cid:18) 1 1 (cid:0) (cid:0) (cid:12) (cid:12) T 1 + (cid:30) (cid:13) (cid:19) 1 + (cid:13) : In thesameway,wecan findthepenultimatepositivecapitalstock, k T (cid:0) 1 : ( 1 k (cid:0) T (cid:0) Æ 1 T ) (cid:0) 2 = (cid:18) (cid:12) T 1 (cid:0) 1 (cid:0) (cid:0) (cid:12) (cid:12) T T (cid:19) (cid:18) 1 1 (cid:0) (cid:0) (cid:12) (cid:12) T 1 + (cid:30) (cid:13) (cid:19) 1 + (cid:13) : Denote this capital stock k ( T ) ; for all capital k > k ( T ) the planner’s optimal policy is to consume a portion of the capital stock and exert zero effort. At some capital stock 0 < (cid:20) < k ( T ) , the planner begins again to work. For capital stocks k ( T ) > k > (cid:20) the planner’s optimal policy is to exert no effort and consume the capital stock entirely, setting k 0 = 0 . For capital sotcks k < (cid:20) the planner’s optimal policy is to work hard and set k 0 (cid:21) k ( T ) . By the envelope theorem we knowthat,forsomesmall (cid:15) > 0 : d d d k d k V V ( ( k k ) ) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) k k = = (cid:20) (cid:20) + (cid:0) (cid:15) (cid:15) = = @ @ F F ( @ ( @ k k k k ; ; 0 k ) ; 0 ) : From lemma4,weknow,when k 0 islargerelativeto k (as is thecase here), that: @ F ( @ k k ; 0 ) > @ F ( @ k k ; k 0 ) : Atthehighercapitalstock,theplannerdoesnotwork,whileatthelower,hedoes. Hence in a local region around (cid:20) the planner is a risk lover. Hence the planner wouldprefera fairgamblewithexpectedvalue (cid:20) than (cid:20) withcertainty. 67

B Comparison with Christiano and Harrison Following section 6.2 of Christiano and Harrison, assume a social planner maximizes: 1 t= X 0 (cid:12) t [ l o g ( c t ) + (cid:27) l o g ( 1 (cid:0) n t ) ] ; (B.1) where (cid:27) > 0 and c t ; n t (cid:21) 0 . Thelawofmotionforcapitalis: k t+ 1 (cid:20) ( k t ) (cid:11) ( n t ) 2 + ( 1 (cid:0) Æ ) k t (cid:0) c t ; (B.2) and k 0 isgiven. We generalize CH by allowingforeither diminishingor constant returnsto capital 0 (cid:20) (cid:11) (cid:20) 1 . Under constant returns to capital (cid:11) = 1 , CH show that the optimal labor supplyisconstant. ItisusefultoreviewCH’sderivationoftheoptimalallocation. Withoutsatiation,equation(B.2)holdswithequalityandweuseittosubstitute c t outoftheobjective,obtaining: 1 t= X 0 (cid:12) t (cid:8) l o g (cid:2) k t ( n t ) 2 + ( 1 (cid:0) Æ ) k t (cid:0) k t+ 1 (cid:3) + (cid:27) l o g ( 1 (cid:0) n t ) (cid:9) : (B.3) Defining (cid:21) t as thegross growthrateofcapital, k t+ 1 = k t , thisexpressionbecomes: 1 t= X 0 (cid:12) t (cid:8) l o g ( k t ) + l o g (cid:2) ( n t ) 2 + ( 1 (cid:0) Æ ) (cid:0) (cid:21) t (cid:3) + (cid:27) l o g ( 1 (cid:0) n t ) (cid:9) : (B.4) CH show that the discountedsum of the log capital stock can beexpressed as afunctionof k 0 and the f (cid:21) t g 1 t= 0 sequence:5 1 t= X 0 (cid:12) t l o g ( k t ) = 1 1 (cid:0) (cid:12) l o g ( k 0 ) + 1 (cid:12) (cid:0) (cid:12) 1 t= X 0 (cid:12) t l o g ( (cid:21) t ) : (B.5) 5Rearrangingtheelementsofaninfinitesumrequiresthesumtobewelldefinedforappropriate sequences,whichholdsinthiscase. 68

Substituting(B.5) into(B.4), thesocialplanner’sobjectivebecomes: (B.6) 1 1 (cid:0) (cid:12) l o g ( k 0 ) + 1 t= X 0 (cid:12) ( 1 t (cid:26) (cid:0) l o Æ g ) (cid:2) (cid:0) (cid:0) n (cid:21) t (cid:1) t (cid:3) 2 + + 1 (cid:12) (cid:0) (cid:12) l o g (cid:0) (cid:21) t (cid:1) + (cid:27) l o g (cid:0) 1 (cid:0) n t (cid:1) (cid:27) ; subjectto n t (cid:21) 0 .6 The dynamic problem in equation (B.6) can be converted into a sequence of staticproblems. Begin by noticingthat consumptionin period t can be written as aproduct ofthepreceding valuesofcapital growth (cid:21) t , theinitialcapital stock k 0 , and thecurrent consumption-capitalratio, c t = k t : c t = = (cid:18) (cid:18) c k c k t t t t (cid:19) (cid:19) k (cid:21) ; t t(cid:0) 1 (cid:21) t(cid:0) 2 (cid:1) (cid:1) (cid:1) (cid:21) 0 k 0 : Use the budget constraint (B.2) to write consumption c t as a function of current and futurecapital, k t ; k t+ 1 . Substitutingproduces: c t = (cid:2) ( n t ) 2 + 1 (cid:0) Æ (cid:0) (cid:21) t (cid:3) (cid:21) t(cid:0) 1 (cid:21) t(cid:0) 2 (cid:1) (cid:1) (cid:1) (cid:21) 0 k 0 : Thetimezero welfare contributionofconsumptionat time t is then: (cid:12) t l o g ( c t ) = (cid:12) t l o g ( k 0 ) + l o g (cid:2) ( n t ) 2 + 1 (cid:0) Æ (cid:0) (cid:21) t (cid:3) + t(cid:0) j = X 1 0 l o g ( (cid:21) t ) ! The separability between differently dated choice variables in (B.6) is a result of constantreturnsto capitalin productionand logpreferences overconsumption. WemodifyCHbyassumingdiminishinginsteadofconstantreturnstocapital. CH find thisassumptionuseful, inpart, becauseit deliversaclosed form solution for the optimal resource allocation. Linear production in capital is often used in endogenous-growth models; by contrast, business cycle modelling in an animal 6Bounds on (cid:21) t , which reflect the requirement that c t ; k t (cid:21) 0 , never bind because of Inada conditions. 69

spiritscontextmorecommonlyusesdecreasing returns tocapital.7 Intuitively, if production is linear in capital, an oscillating path for labor is less likely to be optimal. Suppose the planner is trying to improve on a constant laborsupplyallocationbytakingadvantageofincreasingreturnstolabor. Imagine that, instead of supplying the same amount of labor for two periods in a row, the planner increases labor supply this period in order to finance consumption both this period and next, when the planner decreases labor supply. By concentrating labor effort in the first period, the planner takes advantage of increasing returns to scale, achieving the same output (in present value) at lower utility cost (again, in present value). However, the capital stored by the planner this period (in order to finance consumption next period) increases the marginal productivity of labor next period. Moreover, this increase is always independent of the current level of the capital stock, because production is linear in capital. Thus, with the higher labor productivity in the next period, the planner faces an additional incentive to work.8 Without diminishing returns to capital, the effect on the productivity of futurelaborgrowslinearlyas greater savingsisundertaken. Letuscontinuecharacterizingtheoptimalallocationwhen (cid:11) = 1 . Theplanner chooses (cid:21) t = (cid:21) ; n t = n forall t (cid:21) 0 . tomaximize: l o g (cid:0) n 2 + 1 (cid:0) Æ (cid:0) (cid:21) (cid:1) + 1 (cid:12) (cid:0) (cid:12) l o g ( (cid:21) ) + (cid:27) l o g ( 1 (cid:0) n ) ; subjectto n (cid:21) 0 . CH showthat,given n ,theproblemisstrictlyconcavein (cid:21) ,and concentratetheobjectivefunctionintosolelya functionof n : L ( n ) = 1 1 (cid:0) (cid:12) l o g (cid:0) n 2 + 1 (cid:0) Æ (cid:1) + (cid:27) l o g ( 1 (cid:0) n ) : 7Asan example,mostofthe paperswe cite inourintroductionusemodelswith diminishing returnstocapital. 8Thisargumentmayseemtosuggestthattheoptimalresourceallocationinvolvesinfinitelabor supply and utility. There are two things preventing this. First, CH householdsface high utility penalties as labor grows–which bounds the amount of labor effort in the presence of arbitrary increasingreturns. Second,thediscretetimeformulationalsolimitstheusefulnessofextremely high labor allocations. In essence, the household would like to work very hard for a very brief period of time, but the discrete-time formulationforces the household to choose its labor effort choiceforanentireperiod. 70

Figure16plotsthefunction L . Thelocalmaximumis reached at ~n , where: ~n = 1 2 h (cid:8) + p (cid:8) 2 (cid:0) 4 (cid:24) i ; (cid:8) = 2 + (cid:27) 2 ( 1 (cid:0) (cid:12) ) ; (cid:24) = (cid:27) ( 2 1 + (cid:0) (cid:27) (cid:12) ( ) 1 ( 1 (cid:0) (cid:0) (cid:12) Æ ) ) (B.7) Thegloballyoptimallaborinputistimeinvariant,independentofthecapitalstock and equalseither n = 0 or n = ~n . Figure 16 is drawn so that L ( ~n ) > L ( 0 ) . This inequality may be reversed for some parameterizations, in which case not working is optimal. In the zero labor optimal allocation, the household consumes a fraction of the undepreciated capital stock each period. In the neoclassical model, Inada conditions typically ruleouttheoptimalityofzerolabor;however,withpreferences(B.1)andresource constraint (B.2), the marginal product of labor is zero and the marginal disutility ofworkis finiteat n = 0 . Next, we seek to show that it is possible for L ( ~n ) < L ( 0 ) . Let L ( 0 ) G = L ( ~n ) (cid:0) . Ifwe define x = 1 (cid:0) Æ ; z = 1 (cid:0) (cid:12) ; then G isgivenby: G ( x ; z ; (cid:27) ) = z (cid:0) 1 l o g " (cid:18) p ~n x (cid:19) 2 + 1 # + (cid:27) l o g ( 1 (cid:0) ~n ) where ~n is defined by (B.7). Using the envelope theorem, @ G = @ x < 0 . An increaseintheretentionratemakesitmorelikelythat L ( ~n ) < L ( 0 ) andzerolabor is optimal. The maximum value we can select for x , and still have real-valued ~n and 0 (cid:20) Æ < 1 , is x ? ( z ; (cid:27) ) = m i n f 1 = [ (cid:27) z ( 2 + (cid:27) z ) ] ; 1 g . If (cid:27) z ( 2 + (cid:27) z ) > 1 , then: ~n = 2 + 1 (cid:27) z : In thiscase: G [ x ? ( z ; (cid:27) ) ; z ; (cid:27) ] < 0 iff: l o g (cid:20) 2 ( 2 1 + + (cid:27) (cid:27) z z ) (cid:21) < (cid:27) z l o g (cid:18) 2 1 + + (cid:27) (cid:27) z z (cid:19) To illustrate the possibility of zero labor being optimal, assume (cid:27) z = 1 . Then G < 0 since l o g ( 4 = 3 ) < l o g ( 3 = 2 ) . The optimality of such a corner solution is more likely for larger values of (cid:27) , which implies a greater disutility of work, and for lower values of Æ , which 71

increases theusefulnessofcapital forstorage. Next, consider the case of diminishing returns to capital. Since the model no longer admits a closed form solution, we take a stand on parameter values. The parameters ( (cid:27) ; (cid:12) ; Æ ) = (cid:0) 2 : 0 ; 1 : 0 3 (cid:0) 1 = 4 ; 0 : 0 2 (cid:1) match those chosen by CH. Whereas CH choose (cid:11) = 1 , we set (cid:11) = 1 = 3 . We computethe optimal solutionby discrete discounted dynamic programming on a finite capital grid using value function iteration. Figure 17 plots the optimal policy functions for investment(next period capital),laborandconsumptionasafunctionofthecurrentperiodcapitalstock. Note the discontinuous drop in labor at approximately ~k (cid:25) 6 : 2 . At low capital levels, laborinputishighinordertoincreasethecapitalstock,whichincreasesthefuture marginal product of labor. For sufficiently high levels of capital k > ~k , however, labor input is set equal to zero. Because of diminishing returns to capital, it is costly in terms of labor effort to produce the output necessary to maintain a high capital stock. Instead, the social planner halts production and allows the capital stocktodeclineduetodepreciation andtheconsumptionofexistingcapital. Next, note the associated discontinuity in the capital policy, which occurs at the same capital level as the labor policy. For k < ~k , the capital policy function liesabovethe45degreeline;hence, thesocialplannerengagesinnetinvestment. For any 0 < k 0 < ~k , in a finite number of periods the capital stock will leave the ( 0 ; ~ k ) region for the first, but not the last time. Once the capital stock is greater than ~k , the capital policy function lies below the 45 degree line. Instead of net investment, there is disinvestment. In this region, since labor is zero, any consumption comes from undepreciated capital. For this reason as well as standard depreciation,thecapitalstockfalls. Highlevelsofthecapitalstockare expensive to maintain in terms of labor effort and the social planner finds it optimal to shut down production and consume part of the capital stock. Eventually, the capital stockfallsenough toreturn tothe ( 0 ; ~k ) region. Theoptimalcapitalpolicyexhibitsendogenouscycleswithperiodsofpositive netinvestmentandhighlaborinput,eachofwhichisfollowedbydecliningcapital and zero labor. These cycles are displayed in figure 18 below. The policy of production bunching in the presence of increasing returns to labor is intuitive: intenselaborsupply—whenthemarginalproductoflaborishigh—isfollowedby a period of labor inactivity,as is the case in Murphy,Shleifer and Vishny (1989). Thereareother, slightdiscontinuitiesinthecapital policyfunctionbesidesthatat k = ~k , althoughtheyaremoredifficulttosee. The consumption plan associated with the optimal investment and labor policies requires that the planner eats part of the existing capital stock. The optimal 72

allocation involving labor cycles requires the household to consume out of the undepreciated capital stock in periods where labor equals zero. This, in turn, requiresthatthehouseholdtransformcapitalbackintoconsumptiongoods. Thatis, investment must be reversible. If investment were irreversible, by contrast, zero labor would not be optimal because consumption would be zero (and marginal utilityinfinite)inthoseperiods. Oneinterpretationforthisreversibilityisthatthe capitalstockincludesfinalgoodsinventories,aswellasequipmentandstructures. 100 d=0.02 s =2 0 d=0 s =60 d=0.5 s =0.5 −100 0 1 n )n(L Christiano−Harrison: b =0.99264 Figure 16: Function L ( n ) for various parameter combinations. The dashed horizontallinesgive L ( 0 ) foreach parametercombination. 10 8 6 4 2 0 0 2 4 6 8 10 Capital Today worromoT latipaC CH: a =1/3 d=0.02 b =0.99264 s =2 0.5 0 0 1 2 3 4 5 6 7 8 9 10 Capital Today a. Investmentpolicy troffE CH: a =1/3 d=0.02 b =0.99264 s =2 b. Laborpolicy Figure17: Christianoand Harrison modelwith (cid:11) < 1 . 73

1 0 troffE robaL 10 5 0 latipaC 0.3 0 0 20 40 60 80 100 noitpmusnoC Time Figure18: Dynamicsin theChristianoand Harrison modelwith (cid:11) < 1 . 74

References Baxter, M. and R. G. King (1991). Productive externalities and business cycles.InstituteforEmpiricalMacroeconomicsDiscussionPaper53,Federal Reserve Bank ofMinneapolis. Benhabib,J.andR.Farmer(1994).Indeterminacyandincreasingreturns.Journalof EconomicTheory63(1), 19–41. Benhabib,J.andR.Farmer(1996).Indeterminacyandsector-specificexternalities.Journalof MonetaryEconomics37(3),421–443. Blinder, A. A. and L. J. Maccini (1991). Taking stock: A critical assessment of recent research on inventories. Journal of Economic Perspectives 5(1), 73–96. Boldrin, M. and A. Rustichini (1994). Growth and indeterminacy in dynamic modelswithexternalities.Econometrica62(2), 323–342. Christiano, L. J. and S. G. Harrison (1999). Chaos, sunspots, and automatic stabilizers.Journalof MonetaryEconomics44(1), 3–31. Farmer, R. and J.-T. Guo (1994). Real business cycles and the animal spirits hypothesis.Journalof EconomicTheory63(1), 42–72. Farmer, R. and J.-T. Guo (1995). The econometrics of indeterminacy: An appliedstudy.Carnegie-RochesterConferenceSeries onPublicPolicy43(0), 225–271. Guo,J.-T.andK.Lansing(1998).Indeterminacyandstabilizationpolicy.Journalof EconomicTheory82(2), 481–490. Guo,J.-T.andK.Lansing(2001).Fiscalpolicy,increasingreturnsandendogenousfluctuations.ForthcominginMacroeconomicDynamics. Judd, K. L. (1999). Numerical Methods in Economics. Cambridge, MA: The MIT Press. Krugman, P. (1999). The role of geography in development.International RegionalScienceReview22(2),142–61. Lehnert, A. (1998). Asset pooling, credit rationing, and growth. Finance and EconomicsDiscussionSeries 1998-52,Federal ReserveBoard. Lucas, Jr.,R. E. (1993).Makingamiracle.Econometrica61(2),251–272. 75

Murphy, K. M., R. W. Vishny, and A. Shleifer (1989a). Building blocks of market-clearingbusinesscyclemodels.InNBERMacroeconomicsAnnual, pp. 247–287.Cambridge, MA:MITPress. Murphy, K. M., R. W. Vishny, and A. Shleifer (1989b). Industrialization and thebigpush.Journalof PoliticalEconomy97(5),1003–1026. Paulson, A. L. and R. Townsend (2000, November). Entrepreneurship and financial constraintsin Thailand.Manuscript,UniversityofChicago. Phelan,C. andR. M.Townsend(1991).Computingmulti-period,informationconstrainedoptima.ReviewofEconomicStudies58(5), 853–882. Pontryagin, L., V. Boltyanski, R. Gamkrelidze, and E. F. Mishchenko (1964). TheMathematicalTheoryofOptimalProcesses.NewYork,NY:Pergamon Press. Prescott,E.andR.Townsend(1984a).Generalcompetitiveanalysisinaneconomywithprivateinformation.InternationalEconomicReview25(1),1–20. Prescott, E. and R. Townsend (1984b). Pareto optimaand competitiveequilibriawithadverseselectionand moralhazard. Econometrica52(1), 21–45. Romer, P. (1986). Cake eating, chattering, and jumps: Existence results for variationalproblems.Econometrica54(4), 897–908. Stokey, N. L. (1981). Rational expectations and durable goods pricing. Bell JournalofEconomics12(1), 112–128. Stokey, N. L. and R. E. Lucas, Jr. (1989). Recursive Methods in Economic Dynamics.Harvard UniversityPress. Subrahmanyam,A.andS.Titman(1999).Realeffectsoffinancialmarkettrading: Market crises and the going public process. Manuscript, College of BusinessAdministration,UniversityofTexasat Austin. 76