Asset Pooling, Credit Rationing, and Growth

ASSET POOLING, CREDIT RATIONING, AND GROWTH Andreas Lehnert (cid:3) Board of Governorsof theFederalReserveSystem MailStop93 WashingtonDC,20551 (202)452-3325 alehnert@frb.gov December8,1998 Abstract I study the effect of improved financial intermediation on the process of capitalaccumulationbyaugmentingastandardmodelwithageneralcontract space. With the extra contracts, intermediaries endogenously begin using roscas, or rotating savings and credit associations. These contracts allow poor agents, previously credit rationed, access to credit. As a result, agents work harder and total economy-wide output increases; however, these gains come at the cost of increased inequality. I provide sufficient conditions for the allocations to be Pareto optimal, and for there to be a unique invariant distribution of wealth. I use numerical techniques to study more general models. Journal of Economic Literature classification numbers: O16,E44,G20,G33. (cid:3) The views are expressed are mine and do not necessarily reflectthose of the Board ofGovernorsoritsstaff. Thispaperisasubstantiallyrevisedversionofmydissertation. I thank Robert Townsend, Lars Hansen, Derek Neal, and MaitreeshGhatak for several years of encouragement and support. I also thank Andrew Abel, Mitch Berlin, Ethan Ligon, DeanMaki, SteveOliner, Wayne Passmore and NedPrescottfor helpfulsuggestions. IhavealsobenefittedfromthecommentsofseminarparticipantsattheUniversity ofChicago,UIC,IowaState,Rice,Wharton,UniversityofNorthCarolina,Tufts,University of Virginia, and the Federal Reserve Banks of Richmond, Philadelphia and Kansas City as well as the Board of Governors of the Federal Reserve System. Financial supportfromtheUniversityofChicago,theHenryMorgenthaufoundationandtheNorthwesternUniversity/UniversityofChicagoJointCenteronPovertyResearchisgratefully acknowledged. Anyremainingerrorsaremine.

Does financial intermediation directly cause growth, or is financial intermediation merely a necessary adjunct to growth? In this paper I identify a channel by which a nation's financial structure may directly affect its development experience. I augment a standard capital-accumulation model with a general contracting space. Armed with these extra contractual possibilities, financial intermediaries will endogenously arrange poorer agents into asset-pooling groups, which mimic one type of rosca (rotatingsavingsandcreditassociation)commonlyobservedinthedevelopingworld. Roscashelpagentsovercomecreditrationing,increasingthe demand for capital. The market-clearing interest rate increases, as does the average effort level. Output increases, but at the cost of increased inequality. Economies with the extra contracts grow faster to an invariant distributionofwealthwithbothahighermeanandgreaterinequalitythan economieswithout them. The fact that financial intermediation, particularly asset-pooling contracts likeroscas,contributestoinequalitymaybecounter-intuitive. Inmymodel there are two main reasons for this effect. First, asset-pooling groups causethemarket-clearinginterestratetoincrease,thusincreasingthepremium to wealth. When the interest rate is higher, differences in wealth result in larger differences in consumption. Second, asset-pooling groups allow poor agents to leave their low-return, but safe, option for a highreturn, but risky, option. Because the market-clearing interest rate falls as wealth increases, these factors combine to produce Kuznets-style dynamics in the distribution of wealth, in which inequality is initially increasing and then decreasing. Note that the effect of wealth inequality depends 1

cruciallyonthefinancialmarketstructure. Withouttheasset-pooling contracts, inequality may reduce output. With them, I provide sufficient conditions forwealthinequality tohave noeffect onoutput atall. Theasset-poolingcontractsthatemergeendogenouslymaybeinterpreted as one-period roscas. Financial intermediaries will pool the wealth of many agents of the same wealth, assign the pool to a certain fraction of thecontributors andthenmakethemfurtherloans(ifneeded). Becausein this paper all agents of a given wealth will be identical and live for only one period, the pooled assets are divided with a lottery. Such contracts are known as lot roscas and are observed in the developing world, see for examplethe reviewsof Besley,Coate andLoury (1993,1994). Further, ina studyof Mexicanfinancialinstitutions, Mansell-Carstens (1993)findsthat lot roscas are used by, among others, Volkswagon de Mexico's consumerfinance arm. The extra contractual possibilities may also be interpreted as a joint stock company. Allagents (of the same wealth) trade their wealthfor one share inanenterprisejointlyownedbythemall. Withthetotalequityfromthese shares,theenterpriseeitherdirectlypurchasescapitalinputsorapproachesabank for further debtfinancingof evenmore capitalinputs. Acertain proportionoftheinvestors,chosenbylottery,aredesignatedasmanagers. Theenterpriseallocatestheaccumulatedcapitaltothemanagersforusein theirprojects. Agents,inmymodel,mayonlyworkontheirownprojects, and their labor effort is privately observed only by them. As a result, the correct level of labor effort is induced with an incentive-compatible 2

“managerial compensation contract” that rewards those managers whose projects succeed and punishes those whose projects fail. The remaining shareholders,who were notselectedasmanagers,become residualclaimantsanddivideequallytheoutputremainingafterthemanagersarecompensated and the bank repaid. From the point of view of an individual agent, an equity share in the enterprise represents a lottery ticket with a known probability of success. From the point of view of the enterprise as awhole,theprobabilitythatanyoneshareholderisdesignatedamanager isthe proportion whose projects maybe funded. Returning to the interpretation of input lotteries as roscas, Besley, Coate andLoury (1993,1994)further findthat roscas ingeneralare Pareto-dominated by credit markets. In contrast, I provide sufficient conditions for allocations with asset-pooling contracts to be Pareto-optimal. This difference stems from the fact that in this paper, roscas emerge as an endogenous response to credit rationing, and are part of a larger credit system. The winners of the asset-pooling lottery may go on to get loans from financialintermediaries toaugment theirpooledassets.1 The model in this paperbuilds upon the work of Bannerjee andNewman (1993), Piketty (1997) and especially Aghion and Bolton (1997). These papers study the effect of, and the evolution of, the distribution of wealth in development. In this paper, I provide sufficient conditions for the distri- 1ItisalsoworthnotingthatBesley,CoateandLouryconsidermulti-periodroscas,in whichagentsmustbepreventedfromdefecting. Inthispaper,roscaslastforoneperiod only,asifagentscouldnotbepreventedfromdefecting. 3

bution ofwealthto converge toaunique invariant distribution, no matter what the initial distribution. Thus with the richer contract space there are no“poverty traps.” Furthermore, with the extra contracts, I provide sufficientconditions forthe distribution ofwealthtohavenoeffectonequilibrium prices or aggregate output. This difference stems from the fact that credit rationing provides the main mechanism, in those papers, by which distributions of wealthaffect macroeconomic variablessuch asprices and output. Inthispaper,lotterybasedassetpoolingcontractsprovideamechanismtoovercome creditrationing. The analysis proceeds as follows: I define a contract space based on the work of Prescott and Townsend (1984a, b), in which contracts are seen as lotteries over possible outcomes. I then show how this abstract lottery space can be interpreted as a sequence of familiar contracts, and I solve analyticallyamodelbasedontheworkofAghionandBolton(1997). Ithen solve a set of richer models numerically, using a variant of the techniques ofPhelanandTownsend(1991). Section 1 below defines the notation, contract space and structure of the model. Section 2 specifies the equilibrium concept and some preliminaryresults aboutassetpooling. Section 3presentsanalytic resultsfrom a model with risk neutrality and lumpy capital (fixed project size). I show that, in poor economies, asset-pooling lotteries increase output and the market-clearing interest rate; further, I provide sufficient conditions for optimality and a convergence result. Section 4 presents numerical results foramodelwitharichertechnologyandriskaversion. Section5concludes 4

thispaper. 1 The Model In this section I describe the preferences, technology and endowments of agentsinthemodel,thenatureofthecontractswhichintermediariesoffer to agents and how goods are stored from one period to the next. Here andfor the rest of this paper,objects which are normally considered to be continuous (for example, consumption), are constrained to live in finite sets [following Prescott and Townsend (1984a, b)]. This provides simpler notationandanalysis,andonecanimagineallowingthenumberofpoints to grow arbitrarily large [see PhelanandTownsend(1991)]. 1.1 Preferences, Technology and Endowments Each agent lives for one period and produces exactly a single successor agent at the end of the period, towards which it is altruistic in a very special way: agents get utility directly from the amount bequested, not the utility value of bequests to the next generation. (This is often called “warm-glow” altruism.) Agents get a lump of consumption (cid:28) at the end of the period, where (cid:28) lies in the set T = f (cid:28) 1 ; : : : ; (cid:28) n T g , and (cid:28) = 0, so that 1 there is limited liability in the sense of Sappington (1983). Agents split thislumpbetweenown-consumptionandbequeststotheirsuccessorgen- 5

eration. One can model this choice directly, but here I just assume that agents bequeath a constant fraction s of their consumption lump (cid:28) , and have indirect utility over (cid:28) given by u ( (cid:28) ), where u 0 > 0, u 00 (cid:20) 0. While (cid:28) is public (that is, observed costlessly by all agents), the division between own-consumption and bequests is private (that is, observed only by the agent). Agents also exert private labor effort, z in Z = f z 1 ; : : : ; z n Z g , where z = 0. 1 Private labor effort may be exerted on the agent's own technology only. Effort produces disutility of (cid:18) ( z ), where (cid:18) 0 > 0and (cid:18) 00 (cid:21) 0. Agentsthenhavepreferencesoverconsumptiontransfers (cid:28) in T andeffort inputs z in Z of: U (cid:28) z = u ( (cid:28) ) (cid:0) (cid:18) ( z ) : All agents have access to a back-yard technology which maps inputs of privatelaboreffort z andpublicproductivecapital k intoaprobabilitydistribution over outputs q . Capital k has to lie in the set K = f k 1 ; : : : ; k n K g (where k = 0), and output 1 q has to lie in the set Q = f q 1 ; : : : ; q n Q g . Both capital and output are public, and output may be costlessly confiscated (for example, by an intermediary). Inputs are timed so that capital is added first, before the agent decides on labor effort. Thus given inputs of effort z and capital k , the technology, P ( q j z ; k ), specifies the probability of realizing a particular output q . In addition, for each possible input 6

combination z ; k in Z (cid:2) K ,the technology must satisfy: P ( q j z ; k ) (cid:21) 0 ; and X q P ( q j z ; k ) = 1 : Capital is consumed entirely in the productive process. In the numerical section,thetechnologywillhavetosatisfythestrictercondition P ( q j z ; k ) > 0all q ; z ; k in Q (cid:2) Z (cid:2) K . Thiswillpreventinfinite likelihoodratios. There isacontinuum ofagents ofunit mass,aproportion a ofwhom are endowed at the beginning of the period with one of n A levels of wealth a in A = f a 1 ; : : : ; a n A g , where for each a , a (cid:21) 0 and P a a = 1. Wealth is in the form of capital, is public and may be costlessly transported among agents. Define to be the vector ofpopulation weights, [ a 1 ; : : : ; a n A ]. 1.2 Contracts The contract space studied by Prescott and Townsend (1984a, b) uses lotteries to span non-convexities arising from moral-hazard constraints. For thisreason,itisveryusefulinthispaper. Contractsarespecifiedasweights on the linear space of all of the possible combinations of consumption transfers,output,effortandcapital,conditionalonwealth. Fromthepoint ofviewoftheeconomyasawhole,becausethereisacontinuumofagents, the contract weights are fractions of agents who will receive a particular combinationofconsumption,output, effortandcapital. Fromthepointof view of a particular agent, they are the probability of receiving a particular combination. In section 3 below, I show how contracts in this abstract 7

space maybeinterpreted inmore familiarterms. The space of valid contracts will be a linear space subject to some linear constraints. The constraints are, first, that the contracts form a validset of probabilities; second, that they are Bayes compatible with the underlying technology; and third, that they are incentive compatible with respect to deviations in effort once capital has been announced. Let the linear space L be the Euclidean space of dimension n T n Q n Z n K . A contract x ( (cid:28) ; q ; z ; k ) mustlie inthe space X ,where: X = (cid:26) x 2 L ; suchthat: x ( (cid:28) ; q ; z ; k ) (cid:21) 0 ; all (cid:28) ; q ; z ; k ; (cid:28) X q z k x ( (cid:28) ; q ; z ; k (C1) ) = 1; X (cid:28) x ( (cid:28) ; q ; z ; k ) = P ( q j z ; k ) X (cid:28) q x ( (cid:28) ; q ; z ; k ) all( q , z , k )in Q (cid:2) Z (cid:2) K (C2) ; X (cid:28) q x ( (cid:28) ; q ; z ; k ) (cid:20) P ( q j ˆ z ; k ) P ( q j z ; k ) U (cid:28) ˆ z (cid:0) U (cid:28) z (cid:21) (cid:20) 0 all z ; ˆ z ; k in Z (cid:2) Z (cid:2) K ) : (C3) Becausecontractscanbeviewedasjointlotteriesovereverypossiblecombination of transfers T , output Q , effort Z and capital K , the constraints (C1) to (C3) can be thought of as restrictions on those lotteries. In particular, equation (C1) requires that contracts form valid lotteries, that is, that they sum to unity and are non-negative. Equation (C2) requires that thecontracts respecttheunderlyingprobabilitiesgivenbythetechnology, P ( q j z ; k ). Finally,equation(C3)istheincentive-compatibilityconstraint; it requires that, for every assigned effort level z , the agent not prefer some alternative effort, ˆ z . For more onthis constraint, see the appendix. 8

Notice that the contract space X does not depend on the wealth a of an agents. In the next section I introduce financial intermediaries who will offer the best possible contract in X to agents, subject to a zero-profit condition. Thiszero-profit condition willdependonagents' wealth. Finally, it is convenient to assume that A = s T , so that if an agent is assignedthe j -thelementof T , (cid:28) j ,hebequeaths s (cid:28) j ,whichis(byassumption) exactly equalto a j ,the j -th elementof A . 2 Equilibrium and Pooling Contracts Inthis section Ipresent the structure offinancialintermediation, the equilibrium concept, and explanation of the three different types of lotteries usedinthepaperandsome preliminaryresults ontheuseofpoolingcontracts. 2.1 Competition in Financial Intermediation Zero-profit and costless financial intermediaries (called banks) will compete to attract depositors by offering the best possible contract that generates non-negative profits in expectation. The agent's assets are public and are placed completely on deposit with the intermediary. The capital input can be observed and controlled by the financial intermediary. Becauseagentsliveforonlyoneperiod,contractsaresingle-periodtoo. Both 9

sides commit costlessly to the contract. Once contracts have been signed, the intermediary can confiscate output costlessly. Intermediaries take as givenarisk-free rate ofreturnonaninterbankloanmarket. Becausethere ispurelyidiosyncraticuncertainty(whichthebankssmoothawaybycontracting with groups of agents), banks can borrow and lend freely at this rate onbehalfoftheir depositors. Becausebankscompetewithoneanothertoattractdepositors,abankwill beunabletoachievezeroprofitsbysubsidizingtheconsumptionofagents of one wealth type with the output of agents of some other wealth type. Banksattempting such astrategy would attract only the subsidizedagent type. Financial intermediation, therefore, acts as if agents of each wealth type form a single bank, which makes zero profits. In the next section, I present sufficient conditions for this structure to yield the Pareto-optimal allocations. This will generally not be the case, because the social planner is not bound to respect a zero-profit condition on each wealth type. (See Holmstro¨m (1982) for the classic work on this topic. This effect can motivate acreditsubsidy program for the poor.) At the beginning of the period, one can imagine financial intermediaries writing complicated contracts combining insurance and production with agents. First, the intermediaries accept the agent's inherited stock of productive wealth on deposit, which they then place on the interbank loan market at rate (cid:26) . Next, intermediaries determine how much capital the agentshouldbeallocated. Becausetheycanuseinputlotteries,thischoice will be convex. Intermediaries borrow this amount on the interbank loan 10

market. The contract also specifies how much the agent will be given to consume and bequeath conditional on inputs and outputs. If the agent is assigned some non-zero effort level, these consumption assignments will have to satisfy the incentive-compatibility constraint. Idiosyncratic uncertainty vanishes in the continuum, the intermediaries collect the output and distribute it among the agents in order to honor their commitments. Finally,agents consume andbequeath(by storage). Let W ( a ; (cid:26) ) be the expected utility of an agent of wealth a when the interbank interest rate is (cid:26) . Banks must compete to attract customers, so they offer contracts y a ( (cid:28) ; q ; z ; k ) to agents of wealth a in A to maximize this expectedutility: W ( a ; (cid:26) ) (cid:17) max y a ( (cid:28) ; q ; z ; k ) 2 X (cid:28) X q z k y a ( (cid:28) ; q ; z ; k ) U (cid:28) z : (1) Thismaximizationmust proceedsubject tothe zero-profit constraint: (cid:28) X q z k y a ( (cid:28) ; q ; z ; k )( q + (cid:26) a ) (cid:0) (cid:28) X q z k y a ( (cid:28) ; q ; z ; k )( (cid:28) + (cid:26) k ) (cid:21) 0 : (2) As an accounting convention, the bank puts assets a entirely on the interbank market, earning (cid:26) a , and then borrows some amount k on behalf of itsagents,paying (cid:26) k . Competition amongbankswilldriveprofits tozero. Notice that this zero-profit condition must hold for each wealth type a . The social planner would find it possible to violate these constraints for individualagenttypes,while respecting the resource constraint. Market clearing requires that capital demanded for use as inputs must 11

equalthe aggregate wealthinthe economy. Thus: X a a (cid:28) X q z k y a ( (cid:28) ; q ; z ; k )( k (cid:0) a ) = 0 : (3) This,combinedwiththe banks' zero-profit condition (2),impliesthat: X a a (cid:28) X q z k y a ( (cid:28) ; q ; z ; k )( q (cid:0) (cid:28) ) = 0 ; (4) whichsimplystates that total consumption must equaltotal output. Defineanequilibriumas: Definition Given an initial distribution of wealth , an equilibrium is a se- 0 quence ofinterbank loan rates,contracts andwealthdistributions: n (cid:26) t ; f y a t ( (cid:28) ; q ; z ; k ) g a 2 A ; a t o 1 t =0 ; thatsatisfythe followingconditions: 1. For all t (cid:21) 0, (cid:26) t and f y a t ( (cid:28) ; q ; z ; k ) g a 2 A satisfy: (a) 0 < (cid:26) t < 1 . (b) Given (cid:26) t ,thesetofperiodt contracts f y a t ( (cid:28) ; q ; z ; k ) g a 2 A in X solvethe bank's problem of maximizing the agent's expected utility, equation (1),subjecttothenon-negative profitcondition,equation(2),foreach agent type a in A . (c) Bank profitsare zerofor eachwealthtype. (d) At (cid:26) t the interbankloan marketclears,satisfyingequation (3). 12

2. For all t (cid:21) 0, and for each wealth type, agents who receive a transfer of (cid:28) bequeath s (cid:28) . 3. Themassof agents withwealth a 0 inperiod t +1isgivenby: a t 0 = +1 X a a t q X z k y a t ( (cid:28) = a 0 = s ; q ; z ; k ) : 2.2 The Nature of Lotteries There are three typesof lotteries presentinthis model. The first, andleast interesting type, are grid lotteries. These emerge as financial intermediariesattempttospanthediscontinuitiesintheconsumptionvector, (cid:28) . Grid lotteries appear only because variables that are normally taken as continuous are required to be discrete, and I attempt to minimize their effect by providing a dense grid over consumption in the numerical experiments. In section 3 below I assume that the grid is dense enough to be treated as aninterval,sothatgridlotteriesdisappearentirely. Thesecondtypeareinput lotteries. These are lotteries over assignments of capital andeffort. At each outcome of these lotteries, there is a separate schedule of consumption assignments conditional on output, and, if a non-zero effort level is assigned, these must satisfy an ex post incentive compatibility constraint. Thethirdandfinaltypeoflotterypresentisastandardequilibriumlottery formedasaconvex combination ofcontracts andusedtoclearthe market for capital. In the discussion that follows I concentrate on the effect of the second of these three types: input lotteries. From the point of view of an individual 13

agent,thecontractspecifiesaprobability,say30%,ofrealizingaparticular capitallevel. From the point ofviewofabank,the contract specifieswhat fractionofagentsofaparticularwealthwillbeallocatedthatcapitallevel. These contracts allow the bank to concentrate the wealth of many poor agents into the hands of a subset, making them richer. This is tantamount to allowing banks to violate the zero-profit condition with respect to individual agents, while satisfying it for all agents of the same wealth. Thus thesecontractsallowintermediariestopoolthewealthofmanyagentsand distribute the pool amongasmallergroup ofagents ofthe samewealth. An agent who loses the input lottery, or wins the input lottery but suffers thelowoutput,isnotnecessarilyconsignedtozeroconsumption. Thereis scope in the contractual structure to allow ex post transfers back from the fortunate agents who won the input lottery and realized high output to those agentswhoeither lost the inputlottery orsuffered the lowoutput. To study the effect of input lotteries, I briefly outline how to construct banks' optimalpolicieswithout them(more detailcanbe foundinthe appendix). Financial intermediaries are now restricted to contracts that assign inputs with 100% ex ante certainty. Let W ( NL a ; (cid:26) j z ; k ) be the utility of an agent with wealth a when the interest rate is (cid:26) who is assigned inputs ( z ; k ) in Z (cid:2) K with certainty. A financial intermediary offering this input combination chooses contracts y a ( NL (cid:28) ; q j z ; k )that satisfy: W ( a ; (cid:26) j z ; k ) (cid:17) max y a ( NL (cid:28) ; q j z ; k ) (cid:0) (cid:18) ( z )+ X (cid:28) q y a ( NL (cid:28) ; q j z ; k ) u ( (cid:28) ) : (5) Contractsmustsatisfyversionsoftheconstraints(C1)through(C3). These 14

are detailed in the appendix. The maximization proceeds subject to the bank'szero-profit constraint: (cid:26) ( a (cid:0) k )+ X (cid:28) q y a ( NL (cid:28) ; q j z ; k )( q (cid:0) (cid:28) ) (cid:21) 0 : (6) Note that thisconstraint must holdseparatelyfor each z ; k combination. Given values of W ( NL a ; (cid:26) j z ; k ) as determined above, for a borrower of particularwealth a whentheprevailinginterest rateis (cid:26) ,the bankchooses an inputcombination z ; k suchthat: W ( NL a ; (cid:26) ) = max ( z ; k ) 2 Z (cid:2) K W ( NL a ; (cid:26) j z ; k ) : (7) The bank is forced to assign an input combination with certainty. It picks the one that produces the best utility for its borrower. Because W ( NL a ; (cid:26) ) is formed from a constrained version of the maximization that produced W ( a ; (cid:26) ), it must be the case that W ( a ; (cid:26) ) (cid:21) W ( NL a ; (cid:26) ). This construction leadsdirectly to the observation that banks would always, if allowed, use input lotteries. Notice that the market-clearing price (cid:26) will be affected by the presence or absence of input lotteries. Controlling for these general equilibriumeffects,itisnotalwaysnecessarilythecasethatallagentswill be made better off by the introduction of input lotteries. This point is consideredingreater detailinthe nextsection. 15

3 An Example with Lumpy Capital In this section I specify preferences, endowments and technology following Aghion and Bolton (1997), and solve analytically for the optimal contracts with and without input lotteries. Without input lotteries, the combination of moral hazard and limited liability [in the sense of Sappington (1983)] produces credit rationing. There is a threshold wealth required to get loans. This credit-market failure produces a non-convexity in the agent's expected utility. With the addition of input lotteries agents with wealth below the threshold can trade their wealth for a fair lottery over zero wealth and some high wealth above the threshold. I provide necessaryconditionsforthisextracontracttoincreasethemarket-clearinginterest rate, produce Pareto-optimal allocations, increase total economy-wide output and for the distribution of wealth to converge to a unique invariant distribution. The results in this section depend on several convenient assumptions, including risk-neutral agents and a special technology. Further, I do not here characterize the invariant distribution of wealth withoutlotteries. InthenextsectionIusenumericaltechniquestocomparethe outcomes with and without input lotteries in a model which relaxes the assumptions on technology and preferences. In addition, I can compute the dynamicsboth with andwithout lotteries. 16

3.1 Economic Environment The economic environment is familiar: output Q can take on two values, f 0 ; 2 g ,where anoutput of zero means the project has“failed”;productive capital K is also limitedto two values, f 0 ; 1 g ;while effort Z isassumed to beadensegridontheinterval[0 ; 1],sothatcontractscanbewrittenessentiallytreatingeffortascontinuous. Agentsareriskneutral,sothatinprinciple,transfers T couldbelimitedtotwovalues. However,topreventgrid lotteries from affecting the evolution of the distribution of wealth, transfers are also assumed to be densely gridded. The consumption transfer grid T ison the interval [0 ; 2 = (1 (cid:0) s (cid:11) )],where (cid:11) is a preference parameter (see below) that will also turn out to be the highest market-clearing interest rate. Assets A are assumed to satisfy A = s T . The savings rate s will also have to be constrained. See section 3.5 below. Finally, the choice of k = 1 and 2 q = 2 is merely to conserve on notation. All of the following 2 results go through with more generalvalue for k and 2 q . 2 Agents have preferences that are linear in consumption transfers and quadraticineffort, so that: U (cid:28) z = (cid:28) (cid:0) z 2 (cid:11) ; where 0 < (cid:11) < 1. The technology exhibits strong complementarity between capital and labor effort: P ( q = 2 j z ; k ) = 8 < : z if k (cid:21) 1 0 if k < 1. 17

3.2 Contracts An element y of the contract space X can be interpreted as a probability massfunctionoverallpossibleeventsintheeconomythatsatisfiescertain conditions. Itispossibletorewritethejointprobabilityofaparticularoutcome y ( (cid:28) ; q ; z ; k ) as a sequence of conditional probabilities. (The contract y and all its component sub-lotteries in the discussion that follows are of course conditional on wealth a and the interest rate (cid:26) , but this notation is suppressedhereforclarity.) Thuslet (cid:24) betheprobabilityofbeingassigned thehighcapitallevel. Thiscanbeformedfromtheunderlyingcontract by integrating over allthe other variables: (cid:24) (cid:17) (cid:28) X q z y ( (cid:28) ; q ; z ; k = 1) : Withprobability1 (cid:0) (cid:24) theagentisnotassignedcapital. Giventheextreme form of the technology, it makes no sense to assign him the high effort level because the low output is certain. Call (cid:28) the agent's assigned con- 0 sumption inthatcase,so: (cid:28) 0 (cid:17) X (cid:28) (cid:28) y ( (cid:28) ; q = 0 ; z = 0 ; k = 0) : Assumefurtherthatanydesired (cid:28) isalwaysanelementof 0 T ,sothat,conditionalonlosing theinputlottery [andthusthecertainrealizationoflow capital,loweffortandlowoutput( k = 0 ; z = 0 ; q = 0)],thecontractassigns a consumption transfer (cid:28) with certainty. This is the same as assuming 0 that there are no grid lotteries required to realize an expected consumption transfer of (cid:28) . 0 18

If the agent is lucky (that is, is assigned the high capital input), then the agent will be assigned the high effort level with certainty. To see this, note that uncertainty over effort assignments would cost the agent utility (because (cid:18) 00 ( z ) > 0), would not increase expected output once capital had been assigned and would not help overcome the incentive compatibility constraint, because condition (C3) must hold after the resolution of any effort lottery. Therefore, conditional on assigning the high capital level, the bank will assign a single effort level with certainty, assuming that the the target effort is an element of Z . For convenience, assume that this is always the case. (Even in the numerical work, it is possible to begin with onespecificationofgridelements,andthenrecursivelyadjustthegridsby adding the expected value of any grid lotteries as an element of the grid.) Callthe assignedeffort z . Thus: ICC y ( z = z ICC j k = 1) = 1 ; and: y ( z 6 = z ICC j k = 1) = 0 : Depending on the outcome of the project, the bank will transfer some amount of the consumption good to the agent. Define (cid:28) to be the transfer conditional on the high output, q = 2, and (cid:28) to be the transfer conditional onthelowoutput, q = 0,inthesamewaythat (cid:28) wasdefinedabove. Once 0 again, assume that T contains the right elementsto avoidgrid lotteries. Thechoiceofcontract y canthusbeboileddowntoachoiceoftheparameters f (cid:24) ; z ICC ; (cid:28) 0 ; (cid:28) ; (cid:28) g foreachwealthtype a in A . Todescribevalidcontracts, these parameters must satisfy conditions (C1) to (C3). The requirement 19

that the contract be avalid lottery, (C1),can be satisfied if0 (cid:20) (cid:24) (cid:20) 1,if z ICC isin Z . andif (cid:28) 0 ; (cid:28) ; (cid:28) areallin T . Therequirementthatthecontractrespect the underlying technology, (C2),canbe satisfiedbyrequiring that: X (cid:28) y ( (cid:28) ; q = 2 ; z = z j ; k = 1) = z j (cid:24) ; all z j in Z ,and: X (cid:28) z y ( (cid:28) ; q = 2 ; z ; k = 0) = 0 : The final requirement is that effort assignments be incentive compatible. Usingthe incentive compatibility constraint (C3) above,it is easy to show that,iftheagenthasbeenassignedeffort z andiscontemplatingalower ICC effort level, ˆ z < z ,incentive compatibility requires that: ICC z ICC (cid:20) (cid:11) ( (cid:28) (cid:0) (cid:28) ) (cid:0) ˆ z : Assume that the nearest point in Z less than z is ICC z ICC (cid:0) h . This is the largest possible deviation, ˆ z . Thus the incentive compatibility constraint requiresthat: z ICC (cid:20) (cid:11) ( 2 (cid:28) (cid:0) (cid:28) )+ h 2 : Assume that the grid over effort assignments Z is so dense that we can take h to be zero. (Other interpretations are that the bank has to satisfy the “true” incentive compatibility constraint, formed when h = 0, or is not exactly sure where the nearest grid point is.) Thus, given (cid:28) and (cid:28) , the highesteffort that maybeassigned is: z ( ICC (cid:28) ; (cid:28) ) = (cid:11) ( 2 (cid:28) (cid:0) (cid:28) ) : (8) 20

Notice that if the agent receives the project's payoff, so that (cid:28) = 2 and (cid:28) = 0, incentive-compatible effort is z = ICC (cid:11) , which is also the first-best effort level. Theagent's expectedutility w from acontract f (cid:24) ; z ICC ; (cid:28) 0 ; (cid:28) ; (cid:28) g is: (cid:24) (cid:20) z ICC (cid:28) +(1 (cid:0) z ) ICC (cid:28) (cid:0) z 2 ICC (cid:11) (cid:21) +(1 (cid:0) (cid:24) ) (cid:28) 0 : Using the incentive compatibility condition (8) to substitute out z in ICC termsof (cid:28) ; (cid:28) ,this canbe rewritten as: w ( (cid:24) ; (cid:28) ; (cid:28) ; (cid:28) ) = 0 (cid:24) h (cid:11) ( 4 (cid:28) (cid:0) (cid:28) )2 + (cid:28) i +(1 (cid:0) (cid:24) ) (cid:28) 0 : (9) Finally, the bank must satisfy a zero-profit condition. Assuming that the interbank interest rate is (cid:26) , the bank's net revenues from a contract to an agentof wealth a are: R ( (cid:24) ; (cid:28) ; (cid:28) ; (cid:28) 0 j a ; (cid:26) ) = 2 (cid:24) z ICC (cid:0) (cid:28) (cid:24) z ICC (cid:0) (cid:28) (cid:24) (1 (cid:0) z ) ICC (cid:0) (1 (cid:0) (cid:24) ) (cid:28) 0 (cid:0) (cid:26) (cid:24) + (cid:26) a ; substituting z = ICC (cid:11) ( (cid:28) (cid:0) (cid:28) ) = 2gives: = (cid:11) (cid:24) ( (cid:28) (cid:0) (cid:28) ) (cid:0) (cid:24) h (cid:11) ( 2 (cid:28) (cid:0) (cid:28) )2 + (cid:28) i (cid:0) (1 (cid:0) (cid:24) ) (cid:28) + 0 (cid:26) a (cid:0) (cid:26) (cid:24) : (10) Where the cost of capital, (cid:26) , is multiplied by the probability of assigning capital (or the proportion of agents of wealth a who are assigned capital), (cid:24) ,to calculate the cost of funds. Thegeneralproblemofthebankinthisenvironmentcanbecastaschoosing contracts (for each borrower type a in A ) of f (cid:24) ( a ) ; (cid:28) ( a ) ; (cid:28) ( a ) ; (cid:28) ( 0 a ) g a 2 A to maximize utility (9) subject to the condition that revenues, defined in equation(10),bezero for eachborrower type. 21

3.3 Analysis Without Input Lotteries Prohibiting input lotteries is tantamount to forcing the bank to choose a level of (cid:24) of either 1 (capital assigned with certainty) or 0 (capital denied with certainty). If the bank chooses not to assign capital, then its maximizationproblem degeneratesto: max (cid:28) 0 (cid:28) 0 ; subject to: (cid:26) a (cid:0) (cid:28) 0 (cid:21) 0 : The optimal value of the transfer, (cid:28) ? ( 0 a ), is clearly just (cid:26) a , and assigned effort iszero. Thuswrite the agent'sexpectedutility inthiscase as: W ( NL a ; (cid:26) j (cid:24) = 0) = (cid:26) a : If the bank assigns capital with certainty ( (cid:24) = 1) then it chooses contracts to maximize: (11) W ( NL a ; (cid:26) j (cid:24) = 1) = max (cid:28) ;(cid:28) 2 T (cid:11) ( 4 (cid:28) (cid:0) (cid:28) )2 + (cid:28) subject to: (cid:26) ( a (cid:0) 1)+ (cid:11) ( (cid:28) (cid:0) (cid:28) ) (cid:0) (cid:11) ( 2 (cid:28) (cid:0) (cid:28) )2 (cid:0) (cid:28) = 0 : Because the smallest element of T is zero [there is limited liability in the senseofSappington(1983)],foragentswithwealth a belowthe minimum capitalscale ( a < 1),the optimal transfer whenthe project fails, (cid:28) ? ,willbe zero, and the optimal transfer when the project succeeds, (cid:28) ? , will be less than the high output, 2. This in turn, through the action of the incentive compatibility constraint, equation (8), means that effort supplied under the contract, z , will be below the first-best amount, ICC (cid:11) . Thus the optimal 22

transfer policiesare: (cid:28) ( a ) = 8 < : 0 0 (cid:20) a (cid:20) 1 (cid:26) ( a (cid:0) 1) a (cid:21) 1 (cid:28) ; ( a ) = 8 < : 1+ q 1 (cid:0) 2 (cid:26) (cid:11) (1 (cid:0) a ) 0 (cid:20) a (cid:20) 1 2+ (cid:26) ( a (cid:0) 1) a (cid:21) 1 : From this, one cansee that the supplyof laboreffort is: z ( ICC a ) = 8 > < > : (cid:11) 2 (cid:20) 1+ q 1 (cid:0) 2 (cid:26) (cid:11) (1 (cid:0) a ) (cid:21) 0 (cid:20) a (cid:20) 1 (cid:11) a (cid:21) 1 : (12) These results alsopoint towards the credit-rationing result of Aghion and Bolton (1997), namely, that a threshold wealth is required to obtain loans. For borrowers of wealth a < 1, (cid:28) ( a ) is real only if a > a ? ( (cid:26) ), where the threshold wealth a ? ( (cid:26) )is: a ? ( (cid:26) ) = 1 (cid:0) (cid:11) 2 (cid:26) : Thusthe expectedutility of anagentwhoisassigned capitalis: W ( NL a ; (cid:26) j (cid:24) = 1 ; a (cid:21) a ? ( (cid:26) )) = (cid:11) 4 " 1+ r 1 (cid:0) 2 (cid:11) (cid:26) (1 (cid:0) a ) # 2 : Agentswithwealthbelowthethreshold a < a ? cannotcrediblycommit to work hardenough tomake aloanworthwhile atanyinterest rate. This analysis implies a maximum and a minimum possible value for the market-clearingprice, (cid:26) . If (cid:26) (cid:20) (cid:11) = 2thenallagentswillbeassignedcapital. Thuslet (cid:26) = min (cid:11) = 2. Onthe other hand,if (cid:26) (cid:21) (cid:11) thenevenrich agents will be atbestindifferent about operating the technology. Thus let (cid:26) = max (cid:11) . 23

Banks will assign capital to agents with wealth above the threshold required to get loans only if their expected utility is greater with capital. Thus the expected utility of an agent of wealth a , when the interest rate is (cid:26) ,inaworldwithout asset-pooling lotteries, W ( NL a ; (cid:26) ), is: W ( NL a ; (cid:26) ) = 8 < : W ( NL a ; (cid:26) j (cid:24) = 0) if a < a ? ( (cid:26) ) max f W ( NL a ; (cid:26) j (cid:24) = 0) ; W ( NL a ; (cid:26) j (cid:24) = 1) g if a (cid:21) a ? ( (cid:26) ). Figure 1 plots the expected utility of an agent when input lotteries are prohibited,andalsowhentheyare allowed. (See the nextsection.) Notice the clear non-convexity in the expected utility schedule W at NL a ? (here a ? = 1 = 3): with input lotteries, agents are able to convexify around this region. 3.4 Analysis with Input Lotteries Inputlotterieswillallowbankstowrite contracts thatallowpooragents– that is, agents with wealth below the threshold required to get loans – access to capital. Banks will pool the assets of all agents of the same wealth andconcentrateitinthehandsofaselectedsubgroup. Thissubgroupwill be chosen at random, because all agents are completely identical. This type ofcontract replicates the lotroscastudiedby Besley,Coate andLoury (1993, 1994). The input lottery will smooth out the non-convexity in expected utility as a function of wealth, in a fashion identical to the “gambling for life” literature.2 Sadler (1998) studies a version of this problem, 2SeeRosen(1997)forfurtherreferencestothisliterature. Ithaslongbeenunderstood thatnon-convexities, or “indivisibilities,” in choice sets provide a motive for gambling. 24

Expected Utility 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 wealth a W Lotteries No Lotteries Figure 1: Expected utility with (dashed) and without (dotted) input lotteries when (cid:11) = 0 : 95 and (cid:26) = 0 : 7125. 25

and shows that even risk-averse agents would be willing to take gambles when faced with credit-market non-convexities. Agents who are not credit rationed, and who would have been assigned credit anyway, will alsoavailthemselvesoftheselotteries. Infigure 1theseagentsarelocated above the non-convexity but belowthe eventualtarget wealth. Thestructure ofthe problem isthe same asinthe previous section,except that now banks may use a further control variable. As in the previous section,thenon-negativityconstraintontransferswillbebindingforpoor agents,sothatagentswithwealthbelowunitygetapositive transferonly if they realize the high output. As before, (cid:28) = 0. Without input lotteries, the transfer conditional on not being assigned the high capital level, (cid:28) , 0 was just (cid:26) a . Now, with input lotteries, agents will prefer to concentrate all of their wealth into the state of the world in which they win the input lottery, so (cid:28) = 0. (This result depends on risk neutrality.) From equation 0 (10),write the bank'srevenue function as: R ( (cid:24) ; (cid:28) j a ; (cid:26) ) = (cid:11) (cid:24) (cid:28) (cid:0) 1 2 (cid:24) (cid:11) (cid:28) 2 + (cid:26) a (cid:0) (cid:26) (cid:24) : The bank's zero-profit condition is thus R ( (cid:24) ; (cid:28) j a ; (cid:26) ) = 0. Divide both sides ofthis equationbythe probability of beingassignedcapital, (cid:24) ,to form: (cid:11) (cid:28) (cid:0) (cid:11) (cid:28) 2 + 2 (cid:26) a (cid:24) (cid:0) (cid:26) = 0 : The term a = (cid:24) canbe thought of as the “target wealth”of the input lottery: alloftheagent'swealthisconcentratedintothestateoftheworldinwhich Inanothercontext,firmsthatareinfinancialdistresswillundertakeriskyprojectssothat insomestates,atleast,theyarenotbankrupt. 26

hewinstheinputlottery. Thelesslikelythisstateis,thegreaterhiswealth in it. Because (cid:24) is also the proportion of agents of wealth a who win the input lottery, a = (cid:24) can also be thought of as the wealth transfer from the pooling group asawhole tothose agents designatedasmanagers. By substituting in from equations (9) and (10) the bank's maximization problemis: (13) W ( a ; (cid:26) ) = max (cid:28) ;(cid:24) 2 [0 ; 1] (cid:24) (cid:11) 4 (cid:28) 2 +(1 (cid:0) (cid:24) ) (cid:1) 0 subject to: (cid:28) 2 (cid:0) 2 (cid:28) 2 + (cid:11) (cid:26) (cid:18) 1 (cid:0) a (cid:24) (cid:19) = 0 : Anotherwayto thinkaboutthisproblemisasatwo-stage contract. Inthe first stage, banks concentrate a poor agent's wealth a into an amount a = (cid:24) with probability (cid:24) , and zero with probability 1 (cid:0) (cid:24) . In the second stage, the input lottery outcome has been realized, and there are two possibilities. Either the agent was lucky and won the input lottery, and now has wealth a = (cid:24) , or the agent was unlucky and lost the input lottery and now has wealth zero. In either case, the bank then writes contracts with the agents as if there were no input lotteries. This tremendous simplification is entirely due to the assumption of risk-neutrality and limited liability. If agents were risk averse they would want insurance against the possibility of losing the input lottery, so (cid:28) could be non-zero, in which case this 0 derivation does not go through. In the next section, I solve a range of numericalexampleswithrisk-aversepreferencesandfindsimilardifferences betweeneconomies withinput lotteries andeconomieswithout them. 27

Thusthe bank'sproblem (13)mayalsobe written as: W ( a ; (cid:26) ) = max (cid:24) (cid:24) W ( NL a = (cid:24) ; (cid:26) j (cid:24) = 1)+(1 (cid:0) (cid:24) ) W (0 NL ; (cid:26) j (cid:24) = 0) : The wealth variable a and the choice variable (cid:24) can be replaced by the targetwealthofthegamble, a = TARG a = (cid:24) . Fromthisitfollowsthatthetarget wealth does not vary with own-wealth. Thus all agents who engage in an input lottery are seeking the same target wealth, poorer agents merely havealower probability of achievingit. Solving the bank's problem (13) above in terms of the target wealth, a , TARG revealsthat the optimaltarget is: a ( TARG (cid:26) 2 ) = (cid:11) (cid:26) (cid:0) 1 : (14) The associated lottery probability, (cid:24) ( a ; (cid:26) ), is a = a ( TARG (cid:26) ). Notice that at the highest-possible interest rate, (cid:26) , the target wealth is unity, which is exmax actly the amount of capital required to operate the technology. At this interest rate, all agents with wealth a (cid:20) 1 will be in a pooling group, and (cid:24) ( a ; (cid:26) m a x ) = a ,sotheywillusecapitalequal,inexpectedvalue,totheirown wealthlevel. For allinterest rates below (cid:26) ,the target wealthwillbe bemax lowunity,sothatevenluckyagentsintheasset-poolinggroupwillstillbe net borrowers. Also, there will be some agents with wealth a TARG < a < 1 whowillbe netborrowers but willnotbe inapooling group. Banks will never assign rich agents (agents with wealth a > 1) to pooling groups. As long as the interest rate satisfies (cid:26) (cid:20) (cid:26) , banks will set max (cid:24) = 1 for richagents. 28

With this formulation, it is now possible to write down W ( a ; (cid:26) ), the expected utility of an agent with wealth a at interest rate (cid:26) when input lotteriesare permitted. Thus: W ( a ; (cid:26) ) = 8 < : ( a = a ) TARG W ( NL a TARG ; (cid:26) ) if0 (cid:20) a (cid:20) a TARG W ( NL a ; (cid:26) ) if a (cid:21) a . TARG Notice that W ( a ; (cid:26) )islinearinwealthbelow a . TARG 3.5 Effect of Lotteries In the following results, I will require that the richest individual in the economybeunabletofinancetheprojectoutofownfunds. Onecaninterpret this is as requiring that the economy be “poor” or that the project be largerelativetohouseholdwealth. Let a bethewealthoftherichestinmax dividualintheeconomy. Inthepropositionsbelow,Irequirethat a max (cid:20) 1. Ialsoprovidesufficientconditionsfortheuniqueinvariantdistributionof wealthto satisfy a max (cid:20) 1,sothat allofthese results willholdeventually. Proposition 1(InterestRate) For any distribution of wealth such that a max (cid:20) 1, the market-clearing interest rate with input lotteries will be greater than the market-clearing interest rate withoutinputlotteries. Proof: Seethe appendix. 29

Proposition 2(AggregateOutput) Foranydistributionofwealth thatsatisfies a max (cid:20) 1,theequilibriumaggregate output ishigherwithinput lotteriesthanwithoutthem. Proof: Seethe appendix. Proposition 3(ParetoOptimality) For any distribution of wealth such that a max (cid:20) 1, the equilibrium allocation with input lotteries produces utilities ! ( a ) that are Pareto optimal. The shadow value of capital of the social planner is the market-clearing equilibrium interest rate. Proof: The proof is in the appendix. Although algebraically complex, it is conceptually straightforward: the equilibrium generates a set of expected utilities by wealth. When plugged into the social planner's problem as promisedutilities, the social plannerrealizesazerosurplus. Now consider the dynamics of this model. If the savings rate is above a critical level, the presence of lotteries merely accelerates growth to the same distribution studied by Aghion and Bolton (1997). In the complementary case, in which savings is low, it is a simple procedure to characterize the invariant distribution with lotteries. A subsistence technology (suppressed until now for expositional clarity) is required for nondegeneratedynamics. Withoutit,zerowealthbecomesanabsorbingstate. Assume that any agent may completely abjure intermediation and place all capital and zero labor into a backyard technology in exchange for an (cid:15) probability of realizing the high output. This is merely the most conve- 30

nient form of the subsistence technology. Nothing crucial dependson the assumption that own-capital is completely absorbed by the subsistence technology. Proposition 4(Convergence) If the savings rate s satisfies s (cid:20) 1 = 2 and if there is a subsistence technology as defined above, then, for any initial distribution of wealth, the equilibrium price converges to (cid:26) and the distribution of wealth converges to max ? , in which a proportion ? have wealth 0 a = 0 and the remaining proportion ? have wealth 1 2 s ,where: ? 1 = 0 (cid:0) 2 (cid:11) s 1 (cid:0) 2 (cid:11) s + (cid:15) ; and: ? = 1 (cid:15) 1 (cid:0) 2 (cid:11) s + (cid:15) : Proof: Seethe appendix. 4 An Example with Risk Aversion In this section I solve numerically a model with risk averse agents and multiple input choices. I use a version of the linear programming-based techniques of Phelan and Townsend (1991) to solve for the competitive equilibriawithandwithoutasset-poolinglotteries. (SeealsoPrescott(1998) for recent developments in this literature.) I can compare the resulting timepathsofwealthheterogeneityandmarketclearinginterestratesalong the transitions to the steady-state to determine the effect of adding asset- 31

pooling lotteries. I find that asset-pooling lotteries cause faster growth to a higher steadystate aggregate capital level and invariant distributions of wealth which feature greater inequality. Economies with lotteries are more unequal for a variety of reasons. First, for the same distribution of wealth, the market-clearing interest rate is higher, so that small differences in wealth translate into larger differences in average consumption. Second, agents are in general assigned higher effortwithlotteries,sothattheincentivecompatabilityconstraintrequires a greater variation in consumptions conditional on output. Third, the lotteries themselves promote inequality directly by rewarding lucky agents and punishing unlucky agents. If, without lotteries, a class of agents are “poorsavers”,thentheywillconsumeandbequeathequally. If,byadding lotteries, that same class of agents enter a pooling group, then the lucky oneswillconsume more thanthe unlucky ones. Economists have generally known that the complementarity between labor effort and capital affects optimal contracts.3 To study this effect, I specify the technology to be CES and vary the complementarity parameter. Generally, I find that richer agents wish to supply less effort than poorer agents. If capital and labor effort are complements, this means that poor agents should be assigned capital, while if they are substitutes, 3See,asonlyoneexampleinalargeliterature,Dupor(1998). Inthecontextofmoralhazardconstrainedcontracting,Lehnert,Ligon,andTownsend(1998)considertheeffect ofcomplementarityinamodelinwhichcapitalisnotaccumulated. 32

poor agents should not be assigned capital. Thus the capital allocation curve, that is, how assigned capital varies with wealth, depends critically on the technological complementarity between capital and labor. One interpretationofso-calledtrickle-downdynamicsisthatthecapitalallocation curve is, in poor economies, steeply upward sloping, while in developed economies it is flatter. Those dynamics arise here if capital and labor are substitutes. Ifcapitalandlabor are complements, thenthere may be trickle-updynamics, in which the capital allocation curve slopes down, and flattens as the economy develops. That is, if capital and labor are complements, then in pooreconomies,arichagentdoesnowork,isassignednocapitalandconsumes the rental value of his wealth. That same rich agent, in a relatively richereconomy withalowerrisk-free rate,might, incontrast, beassigned effort and capital, and consume both his (lower) rental income and the proceedsof hisproductive process. 4.1 Parameter Values Agentsare risk-averse, with autility function givenby: U (cid:28) z = 2 p (cid:28) (cid:0) 1 4 z : Effortislimitedtotwovalues,sothat Z = f 0 ; 0 : 9 g . Outputcanalsotakeon onlytwovalues, Q = f 0 ; 2 g . Capitalcantakeononeoffivevalueslinearly spaced between 0 and 1, K = f 0 ; 0 : 25 ; 0 : 50 ; 0 : 75 ; 1 g . The technology is 33

chosen to mimic a standard CES production function in expected value, withtheaddedconstrainttheprobabilityofsuccessorfailureneverbetoo highor too low. Thus: g (cid:11) ( z ; k ) = ( z (cid:11) + k (cid:11) )1 = (cid:11) ; and: P (cid:11) ( q = 2 j z ; k ) = 8 > > > < > > > : 0 : 05 if g (cid:11) ( z ; k ) (cid:20) 0 : 05 g (cid:11) ( z ; k ) if 0 : 05 (cid:20) g (cid:11) ( z ; k ) (cid:20) 0 : 95 0 : 95 if g (cid:11) ( z ; k ) (cid:21) 0 : 95 : As part of the numerical experiment, I calculate equilibrium sequences and transition paths with and without lotteries for eight different values of (cid:11) : (cid:11) = f (cid:0) 100 ; (cid:0) 1 ; (cid:0) 0 : 5 ; (cid:0) 0 : 1 ; 0 : 1 ; 0 : 5 ; 1 ; 100 g : Negative values of (cid:11) mean that capital and labor effort are complements (botharerequiredtorealizehighoutput),whilepositivevaluesmeanthat capital and labor effort are substitutes (either can be used to realize the highoutput). Theextremevaluesof (cid:11) ,-100and100approximateaperfect complements (Leontieff) technology and a super-substitutes technology. The choice of (cid:11) = 100 is unusual and deserves explanation. As (cid:11) grows, the technology converges to the maximumoperator, so that: lim (cid:11) ! 1 g (cid:11) ( z ; k ) = max f z ; k g : This is a quasiconvex function, and is seldom used. It is useful here, however,because it allowsagents to realize the highoutput witheitherahigh capitalorahigheffort level. Underthe more standard formulation ofperfectsubstitutes, (cid:11) = 1,the technology is: g (cid:11) ( =1 z ; k ) = z + k : 34

Noticethat,althoughtheirmarginalcontributiontooutputisindependent of the other input level, both capital and labor are required to make sure ofthe high output. The savings rate is fixedat s = 0 : 3. Inthe numericalwork, I foundit more convenient to have intermediaries assign utility conditional on outcomes. Each assigned utility has an associated transfer and bequest policy. I providedagrid of 81linearly-spacedutility points. Thisis equivalentto having 81 nonlinearly-spaced transfer and bequest points, with a denser concentration of points near the low end of transfers (where the utility function is more curved). Because of computational constraints, richer specifications of the technology, which feature more effort and capital points, must come at the cost of a sparser grid over transfers T . This introduces undesirablegridlotteriesintothecomputedsolutions. Thechoice oftechnology here sacrifices some measure of technological verisimilitude in favor ofavery densegrid overconsumption transfers, T . 4.2 Results I beginwith adetailedanalysisof the case when (cid:11) = (cid:0) 0 : 5(a typical case), andthendiscuss the results across allvaluesof (cid:11) .4 In figures 2 and 3 I plot the evolution of the distribution of wealth with 4The equivalent results from all the other values of (cid:11) are suppressed to save space. Theyareavailabletotheinterestedreader. 35

and without lotteries for the case when (cid:11) = (cid:0) 0 : 5; while in figures 4 and 5 I plot the evolution of the total amount of wealth and the market clearing interest rate in both economies. Both economies begin with all agents endowed with zero capital. Because the minimum probability of the high output is 0.05, 5% of these agents get the high output, and the resulting output is distributed equally to all agents, because there is no moral hazard. Once there is a little bit of capital in the economy, differences begin to emerge betweenthe lottery andthe no-lottery economies. These differences are initially small but cumulative. The no-lottery economy remains relatively poor with a concentrated wealth distribution, while the lottery economy is richer, with a less concentrated wealth distribution. Notice that the market-clearing interest rate, in figure 5, is initially greater in the economywithlotteries. Eventuallythelotteryeconomybecomessomuch richer than the no-lottery economy that the market-clearing interest rate inthe lottery economy falls wellbelowthe no-lottery economy. Theinvariantdistributionsofwealtharrivedatbybotheconomiesaredisplayed in figure 6. In figure 7 I plot the Gini coefficient (a common scalar measure of inequality) over time for both economies. Without lotteries, inequality rises steadily as the economy converges to the invariant distribution. With lotteries, there is an early surge in inequality, which then peaksandmoderatesslightly. Thiseffectismuchmore dramaticfor other values of (cid:11) . When (cid:11) = (cid:0) 1, for example, the Gini coefficient peaks near 0.23 before falling to its steady-state level of 0.125. This accords well with the Kuznets hypothesis about inequality over the development cycle. For a careful microeconomic decomposition of inequality over time in Thai- 36

land,see Jeong(1998). In figure 8 I plot the evolution of aggregate capital from several different initial distributions. These different initial distributions feature different average wealth levels. Two of them begin with more capital than the steady state, so over time capital falls. The distributions converge to the sameinvariant distribution. Repeatingthisanalysisforallvaluesof (cid:11) yieldsasteady-statecapitallevel and market-clearing interest rate for each. These are displayed in figures 10 and 11. Notice that the solutions are close at the extreme values of (cid:11) , but differ markedly in between. Notice also that economies with substitutestechnologies(highvaluesof (cid:11) )arerichernomatterwhatthefinancial structure. Thisisbecausetheproductionpossibilitiessetislargerwheneither capital or labor may be used to achieve the high output. In all cases theeconomywithlotteriesfeaturesahighersteady-stateaggregatewealth levelthan the economy without them. In general, invariant distributions in economies with lotteries feature greater inequality than in economies without lotteries. In figure 9 I plot the Ginicoefficients from the invariant distribution of wealthat eachvalue of thecomplementarity parameter (cid:11) . Notice thaninsevenoftheeightcases, the Gini measure of inequality is higher with lotteries than without them. Only when (cid:11) = 0 : 1 is the lottery economy more equal than the no-lottery economy, and even there they are close. Note also that in most cases the Gini coefficients lie between 0.05 and 0.20, well below the estimates of 37

moderndevelopedeconomies, whichlie between0.4and0.6. Distribution with Lotteries ( a =−0.5) 1 0.8 0.6 0.4 0.2 0 5 10 Time: t 15 0 0.1 0.2 0.3 0.4 0.5 Wealth: A y :noitroporP t Figure 2: Evolution of the distribution of wealthwith inputlotteries. Distribution, no Lotteries ( a =−0.5) 1 0.8 0.6 0.4 0.2 0 5 0.1 10 0.05 Time: t 0 Wealth: A y :noitroporP t Figure 3: Evolution of the distribution of wealthwithout inputlotteries. 38

Wealth ( a =−0.5) 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 25 Time: t a :htlaeW latoT t Lotteries No Lotteries Figure4: Evolutionofaggregatewealthwith andwithout input lotteries. a Interest Rate ( =−0.5) 3 2.8 2.6 2.4 2.2 2 0 5 10 15 20 Time: t r :etaR tseretnI t Lotteries No Lotteries Figure 5: Evolutionofmarket-clearinginterest rate withandwithout input lotteries. 39

a Invariant Distribution ( =−0.5) 0.5 0.4 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 Wealth: a a y :noitroporP Lotteries No Lotteries Figure 6: Invariant distributions of wealth with andwithout inputlotteries. Gini (a =−0.5) 0.25 0.2 0.15 0.1 0.05 0 0 5 10 15 20 Time: t )t( G :iniG a Lotteries No Lotteries Figure 7: Evolution of the Gini coefficient over time,withandwithout input lotteries. 40

a Wealth Evolution ( =−0.5) 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 20 25 Time: t a :htlaeW latoT t Lotteries No Lotteries Figure 8: Aggregate capital levels over time from manydifferent initialdistributions. Steady−State Gini Coefficients 0.25 0.2 0.15 0.1 0.05 0 −100 −1 −0.5 −0.1 0.1 0.5 1 100 a G a Lotteries No Lotteries Figure 9: Gini coefficients of the invariant distributionsofwealthwithandwithoutlotteriesfordifferentvaluesofthecomplementarity parameter (cid:11) . 41

Mean Wealth Levels 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 −100 −1 −0.5 −0.1 0.1 0.5 1 100 a htlaew naeM Lotteries No Lotteries Figure 10: Terminal aggregate wealth levels with and without lotteries for different valuesof the complementarity parameter (cid:11) . Terminal Interest Rates 3 2.5 2 1.5 1 0.5 0 −100 −1 −0.5 −0.1 0.1 0.5 1 100 a r :setaR tseretnI ¥ Lotteries No Lotteries Figure 11: Terminal interest rates with and without lotteries for different values of the complementarity parameter (cid:11) . 42

5 Conclusion Group lending is usually taken to mean a joint-liability credit contract. This sort of lending is preferable because group members have an incentivetoencourageothersintheirgrouptorepayloansandmayhavemeans topressureormonitortheirpeersnotavailabletooutsideinstitutions. (See Ghatak and Guinnane (1998) for an excellent survey of this literature.) In this paper I have identified a subtle variant on this common and interesting contract. This paper concentrated on the ability of a group to pool its assets. The value of the summed assets is greater than the summed value oftheassetsbecauseofnon-convexitiesbuiltintothetechnologyandarisingfrom endogenous credit rationing. Mostresearchersagreethatcreditmarketimperfectionsplayanimportant role indevelopment. Recentmodelshavefocused onthe productive costs of wealth inequality and the possibility that the poor might be trapped in poverty forever. This paperadopted the same general framework used in the literature (acapital market with amoral hazard problem) but allowed financial intermediaries to write very general contracts, based on lotteries, with borrowers. The asset-pooling contracts that then endogenously emerged closely resemble roscas, or rotating savings and credit associations. This contractual innovation produced a host of interesting results. Lotteries interacted with credit markets to allow poor agents to escape the effectsofcreditrationing,andtheinvariantdistributions ofwealthfeatured 43

a higher mean but also increased inequality with lotteries. Input lotteries could produce Pareto optimaloutcomes, but not necessarily Pareto dominate allocations without lotteries. How can we use these results in thinking about economic development? Because lotteries act as a pooling device, this paper can be thought of as discussing the consequences of pooling mechanisms in development. Thereisplentyofevidencethatroscasplayanimportantroleindeveloping societies. Similar institutions, among them the familiar building societies, played an important role during the industrial revolution in developed countries [see Landes (1969)]. Alternatively, these results can be thought ofaspointing towards the effectof betterfinancialintermediation. Given the structure of competition among financial intermediaries, this paper featured no barrier to contracts other than an endogenous moral hazard constraint. Pooling contracts could be victims of a host of other problems: they could be prohibited by government fiat (perhaps for domestic political reasons); some cost to financial contracting (not modeled in this paper) could further constrain contracts between borrowers and lenders; or there could be a commitment problem, with either agents or intermediaries allowed to renege on their obligations. When examining institutions as they exist in developing countries we have to keep this list ofcalamitiesinmind. Presented with two otherwise identical nations, differing only because in one (for the reasons outlined above) asset-pooling groups do not exist, 44

whileintheothernationtheydoexist,wewouldexpecttheformernation togrowmoreslowlyandsettledowntoalowercapitallevelthanthelatter nation. The nation without pooling groups, however, would feature less inequalitythanthe nation withpooling groups. 45

Appendix The Incentive Compatibility Constraint The incentive compatibility constraint in the contract space X , (C3), can be derivedfollowingPrescottandTownsend(184a,b)andPhelanandTownsend(1991). One can think of the contract as specifying a conditional sub-lottery over consumption (cid:28) upon the realization of output q with probabilities x ( (cid:28) q j ). Thus for expectedutilitygivenanassignedeffort z andcapital k todominatetheexpected utilityfromacontemplateddeviationineffortto ˆ z , x ( (cid:28) q j )mustsatisfy: X (cid:28) q x ( (cid:28) q j ) P ( q z j ; k ) U (cid:28) z (cid:21) X (cid:28) q x ( (cid:28) q j ) P ( q j ˆ z ; k ) U (cid:28) ˆ z : Thismayberewrittenas: X (cid:28) q x ( (cid:28) q j ) P ( q z j ; k ) U (cid:28) z (cid:21) X (cid:28) q x ( (cid:28) q j ) P ( q z j ; k ) P ( q j ˆ z ; k ) P ( q z j ; k ) U (cid:28) ˆ z : Multiplying by the marginal probability of a particular assignment ( z ; k ) produces: X (cid:28) q x ( (cid:28) ; q ; z ; k ) U (cid:28) z (cid:21) X (cid:28) q x ( (cid:28) ; q ; z ; k ) P ( q j ˆ z ; k ) P ( q z j ; k ) U (cid:28) ˆ z : Whichis,ofcourse,exactlytheconstraintinequation(C3)above. Thereareafewsubtletiestotheincentivecompatibilityconstraintasusedinthis paper. The order of inputs, for example, makes a critical difference. The model assumes that capital is applied before effort, so that the agent knows k before selecting z . If capital k were selected after effort z , so that the agent could only knowthedistributionofpossiblevaluesofcapitalwhenchoosingeffort,thenthe incentivecompatibilityconstraintwouldbe: (cid:28) X q k x ( (cid:28) ; q ; z ; k ) U (cid:28) z (cid:21) (cid:28) X q k x ( (cid:28) ; q ; z ; k ) P ( q j ˆ z ; k ) P ( q z j ; k ) U (cid:28) ˆ z : When choosing z , with k not known, the agent must use the contracted probability distribution x in determining the expected utility values of various plans. Notice that, because there are now only n 2 Z constraints, as opposed to n 2 Z n K in 46

(C3) above, X would be a larger set. The extra choices lead to solutions that are weakly better. On the other hand, it seems more natural to have effort supplied conditional on a particular capital input to the technology, and it is the usual specificationintheliterature. Because the capital input is public, the suggested capital input level does not needtobeinduced. If,however,capitalwereprivate,sothat,forexample,agents were free to reinvest any capital transfers anonymously in banks before the resolution of production uncertainty, then suggested capital levels would have to beinduced. Thebenefittodeviationsincapitallevelwouldbepurelypecuniary. Optimalcontractssubjecttothis“inputdiversion”constraintarestudiedinmuch greaterdetail by Lehnert,Ligon, and Townsend(1998), who find that it can dramatically alterinputuse. Contracts Without Input Lotteries Let W ( NL a ; (cid:26) z j ; k ) be the expected utility of an agent with wealth a when the market-clearing interest rate is (cid:26) , who is assigned input combination ( z ; k ) with certainty. Let W ( NL a ; (cid:26) )betheexpectedutilityofanagentwhenthebankhaschosenthebestinputcombination( z ; k ). Thushebankchoosescontracts y a ( NL (cid:28) ; q z j ; k ) tosolve: W ( NL a ; (cid:26) z j ; k ) (cid:17) max y a ( NL (cid:28) ; q j z ; k ) (cid:0) (cid:18) ( z )+ X (cid:28) q y a ( NL (cid:28) ; q z j ; k ) u ( (cid:28) ) : Themaximizationproceedssubjecttothebank'szero-profitconstraint: (cid:26) ( a (cid:0) k )+ X (cid:28) q y a ( NL (cid:28) ; q z j ; k )( q (cid:0) (cid:28) ) (cid:21) 0 : Notethatthisconstraintmustholdseparatelyforeach z ; k combination. The distribution over outputs is determined by the choice of non-stochastic inputs. Thus foreach q in Q , givena choice of inputs z ; k , the Bayescompatibility constraint(C2)becomes: X (cid:28) y a ( NL (cid:28) ; q z j ; k ) = P ( q j z ; k ) ; all q ; z ; k in Q (cid:2) Z (cid:2) K : (15) 47

Finally,theassignedeffortmustbeincentive-compatible,sothatifinputs z ; k are assigned,thecontract y a ( NL (cid:28) ; q j z ; k )mustsatisfy,forallpossibledeviations ˆ z in Z : (16) X (cid:28) q y a ( NL (cid:28) ; q z j ; k ) n u ( (cid:28) ) (cid:0) (cid:18) ( z ) P (cid:0) ( q j ˆ z ; k ) P ( q z j ; k [ ) u ( (cid:28) ) (cid:0) (cid:18) (ˆ z )] (cid:27) (cid:21) 0 ; all q ; z ; k in Q (cid:2) Z (cid:2) K . Noticethattheremaybenocontract y a ( NL (cid:28) ; q z j ; k )foraparticularcombination z ; k that satisfies conditions (6), (15) and (16). If this is the case, let W ( NL a ; (cid:26) z j ; k ) (cid:0) 1 (cid:17) . Clearly,thereisatleastonecontractthatdoessatisfyconditions(6),(15)and (16), namely, onethat assignsthelowesteffort and capital level, z = 0 ; k = 0 and hastransfersthatequaltheoutputrealizations, (cid:28) = q . Proof of Proposition 1 Theequilibriumwithlotteriesiseasytocalculate. At (cid:26) theaggregatedemandfor capitalwithlotteriesis: K d ( (cid:26) ) = X a a (cid:24) ( a ; (cid:26) ) : That is, if banks assign a proportion (cid:24) ( a ; (cid:26) ) of each wealth type capital, then the aggregatedemandforcapitalistheweightedsumoftheproportions. Fromequation(14) above,itisclearthat a ( TARG (cid:26) ) = 1,so max (cid:24) ( a ; (cid:26) ) = max a . Thus,withlotteries: K d ( (cid:26) ) = X a a a : But this is just the aggregate quantity of capital in the economy. Further, if (cid:26) (cid:26) < ,then max (cid:24) ( a ; (cid:26) ) > a ,sotheaggregatedemandexceedsaggregatesupply. Without lotteries, in contrast, when (cid:26) = (cid:26) no agents with wealth max a < 1 will operate the technology. If there is any capital in the economy then the capital markethasnotcleared. Thustheequilibriuminterestratewithoutlotteriesmust bestrictlylessthan (cid:26) . max 48

Proof of Proposition 2 Thestrategyhere is toshowthatoutputwithlotteriesattains thefirst-bestlevel, andthatoutputwithoutlotteriesmustfallshortofthislevel. Withlotteries,from proposition 1, the equilibrium interest rate must be (cid:26) . Hence from equation max (14) itis clearthatthetargetwealthis a ( TARG (cid:26) ) = 1andtheprobability ofwinmax ning the input lottery, (cid:24) ( a ; (cid:26) ) is just max a . From equation (12), it is clear that the effortassignedthoseagentswhowinthelotterywillbe z = ICC (cid:11) . Thuseachagent hasaprobability a (cid:11) ofrealizingthehighoutput(inthiscase,2). Henceaggregate economy-wideoutputwithlotteries, Q ,is Q =2 (cid:11) X a a a ; orsimply2 (cid:11) a ,where a isthetotalamountofcapitalintheeconomy. Noticethat this is the first-best amount of output and that each unit of capital is used in a projectinwhichthesuppliedeffortis (cid:11) . Next, note that, without lotteries, at least some units of capital must be used in projects in which the supplied effort is below (cid:11) . From equation (12) above, it is clearthat,withoutlotteries,assignedeffortcanbe (cid:11) foragentswithwealth a (cid:20) 1 onlyif: (cid:11) 2 " 1+ r 1 (cid:0) 2 (cid:11) (cid:26) (1 (cid:0) a ) # = (cid:11) : Thisistrueiff: (cid:26) (1 (cid:0) a ) =0 : Inotherwords,assignedeffortwithoutinputlotteriesislessthan (cid:11) unlesseither (cid:26) = 0 or a (cid:21) 1. Because theminimum possible interestrate is (cid:26) = min (cid:11) = 2, which isgreaterthanzero,thismeansthatitisimpossibletoassignagentswithwealth below unity an effort of (cid:11) . If there are any agents with wealth strictly less than unity,thenoutputmuststrictlybelessthan2 (cid:11) a . Proof of Proposition 3 Fromproposition1 above weknowthat the equilibrium interestrate must be (cid:11) ; further,fromequation(13)weknowthattheexpectedutilityofanagentofwealth a is a (cid:11) . 49

Theplannermustalsochoosecontractswhichliein X . Here,theplanner'sproblemiswrittenwiththeincentivecompatibilityconstraintandtheBayes' compatibilityconstraintexplicitlyformulatedforconvenience: max (cid:25) a 2 L X a a (cid:28) X q z k (cid:25) a ( (cid:28) ; q ; z ; k )( q (cid:0) (cid:28) ) ; subjectto: (cid:28) X q z k (cid:25) a ( (cid:28) ; q ; z ; k ) U (cid:28) z = ! ( a ) ; all a in A (P1) , X a a 2 4 0 @ (cid:28) X q z k (cid:25) a ( (cid:28) ; q ; z ; k ) k 1 A (cid:0) a 3 5 (cid:21) 0 ; (P2) X (cid:28) q (cid:25) a ( (cid:28) ; q ; z ; k ) (cid:20) U (cid:28) z (cid:0) P ( q j ˆ z ; k ) P ( q j z ; k ) U (cid:28) ˆ z (cid:21) (cid:21) 0 ; (P3) all z ; ˆ z ; k in Z (cid:2) Z (cid:2) K ,andthefinalconstraint: X (cid:28) (cid:25) a ( (cid:28) ; q ; z ; k ) = P ( q j z ; k ) X (cid:28) q (cid:25) a ( (cid:28) ; q ; z ; k ) : (P4) Nowreplacethesechoicevariables withthefamiliarchoices: f (cid:24) ( a ) ; (cid:28) ( 0 a ) ; (cid:28) ( a ) ; (cid:28) ( a ) g a 2 A ; usedintheequilibriumanalysisabove. Asbefore,wereplacetheincentivecompatibilityconstraint(P3)withtheconditionthatassignedeffortbeequalto (cid:11) ( (cid:28) (cid:28) (cid:0) ) = 2. ThepoliciesareBayescompatiblewiththeunderlyingprobabilitydistributionifthehighoutputistakentooccurwithprobability (cid:24) z . Imagine that the planner has committed to provide an expected utility of x and nocapitalwithcertainty. Letthefunction D ( x )beplanner'ssurplusinthiscase: D ( x ) (cid:17) max (cid:28) 0 (cid:0) (cid:28) 0 ; subjectto: (cid:28) = 0 x : Itiseasytoseethat D ( x )= (cid:0) x . Nowconsidertheplanner'ssurplusifsheassigns 50

capitaltotheagentwithcertainty: D ( x ) (cid:17) max (cid:28) ;(cid:28) ;z ICC z ICC (cid:28) +(1 (cid:0) z ) ICC (cid:28) ; subjectto: z = ICC (cid:11) ( 2 (cid:28) (cid:0) (cid:28) ) z ; ICC (cid:28) +(1 (cid:0) z ) ICC (cid:28) (cid:0) z 2 ICC (cid:11) = (cid:28) ; (cid:28) (cid:21) x ; 0 : For x (cid:21) 0onlythenon-negativityconstrainton (cid:28) willbind. Beginbysubstituting outtheincentivecompatibilityconstraint. Theconstrainedoptimizationproblem thenbecomes: D ( x )= max (cid:28) ;(cid:28) L ( (cid:28) ; (cid:28) ; (cid:21) 1 ; (cid:21) ) 2 ; where: L ( (cid:28) ; (cid:28) ; (cid:21) 1 ; (cid:21) )= 2 (cid:11) ( (cid:28) (cid:0) (cid:28) ) (cid:0) (cid:11) ( (cid:28) (cid:0) (cid:28) )2 (cid:0) (cid:28) + (cid:21) [ 1 (cid:11) ( (cid:28) (cid:0) (cid:28) )2 (cid:0) (cid:28) (cid:0) x ]+ (cid:21) 2 (cid:28) : Thefirst-orderconditionsformaximizationrequirethat: (cid:21) =1 1 (cid:0) (cid:21) 2 ; and: (cid:21) = 2 (cid:0) 1+ p (cid:11) = x : If the non-negativity constraint does not bind, so that (cid:21) = 0, promised utility 2 mustsatisfy: x (cid:21) ! ? ; where ! ? (cid:17) (cid:11) . Here ! ? is a “critical utility” which will play an important role later. Thus for agents with promised utility below the critical utility, x < ! ? , we know that the non-negativityconstrainton (cid:28) will bind. Hencewecansplittheoptimalpolicies intotwosections: (cid:28) ( x ) =0 ; and (cid:28) ( x ) =2 p x = (cid:11) if 0 (cid:28) (cid:20) x (cid:20) ! (cid:3) ; ( x ) = x (cid:0) ! (cid:3) ; and (cid:28) ( x ) = x (cid:0) ! (cid:3) +2 if x (cid:21) ! (cid:3) : Thuswecanwrite D ( x )as: D ( x j 0 (cid:20) x (cid:20) ! (cid:3) )= 2 p (cid:11) x (cid:0) 2 x (17) D ( x x j (cid:21) ! (cid:3) )= ! (cid:3) (cid:0) x : 51

Planner’s conditional surpluses 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 0.2 0.4 0.6 0.8 1 w ) (w D Figure 12: The functions D ( ! ) (solid) and D ( ! )(dotted) when (cid:11) = 0 : 5. Note that for x (cid:20) ! (cid:3) , D has an increasing component. This produces the wellknownupward-slopingportionofthe Pareto frontierso familiar in modelswith moral hazard constraints [see e.g. Phelan and Townsend (1991)]. In figure 12 I displaytypicalschedulesof D ( ! )and D ( ! ). Wecannowcompletelyrewritetheplanner'sproblem: max (cid:24) ( a ) ; ! ( a ) ; ! ( a ) X a a (cid:16) (cid:24) ( a ) D ( ! ( a ))+(1 (cid:0) (cid:24) ( a )) D ( ! ( a )) (cid:17) ; (18) subjectto: (cid:24) ( a ) ! ( a )+(1 (cid:0) (cid:24) ) ! ( a ) = ! ( a ) X a a (cid:24) ; ( a ) = ! ; ! (cid:21) a ; 0 : Thus the planner can be seen as choosing a joint lottery over capital and utility assignments. Because D ( x ) > D ( x ) the planner finds it cheaperto assignhigher utilityalongwithhighercapitallevels. The planner's optimal plan will be to set ! = 0, ! = ! (cid:3) and then adjust (cid:24) ( a ) to matchthepromisedutilityoftheagent,sothat (cid:24) ( a )= ! ( a ) = ! (cid:3) . 52

To see this, consider a planner who chooses some values ! ( a ) ; ! ( a ) for an agent with promised utility ! ( a ). To satisfy the promise-keeping constraint, (cid:24) ( a ) must thensatisfy: (cid:24) ( a ) = f ( a ) (cid:17) ! ( a ) (cid:0) ! ( a ) ! ( a ) (cid:0) ! ( a ) : Notethatthederivativesof f withrespectto ! and ! are: @ @ f ! = (cid:0) ! f (cid:0) ! and @ @ f ! = (cid:0) 1 ! (cid:0) (cid:0) f ! : Rewritetheplanner'sproblem(18)intermsof f ,removingthepromise-keeping constraint. Let (cid:22) be the multiplier associated with the resource constraint. The first-ordernecessaryconditionsforoptimalityare: 2 p (cid:11) ! ( a ) (cid:0) ! ( a ) (cid:0) ( ! ( a ) (cid:0) ! ( a )) r ! (cid:11) ( a = ) (cid:22) (19) 2 p (cid:11) ! ( a ) (cid:0) ! ( a )= (cid:22) (20) X a a f ( a )= a : (21) Combiningequations(19)and(20)immediatelyproduces: ! ( a ) = (cid:11) ; and: (cid:22) = (cid:11) : Noticethattheshadow-priceofresources, (cid:22) , istheequilibrium interestrate, (cid:22) = (cid:26) ? . Tofind ! ( a )considertheresourceconstraint,equation(21): X a a ! ( a ) (cid:0) ! ( a ) ! ( a ) (cid:0) ! ( a = ) X a a a : By proposition 1 we know that the competitive equilibrium produces expected utilities: ! ( a ) = a (cid:11) ; or: ! ( a ) = a w ? : Combinedwiththeearlierresultthat ! ( a ) = w ? ,theresourceconstraintbecomes: X a a a w ? (cid:0) ! ( a ) w ? (cid:0) ! ( a = ) X a a a : 53

Because a max (cid:20) 1, ! ( a )= 0all a . Thisinturnprovidesapolicyfor (cid:24) : (cid:24) ( a ) = a : Sothatanagent'swealthdeterminestheprobabilityofgettingthecapitalassignment. Finally,wemustdeterminethevalueoftheplanner'ssurplus. Because ! ( a ) = w ? and ! ( a ) = 0, and because D ( w ? ) = 0 and D (0) = 0, we see immediately that the planner's surplus must be zero. The planner does not have a positive surplus at the expected utilities generated by the competitive equilibrium. Hence the equilibriumallocationsmustlieontheParetofrontier. Proof of Proposition 4 In this discussion, let a 0 ( a ; (cid:26) ) be the bequest of an agent of wealth a when the interest rate is (cid:26) . By assumption, s (cid:26) < 1, so successful (and rich) agents with wealth a > 1bequeath: a 0 ( a (cid:21) 1 ; (cid:26) j success) = s [2+ (cid:26) ( a (cid:0) 1)] < a : Unsuccessful(andrich)agentswithwealth a > 1bequeath: a 0 ( a (cid:21) 1 ; (cid:26) j failure) = s (cid:26) ( a (cid:0) 1) < a : The richest agent in the economy will eventually have wealth a = 2 max s (cid:20) 1. From this point forward the interest rate will be (cid:26) from proposition 1 above. max In figure 13 I display the wealth transitions with lotteries. Agents with wealth below a f willchoosetoforgointermediationandusethesubsistencetechnology, where a f = 2 (cid:15) = (cid:11) . Nowconsider the unique invariant distribution to which all initial distributions converge. From the discussion above, it is clear that the richest member of the economy, with wealth a will eventually satisfy max a max (cid:20) 1 and the equilibrium discussion from above will hold, so that (cid:26) ? = (cid:26) . Say that this occurs in some max period,called,arbitrarily,period t =0andthatthedistributionofwealthis ( 0 a ). Of all agents with wealth a < a f , a proportion (cid:15) will bequeath a 0 = 2 s while the remaining1 (cid:0) (cid:15) bequeath a 0 =0,andofallagentswithwealth a (cid:21) a f ,aproportion a (cid:11) willbothwintheinputlotteryandhave thehighoutput,sobequeathing a 0 = 2 s whiletheremaining1 (cid:0) a (cid:11) bequeath a 0 =0. Thusinthenextperiod, t =1,there 54

Wealth transitions with lotteries 2 1.5 1 0.5 0 0 0.5 1 1.5 2 wealth a ’a stseuqeB 45−degree line High Bequest Low Bequest Figure 13: Transitions when (cid:11) = 0 : 95, (cid:15) = 0 : 05and s = 0 : 5. willbeonlytwotypesofagents: agentswithwealth0andagentswithwealth2 s . Furthermore, agents will transition only between these two wealth levels. The Markovtransitionmatrixforthesestates,followingHamilton(1994), is: P = (cid:20) 1 (cid:0) (cid:15) 1 (cid:0) 2 (cid:15) (cid:11) s 2 (cid:11) s (cid:21) : Herethecolumns of P give today'sstatewhile therowsgive tomorrow's. Thus, given that an agent has zero wealth, the probability of remaining at zero wealth is1 (cid:0) (cid:15) whiletheprobabilityoftransitingto2 s is (cid:15) . Inthesameway,giventhatan agenthas2 s ,theprobabilityoffallingtowealth0is1 (cid:0) 2 (cid:11) s ,whiletheprobability ofremainingthereis2 (cid:11) s . Bysolving theeigenproblemassociatedwith P , one can find theeigenvectorassociated withthe unit eigenvalue. This gives the distributionof wealthbetween thetwopossiblewealthstates. Itis: ? = (cid:18) 1 (cid:0) 2 (cid:11) s 1 (cid:0) 2 (cid:11) s + (cid:15) ; (cid:15) 1 (cid:0) 2 (cid:11) s (cid:19) : Theaggregatesteady-statewealthinthiseconomyisthus a 1 =2 s (cid:15) = (1 (cid:0) 2 (cid:11) s + (cid:15) ). 55

Numerical Techniques Because the objective function (1), the contract constraints (C1), (C2), and (C3) andthezero-profitcondition(2)arealllinearinthechoiceobjects y a ( (cid:28) ; q ; z ; k ),for a given set of parameters, wealth a and interest rate (cid:26) it is in principle a simple matter to calculate the optimal contract. Given the optimal contracts and the distribution of wealth at time t , one can search for the market-clearing interest rate, (cid:26) ? t , and next period's distribution of wealth. I now describe this process in greaterdetail. Let N = n T n Q n Z n K be the length of the contract vector. Each position along the vector corresponds to a unique event, that is, a combination of consumption transfer (cid:28) ,output q ,effort z andcapital k . Thusthelastentryinthevectormight correspond to the event that the highest capital and effort levels are assigned (and used), that the highestoutputis realized and that the highestpossible consumption transfer is made. For concreteness, assume that variables are ordered as T ; Q ; Z ; K , so that the first n T elements of the event space correspond to the events f T ; q 1 ; z 1 ; k 1 g , the next n T elements to f T ; q 2 ; z 1 ; k 1 g and so on. Let T e be the1 (cid:2) N vectoroftransfervaluesateachpointintheeventspace: T e (cid:17) 1 0 N (cid:0) n T (cid:10) T 0 : Here1 m isthe m (cid:2) 1vectorofunitsand (cid:10) denotestheKroneckerproduct. Define Q e ; Z e ; K e inthesamefashion. ObjectiveandConstraints I now describe how to form the linear programming objective vector and constraintmatrixforaparticularcombinationofwealthandtheinterestrate,( a ; (cid:26) ). Thebank'sobjectivefunction,equation(1),istheutilityoftheagentateachpoint alongtheeventvector. Thiscanbewrittenas: C OBJ (cid:17) u ( T e ) (cid:0) (cid:18) ( Z e ) : Thebank'szero-profitconditionwillvarydependingontheagent'swealth a and theinterestrateunderconsideration, (cid:26) . Thuslet: C ( ZP a ; (cid:26) ) (cid:17) Q e (cid:0) T e + (cid:26) ( a (cid:0) K e ) : 56

Bankswillberequiredtomakezeroprofitsinequilibrium,solet B =0 ZP : Most linear-program solvers easily constrain the choice variables to be positive. Tosumoverallcontractweights,let: C = 1 1 0 N : Thecontractweightsmustsumtounity,solet B = 1 1 : TheBayes'consistencyconstraint,(C2),willtaketheformofaseparateconstraint for each q ; z ; k combination. For a particular combination of inputs and output, let: C ( 2 q ; z ; k ) (cid:17) i( q ; z ; k ) (cid:0) P ( q z j ; k )i( z ; k ) : Here i( q ; z ; k ) is a 1 (cid:2) N vector with unit values only where Q e = q , Z e = z and K e = q , with zeros elsewhere. i( z ; k ) is defined in a similar fashion. There will be n Q n Z n K separateconstraintsofthisform. Foreachconstraint,therighthand sidemustbezero,solet B beacolumnvectorofzerosoflength 2 n Q n Z n K . Finally, the incentive compatibility constraint, equation (C3), must hold separatelyforeachcombinationofassignedeffort, z ,assignedcapital, k andpotential deviationineffort, ˆ z . Foraparticularcombination z ; k ; ˆ z ,let: C ( 3 z ; k ; ˆ z )= i( z ; k ) (cid:26) P ( q j ˆ z ; k ) P ( q j z ; k [ ) u ( T e ) (cid:0) (cid:18) (ˆ z )] (cid:0) [ u ( T e ) (cid:0) (cid:18) ( z )] (cid:27) : Therewillbe n 2 Z n K oftheseconstraints,althoughseveralwillnotbebinding(see the discussion above). For a particular input assignment to be incentive compatible, the right hand side must be less than or equal to zero, so let B be a 3 column-vectorofzerosoflength n 2 Z n K . Thesefinalconstraintswillbeinequality constraints,allotherswillbeequality. Let C ( a ; (cid:26) )bethe M (cid:2) N matrixofconstraintsonthelinearprogramwhenwealth is a andtheinterestrateis (cid:26) . where M =1+1+ n Q n Z n K + n Z n K n Z . Thus: C ( a ; (cid:26) ) = 0 B B B B B B B B B B B B @ C ZP C 1 C ( 2 q 1 ; z 1 ; k ) 1 . . . C ( 2 q n Q ; z n Z ; k n K ) C ( 3 z 1 ; k 1 ; z ) 1 . . . C ( 3 z n Z ; k n K ; z n Z ) 1 C C C C C C C C C C C C A : 57

Inthesameway,let B bethevectorofright-hand-sizevalues: B = 0 B B @ B ZP B 1 B 2 B 3 1 C C A : OptimalContracts Acontractisan N (cid:2) 1vectorofweightsonevents. Itwillbechosentomaximize the objective function, subject to the constraints outlinedabove. Here, let y ( a ; (cid:26) ) denote the optimal contract for an agent of wealth a at the interest rate (cid:26) . It is formedfromthelinearprogram: max y ( a ;(cid:26) ) C OBJ y ( a ; (cid:26) ) ; (YP) subjectto: C ( a ; (cid:26) ) y ( a ; (cid:26) ) 5 B : Where the first 2 + n Q n Z n K constraints are equality and the remaining n 2 Z n K are inequality. Mostlinear program solvers allow one to specify individual constraintsasequalityorinequalitywithease. Solving for all wealth levels a in A at a given interest rate (cid:26) gives the demand for capital by each wealth type. Combined with a distribution of wealth (an n A -vector),thisimpliesanaggregatedemandforcapital: K d ( (cid:26) ; )= X a a [ K e y ( a ; (cid:26) )] : Asabove, a isthepopulationmassofwealth a . Equilibrium Givenadistributionofwealth. ,equilibriumisaprice, (cid:26) ? ,andasetofcontracts for each wealth type, y ( a ; (cid:26) ? ), at which aggregate demand and supply of capital areequalandbanksaresolvingtheiroptimizationproblem,(YP): K d ( (cid:26) ? ; ) = 0 A ; and: y ( a ; (cid:26) ? )solvesYP. 58

It is possible to calculate an upper bound on (cid:26) ? , at which no agent, no matter howwealthy,woulddemandanycapitalatall. Thelowerboundmaybetakenas zero. Thusfindinganequilibrium becomesamatterofsearchingovercandidate valuesof (cid:26) untilthecapitalmarketclears. Ateachcandidatevalue,ofcourse,the entireproblem(YP)mustbesolvedbeforethedemandforcapitalmaybefound. Inpractice I usedcombinations ofbothagrid search(overmany predetermined valuesof (cid:26) )andabisectionapproach. Thereis onefinalfillip to thisproblem. Abisectionalgorithm, forexample, may convergetoaneighborhoodof (cid:26) inwhich,atacandidatevalue (cid:26) ,demandistoo 0 highandatacandidatevalue (cid:26) demandistoolow. Thedistancebetween 1 (cid:26) and 0 (cid:26) canshrinktothelevelofmachineprecisionwithoutfindingavalueof 1 (cid:26) which clears the market. This happens when there is some critical value of (cid:26) at which a large proportion of agents in the economy suddenly switch from demanding capital to not demanding capital. In that case, I set (cid:26) ? to the midpoint of (cid:26) and 0 (cid:26) and find a value, 1 (cid:17) , such that the convex combination of contracts clears the market. Thatis,Idefine: y ( a ; (cid:26) ? ) = (cid:17) y ( a ; (cid:26) )+(1 0 (cid:0) (cid:17) ) y ( a ; (cid:26) ) 1 ; where (cid:17) issuchthat: (cid:17) K d ( (cid:26) 0 ; )+(1 (cid:0) (cid:17) ) K d ( (cid:26) 1 ; ) = 0 A : Thisisthestandardequilibriumlottery,andisnottobeconfusedwitheithergrid lotteriesorinputlotteries. Dynamics Given an equilibrium price (cid:26) ? and set of contracts y ( a ; (cid:26) ? ), and a starting distribution of wealth, t , it is a simple matter to calculate the distribution of wealth inthe nextperiod, t . Theassumptionthat +1 A = s T will be particularly useful here. Anagentgivenaconsumptiontransferof (cid:28) j in T willbequeath s (cid:28) j ,whichis just(bythisassumption) a j ,the j -thelementof A . Thusallagentsinperiod t +1 born with wealth a j must have had parents who were allocated a consumption transferof (cid:28) j . Findingthemassofagentsat a j inperiod t +1isthusasimplematterofaddingupall oftheagentsallocated consumptiontransfersof (cid:28) j inperiod t . 59

Themassofagentsinperiod t +1withwealth a j in A willbegivenby: a t j = +1 X a a t [i( (cid:28) = (cid:28) j ) y t ( a ; (cid:26) ? t )] : Contracts y and price (cid:26) ? must now be indexed by time, so that y t ( a ; (cid:26) ? t ) is the optimalcontractinperiod t foragentsofwealth a ,atthemarketclearinginterest rate in period t , (cid:26) ? t . Repeating this analysis for all possible wealth levels a in A givesthecompletedistributionofwealthinperiod t , t . +1 ComputationsWithout InputLotteries The procedure without input lotteries is similar to that outlined above, so I will onlybrieflycoverthedifferencesbetweenthetwo. Wheninputlotteriescannotbe used,optimalcontracts betweenbanks and agentsare foundas the solutiontoa two-stagealgorithm. Inthefirststage,thebanktakesasgiventhechoiceofinputs ( z ; k ) and finds the best, feasible, incentive-compatible contracts y ( NL a ; (cid:26) ; z ; k ). If such a contract does not exist (for example, the high effort level, low capital is assigned to an agent with zero wealth) I set the associated expectedutility level to a large negative number. I then choose the value of ( z ; k ) associated with the largest expected utility. The optimal contract is thus a pair of objects: an input choice ( z ; k ) ? and an ex post contract over T and Q , y ( NL a ; (cid:26) ; ( z ; k ) ? ). Everything elsethenproceedsasdescribedearlier. 60

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