contractual structure and wealth accumulation · wealth over time, so wealth distributions at any...

Contractual Structure and Wealth Accumulation

By DILIP MOOKHERJEE AND DEBRAJ RAY*

Can historical wealth distributions affect long-run output and inequality despite“rational” saving, convex technology and no externalities? We consider a model ofequilibrium short-period financial contracts, where poor agents face credit con-straints owing to moral hazard and limited liability. If agents have no bargainingpower, poor agents have no incentive to save: poverty traps emerge and agents arepolarized into two classes, with no interclass mobility. If instead agents have allthe bargaining power, strong saving incentives are generated: the wealth of poorand rich agents alike drift upward indefinitely and “history” does not mattereventually.(D31, D91, I32, O17, Q15)

Whether historical levels of inequality andpoverty constrain the current performance ofeconomies is a matter of considerable interest inthe economics of institutions, growth, and de-velopment [see, e.g., Robert J. Barro (1991),Alberto Alesina and Dani Rodrik (1994), Barroand Xavier Sala-i-Martin (1995), Daron Ace-moglu et al. (1999), Stanley Engerman et al.(1999), and Michael R. Carter and Frederick J.Zimmerman (2000)]. Theoretical literature onthis topic has recently focused on this questionin the presence of capital-market imperfections[Glenn C. Loury (1981), Abhijit V. Banerjeeand Andrew F. Newman (1993), Oded Galorand Joseph Zeira (1993), Lars Ljungqvist(1993), Ray and Peter A. Streufert (1993), andothers]. Barring few exceptions, these theoriesprovide explanations for the evolution, persis-tence, and macroeconomic consequences ofwealth inequality based on technological indi-

visibilities in investment and the absence ofstrategic saving behavior.1 In these models, thepoor cannot escape poverty by investing in rel-evant assets, either via borrowing (owing tocapital-market imperfections), gradual accumu-lation of capital stocks (owing to investmentindivisibilities), or stepped-up savings (owingto habit persistence, myopia, limited rational-ity, or a warm-glow bequest motive in anoverlapping-generations setting).

While it is generally agreed that the poor facesignificant borrowing constraints in most econ-omies, the significance of investment indivis-ibilities or constraints on savings is open todebate. As Loury (1981) showed, historical lev-els of inequality cannot affect long-run outputand inequality if the technology is convex andagents save strategically, if capital markets areentirely missing (for exogenous reasons). Wereexamine this question in a context wherecapital-market imperfections are endogenouslyderived from underlying problems of moralhazard. We continue to maintain Loury’s as-sumptions concerning convexity of the technol-ogy and strategic savings behavior.2

* Mookherjee: Department of Economics, Boston Uni-versity, 270 Bay State Road, Boston, MA 02215 (e-mail:[email protected]). Ray: Department of Economics, NewYork University, 269 Mercer Street, 7th Floor, New York,NY 10003 (e-mail: [email protected]). This paper isbased on an earlier paper titled “Tenancy, Saving Incentivesand Wealth Dynamics.” We are very grateful to three anon-ymous referees for useful comments on the previous draft,Ethan Ligon for useful discussions, and seminar participantsat the University of Texas–Austin, Cornell University, Flor-ida International, University of Illinois, London School ofEconomics, University of Michigan, Yale University, Uni-versity of Wisconsin, the 1999 NEUDC, and members ofthe MacArthur Network Inequality and Economic Perfor-mance for useful comments. This research was funded bythe National Science Foundation (Grant No. SES-97909254)and the MacArthur Foundation.

1 There is also a literature, exemplified by the workof Paul M. Romer (1986), Roland Benabou (1996a, b)and Steven N. Durlauf (1996), which generates—viaexternalities—increasing returns (and therefore historydependence) at the societal level while retaining convex-ity at the individual level.

2 A related paper of ours (Mookherjee and Ray, 2000)shows that nonconvexities at the individual level can arisefrom pecuniary externalities in human capital-accumulationdecisions when capital markets are missing, agents savestrategically and there are no indivisibilities.

818

In the presence of limited liability, moralhazard problems give rise to credit constraintsfor poor agents: they obtain positive but limitedaccess to credit.3 Credit access depends on thewealth of borrowers, which defines the amountof collateral they can post. Wealth accumulationwould allow increases in future collateral, pro-viding a motive for poor agents to save. Thepresence of uncertainty in returns from produc-tion potentially creates a supplementary precau-tionary motive for saving. On the other hand,the presence of moral hazard endogenously cre-ates nonconvexities in the returns to asset accu-mulation, possibly restricting saving incentives.

In order to focus on the interplay betweenthese various effects, we simplify the model ina variety of dimensions. We exogenously fix theset of agents (borrowers, workers, tenants, en-trepreneurs) and principals (lenders, employers,landlords, financiers), and assume that (i) agentsoperate a convex production technology, (ii)agent-principal pairs are randomly matched inevery period to negotiate over a short-termfinancial (credit, or interlinked credit-cum-tenancy) contract, and (iii) there are no exter-nalities across agents. So as to focus cleanlyon accumulation driven by contractual choice,we assume that agents discount future utility ata rate equal to some exogenous interest rate onsavings. In the absence of any moral hazard, itwill be optimal for every agent to maintain theirwealth over time, so wealth distributions at anylater date will exactly mirror the initial wealthdistribution. It follows that wealth accumulation

incentives in our model will be driven entirelyby the nature of contractual distortions resultingfrom moral hazard.

We find that the pattern of wealth accumula-tion depends critically on the allocation of bar-gaining power. We consider two extreme cases:one in which principals, and the other in whichagents, have all the bargaining power (i.e., oneparty makes a take-it-or-leave-it contract offerto the other). Poverty traps arise when princi-pals have all the bargaining power.4 Owing totheir lack of collateral, poor agents are provideda “floor” contract awarding them rents (i.e., apayoff in excess of their outside opportunities),simply in order to provide adequate effort in-centives: this “support system” is progressivelywithdrawn by principals as the agents becomewealthier. This effectively creates a 100-percentmarginal tax on limited degrees of wealth accu-mulation by the poor. At sufficiently high levelsof wealth, however, the contractual rents disap-pear, as agents can post sufficient collateral andhave high outside options (which rise withwealth): hence wealthy agents do recover someof the benefits of their saving. Consequently, asharp nonconvexity in returns to savingsemerges endogenously. For a range of suitableparameter values, this nonconvexity causes thepoor to not save at all, precipitating a povertytrap. Wealthy agents, in contrast, have strongincentives to maintain wealth; consequently(conditional on not sliding into poverty) theirwealth drifts upward. The wealth distribution isthus progressively polarized into two “classes”that grow further apart, with no interclass mo-bility. In this case long-run productivity anddistribution depend strongly on wealth distribu-tions in the distant past.

In the alternative setting where agents haveall the bargaining power, strong incentives tosave are generated at all wealth levels. All thebenefits of incremental wealth, including thosearising from the relaxation of credit constraints,accrue entirely to agents—poor and rich alike.This is supplemented by the precautionary sav-ings motive. Irrespective of the potential non-convexities that may arise from incentiveconstraints, no poverty trap can exist in such a

3 Such models have been widely employed in differentareas in recent literature. In the development economicsliterature, they have been employed to study effects of landreforms, contract regulations, subsidized credit, and publicemployment programs [Mukesh Eswaran and Ashok Kot-wal (1986); Sudhir Shetty (1988); Bhaskar Dutta et al.(1989); Karla Hoff and Andrew B. Lyon (1995); Banerjee etal. (1997); and Mookherjee (1997a, b). In the macroeco-nomics and finance literature, they have been used to ex-plain the presence of borrowing constraints, why externalfinance is more expensive than internal finance and whyincome distribution may have a role in explaining outputfluctuations [Ben Bernanke and Mark Gertler (1990); Galorand Zeira (1993); Hoff (1994); Bengt Holmstrom and JeanTirole (1997)]. Similar models have been used to explainwhy forms of industrial organization may depend on wealthinequality, and why worker cooperatives may occasionallyperform superior to capitalist firms [Banerjee and Newman(1993); Samuel Bowles and Herbert Gintis (1994, 1995),Patrick Legros and Newman (1996); Banerjee et al. (2001)].

4 This result can be viewed as an alternative formaliza-tion of the Amit Bhaduri (1973) hypothesis that rent extrac-tion motives of landlords precipitate poverty traps for theirtenants.

819VOL. 92 NO. 4 MOOKHERJEE AND RAY: CONTRACTUAL STRUCTURE AND WEALTH

setting; agents accumulate wealth indefinitely,irrespective of initial conditions. In this case,therefore, wealth levels exhibit considerable up-ward mobility, and historical wealth distribu-tions are irrelevant in the long run.

Section I introduces the model. Section IIconsiders the case where principals have all thebargaining power; Section III considers the re-verse situation. Sections IV and V respectivelydiscuss possible extensions of the model, andrelated literature. Finally, Section VI concludes.

I. The Model

A. Matching, Bargaining, and Power

There is a large population of principals(e.g., landlords, lenders) and agents (e.g., ten-ants, borrowers) randomly matched in pairs atthe beginning of any given period. A matchedprincipal–agent pair selects a contract. At theend of the period, the match is dissolved. Thenthe next period arrives and the story repeatsitself, ad infinitum. It is simplest to initiallyconsider the setting where a matched pair do nothave the option of seeking out alternative part-ners to contract with in the same period, shouldthey happen to disagree on the contract. Hencethe outside options of either partner are given byautarky in the current period, followed by theprospect of being matched with a fresh partnerin future periods.

There are two key assumptions embodiedhere. One is that a matched principal–agent pairdo not expect to be matched in future periods,so they enter into a short-term contract. Onepossible interpretation of the model is that theseagents represent successive generations of aninfinitely long-lived dynasty with altruistic pref-erences a la Barro (1974), in which case eachperiod corresponds to a distinct generation. Theimplications of allowing long-term contractswill be discussed in Section IV.

The other important assumption is that in anygiven period a matched pair do not have theoption of seeking out alternative partners withinthe same period. In Section I, subsection F weexplain why this is inessential: the same resultswill continue to hold with strategic search forcontracting partners.

Returning to the simple setting describedabove, the payoff of agents from a contract isanalogous to the one-period return in an optimal

growth exercise. Their wealth will represent astate variable, whose evolution will depend onoutcomes of any given period. The agents willbehave like Ramsey planners, maximizing adiscounted sum of utilities and accumulating (ordecumulating) wealth in the process. We as-sume there is no analogous state variable for theprincipals. Since they do not expect to contractwith the same agent in future periods, theysimply maximize net returns in the currentperiod.

A matched principal–agent pair will have ac-cess to a “utility possibility frontier,” represent-ing efficient payoff combinations achievable tothem from incentive-compatible contracts. Par-ticipation constraints correspond to the optionof remaining in autarky in the current period,followed by the continuation value from nextperiod onwards when the agent (or principal) ismatched with a fresh partner. The precise de-scription of the relevant constraints will be pro-vided below.

We identify the allocation of power with thechoice of a particular point on the (constrained)utility possibility frontier. Equivalently, onemight identify the allocation of power with theimplicit welfare weights on the utility of the twoparties, which determine allocation of the sur-plus available after meeting the participationconstraints. (To be sure, “power” also resides inthe determination of these participation con-straints or outside options. For instance, thesewould be influenced by the ability to strategi-cally choose a contracting partner, a topic wepostpone to Section IV, subsection B.)

The literature on noncooperative bargaining[summarized in Martin J. Osborne and ArielRubinstein (1990) or Abhinay Muthoo (1999)]provides a number of factors that affect theallocation of power (as defined above). Theseinclude factors that range from the individual(impatience, risk aversion) to the social andinstitutional (mirrored, for example, in the spec-ification of the bargaining protocol). In addi-tion, third parties may be persuaded to changetheir dealings with the other party in case of abargaining breakdown (as in Kaushik Basu,1986). To these factors one might add featuresthat are less frequently modeled but just aspertinent, especially in developing countries:literacy, negotiation skills, access to legal re-sources, and contractual regulations. We treatall these as exogenous “institutional” factors,

820 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2002

and study their impact on productive efficiencyand the wealth accumulation process.

In particular, we will study two extreme al-locations of bargaining power, corresponding tosituations where one party receives a zero im-plicit welfare weight relative to the other. Ef-fectively, thus, either the principal alwaysmakes a take-it-or-leave-it offer to the agent, orvice versa.

B. Projects and Payoffs

Turn now to a detailed description of thepossible outcomes from any given matched pairof agent (A) and principal (P). A is a worker,entrepreneur, or tenant operating a project forwhich P leases out relevant assets and providesfinance. The project can be operated at a scale �lying between 0 and 1. At scale � the projectinvolves an upfront cost of �f, and yields areturn �R (where R � f ) with probability e.With probability 1 � e, the project fails togenerate a return. The probability e is affectedby the (noncontractible) effort exerted by theagent at that date. Without loss of generality, weidentify e with effort as well as the successprobability. Notice there are constant returns toscale, subject to a constraint on the scale of theproject, arising from limits on the time or atten-tion of A. In particular, the production technol-ogy is convex.

A’s current payoff is given by a von-Neumann-Morgenstern function u(c) � D(e),where c denotes consumption and e effort. Theutility function u is continuous, strictly increas-ing, and concave (note that linear utility is in-cluded as a special case). We impose limitedliability: consumption must be nonnegative. Ac-cordingly, u(0) is the floor consumption utilityand we normalize it to zero.

On the disutility of effort we impose thefollowing restrictions: D is strictly increasing,strictly convex, satisfying D(0) � D�(0) � 0,lime31D�(e) � �, and D�(e) � 0 for alle � 0.

The principal is risk-neutral, and his payoff issimply expected return net of consumption pay-ments to the agent (a formal description followsshortly).

Finally, A discounts future utility using thefactor �. To abstract from time trends createdpurely from time preference, we impose thecondition �(1 � r) � 1, where r is some

(exogenously given) risk-free interest rate. Palso discounts future utility, but with randommatching of partners at every date this discountrate will not play any role in the model.

C. Contracts

Once matched, P and A negotiate a (short-term) contract. Without loss of generality wemay suppose that P pays for the setup cost exante and receives outcome-contingent paymentsfrom A upon completion of the project.

A’s ex ante wealth is assumed to be observ-able and collaterizable; the contract can be con-ditioned on this wealth.5 In particular, A’s expost liability is limited to the sum of the projectreturns and collateral. Thus, given startingwealth, contractual payoffs can be identifiedwith the end-of-period wealth of the agent,which we represent by x in the failure state andy in the success state.

Specifically, if ps and pf denote A’s paymentsin success and failure states respectively, thenx � w � pf and y � �R � w � ps, whilelimited liability requires ps � w � �R andpf � w. Then P’s net returns are pf � �f �w � x � �f and ps � �f � �R � w � �f �y under failure and success respectively, whilethe limited liability constraint reduces to:

LL: x, y � 0.

To summarize, then, given some initialwealth w, a contract is identified by the collec-tion (�, x, y), where � is the scale of theproject, and x and y are end-of-period wealthssatisfying (LL).

D. Consumption-Saving Decisions

Let wt be starting wealth at some date t andlet (�t, xt, yt) be a contract. The agent’s wealthat the end of the period is zt � { xt, yt}. A then

5 The assumption that wealth can be observed is impor-tant for the results, or at least for the methodology wefollow. If wealth is imperfectly observed (perhaps at somecost), or not observed at all, the analysis would be far morecomplicated. For instance, wealth revelation constraintswould have to be additionally incorporated, and the princi-pal could decide on a wealth investigation strategy. Agents’saving incentives would be affected since they may be ableto “hide” future endowments.


chooses consumption ct � [0, zt]. The result-ing saving zt � ct is invested at the risk-freerate r � 0, resulting in a level of wealthwt � 1 � (1 � r)( zt � ct) at the beginning ofdate t � 1.

Figure 1 depicts the sequence of events thatwe have described so far.

E. Equilibrium

We study Markov equilibrium (for any givenallocation of bargaining power), in which con-tracts and agent effort are only conditioned onthe ex ante (start-of-period) wealth of agents,and in which A’s consumption strategy dependson ex post (end-of-period) wealth. Thus we em-ploy the notation (�(w), x(w), y(w)) for a“contract function,” e(w) for the effort function,and c( z) for the consumption function. Refer tothis collection as an environment.6

An environment induces a function V(w), thepresent value utility for A at the beginning ofany date when he has wealth w. We may thinkof this as A’s ex ante value function. Then A’sconsumption strategy may be described as fol-lows. Let w� denote his ex ante wealth for thenext date. This corresponds to saving �w�, se-lected to solve the following problem:

(1) Bz � max0 � �w� � z

�uz � �w� � �Vw��.

Here B( z) denotes A’s present value utility atthe end of any date in which he attains end-of-period wealth z. (This is after project returnshave been realized and contractual payments, ifany, have been made.) Call this the ex postvalue function.

Notice that ex ante and ex post value func-tions are related to each other as follows:

(2) Vw � ewByw

� 1 � ewBxw � Dew.

Observe that the specifications (1) and (2) au-tomatically embody what one might call thesaving incentive constraint (SIC):

(SIC) At every date, the agent solves 1.

But this is not the only restriction to be im-posed. There are the familiar incentive and partic-ipation constraints, which we now describe. Saythat the effort incentive constraint (EIC) is satis-fied if specified effort e at every wealth levelmaximizes eB(y) � (1 � e)B(x) � D(e), yielding

(EIC) D�e � max By � Bx, 0�.

Then there are participation constraints. Theagent’s participation constraint (APC) is met atwealth w if

(APC) eBy � 1 � eBx � De � Bw

while the principal’s participation constraint(PPC) is satisfied if

(PPC) w � �f � e�R � y � 1 � ex � 0.

If, under some environment, a contract-effortpair satisfies (EIC), (APC), and (PPC), wewill say that it is feasible (relative to thatenvironment).

We are now in a position to define an equi-librium. The concept varies depending on whohas the power to make a take-it-or-leave-it offer.

When principals have all the bargainingpower, say that an environment is a P-equilib-

6 The term intends to capture the fact that this describesthe ongoing practice elsewhere in the economy when aparticular principal–agent pair is matched.

FIGURE 1. SEQUENCE OF EVENTS


rium if (a) (SIC) is satisfied, and (b) for everyw, the stipulated contract-effort pair (�(w),x(w), y(w)) and e(w) maximizes P’s payoffw � �f � e[�R � y] � (1 � e) x overthe set of contracts that are feasible relative tothis environment.

When the allocation of bargaining power isreversed, the definition is modified as follows.An environment is an A-equilibrium if (a)(SIC) is satisfied, and (b) for every w, thestipulated contract-effort pair maximizes A’s payoffeB(y) � (1 � e) B( x) � D(e) over the set ofcontracts that are feasible relative to thisenvironment.

Notice that both P- and A-equilibria selectcontractual sequences that are on the con-strained Pareto frontier of equilibrium payoffs.In addition, observe that every feasible contractwith � � 0 but less than 1 can be weaklyimproved (for both P and A) by one with either� � 1 or � � 0.7 Hence there is no loss ofgenerality by restricting � to lie in {0, 1}. Fromnow on we shall do so, recalling only that � wasintroduced in the first place to clarify that thereare no technological indivisibilities that driveour analysis.

Notice that � � 0 corresponds to no contractbeing offered at all, so such a case correspondsto e � 0 and x � y � w.

F. Strategic Choice of Contracting Partner

We shall argue now that the same character-ization of equilibrium applies when matchedpartners have the option of searching for otherpartners to contract with in the same period,provided that there is no restriction on the num-ber of agents that a principal can contract with(i.e., each principal has sufficient assets so that“capacity constraints” do not bind).

Assume that each agent can contract with atmost one principal. Given the absence of capac-ity constraints for a principal, contracting withany agent does not crowd out the opportunity tocontract with other agents. Hence there are noexternalities across agents: the market for con-tracts with agents of a specific wealth level is

independent of the market for contracts withagents of any different wealth level. Moreover,the utility attained by any agent will be a func-tion only of her own wealth.8

In this setting, consider the following searchscenario. Each principal “posts” a contract offerconditioned on the agent’s wealth. Initiallyagents are randomly assigned to principals. Anagent has the option of either accepting thecontract posted by the principal she is assignedto, or searching for another principal, perhaps atsome cost. If she decides to search, she visitsanother principal randomly selected from thepool of principals. Then the agent has the optionof accepting the contract offer posted by thenew principal, or continuing to search further,and so on.

It is easily checked that in this setting, theA-equilibrium results if there is no search cost,and the P-equilibrium results whenever thesearch cost is positive. In other words, in theabsence of any search frictions the market foragents involves Bertrand competition amongprincipals, resulting in the A-equilibrium; in thepresence of search frictions the market effec-tively reduces to a set of segmented monopoliesthat result in the P-equilibrium. The reasoning isstraightforward, so we only provide a briefoutline.9 If there are no search costs, then eachA can costlessly search indefinitely and visitevery single principal. Hence A can costlesslyfind the principal offering the contract that gen-erates her the highest utility, and the “demandcurve” for any principal is the same as in aBertrand model. On the other hand, if searchcosts are positive, the model reduces to thecelebrated search model of Peter A. Diamond(1971) which results in the monopoly outcome.The argument is as follows: (a) all principalswill offer the same utility to agents of a givenwealth level (otherwise the principal offeringthe contract with the highest utility can offer a

7 If eR � f � 0, then putting � � 1 and keeping x, y,e unchanged preserves feasibility and increases the lenderspayoff, while leaving the borrower’s payoff unchanged. Onthe other hand if eR � f � 0 then the same is true if � isset at 0 and x, y, e are left unchanged.

8 This will no longer be true in the presence of capacityconstraints: agent utility will depend on the distribution ofwealth across the set of all agents in the market: this isdiscussed in Section IV, Subsection B.

9 Note that since the contracts are short term, we can takeas given the continuation payoffs that each party expectsfrom the next period onwards as a function of current-periodoutcomes, and concentrate entirely on the current-periodcontract. Moreover, since there are no externalities acrossagents, we can treat the market for agents with a given levelof starting wealth in isolation from all other agents.


more profitable contract with slightly lower util-ity without losing any clients), and (b) the com-mon utility level offered must be the lowestconsistent with participation constraints corre-sponding to autarky in the current period (oth-erwise each principal can lower the offeredutility level slightly without losing any clients).

In the converse situation in which agents postoffers (and principals search for contractingpartners), it turns out that the A-equilibriumobtains irrespective of the search cost. Thisasymmetry results because each agent can con-tract with at most one principal, while eachprincipal can contract with any number ofagents. Since contracting with any agent doesnot crowd out the prospect of contracting withother agents, a principal treats offers from dif-ferent agents in isolation from one another, andhas an outside option equal to zero profit. Soeach agent will offer a contract which reducesthe principal’s payoff to zero, leading to theA-equilibrium.

Which of the two search models is moreappropriate may well depend on the relativenumber of principals and agents. If there arerelatively few principals and many agents, themodel where principals post contract offerswhile agents search seems plausible. Con-versely, if there are relatively few agents andmany principals, the reverse situation whereagents post offers and principals search wouldbe natural. With positive search costs, theP-equilibrium will arise in the former situa-tion, and the A-equilibrium in the lattersetting.

G. The Static Benchmark

Our model has a particularly simple staticcounterpart, studied in more detail in Mookher-jee (1997a, b).10 In this situation, A simply con-sumes all end-of-period wealth, so that B (the expost value function) is simply replaced by u (theone-period utility function) in all the incentiveand participation constraints described above,and (SIC) is ignored.

It will be useful to consider the properties ofthe optimal static contract when P has all the

bargaining power, and the agent has zerowealth. Then (APC) is implied by (LL) and(EIC), so it is optimal for P to set x � 0, and theproblem reduces to selecting y and e to maxi-mize e[R � y], subject to the constraint thatD�(e) � u( y). Let ( y*, e*) denote the solutionto this problem.11 Note that e* � 0 (and con-sequently y* � 0), since a small increase in yfrom 0 would increase P’s profit. We make thefollowing assumption:

ASSUMPTION [�]: e*[R � y*] � f.

Assumption [�] simply ensures that it is op-timal to offer some non-null contract (in thestatic model) when agent wealth equals zero.The consequences of dropping this assumptionare discussed in Section IV, subsection C.

Mookherjee (1997a, b) analyzes how the al-location of bargaining power and/or A’s wealthaffects the optimal contract and the inducedeffort in this static model. An important impli-cation of the static model is that a P-equilibriumgenerates positive surplus to the agent whenagent wealth equals zero (assuming that [�] ismet). When agent wealth climbs, some—possi-bly all—of the resulting gain is expropriatedby the principal, simply because the (APC) wasnot binding to begin with. In contrast, when Ahas all the bargaining power, all incrementalgains in wealth—and possibly some additionalgains resulting from better incentives—accrueto A.

These results suggest that in the dynamicmodel with wealth accumulation, savings willbe affected in different ways under the twobargaining regimes. This is precisely the ques-tion investigated in this paper.

H. The Dynamic First-Best Benchmark:Efficient Wealth Accumulation

It is also useful to define a benchmark patternof wealth accumulation that results in the ab-sence of moral hazard, the fundamental marketimperfection in the model. If agents’ effort canbe verified by contract enforcers, it can be writ-ten into the contract. Then since each principalis risk neutral, contracts will completely insure

10 The same model is studied in Dutta et al. (1989), whoalso extend it to a dynamic context, but with no statevariable of any kind.

11 Under our assumptions, a maximum exists and isunique.


each agent against the uncertainty in the returnsto their projects. In addition, the agent will berequired to put in the “first-best” level of efforte*, found by maximizing

eR � De

and we can map out the entire first-best frontierby means of a fixed payment to the agent—callit i—satisfying the necessary conditions

De* � ui and i � 0.

The payment i will be determined by specificparticipation constraints and allocation of bar-gaining power.

The resulting Ramsey problem for an agentwith starting wealth w is then:

max �t � 0

�

�tuct

subject to the constraints that w0 � w, and foreach t � 0,

wt � 1 � 1 � rwt � i � ct .

Because we have assumed that �(1 � r) � 1,the solution to this problem is to hold wealthconstant. Hence any patterns of wealth accu-mulation or decumulation that may result inan equilibrium with moral hazard can be in-terpreted as resulting from the presence ofmoral hazard, in conjunction with the main-tained assumptions about relative bargainingpower.

II. Where P Has All the Bargaining Power

A. Wealth Polarization

The purpose of this section is to outline ourmain findings when the principal has all thebargaining power. It turns out to be extremelydifficult to provide a fully general analysis.The main reason for this is that all incentiveand participation constraints for the agentmust now be defined—not by some exog-enously given utility function as in the staticmodel—but by a value function which is fullyendogenous with respect to developmentsat some later date. This creates significant

complications, which we tackle by means oftwo restrictions.

First, recall the static model and Assumption[�]. Under that assumption, the static P-equilib-rium (with w � 0) yields a strictly positivepayoff to the agent; denote this payoff by v*.We impose

ASSUMPTION [�]:

(3)1

� � 1 � �u� R

1 � �� uR� v*.

There are several ways to view this assumption.One might interpret it as stating that the successoutput R is not “too large,” or that the staticfloor payoff v* is not “too small,” or even as arestriction on the discount factor. But [�] isperhaps best viewed as a restriction on the cur-vature of u (given all other parameters), stat-ing that the agent does not have excessivelystrong preferences for consumption smoothing.For instance, [�] is always satisfied when u islinear.

It might be of interest to note that growth ortechnical change does not jeopardize this as-sumption, the reason being that R and v* willtypically move in step. For instance, if u exhib-its constant relative risk aversion � (0, 1) andthe effort disutility function is quadratic (sayD(e) � e2/ 2), [�] reduces to

(4)�

1 � � � 1 � 1 � 1 �

2 � � 1 �

� 2

which depends only on A’s discount factor andhis degree of risk aversion.

Our second restriction is that we study P-equilibria with continuous ex post value func-tions. This class is not empty. Later, we willexplicitly construct one such equilibrium.

PROPOSITION 1: Assume [�] and [�]. In anycontinuous P-equilibrium,

(i) there exist wealth levels w below R wherethe agent earns rents: V(w) � B(w); inparticular this is true at zero wealth. Forany w � R, the agents earns no rents:V(w) � B(w).

(ii) If u is strictly concave, there exists a


threshold z* � R such that A maintainswealth (w(z) � z) if z � z*, and decu-mulates it (w(z) � z) if R � z � z*.If limc3 �[u(c) � cu�(c)] � u(R) thenz* � �.

(iii) There is a poverty trap below R, i.e., zt �R implies almost surely that zT and wT donot exceed R at any subsequent date T, andequal 0 infinitely often.

Of particular interest in this proposition is thedescription of the poverty trap. Once ex postwealth falls below the value R, the propositiontells us that wealth can never escape this bound. Inaddition, agent wealth will visit zero infinitelyoften. The exact behavior of wealth is difficult todescribe at this level of generality: for instance, itis quite possible that following certain values ofz � R the agent accumulates wealth and starts thefollowing period with w(z) � z. What is claimed,however, is that such instances of wealth accumu-lation will be purely temporary.

In the particular P-equilibrium we later con-struct, the poverty trap does take the sharperform in which once z � R at some date, ex antewealth at every subsequent date equals zero.

Turn now to the induced dynamics of wealthin all regions, not just within the poverty trap.Notice that a P-equilibrium generates a first-order Markov process for wealth. Let

W � w � z*�

there is no feasible contract at w}

denote the set of wealth levels at which theagent maintains wealth, and no feasible contractexists. It should be clear that every wealth in Wis absorbing: once the agent arrives at such awealth, he is forced to “retire”; moreover, theagent optimally chooses to maintain this wealthforever thereafter. Note also that by Proposition1, the event of achieving an ex post wealthbelow R is “absorbing,” and W � [0, R] isempty. Thus (provided W is nonempty), theseare two nonoverlapping (ex post) wealth classeswith no mobility between them. We can gofurther to show that all wealths outside thesetwo classes must be transient:

PROPOSITION 2: Assume [�], [�] hold. Con-sider any continuous P-equilibrium, in which

the wealth maintenance threshold z* � �. Thenfrom arbitrary initial wealth w0, with probabil-ity 1 wt either enters a poverty trap or entersW, or converges to �. The probability oftransiting to the poverty trap is positive fromany initial wealth less than z*. In addition, ifW is bounded, the probability that wealth goesto infinity from any w � sup W is strictlypositive.

Hence the long-run wealth distribution mustbe concentrated entirely at most three disjointnoncommunicating wealth classes. If W is non-empty and bounded, then each of these classeswill have positive mass as long as the initialdistribution has support over the whole range ofnonnegative wealths. On the other hand if W isempty, then wealth will either gravitate towardsthe poverty trap or grow arbitrarily large. Ineither case the limit wealth distribution willexhibit a high degree of polarization. Long-runwealth distributions will depend on historicaldistributions, and this dependence is particu-larly manifested in the possible persistence ofpoverty.

B. Explanation and Discussion

The fundamental observation that underliesmany of the arguments is that poor agentswill be offered a “floor” contract that awardsthem utility in excess of their outside option.This, in turn, follows from the limited liabilityconstraint. When such agents acquire wealth,there will be a range over which the earliercontract continues to satisfy the agent’s partic-ipation constraint, and thus remains feasible—implying that it is also the optimal contract fromP’s point of view. It follows that moderate in-creases in wealth are entirely expropriated bythe principals. This imparts an (endogenous)nonconvexity to payoffs that hinders wealthaccumulation.

To illustrate this as starkly as possible, weexplicitly construct a P-equilibrium with a par-ticularly simple structure. This equilibriumyields agent rents only over some interval oflow wealth values, implying the following rela-tion between the ex ante and ex post valuefunctions:

(5) Vw � max V0, Bw�.


PROPOSITION 3: Assume [�] and [�] hold.Then there exists a P-equilibrium with the fol-lowing properties: there is w* � 0 such that

(i) For all w � w*, the static (P-) optimalcontract-effort pair is offered: e(w) � e*,y(0) � y*, x(0) � 0 and the resulting floor(present value) utility is V* � v*/(1 � �).If w � w*, then V(w) � B(w).

(ii) If

(6) limc3�

�uc � cu�c� � v*

there is an infinite increasing sequence ofex post wealth thresholds z1, z2, ...3 z* �� such that

B z � �1 � �k

1 � �u� 1 � � z

1 � �k � � �kV*

if z � � zk � 1 , zk �1

1 � �u1 � � z

if z � z*.

If (6) does not hold, then the agent fullyconsumes all ex post wealth and starts thenext period with zero wealth.

(iii) For any z � z*, the agent dissaves (w(z) �z), while for any z � z* the agents main-tains wealth (w(z) � z). In particular,for any z � z1, the agent saves nothing(w(z) � 0).

(iv) There is a strong poverty trap below w*,i.e., wt � w* implies wT � 0 for all T � twith probability 1.

Apart from assuring existence, this particularP-equilibrium lends itself to easy interpretation.To begin with, note from part (i) that the valuefunction satisfies the following version of (5):V(w) � max{V*, B(w)}. Thus, the ex postvalue function must satisfy a modified Bellmanequation of the form:

(7) Bz � max0 � �w � z

�uz � �w

� � max V*, Bw�].

This allows us to interpret the ex post optimi-zation problem as a Ramsey problem with an“exit option”: at any date (starting from the next

period), the agent can depart with some outsideoption V*. (It is important to realize that this isonly an interpretation that allows us to solve themapping from ex post wealth to the followingperiod’s ex ante wealth, but says nothing about theactual evolution of wealth or contracts thereafter.)

Part (ii) of the proposition solves this modi-fied Ramsey problem. As long as (6) is satisfied,the optimal saving strategy out of ex post wealthis characterized by an infinite sequence ofthresholds z1, z2, ... converging to some finitevalue z*, with the property that for any wealthbelow z1, A dissaves maximally. For interme-diate wealth levels he plans to consume at a ratewhich would run down his wealth in a finitenumber of periods. For levels above z* he fullymaintains ex post wealth.

The resulting ex post value function B is con-tinuous and strictly increasing. It is concave be-yond z* in the wealth maintenance region. Beforethis, however, every threshold constitutes a pointwhere B is kinked and locally nonconvex. Cross-ing one of the thresholds causes the agent to slowdown the rate at which the wealth is decumulated,by consuming less every period, which increasesthe marginal utility of the next increment inwealth. Figure 2 illustrates this function.

If, on the other hand, (6) is not satisfied, thesituation is much easier to describe. Savings outof ex post wealth are always zero, so that B( z)is simply u( z) � �V*. Whether or not (6) ismet depends on the parameters of the problem.12

In both cases, the ex post value function isdefined entirely by the utility V* correspondingto the floor contract. To complete the proof ofequilibrium it suffices to check that the optimalcontract designed by P for zero-wealth agentswhose savings and effort incentives are definedby this value function, generates exactly thesame floor contract. Assumption [�] ensuresthis is indeed the case.13 We therefore have abonafide P-equilibrium.

12 A utility function of the form u(c) � c�, with � � (0,1), always satisfies (6). A linear utility function never does.

13 Specifically, it ensures that R, the maximum returnfrom the project, is less than the first threshold z1 requiredto induce A to save anything. Since R exceeds the maximumpayment that the principal might conceivably make to A(with zero initial wealth) in the event of success, this impliesthat such an agent will invariably consume his entire end-of-period wealth. This reduces the contracting problem to astatic one, so P will indeed offer such agents the samecontract as in a one-period setting. This, in turn, is the floor


The wealth dynamics under this equilibriumtake on a particularly simple form. When initialwealth falls below the threshold w*, the agentreceives the floor contract, in which ex postwealth will surely fall below R. Assumption [�]guarantees that R is less than the first “dissav-ing” threshold z1. It follows that the agent willdissave maximally and start the next period withzero wealth, so that the same story is repeatedad infinitum thereafter.

What happens when the initial wealth ex-ceeds w*? When (6) fails the answer is obvious:the agent will invariably consume all currentwealth, and enter the poverty trap the very nextperiod. When instead there is enough preferencefor consumption smoothing so that (6) is satis-fied, the agent will plan to maintain his wealthfor sufficiently large realizations of ex postwealth. Nevertheless, over any bounded rangeof ex ante wealth levels for which a non-nullfeasible contract exists, the probability of a sin-gle failure is (uniformly) positive; hence, theprobability of any given finite number of suc-cessive failures is also uniformly positive over

the bounded region of starting wealths. More-over, any such failure must result in a drop intomorrow’s ex ante wealth by some discreteamount. Hence from any initial wealth in thisset, a finite string of successive failures will takethe agent into the poverty trap, an event whichtherefore has uniformly positive probabilityover any compact range of wealths for which afeasible contract exists.

To be sure, if the agent happens to land in theinterim at a maintenance wealth level at whichno feasible contract exists, he stays thereforever.

In the case where feasible contracts exist foran unbounded range of wealth levels, similarbut more complicated arguments apply, and theAppendix may be consulted for the details.

Similar phenomena obtain in any P-equilib-rium with continuous value functions. The rea-soning there is more involved. The maindifference in the nature of the equilibrium oc-curs at wealth levels below R, the gross returnfrom the project in the successful state. In ageneral P-equilibrium, the agent could conceiv-ably earn rents anywhere within this region, andin any pattern: in particular, no restriction canbe derived on the slope of the value function Vover this range. Hence no implication can be

contract used to construct the ex post value function, sothese strategies do constitute a P-equilibrium.

FIGURE 2. THE EX POST VALUE FUNCTION B


derived regarding the nature of local savingincentives for poor agents, unlike the simpleP-equilibrium. It is possible, for instance, thatsufficiently poor agents save at a high rate inorder to avail of rents accruing at higher wealthlevels within this region.

What can be shown, however, is that rentscannot accrue to agents with wealth above R.14

Thus the scope of any incentives to accumulatewealth among poor agents must be restricted tothe region [0, R]. Moreover, Assumption [�]implies that given any ex post wealth in thisregion, it is optimal for the agent to save ordissave in such a way as to arrive at a rent-generating contract within [0, R] at some futuredate. The region [0, R] collectively constitutesa generalized poverty trap—once the agent is inthat region he can never escape it, and mustarrive at 0 infinitely often thereafter. The rest ofthe wealth dynamics—for higher wealth levelsoutside this trap region—turns out to be quali-tatively similar to the case of the simple P-equilibrium discussed above.

III. Where A Has All the Bargaining Power

In this section, we show that once the agenthas all the bargaining power, no polarization ofthe wealth distribution is possible, in stark con-trast to behavior exhibited under P-equilibrium.

We are able to establish these results quitegenerally. In particular—and unlike the analysisof P-equilibrium—we impose no parametric re-strictions on the problem. For the most part,however, we do restrict attention to a subclassof equilibrium functions, for which the ex postvalue function B( z) is right continuous; werefer to these below as right-continuous A-equi-libria. Obviously, this class is broader than theclass of all continuous equilibria.

We begin by showing that in any A-equilib-rium, the agent is able to fully internalize allmarginal gains to saving.

PROPOSITION 4: (i) Consider any A-equilib-rium. Let c(z) � z � �w(z) denote theconsumption strategy of the agent. Then atany wealth w:

(8) lim 3 0�

Vw � � Vw

� u�max c yw, c xw�.

(ii) If the equilibrium is right continuous, thenat any w:

(9) lim 3 0�

Vw � � Vw

� �ew1

u�c yw

� 1 � ew1

u�c xw��1

and the consumption strategy has theproperty that at any z:

(10)1

u�c z

� ew z1

u�c yw z

� 1 � ew z1

u�c xw z.

The reasoning underlying these results isquite simple: any increment in ex ante wealth ofthe agent can be rebated to him ex post withoutupsetting the incentive constraints in a way thatjeopardizes the profitability of the principal.The value of increased ex post wealth is, in turn,bounded below by the marginal utility of con-sumption, as the agent always has the option ofimmediately consuming the incremental wealth.Hence the rate of increase of V is boundedbelow by the corresponding marginal utility of(the higher of the two possible values of) con-sumption at the end of the period, as expressedby (8). Right continuity of the value function Bensures that the increased ex ante wealth can berebated to the agent to result in an uniformincrease in ex post value in both states, successand failure, which leaves effort incentives en-tirely unchanged. Hence the expected marginalutility of ex post consumption forms a lowerbound to the return to incremental ex antewealth. Since the optimal consumption-saving

14 This is where the continuity of the value function Bplays a key role.


strategy of the agent trades off current con-sumption against starting the next date with ahigher wealth, the “Euler equation” (10) fol-lows. It incorporates the precautionary motivefor saving that arises from the uncertainty inproject outcome at future dates: roughly speak-ing, the marginal utility of current consumptionis related to the expected marginal utility ofconsumption at the following date. Here it takesthe form of an inequality because of the possi-bility of a corner solution (where the agentconsumes all current wealth), and because weonly have a lower bound on the rate of return tosaving.

Notice that effects of augmenting futurewealth equal the rate of return on savings, plusthe surplus-enhancing benefits of relaxing creditconstraints, which are especially sharp for pooragents. One would therefore expect poor agentsto save aggressively, rendering poverty trapsless likely. In particular, if u is linear and theagent has no preference for consumptionsmoothing, then the rate of return on savinggiven by (8) is everywhere at least as large asthe marginal utility of current consumption. Insuch a context it is always a best response forthe agent to save all current wealth, at least upto the wealth level where the contract itselfattains full efficiency. The contrast with savingspolicies in the simple P-equilibrium is espe-cially striking: there agents with linear utilityfunctions never save anything, ensuring the per-petuation of poverty.

We now use Proposition 4 to establish moregeneral behavior for the class of all concavefunctions, not just the linear ones. Begin withthe idea of a “strong” poverty trap, the exis-tence of which was established for the simpleP-equilibrium. Recall that this is a situation inwhich zero ex ante wealth constitutes an absorb-ing state.

PROPOSITION 5: If u is strictly concave,there cannot be any A-equilibrium with a strongpoverty trap.

(10) can be used (in the case where the equi-librium is right continuous) to gain some intu-ition for this result. If there is a strong povertytrap, the agent must consume his entire ex postwealth in both success and failure states. More-over, his consumption thereafter must follow astationary distribution, exactly equal to the ex

post wealth distribution that results from thecontract offered to a zero-wealth agent. If u isstrictly concave, condition (10) must be violatedin the successful state (the marginal utility ofcurrent consumption is too low relative to theexpected marginal utility of consumption at thesucceeding date, i.e., the agent is saving toolittle).

More generally, (10) implies that consump-tion cannot converge to an invariant distributionwith a bounded support. This also suggests theimpossibility of a more general poverty trap, asdescribed in Proposition 1, in which zero wealthis not an absorbing state, but a recurrent stateinstead. The reasoning outlined in the previousparagraph suggests that from any initial wealthlevel, the stream of possible future consumptionlevels (hence also wealth) will be unboundedabove. Hence, irrespective of initial conditions,all agents can possibly become arbitrarily wealthy.

This suggests something more than just theabsence of a poverty trap (weak or strong), andindeed an “upward mobility” result can be es-tablished for any right-continuous equilibrium.Consider first the case where a feasible contractexists at all wealth levels, and marginal utility isbounded away from zero. Then with probability1 every agent—rich or poor—must eventuallybecome arbitrarily wealthy:

PROPOSITION 6: Assume that u is strictlyconcave, and that u�(�) � 0. Consider anyright-continuous A-equilibrium in which a fea-sible non-null contract exists at all wealth lev-els. Then wealth almost surely converges to �from any initial level.

The outline of the argument is as follows.Under the stated assumptions, (10) reduces tothe statement that 1/u�(ct) forms a submartin-gale; hence ct converges almost surely. In theevent that it converges to something finite,agents cannot be motivated to supply effort bythe prospect of higher levels of consumption,but instead by the prospect of higher levels offuture wealth. This must eventually result inoveraccumulation of wealth, i.e., the agentwould not be pursuing an optimal consumption-saving strategy. Hence consumption must con-verge to infinity almost surely, implying thesame must be true of wealth.

What can we say about the case in which afeasible contract does not exist at all wealth


levels? It is natural to expect that if sufficientlyhigh levels of consumption reduce marginalutility of further consumption to negligible lev-els, agents will “retire” upon achieving certainthresholds of affluence.

PROPOSITION 7: Consider any right-continuous A-equilibrium, in which a feasiblenon-null contract exists if and only if wealth liesbelow some threshold W*. Then from any initialwealth w � [0, W*), the wealth at some futuredate will exceed W* with positive probability.

Hence irrespective of initial conditions everyagent must “retire” with positive probability atsome future date. However, it is difficult toensure that retirement is an absorbing state, soone must entertain the possibility that agentsretire for a finite number of periods and thenreturn to productive activity. In this case “afflu-ent retirement” is a recurrent event for allagents, irrespective of their initial wealth. The netresult is similar to the previous case: the economyexhibits a high degree of upward mobility.

IV. Extensions

In this section we discuss possible extensionsof our model, and their possible implications forthe robustness of our results.

A. Long-Term Relationships

The random-matching assumption enabled usto ignore long-term contracts. Depending on thecontext, this may or may not be good modelingstrategy. If different periods correspond to dif-ferent generations of agents, this is a reasonableapproach: one cannot expect parents to enterinto contracts that bind their children. If insteadthey correspond to different dates in the life ofthe same agent, the assumption may be restric-tive. Moreover, even if only short-term con-tracts are allowed (say, owing to the inability ofparties to commit to long-term contracts), addi-tional dynamic complications from the princi-pal’s side may appear if a principal expects tocontract repeatedly with the same agent in fu-ture periods. This is particularly pertinent to theP-equilibrium, since one context where princi-pals may naturally be thought of as havingdisproportionate bargaining power is when themarket is monopolistic.

Suppose, then, that a single P possesses amonopoly over contracting rights with a set ofagents. Suppose, moreover, that this monopolistowns enough productive assets and wealth to beable to transact simultaneously with all theagents; hence there is no need for agents tocompete with one another for the right to obtaina contract. Then the notion of P-equilibrium isappropriate only if P is completely myopic.Otherwise each agent’s wealth will also consti-tute a state variable for P. He will offer con-tracts with the possible manipulation of theagent’s future wealth in mind. Consequently,the appropriate notion of equilibrium must in-corporate the effect of the current contract onfuture profits of P owing to dependence of thoseprofits on the agent’s future wealth.

In order to analyze the resulting complica-tions, we need to evaluate the effect of agent’swealth on P’s profit. There are two contrastingeffects. On the one hand, with greater wealth,the limited liability constraint is attenuated, per-mitting high-powered incentives and thereforereducing P’s exposure in the event of a projectfailure. On the other hand, a wealthier agent’swillingness to exert effort decreases owing towealth effects in the demand for leisure. Thetrade-off between these two effects is difficult toresolve in general. If the second effect dominates,then P benefits from contracting with pooreragents, and the tendency toward a poverty trap isfurther accentuated by dynamic considerations onP’s side. Conversely, if the first effect dominates,a poverty trap would be less likely.

Suppose we were to abstract from the secondeffect by assuming that the agent’s utility overconsumption is linear. Then wealth has no ef-fect on the demand for leisure. The precedingargument suggests that in this case a povertytrap is least likely. It turns out, however, thatour results on P-equilibria extend to this case:with a single monopolist principal, indefiniterepetition of the static optimal contract and zerosaving by A (at all wealth levels) is a perfectequilibrium (when P makes take-it-or-leave-itoffers). Indeed, if the game has a finite termi-nation date, this is also the unique subgame-perfect equilibrium outcome.15 The contractsoffered by P award rents to A in the form of the

15 When the relationship is infinitely repeated, otherequilibria may conceivably appear. We conjecture thatour analysis would continue to apply if we selected the


floor contract below the threshold w*, and norents for wealthier agents. The agent’s ex antevalue function then reduces to V(w) �max{V(0), w}, inducing A to entirely consumeany wealth at the end of any period, in order toavail of the floor V(0) at the next period. A’sincentives in the dynamic setting are then ex-actly the same as in a static setting; anticipatingthis, it is optimal for P to offer the static optimalcontract at all dates, as the current contract hasno effect at all on A’s future wealth.

The argument of the two preceding para-graphs suggest that reintroducing concavity intothe agent’s (consumption) utility function willonly strengthen the existence of poverty traps.Unfortunately this is not something that wehave formally been able to establish.16

In the reverse situation, there is a single mo-nopolist agent dealing repeatedly with a largenumber of principals. Then our analysis of theA-equilibrium applies exactly. For in such asituation, A will make a take-it-or-leave-it offerwhich will push each P down to their outsideoption, where they earn zero profits. Dynamicconsiderations will arise only on A’s side, andcontracts will be chosen to maximize A’spresent value utility, subject to breakeven con-straints for the principals, as represented in thedefinition of A-equilibrium.

In summary, our results extend to contextswhere long-term contracts are allowed, in thecase where agents have no preference for con-sumption smoothing. And we conjecture thatthey are reinforced if agents do have prefer-ences for smoothing, though this remains to beformally verified.

B. Alternative Models of the Marketfor Contracts

Our model of search for contracting partnerswas based on (i) an asymmetry between the twosides of the market: one side posted offers while

the other side searched for partners; and (ii) theabsence of capacity constraints on the side ofthe principals. We now discuss the implicationsof alternative formulations on both of thesedimensions.

Begin by dropping (i), while retaining (ii).Consider instead a pairwise matching approach(as in Ariel Rubinstein and Asher Wolinsky,1985). Within any given period, there are manysuccessive rounds; if a pair matched in anygiven round fail to agree on a contract, theyremain in the pool of market participants thatare randomly rematched in the next round.Given pairwise matching, if there are a differentnumber of principals and agents, then some ofthose on the “long side of the market” will notbe matched in the current round and wait to bematched in future rounds. Delays in matchingmay result in a small cost to either party, or ashrinkage of the available surplus. This modelpermits the exploration of an alternative sourceof bargaining power, based on outside optionsrather than implicit welfare weights. The out-side options will depend on the imbalance be-tween the number of agents and principals,since this affects the waiting times for each typeof agent, in the event of disagreement with thecurrent partner.

To abstract from the source of bargainingpower studied so far, suppose that the symmet-ric Nash bargaining solution applies to select acontract for any matched pair, with disagree-ment payoffs given by their outside optionsfrom returning to the pool of unmatched partic-ipants at the next round. Suppose, moreover,that there are relatively few principals and manyagents, so the latter are on the long side of themarket. Then the disagreement payoff of anagent decreases with increased imbalance be-tween the number of principals and agents,since this increases the time an agent must waituntil the next match (in the event of a currentdisagreement). Indeed, if the imbalance is suf-ficiently great, the disagreement payoff con-verges to the autarkic utility level. On the otherhand, the disagreement payoff for a principal isalways effectively zero, owing to the absence ofcapacity constraints. It follows that the greaterthe imbalance between the two sides, the moreprofitable will be the selected contract for theprincipal.

In fact, if the agent’s wealth is sufficientlylow, the limited liability constraint for the agent

equilibrium that maximizes the principal’s payoff from theclass of all subgame-perfect P-equilibria.

16 The difficulties stem from discontinuous upwardjumps in the agent equilibrium value functions. Such jumpscorrespond to (local) zones of high-powered incentives thatthe principal may want to offer to the agent. But such zonescan only exist if there are further jumps in the value func-tion at even higher levels of wealth. This sort of bootstrap-ping can possibly be ruled out, but this is beyond the scopeof the present paper (and—presently—its authors).


tends to bind, and the contract selected by theNash bargaining solution will be the profit-maximizing contract.17 But then, just as in thispaper, the effects of small (agent) wealth incre-ments will accrue entirely to the principal, cut-ting down the incentive to save. To be sure, athigher wealth levels the selected contract willdiverge from the profit-maximizing contract,owing to the fact that the latter awards the agentno utility gain relative to her disagreement pay-off (as the participation constraint defined byautarky binds, instead of the limited liabilityconstraint). Now the agent will participate in themarginal gains resulting from any wealth incre-ment. A nonconvexity similar to that in theP-equilibrium in the agent’s value function andsaving incentives reappears in this setting, eventhough the exact equilibrium will diverge fromthe profit-maximizing contract at high wealthlevels.

In contrast, if the principals are on the longside of the market, then as the cost of delayacross successive rounds becomes small theequilibrium will converge to the A-equilibrium.Since each agent expects to be rematched inevery successive round, the delay costs theyencounter are small. With strategies conditionedonly on the current number of unmatched par-ticipants on either side (a standard assumptionmade in the Rubinstein-Wolinsky model), it fol-lows that agents will obtain almost all the sur-plus in such a case, in both absolute andmarginal terms.

Next, we discuss the implications of droppingassumption (ii) concerning lack of capacity con-straints. Each capacity-constrained principalwill try to seek out agents with “profitable char-acteristics.” In particular, the options availableto any agent will depend not only on her ownwealth, but also those of other agents. Thiscomplicates the model considerably, since theappropriate state variable involves the entirewealth distribution across agents.

However, one observation is easy to make:the A-equilibrium must arise whenever each

principal contracts with one agent, there aremore principals than agents, and the principalsengage in Bertrand competition with one an-other (as in the search model in Section I, Sub-section F with zero search costs). The reason isthat an equilibrium must involve every principalearning zero profits (since at least one principal,P1 say, must end up not contracting with anyagent, and thus earn zero profit; any other prin-cipal earning positive profit could be undercutby P1).

The real complications arise in the case withfewer principals than agents: then some agentswill have to stay in autarky, while others obtaina contract. The selection of agents that end upwith a contract will depend on their relativewealth levels: their value functions will beinterdependent in a complex manner. In partic-ular, savings incentives can now appear for low-wealth agents, since an increase in their futurewealth will increase their chances of obtaining acontract in future periods which awards themrents owing to limited liability. So our resultsconcerning the P-equilibrium are unlikely toobtain in this setting.

C. Market Exclusion

Assumption [�] ensured that agents with zerowealth would not be excluded from the market.In its absence, there will exist an interval ofwealth levels such that only the null contract(with � � 0) is feasible.18 Such agents willtherefore be excluded from the market. If the“entry wealth threshold” w� is not too large,19

then at w� the P-optimal contract will award astrictly positive surplus to both parties: this cor-responds to a phenomenon analogous to a ten-ancy ladder (Shetty, 1988).

In this case we conjecture that the povertytrap would take the form of wealth level w�constituting a recurrent state. Agents withwealth below w� would seek to accumulate as-sets up to w� via saving in order to be eligible toenter the market; conversely agents with wealthabove w� would take care to avoid falling beloww� to avoid exclusion. Agents with wealth w�would have no (local) incentives to increase17 This occurs if the agent’s payoff at the profit-

maximizing contract is sufficiently large relative to theagent’s autarkic utility (owing to limited liability). Then theutility gain for the agent at the profit-maximizing contractrelative to the disagreement payoff is large relative to thepayoff gain for the principal, so the profit-maximizing con-tract will be the Nash bargaining solution.

18 Specifically, all agents with wealth less than w� will beexcluded, where w� � f � e*[R � u�1(D�(e*))].

19 The specific condition is e*D�(e*) � D(e*) � u(w� ).


their wealth above w� . In the successful statethey would save just enough to ensure that theystart the next period with exactly w� . A failurewould, however, cause their wealth to fall tozero, whence they would tend to save their wayback to w� in due course, and the same cycle willbe repeated thereafter.

D. Endogenous Entry and Exit

A more ambitious extension of the modelwould endogenously determine entry into theranks of principals and agents. For instance,wealthy agents could become principals andconversely poor principals could becomeagents, and the allocation of bargaining powerat any date could depend on the relative num-bers of principals and agents. If we combine thisapproach with that described in this paper, bothwealth polarization and sustained accumulationmight constitute steady states of such an ex-tended model. Specifically, poverty traps andwealth polarization are consistent with the pres-ence of a few wealthy agents (who becomeprincipals) and a large number of poor agents,which restricts entry into the ranks of the prin-cipals, reinforcing the disproportionate bargain-ing power of the latter. Conversely, sustainedaccumulation is consistent with an ever-presentand large pool of principals (owing to the sig-nificant upward wealth mobility in the equilib-rium) that compete vigorously with one another,causing agents to have most of the bargainingpower.

V. Related Literature

The relation to the imperfect capital-marketmodels of Loury (1981), Banerjee and New-man (1993), and Galor and Zeira (1993) havealready been discussed above. Tomas Piketty(1997) considers a model with diminishingreturns to (divisible) capital, but assumesthere are two levels of effort, and that savingsrates are constant and exogenous. His modelhas no poverty traps, but owing to pecuniaryexternalities has multiple steady states,whereby long-run output can depend on ini-tial inequality. Philippe Aghion and PatrickBolton (1997) use a nonconvex technology,and an exogenous savings rate: the latter isassumed high enough to ensure that wealth isergodic; they focus on inequality dynamics

generated by pecuniary externalities acrossagents in the intermediate term. MaitreeshGhatak et al. (1997) analyze a model with anexogenous investment threshold and endoge-nous savings in a two-period setting. Theyfocus on the effects of credit-market imper-fections on the effort and saving incentives ofyoung agents in a competitive setting whereagents have all the bargaining power.

In contrast to these papers, our model is char-acterized by a convex technology, endogenoussavings decisions in an infinite-horizon frame-work, absence of pecuniary externalities acrossagents, and varying allocations of contractualbargaining power. Whether or not poverty trapsarise depends on the endogenous emergence ofinvestment thresholds, which are determinedexclusively by institutional characteristics ofthe economy.

R. Glenn Hubbard et al. (1994, 1995) showthat social insurance programs with means test-ing discourage saving by poor households, ow-ing to the phase-out of safety nets as theyaccumulate assets. Moreover, they show thatsuch a model can replicate empirical patternsconcerning savings rates across households ofdifferent wealth levels. This mirrors the effectof the “floor” contract available to poor agentsin the P-equilibrium in our model. The maindifference is that our model concerns a laissez-faire market economy rather than the effect ofan exogenous government welfare program. Inparticular it is in the private interest of P’s witha lot of bargaining power to offer contractssimilar to a means-tested social insurance pro-gram which is phased out as agents accumulateassets. This is consistent with the description of“patron-client” relationships characterizing un-equal hierarchical societies, in which the role ofquasi-monopolistic landlords and lenders withtheir clients simultaneously incorporates pat-terns of “patronage and exploitation” frequentlyemphasized by sociologists (e.g., Jan Breman,1974).

Other literature on the dynamics of inequal-ity in a asymmetric information contractingframework includes Edward J. Green (1987),Jonathan Thomas and Tim Worrall (1990), An-drew Atkeson and Robert E. Lucas, Jr. (1992),Cheng Wang (1995), and Christopher Phelan(1998). These papers study efficient insurancewhere agent endowments are private informa-tion and follow an independently and identically


distributed (i.i.d.) process. In this literature,wealth constraints do not play any role andagents cannot save by assumption. Moreoverthe incentive problem arises from private infor-mation (unobservable endowments) rather thanmoral hazard (unobservable effort); and theirfocus is on a social planning problem of design-ing an efficient long-term mechanism. In con-trast, we study a sequence of equilibriumshort-period contracts in a decentralized setting.The results also differ markedly: the insurancemodel tends to generate (almost) all agentsdrifting down into poverty. In the corresponding“competitive” case, our model produces a dia-metrically opposite conclusion: agents’ wealthdrift upwards indefinitely, irrespective of initialconditions.20

VI. Concluding Comments

Our model suggests that poverty traps canarise even when agents are farsighted and savestrategically, and there are no technologicalnonconvexities or externalities. The extent ofwealth mobility then depends in a fundamentalway on the institutional characteristics of theeconomy. In this paper, we have concentratedon one of those characteristics: the unequal al-location of bargaining power. Our principalresult is that incentives for poor agents to accu-mulate wealth are less than in a first-best setting(without moral hazard or uncertainty) if theyhave too little bargaining power, and greaterthan first-best if they have all the bargainingpower. The basic reason is that in the formercase the principal appropriates most of the gainin surplus when agent wealth rises, as the lim-ited liability constraint forces the principals toprovide incentive rents to poor agents. Forexactly the mirror-image set of reasons, anA-equilibrium generates overaccumulation: anincrease in wealth generates an increase in ef-

ficiency in the contract (once again owing to thelimited liability constraint), but this time theagent pockets it all. This “marginal efficiency”improvement may be thought of as an addi-tional rate of return on wealth accumulation.Added to r (the rate of interest), the equilibriumgenerates an “effective” rate of return thatmakes for unbounded wealth accumulation un-der the A-equilibrium.

In future research it would be worthwhile toexplore more fully some of the extensions de-scribed in Section IV, e.g., existence of long-term relationships, capacity constraints on theprincipals’ side, and endogenous entry into theranks of the principals and agents. We alsoabstracted from the possibility of asset sales thatmight permit an agent to purchase the right tobecome a principal. For instance, in an agricul-tural setting a tenant may be able to purchaseland from a landlord for purposes either ofself-cultivation or becoming a landlord. Landmarkets are known, however, to be thin in mostdeveloping countries, a phenomenon whichneeds to be theoretically explained. Mookherjee(1997a) provides one explanation using a staticversion of the model in this paper, based oncredit constraints faced by poor tenants. In adynamic setting, however, such credit con-straints can conceivably be overcome via sav-ing, as stressed by Carter and Zimmerman(2000) in the context of a numerical modelcalibrated to Nicaraguan data. The implicationsof potential land-market transactions in a dy-namic setting deserves to be analyzed in futureresearch.

Finally, it would be worthwhile to exploresome empirical and policy implications of ouranalysis. Is there any evidence suggesting therole of institutional factors affecting relativebargaining power on savings and upward mo-bility? If so, what policy measures might affectthese institutional variables? Social insuranceprograms where benefits are phased out asagents become wealthier may encourage theformation of poverty traps. Poverty-reductionstrategies should perhaps devote greater atten-tion to institutional reforms such as asset re-distributions, legal and literacy reforms thatdistribute contractual bargaining power moreequitably. Detailed policy implications must ul-timately be based on careful empirical analysisof the role of these factors in investmentincentives.

20 This owes partly to the operation of precautionarymotives for “private” saving in our model. A key role is alsoplayed by the limited liability constraint in our model,which ensures that the incentive problem remains nonneg-ligible for poor agents. Despite these differences, it is in-teresting to note that both models are characterized by asimilar “Euler” equation governing the intertemporal distri-bution of consumption. The dynamic properties of the twomodels thus appear to be similar in many respects. We thankEthan Ligon for this observation.


APPENDIX

For the sake of brevity, we omit details atseveral points. A complete Appendix is avail-able at �http://econ.bu.edu/dilipm/wkpap.htm/cswaapp.pdf�

In what follows, Assumptions [�] and [�] aremaintained throughout, though they do not ap-ply to all the lemmas or propositions. The nor-malization u(0) � 0 is also employed withoutcomment. Finally, recall that v* is the payoff tothe agent under the one-shot P-optimal contract,and V* � (1 � �)�1v*.

LEMMA 1: In any P-equilibrium, if y � y�,then B( y�) � B( y) � u( y�) � u( y).

PROOF:Letting c( z) denote A’s optimal consumption

at wealth z, we have B( y) � u(c( y)) ��V(w( y)), while B( y�) � u(c( y) � y� �y) � �V(w( y)), implying B( y�) � B( y) �u(c( y) � y� � y) � u(c( y)) � u( y�) �u( y), the last inequality resulting from the con-cavity of u and c( y) � y.

LEMMA 2: In any P-equilibrium, V(0) �B(0) � 0.

PROOF:Suppose that P selects x � 0 and y � y*(�0)

from the static optimal contract. In response, Awill choose e � e*, since B(y) � B(0) � u(y) �u(0) for all y (by Lemma 1). This will generate aprofit for P no smaller than that of the staticoptimal contract, so Assumption [�] implies thatthere does exist a feasible and profitable contractat w � 0. But any such contract must yield theagent strictly positive return (because either x or yor both must be strictly positive). So V(0) � 0.

But B(0) � u(0) � �V(0) � �V(0). BecauseV(0) � 0, we conclude that V(0) � B(0).

LEMMA 3: Consider the Ramsey problemwith exit option V* depicted in (7).

(i) If

(A1) limc3�

�uc � cu�c� � 1 � �V*

is violated, then the solution is (for all z):

(A2) c z � z, B* z � u z � �V*.

(ii) If, on the other hand, (A1) is satisfied, thesolution is described as follows. Let T( z)equal the optimal date of “ exit” for anagent with initial wealth z, with T( z) � �if the agent never exits. There exists aninfinite sequence { zk}k � 0

� , with 0 � z0 �z1 � z2 � ... , and z* � limk3� zk � �,such that T( z) � k for z � [ zk � 1, zk](with indifference holding for adjacent val-ues of T at the endpoints) , and T( z) � �for all z � z*. Moreover, under [�], wehave R � z1.

The associated value function in case (ii) is

(A3)

B*z �1 � �k

1 � �u� 1 � �z

1 � �k � � �kV*

for all z � �zk � 1 , zk �

�1

1 � �u1 � �z

for all z � z*.

PROOF:Consider the related problem of selecting a real

number x � [1, 1/(1 � �)] to maximize �(x; z) �xu(z/x) � [1 � (1 � �)x]V*. Here x correspondsto (1 � �k)/(1 � �), where the exit date k is treatedas a continuous variable in [1, �). Since it isoptimal to smooth consumption perfectly until theexit date, A will consume until the exit date at thesteady level of c(z) � z(1 � �)/(1 � �k), therebyrunning down wealth to 0 at k, and exiting withV*. This generates the value function

Bz �1 � �k

1 � �u� z1 � �

1 � �k � � �kV*.

Notice that

�x � u� z

x� � xu�� z

x� z

x2 � 1 � �V*

� u� z

x� �z

xu�� z

x� � 1 � �V*

so the concavity of u implies that � is concavein x, for any z.


Note, moreover, that if (A1) fails with a strictinequality, then �x( x; z) � 0 for all x, so thenthe optimal value of x � 1, i.e., k � 1. Then Aconsumes all current wealth and exits at the nextdate, implying (i). [It is easily shown that thesame is the case when (A1) holds as anequality.]

If (A1) is satisfied, there exists m such thatu(m) � mu�(m) � (1 � �)V*. Define z� �m/(1 � �). Then z � z� implies z/x � z�(1 ��) � m for all x � [1, 1/(1 � �)]. Hence�x( x; z) � 0 for all x, so optimal x � 1/(1 ��), or k � �. Conversely, z � z� implies that�x(1/(1 � �); z) � 0, so it is optimal for theagent to exit at some finite date. In particular,z � m implies that optimal x � 1 (so that k �1). And z � (m, m/(1 � �)) implies that theagent must exit at some date k � 1.

To calculate the exact exit date, the concavityof � implies that it suffices to look at the twointeger solutions for k generating values of z/xclosest to m. Every finite k will therefore be anoptimal exit date for some z, and the exactswitch points can be calculated by the conditionof indifference between adjacent exit dates.

Finally, note from Assumption [�] that for allz � R,

1

� � 1 � �u� z

1 � �� uz� v*

because (1 � �)u(c/(1 � �)) � u(c) is non-decreasing in c. Rearranging this inequality andrecalling that V* � (1 � �)�1v*, we see that

uz � �V* � 1 � �u� z

1 � �� 2V*

which means that exit at date 1 is strictly pre-ferred to exit at date 2. By the concavity of �, itfollows that exit at date 1 is uniquely optimal,proving that R � z1.

The reader is reminded that in what follows, acontinuous P-equilibrium refers to a P-equilib-rium with continuous ex post value function B.

LEMMA 4: For any continuous P-equilibrium,let wn be a sequence of wealth levels converg-ing to w, with corresponding contracts ( xn , yn,en) converging to ( x, y, e). Then ( x, y, e) is anoptimal contract for P at w.

PROOF:Recalling that B and D are continuous, this

follows from a simple application of the maxi-mum theorem to the principal’s (constrained)optimization problem.

LEMMA 5: For a continuous P-equilibriumand wealth w, suppose that a non-null contractis offered and V(w) � B(w). Then x(w) �w � y(w). Moreover, for any compact interval[0, w� ], there exists � � 0 such that (i) x(w) �w � � and (ii) y(w) � w � �, whenever w �[0, w� ] and a non-null contract is offered withV(w) � B(w).

PROOF:The first part follows from the need to main-

tain (APC) and (PPC). The uniform versionfollows from Lemma 4.

LEMMA 6: For any continuous P-equilibriumand any compact interval [0, w� ], there is �0 such that whenever w � [0, w� ] and a non-null contract is offered, we have e(w) � ( ,1 � ).

PROOF:That e(w) � (0, 1) follows from the need

to maintain (APC) and (PPC). As in Lemma5, the uniform version is a consequence ofLemma 4.

LEMMA 7: Consider a P-optimal contractfor some continuous B. If (APC) is not bind-ing at some w, so that V(w) � B(w), thenthe P-optimal contract has x(w) � 0 andy(w) � R.

PROOF:Suppose that V(w) � B(w). It can be verified

that the principal’s return conditional on suc-cess must be higher than his return conditionalon failure; that is,

(A4) w � R � yw � w � xw.

With (A4) in hand, and (APC) not binding, itis easy to show that x(w) � 0 (we omit thedetails). Now use this fact along with (A4) toconclude that y(w) � R.

LEMMA 8: If w � R, then V(w) � B(w).


PROOF:Suppose not; then V(w) � B(w). By Lemma

7, x(w) � 0 and y(w) � R � w. But then

Vw � ewByw � �1 � ew�Bxw

� De Byw BR � Bw

which is a contradiction to (APC).

LEMMA 9: If V(w) � B(w), then V(w) � V*.

PROOF:By Lemma 7, if V(w) � B(w), then x(w) �

0. So {e(w), y(w)} � (e, y) solves

(A5) maxe

e�R � he�

subject to eD�(e) � D(e) � B(w) [whichsimply combines (APC) and (EIC)].21 Recallthat the one-shot P-optimal contract can beidentified with the unconstrained solution e* to

(A6) maxe

e�R � h*e�.

Let e(w) be a solution to (A5). Using Lemma1, it can be verified that e(w) � e*. Nowobserve that

(A7) Vw � �V0

� �ewD�ew � Dew�.

Setting w � 0 in (A7), using the fact thatV(0) � B(0) (Lemma 2) so that e(0) � e* (asasserted above), and noting that eD�(e) � D(e)is increasing in e, we conclude that

(A8)

V0 � 1 � ��1�e0D�e0 � De0�

� 1 � ��1�e*D�e* � De*� � V*.

Now consider any w such that V(w) � B(w).With (A8) in mind and again using e(w) � e*,we may conclude that

Vw � �V0

� �ewD�ew � Dew�

� �V* � �e*D�e* � De*� � V*.

Therefore V(w) � V* for all w with V(w) �B(w), as claimed.

In what follows, we consider another Ramseyproblem with exit, this one more general thanthe one described in Lemma 3. Recall that inany equilibrium, we have

(A9) Bz � max0 � �w � z

�uz � �w

� � max Vw, Bw�].

This induces the following exit problem: foreach initial wealth level z, choose an exit dateT( z) � 1 and an exit wealth �( z) such that afterT( z) periods of consumption and saving, theagent simply takes the outside option V(�( z)).To be sure, V(�( z)) � B(�( z)), otherwise theexit interpretation is meaningless. (Notice thatthe existence of such a wealth level is guaran-teed by Lemma 2.) If no wealth w with V(w) �B(w) is ever reached, set T( z) equal to infinity.

Because � � 1/(1 � r) and u is concave, itis optimal to hold consumption constant untilthe exit date. This means that if T � � denotesthe exit date, the sequence of wealths until exitat wealth w is given by the difference equation

(A10)

wt � 1 � 1/��wt �z � �Tw1 � �

1 � �T �where w0 is just z. Indeed, this is the uniquepolicy if u is strictly concave.

LEMMA 10: There exists a finite integer Msuch that for every z � [0, R], exit occurs inthe generalized Ramsey problem at some dateT( z) � M.

PROOF:Consider the special Ramsey problem as de-

scribed in Lemma 3, with the exit option set

21 It is understood that the corresponding value of y willbe read off from D�(e) � B( y) � B(0). Here h(e) �g�1(D�(e)) and h*(e) � gx�1(D�(e)), where g( y) �B( y) � B(0) and g*( y) � u( y) � u(0).


equal to V*. The last statement in part (ii) ofthat lemma assures us that under Assumption[�], it is optimal to exit at date 1, provided z �[0, R]. In particular, date 1 exit is preferred toindefinite maintenance.22 Applying Lemma 9,this immediately shows that T( z) � �. Theuniform bound over a compact set of wealthsinvolves minor technicalities that are omitted.

LEMMA 11: If exit from z � [0, R] occurs inthe generalized Ramsey problem at some wealth�( z) � z, then exit must occur at date 1.

PROOF:Let exit occur at some wealth z� � �( z),

where z� � z. It will suffice to prove that for allT � 2,

(A11) �1 � �T � 1Vz�

�1 � �T

1 � �u� z � �Tz�1 � �

1 � �T �� uz � �z�.

[The reason is that it is optimal to hold con-sumption steady until exit, so that the condition(A11) above is just a sufficient condition forexit at T � 1.]

Suppose, on the contrary, that (A11) is falsefor some T � 2. Because that the agent couldhave spent these T periods running his wealthdown to zero and taking V(0) instead, it must bethe case that

(A12)1 � �T

1 � �u� z � �Tz�1 � �

1 � �T �� TVz� �

1 � �T

1 � �u�z1 � �

1 � �T �� TV*,

where we use Lemma 9 to replace V(0) by V*.Combining the negation of (A11) with (A12)

and eliminating V( z�), we see that

(A13)1 � �T

1 � �u� z � �Tz�1 � �

1 � �T �� uz � �z� �

1 � �T1 � �T � 1

1 � ��T � 1

� �u�z1 � �

1 � �T � � u� z � �Tz�1 � �

1 � �T �� 1 � �T � 1V*.

It is possible to check that the left-hand side of(A13) is nonincreasing in z�. Thus setting z� �0 in (A13), we have

1 � �T

1 � �u� z1 � �

1 � �T � � uz

� �1 � �T � 1V*.

But this contradicts Assumption [�].

LEMMA 12: If exit from z in the generalizedRamsey problem occurs at some wealth level�( z) � z, then for all intervening wealth levelsw until exit, both x(w) and y(w) lie in [0, R](assuming a non-null contract is offered atthose wealths).

PROOF:If u is concave, then

(A14) If z z� � �z, then wt z�

for all 0 t T.

[This is easy to check from (A10) when u isstrictly concave. When u is (weakly) concavethe argument is a bit more complicated, and isomitted.]

It follows that if some intermediate wealthlevel w is attained, then w � z� � �( z) andV(w) � B(w). Now V( z�) � B( z�), so by themonotonicity of B,

(A15) Vz� � Bz� � Bw � Vw.

On the other hand, we know that x( z�) � 0,that y( z�) � R, and that x(w) � 0 (assumingthat a contract is offered at w). So the only way

22 This is true because indefinite wealth maintenancemust be an optimal strategy in the generalized Ramseyproblem without exit, provided the exit option is not taken.


in which (A15) can be satisfied is by havingboth x(w) and y(w) lie in [0, R].

In what follows, we move away from thegeneralized Ramsey problem, which is artificialinsofar as it pertains to planned wealth levels inthe future in the absence of any uncertainty. It isuseful in defining optimal saving decisions butnot the actual evolution of future wealths. Everyreference to a process of wealths is now (unlessotherwise stated) to the actual wealth evolutionof the agent in the P-equilibrium. Notice, how-ever, that for every end-of-period wealth z, to-morrow’s beginning wealth is given exactly bysome policy function induced by the Ramseyproblem.

LEMMA 13: If end-of-period wealth z � [0,R], then the stochastic process of all subsequentwealths in any continuous P-equilibrium mustlie in [0, R] almost surely.

PROOF:Consider any z � [0, R]. Let z denote the

random variable that describes end-of-periodwealth next period, conditional on z today.Recall that z is determined by the conjunc-tion of two processes. First, a choice is madefor next period’s starting wealth; call it w1.This is done by following exactly some first-period solution to the artificial Ramsey prob-lem. Next, a contract may be offered at w1,leading to the random variable z that de-scribes next period’s end-of-period wealth. Ifa contract is not offered, then z is simplyequal to w1.

Suppose, first, that exit takes place at date 1.Then V(w1) � B(w1). By Lemma 8, it must bethat w1 � R. Moreover, by Lemma 7, we havex(w1) � 0 and y(w1) � R, so that z � [0, R]almost surely in this case.

Suppose, next, that exit takes place at somefinite time greater than 1 (these are the only twopossibilities: by Lemma 10, finite exit must oc-cur). Then, by Lemma 11, it must be the casethat w1 � �( z), so that by Lemma 12, bothx(w1) and y(w1) lie in [0, R], if a non-nullcontract is offered. So z � [0, R] once again.Moreover, since w( z) � R (see Lemma 8),w1 � [0, R] as well.

Finally, if no contract is offered at w1, thenz � w1 and by the same argument as in theprevious paragraph, z � [0, R].

LEMMA 14: From any end-of-period wealthz � [0, R], agent wealth must visit 0 infinitelyoften (with probability one) in any continuousP-equilibrium.

PROOF:For any initial end-of-period wealth z, recall

that the policy function (or any selection fromthe policy correspondence) of the artificialRamsey problem gives us next period’s begin-ning wealth—say w1—after which end-of-period wealth z is either w1 (if no contract isoffered), or (if a contract is offered) the randomvariable which takes values y(w) with probabil-ity e(w) and x(w) with probability 1 � e(w).

Define S to be the smallest integer greaterthan M � R/�, where M is given by Lemma 10and � is given by Lemma 5. Pick some agentwith initial wealth z. Consider a sample path inwhich—over the next S periods—there are suc-cesses for this agent whenever contracts areoffered. We claim that the agent’s wealth mustvisit some value w for which V(w) � B(w)within these S periods.

Suppose that the claim is false. Then it mustbe that at any end-of-period wealth zt along thissample path (0 � t � S), beginning wealthnext period must satisfy wt � 1 � zt. [This fol-lows from Lemma 11. If wt � 1 � zt, thenV(wt � 1) � B(wt � 1).] It follows that S cannotcontain a subset of more than R/� periods forwhich a contract is offered. (If it did, then, byLemma 5 and the assumption that only suc-cesses occur, wealth would wander beyond [0,R], a contradiction to Lemma 13.) But then, bythe definition of S, it must contain more than aset of M consecutive periods for which the nullcontract is optimal.

In this case, wealth simply follows the se-quence in the artificial Ramsey problem. ButLemma 10 assures us that within M periods, awealth level w will be reached for whichV(w) � B(w). This proves the claim.

Let �( z) denote the first wealth level forwhich—following the path described above forS periods—V(�( z)) � B(�( z)). Let C( z) de-note the number of times a contract is offereduntil the wealth level �( z) is reached. Considerthe event E( z) in which successes are obtainedeach time, and the first failure occurs at �( z).Define q( z) � C( z) � 1, where is given byLemma 6. Then the probability of the eventE( z) is bounded below by q( z). But E( z) is


Roy Dennis

clearly contained in the event that wealth hitszero starting from z. We have therefore shownthat the probability of this latter event isbounded away from zero in z [because q( z) � S].

The lemma follows from this last observation.

LEMMA 15: In any continuous P-equilibriumwith strictly concave u, w( z1) � z1 for somez1 � R implies w( z) � z for all z � z1.

PROOF:Otherwise there exists z2 � z1 such that

exiting at some future date is optimal at z2,while wealth maintenance is optimal at z1.Hence there exists integer T and z� � R suchthat

(A16) Bz2 �1 � �T

1 � �u� z2 � �Tz�1 � �

1 � �T �� TVz� �

uz2 1 � �

1 � �

while z� � R � z1 implies

(A17)1 � �T

1 � �u� z1 � �Tz�1 � �

1 � �T �� TVz� �

uz1 1 � �

1 � �.

This contradicts the fact that

1 � �T

1 � �u� z � �Tz�1 � �

1 � �T � �uz1 � �

1 � �

is strictly decreasing in z, by the strict concavityof u.

LEMMA 16: In any continuous P-equilibriumwith strictly concave u, w( z) � z for all z � R.

PROOF:We know that either an agent will smooth

wealth indefinitely or (by virtue of Lemma 8)will plan to exit at some wealth not exceedingR. Moreover, by strict concavity, (A10) holds,so that wealth approaches the exit wealth mono-tonically over time. In particular, w( z) � z inthis case.

LEMMA 17: Assume that u is strictly concaveand

(A18) limc3�

�uc � cu�c� � uR.

Then, in any continuous P-equilibrium, thereexists z* � (R, �) such that z � z* impliesw( z) � z and z � [R, z*) implies w( z) � z.

PROOF:The optimality of nonmaintenance of wealth

at z � R implies the existence of T and exitwealth z� � R such that

(A19)1 � �T

1 � �u� z � �Tz�1 � �

1 � �T ��

uz1 � �

1 � �� TVz� � 0.

Since z� � 0, this requires

(A20)1

1 � � � 1 � �Tu� z1 � �

1 � �T �� uz1 � ��

� �TVz� � 0.

But the concavity of u implies that

(A21) 1 � �Tu� z1 � �

1 � �T � � uz1 � �

� �T�z1 � �u�z1 � � � uz1 � ��.

Combining (A20) and (A21), we conclude that

(A22) 1 � �Vz� � uz1 � �

� z1 � �u�z1 � �.

Now we know that V( z�) � B( z�) (by thedefinition of an exit wealth), and for suchwealths w we know that x(w) � 0 and y(w) �R (Lemma 7). Consequently, V( z�) cannot ex-ceed (1 � �)�1u(R). Using this information in(A22), it follows that a necessary condition fornonmaintenance is

uR � uz1 � � � z1 � �u�z1 � �.


Using (A18) and the fact that u(c) � cu�(c) isincreasing in c, define z** such that equalityholds in the expression above. Then w( z) � zfor all z � z**. On the other hand, w( z) � zfor z in a (right) neighborhood of R, sinceAssumption [�] ensures that exit is optimal inthe Ramsey problem for such a neighborhood ofR. Hence there exists z* � (R, z**] defined byinf{ z�w( z) � z} such that it is optimal for theagent to maintain wealth above z* and decumu-late below, this very last observation being aconsequence of Lemma 15.

PROOF OF PROPOSITION 1:Part (i) follows from Lemmas 3 and 8. Part

(ii) follows from Lemmas 15, 16, and 17. Part(iii) follows from Lemmas 13 and 14.

LEMMA 18: Consider any continuous P-equi-librium and any w� � �. Then there exists � �0 such that for any initial wealth w0 � [R, w� )not in W, wt enters [0, R] or the set W atsome date with probability at least �. In partic-ular, if w0 � z* then, wt enters [0, R] at somedate t with probability at least �.

PROOF:Since w0 is not in W, either a contract is

offered at w0, or w0 � z* and a contract is notoffered at w0. Consider the event that a failureresults whenever a contract is offered and theagent’s wealth exceeds R. Note that in such anevent the agent’s wealth falls monotonically(conditional on wealth exceeding R and notentering W ), by virtue of the description of theagent’s saving behavior (if a contract is notoffered then wealth must lie below z*, in whichcase the agent’s starting wealth in the next pe-riod is smaller) and given Lemmas 5 and 8 (incase a contract is offered failure implies theagent’s wealth drops from the beginning to endof the period). Moreover, in this event wealthfalls by at least � every time a contract isoffered. If the number of times a contract isoffered exceeds

K �w� � R

�� 1,

the agent’s wealth must—at some time—fallbelow R. Note also that if a contract is notoffered at some intervening wealth less than

z*, his wealth falls deterministically untilsuch time that it either drops below R orarrives at a level where a contract is offered.Hence with probability at least � � K,where is the uniform lower bound on theprobability of failure given by Lemma 6, theagent’s wealth must eventually either fall be-low R or enter W.

Finally, note that if w0 � z*, notice thatexactly the same event can be constructed toyield the second part of the lemma.

LEMMA 19: Consider any continuousP-equilibrium and an unbounded sequence ofwealth levels wn 3 � for each of which afeasible contract exists. Suppose also that u isstrictly concave and z* � �. Then thereexists W� � z* and a number � � 0 such thatfor any w � W� :

(i) x(w) � z*(ii) e(w) � e� � R/f

(iii) e(w) y(w) � [1 � e(w)] x(w) � w � �.

PROOF:Omitted.

PROOF OF PROPOSITION 2:By the first part of Lemma 18, the probability

of entering either W or the poverty trap isbounded away from zero over any compactinterval of initial wealths of the form [0, w� ]. Bythe second part of that lemma, the probability ofentering the poverty trap is bounded away fromzero if initial wealth is less than z*.

To complete the proof of the proposition, itsuffices to prove that the only remaining limitevent is one in which wealth goes to infinity,and that this has positive probability wheneverinitial wealth w exceeds sup W.

To this end, we make the following claim:there exists w* � 0 and � � 0 such that if w �w*, then Prob(wt 3 �) � �.

To prove this claim, pick any w � w� , wherew� is given by Lemma 19, and consider theevents

Lw � w0 � w; wt � w� for all t�

and

Gw � w0 � w; wt 3 ��.


Using Lemma 19, it is easy to verify that

(A23)

Prob�Gw�Lw� � 1 for all w � w� .

The reason for this is that to the right of w� , theprocess is akin to a Markov process in which,for every w, there are two possible continuationvalues for end-of-period wealth. The expectedvalue exceeds w by some amount boundedaway from zero [see Lemma 19, part (iii)].Moreover, the probability of the success wealthis also bounded away from zero [see part (ii) ofthat lemma]. Finally, since all wealths lie abovez* [part (i) of Lemma 19], next period’s startingwealth equals this period’s ending wealth. Stan-dard arguments then show that conditional onstaying above w� , the process must converge toinfinity almost surely, which is exactly (A23).

The same argument actually reveals some-thing stronger: that if w is large enough (andsufficiently larger than w� ), the process will stayabove w� forever with probability that isbounded away from zero.23 In other words,there exists w* � 0 and � 0 such that for allw � w*,

(A24) Prob�Lw� � .

Combining (A23) and (A24), we may concludethat for all w � w*,

(A25) Prob�Gw� � .

Next, observe that starting from any wealth w �sup W, it is possible to hit a starting wealth thatexceeds w* with probability bounded awayfrom zero (take � as given in Lemma 5, andsimply look at the event in which K successesoccur, where K is the smallest integer exceeding[w* � sup W ]/�). It follows [applying (A25)]that there exists � � 0 such that for all w �sup W,

Prob�w0 � w; wt 3 �� .

Combining this observation with the fact thatover any compact interval, the probability of

entering W or the poverty trap is bounded awayfrom zero, the proof of the proposition iscomplete.

LEMMA 20: If Assumption [�] holds, thene(0) � e* and y(0) � y* maximizes e[R � y]subject to D�(e) � max{B*(y) � B*(0), 0} andy � 0, where B* denotes the ex post value functionin the Ramsey problem with exit option V*.

PROOF:Note that we can restrict the range of feasible

values of y to [0, R], since any y � R is strictlydominated by y � R. By Lemma 3, part (ii), itfollows that y � z1. That is, B*( y) � u( y) ��V(0) � u( y) � B*(0). So B*( y) � B*(0) �u( y), and the problem reduces to the staticoptimal contracting problem.

PROOF OF PROPOSITION 3:Lemma 3 describes the solution B* to the

Ramsey problem with exit option V*. Assum-ing for the moment that B* will be the ex postvalue function under the constructed P-equi-librium, it should be clear—from Lemma 5itself—that (ii) and (iii) of the proposition areimmediately satisfied.

By definition, B* satisfies the functionalequation

B*z � max0 � �w � z

�uz � �w

� � max V*, B*w�].

Thus—if B*( z) is to be an equilibrium—it mustbe that the ex ante value function satisfies

(A26) Vw � max V*, B*w�.

Part (i) of the proposition would follow rightaway from (A26), and part (iv) would onlyrequire the additional assistance of Lemma 7.

So all that remains to be proved is that if theprincipal does take the value function B* asgiven to solve his constrained optimizationproblem, then indeed, the resulting ex antevalue function satisfies (A26). The easy verifi-cation of this claim is omitted.

LEMMA 21: In any A-equilibrium, (PPC)binds at every wealth level w.

23 The formal proof of this result requires a simple cou-pling argument which we omit.


PROOF:Omitted.

LEMMA 22: In any A-equilibrium V is strictlyincreasing.

PROOF:Obviously V is nondecreasing. Suppose there

exist w1, w2 with w2 � w1 such that V(w2) �V(w1). Then { x(w1), y(w1), e(w1)} is feasibleat w2, and hence must also be optimal at w2.But here (PPC) does not bind, which contradictsLemma 21.

PROOF OF PROPOSITION 4:Consider part (i). Divide the proof into three

cases: (a) R � y � x; (b) R � y � x; (c) R �y � x. We illustrate (a), the arguments in theremaining cases being very similar:

Case (a): Take any positive � e[R � ( y �x)], where e denotes the effort assigned at w.Construct a contract ( y � y � /e, x � x), andlet e denote the associated effort response. Thenby construction (PPC) is satisfied at wealth w � by the new contract ( y, x), if the agent wereto continue to select effort e. Since B is strictlyincreasing, the agent’s optimal effort responsee � e. Given R � y � x, (PPC) must continueto be satisfied at e. Hence the new contract isfeasible at wealth w � . Since the effort e isstill available to the agent,

(A27)

Vw � � eBy � 1 � eBx � De.

Note that B( y) � B( y) � u(cs � /e) � u(cs)if cs � c( y) since it is always feasible for theagent to entirely consume any increment in end-of-period wealth. Hence

(A28)

Vw � � Vw � e�u� cs �

e� � ucs �� u�� cs �

e�� u��max cs ,cf� �

e�

where (cs, cf) denotes (c( y), c( x)). The resultthen follows upon dividing through and takinglimits with respect to .

Now turn to part (ii). By assumption,

(A29) Vw � eBy � 1 � eBx � De

where B is right continuous at y � y(w) andx � x(w), with e denoting e(w). Then for smallenough � 0, there exist positive incrementalpayments �y( ), �x( ) that solve the followingtwo equations:

e�y � 1 � e�x � .

By � �y � By � Bx � �x � Bx.

Moreover, �y( ) and �x( ) both tend to 0 as 3 0�, and so does �( ) � B( y � �y( )) �B( y). By construction, the contract y � �y( ),x � �x( ) elicits the same effort response e asthe previous contract; hence it is feasible atwealth w � . Therefore:

(A30) Vw � � Vw � � .

Since B( z � �) � B( z) � u(c( z) � �) �u(c( z)), it follows that for z � y, x:

(A31)

lim 3 0�

�� z u�cz� � 0.

Weighting inequality (A31) by the probabilityof the corresponding outcomes and addingacross the two states, it follows that

(A32) lim 3 0�

�� 1 � � 0

where � denotes

�e1

u�cywz� 1 � e

1

u�cxwz� .

The first inequality in part (ii) of the propositionthen follows from combining (A30) and (A32).

Finally, to establish the second inequality in(ii), note that if this result is false at some z,c( z) must be positive, so it is feasible for theagent to consume a little bit ( � 0) less. Thiswould cause wealth at the following date to be


w( z) � /� instead of w( z). It must thereforebe the case that for every small � 0:

(A33) ucz � ucz �

� ��V�wz �

�� Vwz� .

Taking limits with respect to , and using thefirst inequality in (ii) (which we have alreadyestablished), we obtain a contradiction.

PROOF OF PROPOSITION 5:Suppose there is an equilibrium with a strict

poverty trap, which requires that w( y(0)) �0 � w( x(0)). If u is strictly concave, a stan-dard revealed-preference argument implies thatw� must be nondecreasing. We also know thaty* � y(0) � x* � x(0) is necessary for theagent to exert effort and thus satisfy (PPC).Hence w( z) � 0 and B( z) � u( z) � �V(0) forall z � [0, y*]. So B is differentiable at x*.

If B is right differentiable at y*, then usingu�( y*) to denote the right derivative at y*, (10)of Proposition 4 implies that

1

u�y*� �e0

1

u�y*� 1 � e0

1

u�x*�upon using the hypothesis that c( y*) � y* �c( x*) � x* � 0. Since 1/u� is strictly increas-ing, and e less than 1 [otherwise (PPC) will beviolated], this inequality contradicts y* � x*.

If B is not right differentiable at y*, then notethat the rate of increase of B at y* is boundedbelow by u�( y*), since it is feasible for theagent to entirely consume all incrementalwealth. Now modify the proof of Proposition 4to infer that the inequality in (10) must be strictat y*, which will again generate a contradiction.

LEMMA 23: In any A-equilibrium, let M de-note the set of wealth levels for which a non-nullcontract is offered. Then

(i) if either M is bounded or u�(�) � 0,infw � Me(w) � 0.

(ii) if M is bounded, supw � Me(w) � 1.

PROOF:If (i) is false, there is a sequence wn and

corresponding non-null contracts ( xn , yn , en)

with en 3 0. This implies yn � xn3 0. (PPC)then implies that yn (and hence also xn) must beless than wn , but this contradicts (APC). If (ii)is false we can find a sequence with wn 3 wand en 3 1, so limnyn � �. Since (PPC) bindsfor each n, limnwn � �, contradicting theboundedness of M.

PROOF OF PROPOSITION 6:Since u�(�) � 0, Proposition 4 implies that

1/(u�(ct)) forms a submartingale in any right-continuous A-equilibrium, where ct denotes theconsumption of the agent at date t. Hence1/(u�(ct)) converges almost surely. Since u isstrictly concave, this implies that ct convergesalmost surely to a (possibly infinite valued)random variable c.

We claim that almost surely c � �, implyingthat zt 3 �, and hence that wt 3 �.

Because u�(�) � 0 by assumption, part(i) of Lemma 23 applies. Define m �D�(infw � Me(w)). Then m � 0. Pick any Z �� and integer T � 1/m[B(Z) � B(0)] � 1.Also select any nonnegative integer q. Definethe event

Bq Z � c � �q, q � 1 and zt � Z

for all t�.

Note that zt � 1 � Z implies that wt � W �Z/�. Applying both parts of Lemma 23, thereare bounds e� , e� � (0, 1) for effort levelsarising in any contract corresponding to wealthin [0, W].

Next, select � � (0, (1 � �)m). Note thatsince u(0) � 0, continuity and concavity of uimply that u is uniformly continuous. Hence wecan find � 0 such that �u(c) � u(c�)� � �whenever �c � c�� .

Conditional on wealth w at the beginning ofdate 0, define for any positive integer T theT-step-ahead possible realizations of z, c, and wunder the given A-equilibrium in the followingmanner. Let nt � {s, f} denote the outcome ofthe project t dates ahead, and let nt denote thehistory of project outcomes (nt, nt � 1, ... , n0)between dates 0 and t. Given the equilibriumwe can find the T-step-ahead realizations asfunctions of the history of the outcomes of theproject between 0 and T: zT(nT, w), cT(nT, w),wT(nT � 1, w). Define C(w, T) � {c�c � ct(nt,


w), t � T}, the set of possible realizations ofconsumptions T-steps ahead.

Then define the event

AZ � et � e� , e�

for all t and limt3�

diam Cwt , T � 0�.

LEMMA 24: For any Z and any integer q:

(A34) Prob�AZ�Bq Z� � 1.

We omit the proof since it involves sometechnical details. The basic idea underlying it isquite simple: if the diameter of C(wt, T ) doesnot converge to 0, this means that the varianceof consumption T-steps ahead does not vanish,so consumption cannot converge to a finitelimit.

LEMMA 25: For any Z and any q:

(A35) Prob�AZ � Bq Z� � 0.

PROOF:The proof rests on the following claim.

Claim: Let st (respectively, f t) denote t-step-ahead histories in which the project results in asuccess (failure) in every period. Then for any Zand any q:

Prob�BzTsT, wt � BzT f T, wt � mT � 1

for all t � t* for some t*�AZ � Bq Z] � 1.

To prove this claim, consider any path inA(Z) � Bq(Z), and select t* such that diamC(wt, T) � / 2 for all t � t*. Note that alongany such path, et � e� at all t; hence the effortincentive constraint implies B( z0(s0, w)) �B( z0( f 0, w)) � m for all w. So the inequality

(A36)

Bzksk, wt � Bzk f k, wt � mk � 1

holds for k � 0 for all t. We shall show that ifit holds for k � 1 it holds for k as well.

Use zS and zSS to denote zk � 1(sk � 1, wt),zk(sk, wt), respectively. Similarly use zF and

zFF to denote zk � 1( f k � 1, wt), zk( f k, wt),respectively. And use zFS, zSF to denote zk( f,sk � 1, wt) and zk(s, f k � 1, wt), respectively.

Next note that

VwzS � VwzF

�1

��BzS � BzF�

�1

��uczS � uczF�.

By the induction hypothesis B( zS) � B( zF) �km. Moreover, since t � t*, we have ensuredby construction that

1

��uczS � uczF�

1

��.

Since � � (1 � �)m:

VwzS � VwzF �1

��km � �� km.

Since V(w(zS)) � maxe[eB(zSS) � (1 � e) �B(zFS) � D(e)] and V(w(zF)) � maxe[eB(zSF) �(1 � e) B( zFF) � D(e)], it is evident thatV(w( zS)) � V(w( zF)) � km implies thatmax{B( zSS) � B( zSF), B( zFS) � B( zFF)} �km. Since B( zSS) � B( zFS) � m andB( zSF) � B( zFF) � m, it then follows thatB( zSS) � B( zFF) � (k � 1)m, establishingthe claim.

We are now in a position to prove Lemma 25.In the event Bq(Z), B( zt) is bounded above byB(Z). Since we selected T � (1/m)[B(Z) �B(0)] � 1, the claim above implies that zT(sT,wt) exceeds the upper bound Z for all t � t*.Since given event A(Z), a string of T succes-sive successes will almost surely occur infi-nitely often, it follows that the event A(Z) �Bq(Z) has zero probability.

Combining the results of Lemmas 24 and 25it follows that Prob[Bq(Z)] � 0 for any Z andq. If we define the event Bq � �k�1

� Bq(k) �limkBq(k) that zt is bounded while c lies in [q,q � 1), this implies that Prob[Bq] � 0. Henceif consumption converges to a limit in [q, q �1), almost surely zt 3 �; i.e., wt 3 �. But in


this event Proposition 4 implies that the asymp-totic rate of increase of V will be boundedbelow by u�(q � 1). On the other hand V isbounded above by the value function V corre-sponding to the case where effort disutilityfunction is identically zero, whose asymptoticrate of increase equals u�(�) � u�(q � 1), andwe obtain a contradiction. Hence consumptionconverges to a limit between q and q � 1 withzero probability. Since this is true for all inte-gers q, it follows that almost surely consump-tion will converge to �.

PROOF OF PROPOSITION 7:Proceed in a manner analogous to that in the

proof of the previous proposition. If the resultis false, then wealth and consumption arebounded with probability 1, which ensures that1/(u�(ct)) again forms a submartingale, so ctconverges almost surely. Lemma 23 ensuresthat effort is bounded away from zero and one,so all finite step histories will occur infinitelyoften with probability 1. If the agent receives acontract at all dates, then the agent’s wealthmust be unbounded with probability 1 in orderto provide necessary effort incentives at alldates, and we obtain a contradiction.

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contractual structure and wealth accumulation · wealth over time, so wealth distributions at any...

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