Risk, Incentives, and Contracting Relationships

Transcription

1 Risk, Incentives, and Contracting Relationsips Xiao Yu Wang* October 8, 203 Abstract Te aim of tis paper is to understand te impact of optimal provision of bot risk and incentives on te coice of contracting partners. I study a risky setting were eterogeneously risk-averse employers and employees must matc to be productive. Tey face a standard onesided moral azard problem: mean output increases in te noncontractible input of te employee. Better insurance comes at te cost of weaker incentives, and tis tradeo di ers across partnersips of di erent risk compositions. I sow tat tis eterogeneous tradeo determines te equilibrium matcing pattern, and focus on environments in wic assortative matcing is te unique equilibrium. Tis endogenous matcing framework enables a concrete and rigorous analysis of te interaction between formal and informal insurance. In particular, I sow tat te introduction of formal insurance crowds out informal insurance, and may leave tose individuals acting as informal insurers in te status quo strictly worse o. JEL Classi cation Codes: O, O3, O6, O7 Keywords: Moral Hazard; Assortative Matcing; Risk Saring; Informal Insurance; Formal Insurance; Crowd Out; Endogenous Group Formation *I am indebted to Abijit Banerjee, Ester Du o, and Rob Townsend for teir insigts, guidance, and support. Discussions wit Gabriel Carroll, Arun Candrasekar, Sebastian Di Tella, Cris Walters, and Juan Pablo Xandri were essential to tis researc. Tanks are also due to participants in te MIT Teory, Applied Teory, and Organizational Economics groups. Any errors are my own. Support from te National Science Foundation is gratefully acknowledged. Duke University. xy.wang@duke.edu

2 Introduction Risk is an unavoidable feature of life, but its pervasiveness and te metods wic exist to manage it di er in developed and in developing countries. People in developing countries face ig levels of risk: disease is widespread, te climate is punising, and occupations are azardous (Dercon (2005), Fafcamps (2008)). Instead of steady salaries, incomes are typically igly variable and depend on factors beyond individual control. For example, a farmer s liveliood is bound to te wims of nature, te vagaries of ealt, and te caprice of crop prices. In addition, people in developing countries are more vulnerable to te risks tey face. Tey live close to or at subsistence levels, and tus ave no bu er wit wic to cusion negative socks. Moreover, te formal insurance and credit institutions, credibly backed and enforced by stable governance and legal systems, wic are available to people in developed countries, are notably absent in developing environments (Dercon (2005), Fafcamps (2008)). Consequently, te poor must depend on creative ways of incorporating risk protection into teir interactions wit eac oter (Alderman and Paxson (992), Morduc (995), Dercon (2005), Fafcamps (2008)). For example, two farmers working togeter in a cropping group to grow a risky arvest could smoot eac oter s consumption by agreeing to a saring rule of te income from te realized arvest. Tese farmers could also try to establis saring rules wit outsiders, but committing to a division of pooled income becomes proibitively di cult if te contracting parties do not jointly observe or cannot easily verify income. As a result, te two farmers ave compelling reasons to adapt teir working partnersip wit eac oter to accommodate risk concerns. Tis "costly state veri cation" (Townsend (979)) often causes subsets of people ostensibly matced for oter purposes to incorporate risk management into teir existing relationsips. (See Townsend and Mueller (998) for examples of costly state veri cation and informal insurance in te Indian village of Aurepalle.) Hence, te absence of formal institutions not only induces individuals to rely on teir relationsips wit eac oter, it also causes tese relationsips to address layers of needs. Tese relationsips and te arrangements witin tem operate outside of formal insurance and credit cannels and can terefore be tougt of as "informal institutions". Indeed, Bardan relates in is 980 paper on interlocking agrarian factor markets: "Generalizing from is experience wit te ill peasants of Orissa, Bailey (966) notes: Te watersed between traditional and modern society is exactly tis distinction between single-interest and multiplex relationsips. " In tis paper, I study te implications of tis multidimensionality of interpersonal relationsips arising from te absence of formal institutions by focusing speci cally on te in uence of missing formal insurance institutions on te equilibrium formation of productive partnersips between riskaverse people and subject to one-sided moral azard. A classic example is sarecropping: a farmer lives on and crops a plot belonging to a landowner and pays rent as a sare of te realized arvest. Sarecropping as trived in di erent parts of te world for centuries, and continues to be prevalent in parts of te world today, for example in rice farms in Banglades (Akanda et. al. (2008)) and in Madagascar (Bellemare (2009)). If inputs (including e ort) are di cult to monitor and are noncontractible, owever, te contract tat provides rst-best incentives for te farmer sould be 2

3 te contract tat makes te farmer te residual claimant (te farmer sould pay a xed rent). So wy te sare contract? Stiglitz (974) suggested tat te sare contract emerged because employment relationsips also provided protection from risk, in te absence of formal insurance: wile a xed rent contract would induce te rigt incentives, it would force te agent to bear te full risk of te stocastic arvest. A xed wage contract would fully insure te agent, but te agent would ave no incentive to exert costly e ort. However, wile te e ect of te tradeo between incentive and insurance provision on a contract between a given pair of principals and agents as been studied extensively, te e ect of tis tradeo on te formation of employment relationsips as received far less attention. Tis latter analysis is essential for rigorously understanding te true strengt of informal insurance tat is, te level of risk-saring acieved witin te network of equilibrium employment relationsips wic emerges in te absence of formal insurance and credit institutions, and not merely te risk-saring acieved wit a single, isolated group of individuals. To perform tis analysis rigorously, I build a model of endogenous one-to-one matcing between eterogeneously risk-averse principals and agents wo face a classic moral azard problem. Principals eac own a unit of pysical capital, but ave in nite marginal cost of e ort, wile agents own zero units of pysical capital, but ave te same nite marginal cost of e ort. Pysical capital must be combined wit uman capital in order to produce output. For example, in a sarecropping setting, landowners would be principals, farmers would be agents, and land and farming expertise would need to be combined to produce any sort of arvest. Te distribution of output depends additively on te unobservable and noncontractible e ort exerted by te agent, as well as on te level of riskiness of te environment. More speci cally, an increase in e ort exerted by te agent leads to an increase in mean output but as no impact on te variance, wile te level of riskiness of te environment determines te variance of output. A matced principal and agent can commit ex ante to a return-contingent saring rule of teir jointly-produced output. Te di culty of caracterizing te equilibrium wage scedule in moral azard models in wic bot te principal and te agent are risk-averse (and ave di ering risk attitudes) is well-known. Instead of using a Holmstrom and Milgrom (987) type story were te principal observes some coarser aggregate of output tan te agent does to justify linear contracts in a CARA-Normal framework, I develop a model of one-sided moral azard in wic principals and agents ave CARA utility and returns are distributed Laplace. Te Laplace distribution as two key features: rst, it resembles te normal distribution (for example, it is symmetric about te mean), but as fatter tails, and second, its likeliood ratio is a piecewise constant wit discontinuity at te mean. (Please see Appendix for furter details of te Laplace distribution.) Tis framework is optimally suited to analyzing te formation of equilibrium networks of relationsips subject to one-sided moral azard and risk-saring, since te equilibrium wage cleanly separates incentive and insurance provision. Te equilibrium wage scedule is piecewise linear wit discontinuity at te mean, were te linearity at output levels away from te mean captures e cient risk-saring 3

4 between te risk-averse principal and agent, and te discontinuity at te mean captures incentive provision. (Please see Appendix for a detailed proof and discussion of tis result.) I caracterize su cient conditions for te unique equilibrium matcing between principals and agents to be negative-assortative in risk attitude (te i t least risk-averse principal works wit te i t most risk-averse agent, and so on), and for te unique equilibrium matcing to be positiveassortative in risk attitude (te i t least risk-averse principal works wit te i t least risk-averse agent). Intuitively, te equilibrium risk composition of partnersips is determined by te tradeo between incentive and insurance provision in te following way: a less risk-averse principal wo ires a more risk-averse agent can carge tat agent a risk premium for insurance, but te more tat te principal insures te agent, te less e ort tat agent will exert. Moreover, a more risk-averse agent as a iger e ective marginal cost of e ort in te rst place (because a more risk-averse agent is "more concerned" about te possibility of exerting ig e ort but being unlucky and realizing a low output draw). I sow tat te key determinants of equilibrium group composition are: te curvature of te cost of e ort function, te riskiness of te environment, and te distribution of risk types in te economy. Te curvature of te cost of e ort function in uences te equilibrium matc because it caracterizes te tradeo between insurance and incentive provision across partnersips of di erent risk attitudes. Consideration of an extreme example provides te intuition. Suppose te cost of e ort function were "in nitely convex". Ten, regardless of risk attitude, any agent would exert near zero e ort. But tis means tat no agent is ceaper to incentivize tan any oter agent tat is, all agents are equally expensive to incentivize. Hence, from te principals perspective, incentive provision is e ectively eld xed across agents of di erent risk attitudes, and te matcing becomes driven purely by insurance provision. So, te equilibrium matc will be negative-assortative in risk attitudes, since te less risk-averse principals are di erentially more willing to provide insurance to te more risk-averse agents, wo are di erentially more willing to purcase it. As te cost of e ort function becomes less extreme, owever, te less risk-averse agents become notably ceaper to incentivize tan te more risk-averse agents, and positive-assortative matcing may arise. Te riskiness of te environment is a particularly interesting determinant of te equilibrium matc. It in uences te matcing in two important ways: rst, te safer te environment is, te lower te premium for insurance. Second, te safer te environment, te more informative output is as a signal of agent e ort. In oter words, insurance provision becomes less di erentiated across agents of di erent risk attitudes wen te environment is safer, because even te more risk-averse agents don t need so muc insurance, wile incentive provision more sarply di erentiates across agents of di erent risk attitudes wen te environment is safer, because tere is less room for idden action. Finally, te distributions of risk types in te group of principals and in te group of agents a ect te equilibrium matc. Tis is because e ort is supplied only by te agent, and not by te Tis is in contrast wit te distribution-free result from Wang (202a) wic studied te impact of te tradeo between two informal insurance tecniques (income-smooting and consumption-smooting) on te equilibrium formation of relationsips. 4

5 principal ence, te risk attitude of a principal as a di erent e ect on equilibrium e ort tan does te risk attitude of an agent. Te one-sidedness of moral azard generates an asymmetry in te model wic makes te distribution of risk types important for matcing: equilibrium partnersips depend bot on "witin group eterogeneity", tat is, ow muc risk attitudes vary among principals, and ow muc risk attitudes vary among agents, as well as "across group eterogeneity", tat is, ow di erent te risk attitudes of principals are form te risk attitudes of agents. Loosely, te features of te environment in wic negative-assortative matcing emerges as te unique equilibrium are: a cost of e ort function wic is eiter close to linearity or is extremely convex, a igly risky environment (output is a very noisy signal of e ort, and te premium paid for insurance is ig), and a group of principals wo are distinctly more risk-averse tan te agents. By contrast, te features of te environment were positive-assortative matcing emerges as te unique equilibrium are: a "moderately convex" cost of e ort function, a safe environment (output is a quite precise signal of e ort, and te premium paid for insurance is low), and a group of principals wo are distinctly less risk-averse tan te agents. To be sligtly more speci c, te distribution of risk types in uences te equilibrium matc by a ecting te bounds wic delineate tese "close to linear", "moderatey convex", and "extremely convex" descriptors of te curvature of te cost of e ort function. After establising te main teoretical results, I work troug a numeric example wic demonstrates te policymaking importance of understanding te risk composition of te equilibrium network of partnersips: evaluating policy requires understanding ow people will re-optimize upon its introduction, and understanding ow people will re-optimize in response to te introduction of a policy requires understanding ow people optimize in te status quo. Tat is, in order to predict wat e ect te introduction of a formal institution would ave, it is necessary rst to ave a complete picture of te operations of te equilibrium informal institutions. For example, I discuss te impact of introducing formal insurance. I sow tat suc a policy leads to a "crowding out" of informal insurance, and in particular may leave te providers of informal insurance worse o. Tis framework of endogenous matcing is exactly suited to analyzing concretely and rigorously te "crowding out" e ect, wic is often discussed in te literature. A small body of existing literature examines te relationsip between te insurance and incentive provision tradeo and te endogenous formation of contracting relationsips under one-sided moral azard. Gatak and Karaivonov (203) develop a model of endogenous pairwise matcing between risk-neutral landlords and tenants wo are eterogeneous in ability and must work togeter to be productive. As in te sarecropping model of Eswaran and Kotwal (985), landlords specialize in managerial tasks wile tenants specialize in labor tasks. Production requires te completion of bot kinds of tasks, and inputs are not contractible. A landlord can complete bot tasks erself, or se can "sell te farm" to te tenant, wo ten completes bot tasks, or se can o er te tenant a sarecrop contract, and tey eac complete te task of teir specialty. In te rst-best, te equilibrium matc is positive-assortative in ability, but, if te agent s abilities are substitutes or weak complements in production, Gatak and Karaivonov sow tat te equilibrium matcing 5

6 becomes negative-assortative in te second-best. (Wit su cient complementarity, te equilibrium matcing is positive-assortative in te second-best.) Franco et. al. (20) consider a risk-neutral principal wo requires two (risk-neutral) agents to operate er macinery, were agents di er in teir marginal cost of e ort (ig or low). Te principal must decide wat teams of agents to form low marginal cost wit low marginal cost (positive-assortative), or low wit ig (negative-assortative). Moral azard is double-sided te key new force beind te matcing decision is te principal s inability to condition compensation on an individual agent s contribution. Te main result is tat te super/submodularity of te production tecnology in workers inputs no longer drives assortative matcing in te presence of tis doublesided moral azard. Wen te production tecnology is modular, so tat in te absence of moral azard tere is no matcing prediction, Franco et. al. sow tat te presence of moral azard can still lead to positive- or negative-assortative matcing, depending on te optimal compensation sceme, wic depends on te types and output of te team. For example, if types exerting iger input are rewarded more according to te sceme, ten negative matcing is optimal, to reduce te likeliood of "accidental payment", tat is, paying an individual wo exerts low e ort for te ig e ort exerted by is partner. Wen te tecnology exibits complementarities, a scenario is described were increasing complementarity does not lead to positive-assortative matcing because of te double-sided moral azard. Serfes (2008) studies a setting were risk-neutral principals owning exogenously-assigned projects matc wit risk-averse agents, were te projects of principals vary in riskiness (riskier projects ave iger mean and iger variance), and te risk aversion of agents also varies. A principal and an agent jointly produce output, were output depends additively on unobservable and noncontractible e ort exerted by te agent. Serfes sows tat te equilibrium matc is often not globally assortative in risk attitude, and tat te relationsip between te riskiness of te environment and te power of te contract can be ambiguous, due to te endogeneity of matcing. Importantly, te approac to studying te tradeo between incentive and insurance provision in tis paper is markedly di erent from approaces in papers suc as te one by Serfes, were principals are assumed to be risk-neutral, but own projects wic vary in riskiness. Wile suc a model generates a tractable equilibrium wage (because risk-neutrality ensures linearity), te endogenous assignment problem of eterogeneously risk-averse agents to risk-neutral principals wit eterogeneously risky projects is really answering te question, "Wat risky project is a riskaverse agent assigned to wen te output of te project depends on te noncontractible e ort of te agent, and a risk-neutral insurer is available wo sells insurance?" Tat is, tis approac can be tougt of as focusing on te impact of a formal institution on equilibrium activities undertaken in a village, wen tere is some sort of monitoring problem. By contrast, my paper builds a model of endogenous matcing between eterogeneously risk-averse principals and eterogenously risk-averse agents, were only one project is available (for example, weat is te only crop tat can be grown). Tis approac focuses on te emergence, structure, and performance of informal insurane institutions in te status quo. Hence, te rst question tat tis paper answers is, "How 6

7 well-insured are eterogenously risk-averse individuals wen a lack of formal institutions puses teir interpersonal relationsips to address multiple needs, including te need for risk protection?" Legros and Newman (2007) studied more generally te problem of endogenous matcing under nontransferable utility. Tey present tecniques to caracterize stable matcings in nontransferable utility settings by generalizing te Sapley and Subik (972) and Becker (974) supermodularity and submodularity conditions for matcing under transferable utility. Under nontransferable utility, te indirect utility of eac member of te rst group given a partnersip wit eac member of te second group can be calculated, xing te second member s level of expected utility at some level v. Ten, tis indirect utility expression, wic depends on bot members types and v, is analyzed for supermodularity and submodularity in risk types. A number of papers ave attempted to empirically detect te inverse relationsip between te riskiness of te environment and te power of te contract predicted by te basic principal-agent model wit risk and moral azard. For example, Allen and Lueck (992) study sarecropping relationsips in te American Midwest in 986, and observe tat te strengt of dependence of a sarecropper s rent contract on realized arvest is positively correlated wit te riskiness of te crop grown. From tis, tey conclude tat risk is not a problem for farmers, since risk considerations do not appear to in uence contract design. However, if more risk-averse farmers work for less risk-averse landowners, cultivating safer crops under low-powered contracts, wile less riskaverse farmers work for more risk-averse landowners, cultivating riskier crops under ig-powered contracts, te same empirical observations would emerge, but risk concerns would be playing a signi cant role in contract design, troug te unaccounted-for cannel of contracting partner coice. Alternatively, we expect formal institutions to be stronger in te United States ence landownerfarmer relationsips may not be as multidimensional as tey are in developing countries. Tis would imply tat tis is not te rigt dataset to test for te teoretically-predicted relationsip between risk and incentives. Te remainder of te paper proceeds as follows. Te next two sections present te model and te results. Section 4 works troug a ypotetical policy example and sows tat te evaluation of policy may cange drastically if it accounts for te endogenous network response. Section 5 concludes. 2 Te Model In tis section, I introduce a framework designed to analyze te formation of equilibrium contracting relationsips under one-sided moral azard wen te contract balances insurance provision wit incentive provision. Te framework consists of te following elements: Te population of agents: te economy is populated by two groups of agents, G and G2, were jgj = jg2j = Z, Z a nite, positive integer. Call members of G "principals" and members of G2, "agents". All principals own one unit of pysical capital, but ave in nite marginal cost of 7

8 e ort; all agents lack pysical capital but ave nite and identical marginal costs of e ort. Te cost of e ort for all agents is c(a), c(a) > 0, c 0 (a) > 0, c 00 (a) > 0. Principals and agents bot ave CARA utility, u(x; r) = e rx, were individuals di er in teir degree of risk aversion. Let r represent a principal, and r 2 represent an agent. Principals and agents are identical in all oter aspects. Tere are no assumptions on distributions of risk types. 2 Te risky environment: A principal-agent partnersip can only produce positive output if one unit of pysical capital is combined wit uman capital. For example, a landowner owns land, and a farmer as agricultural experience and skill, and a successful arvest requires te landowner and farmer to combine teir capital. Output is given by Rja = a + ", were te riskiness of te environment is captured by " f ", an exogenously-given, well-de ned, di erentiable probability distribution function wit support on ( doesn t a ect te variance of returns. ; ). Te e ort of an agent, a, increases te mean but Information and commitment: All agents know eac oter s risk types. In te rst-best environment, an agent s e ort is bot observable and contractible. In te second-best environment, te agent s e ort is neiter observable nor contractible. A given matced pair (r ; r 2 ) observes te realized output of teir partnersip, and is able to commit ex ante to a return-contingent saring rule s(r p2 ), were R p2 is te realized return of (r ; r 2 ) s joint project p 2. More precisely, s(r p2 ) speci es te wage paid to te agent r 2 wen te realized return is R p2, were s : R! R (tere are no limited liability assumptions). In order to be feasible, te income te principal r receives must be less tan or equal to R 2 s(r p2 ). Since all individuals ave monotonically increasing utility, r s sare will be equal to R 2 s(r p2 ). Te equilibrium: An equilibrium is:. A matc function (r ) = r 2, were () assigns eac r to at most one agent r 2, and distinct people ave distinct partners. Moreover, te matcing pattern described by () must be stable. Tat is, it must satisfy two properties: (a) No blocks: no unmatced principal and agent sould be able to write a feasible wage contract suc tat bot of tem are appier wit eac oter tan tey are wit te partners assigned to tem by. (b) Individual rationality: eac agent must receive a iger expected utility from being in te matc () tan from remaining unmatced. 2 Of course, in reality, types are multidimensional, and matcing decisions are not exclusively based on risk attitudes. It is wort noting tat te model can account for tis. For example, kinsip and friendsip ties are important, in large part because of information (tey know eac oter s risk types), and commitment (tey trust eac oter, or can discipline eac oter). Kinsip and friendsip ties would enter into tis teory in te following way: an individual would rst identify a pool of feasible risk-saring partners. Tis pool would be determined by kinsip and friendsip ties, because of good information and commitment. Following tis, individuals would coose risk-saring partners from tese pools. Tis coice would be driven by risk attitudes, as addressed in tis bencmark wit full information and commitment. Tus, tis teory can be tougt of as addressing te stage of matcing tat occurs after pools of feasible partners ave been identi ed. 8

9 2. A set of saring rules and e ort coices by te agents, one saring rule and one e ort coice for eac matced pair. Te agent cooses an e ort level wic is optimal for er se sould not be able to coose a di erent e ort level and become better o. Furtermore, no pair sould be able to coose a di erent saring rule wic leaves bot partners weakly better o, and at least one partner strictly better o (te agent cooses e ort optimally in response to te saring rule). Matcing patterns: It will be elpful to introduce some matcing terminology. Suppose te people in G and in G2 are ordered from least to most risk-averse: fr j ; rj 2 ; :::; rj Zg, j 2 f; 2g. Ten "positive-assortative matcing" (PAM) refers to te case were te i t least risk-averse person in G is matced wit te i t least risk-averse person in G2: (ri ) = r2 i, i 2 f; :::; Zg. On te oter and, "negative-assortative matcing" (NAM) refers to te case were te i t least risk-averse person in G is matced wit te i t most risk-averse person in G2: (ri ) = r2 Z i+, i 2 f; :::; Zg. To say tat te unique equilibrium matcing pattern is PAM, for example, is to mean tat te only wic can be stable under optimal witin-pair saring rules and projects is te matc function wic assigns agents to eac oter positive-assortatively in risk attitudes. 3 Te Results 3. Te First-Best It will be useful to begin by solving te rst-best problem, wen te agent s e ort is observable and contractible. Te rst step to caracterizing te equilibrium network of relationsips is to caracterize wat appens in a given relationsip. Suppose principal r is matced wit agent r 2, and tat te returns of te risky project R are distributed according to some general density function f: Rja = a + "; " f(") Assume te random variable R as a well-de ned cumulant generating function 3. Denote te agent s sare of realized output R by s(r). Ten, te pair s equilibrium saring rule, given tat r 2 receives expected utility at least e v for some xed level v, solves te following problem: Z max a;s(r) e r [R s(r)] f(r a)dr s:t: (IR) Z e r 2[s(R) c(a)] f(r a)dr e v Te equilibrium saring rule and e ort are described in te following lemma. 3 Recall tat te cumulant-generating function is te log of te moment-generating function. 9

10 Lemma Te optimal rst-best contract of a principal-agent pair (r ; r 2 ), were s(r) denotes te agent s sare, is: s F B(R) = r R + KF r + r B(r ; r 2 ; v) 2 0 KF B(r ; r 2 ; v) = v + c(c 0 ()) + r 2 r 2 a F B = c 0 () Z e r r 2 R r +r 2 f(r c 0 ())dra (Te proof is in Appendix 2.) Te rst-best contract as several notable features. First, equilibrium e ort level is te same in any possible pair tis is because e ort is contractible. Hence, any principal-agent pair cooses te e ort wic "maximizes te pie", and ten e ciently sares te risk of tat pie. Te e ort wic "maximizes te pie" is te e ort level wic equates te marginal bene t of e ort (te marginal impact on mean output, ) wit te marginal cost of e ort exertion, c 0 (). Second, te equilibrium wage is linear. Tis is unsurprising again, e ort is contractible, so te saring rule needs only to provide insurance, and not incentives. Moreover, te less risk-averse individual receives a sare tat is more eavily dependent on output realization. Now tat we ave caracterized te optimal saring rule and equilibrium e ort witin a matced pair r and r 2, we can solve for te equilibrium network of relationsips. Intuitively, since we know tat any possible matced pair cooses te same e ort level, we expect negative-assortative matcing to arise as te unique equilibrium. Tis is because we know from Sculofer-Wol and Ciappori and Reny tat endogenous matcing under pure ex post risk management results in unique negativeassortative matcing (tere is no moral azard, and no scope for ex ante risk management matced pairs are not able to coose wat risk tey face). Negative-assortative matcing arises because te least risk-averse individuals are di erentially willing to provide insurance, wile te most risk-averse individuals are di erentially willing to pay for it. To formalize tis intuition, we need to identify a metod for caracterizing te equilibrium matc. A callenge is posed by te eterogeneity of risk-aversion in agents, wic makes tis a model of matcing under nontransferable utility. Tat is, te amount of utility experienced by an agent wit risk aversion r from consuming one unit of output di ers from te amount of utility an agent wit risk aversion r 2 experiences from one unit of output. Tus, we cannot directly apply te Sapley and Subik (962) result on su cient conditions for assortative matcing in transferable utility games. It will be elpful to review brie y tat environment and result. Consider a population consisting of two groups of risk-neutral workers, were all workers ave utility u(c) = c. Let a denote te ability of workers in one group, and a 2 denote te ability of workers in te oter group. Te production function is given by f(a ; a 2 ), wic can be tougt of as: "te size of te pie generated by matced workers a and a 2 ". Ten, positive-assortative matcing, and d2 f da da 2 d 2 f da da 2 > 0 is a su cient condition for unique < 0 is a su cient condition for unique negative-assortative 0

11 matcing. My approac ere will be to identify te function in tis model of nontransferable utility wic is analogous to te Sapley and Subik production function f(a ; a 2 ). In Proposition 2 below, I prove tat expected utility is transferable in tis model instead of tinking about moving "ex post" units of output between agents, we sould instead tink about moving "ex ante" units of expected utility. I sow tat te sum of te certainty-equivalents CE(r ; r 2 ) of a given matced pair (r ; r 2 ) is te analogy to te joint output production function in te transferable utility problem. Te sum of te certainty-equivalents of a matced pair is "te size of te expected utility pie generated by matced agents r and r 2 ", and su cient conditions for positive-assortative and negative-assortative matcing correspond to conditions for te supermodularity and submodularity of CE(r ; r 2 ) in r ; r 2. More tecnically, expected utility is transferable in tis model because te expected utility Pareto possibility frontier for a pair (r ; r 2 ) is a line wit slope under some monotonic transformation. Proposition 2 Expected utility is transferable in tis model. Proof. Using te optimal saring rule and equilibrium e ort from Lemma, we can write te expressions for te certainty-equivalent of a principal r and of an agent r 2 wo are matced wit eac oter: CE r = v + log E e r 2 r r 2 r r 2 r +r 2 R ja = c 0 i () c(c 0 ()) CE r2 = v r 2 Hence, it is clear te cost to te principal r of increasing te certainty-equivalent of er agent r 2 by one unit is exactly one unit (and vice versa). Tat is, expected utility is transferable in te model. Tis tells us tat te sum of certainty-equivalents for a matced principal and agent in tis model can be tougt of as te function wic is analogous to te joint output function in te Sapley and Subik transferable utility setting: CE(r ; r 2 ) = r + r 2 log E e r r 2 r +r 2 R ja = c 0 i () c(c 0 ()) It will be elpful to make two observations at tis point. First, we can de ne te representative risk aversion of a matced pair (r ; r 2 ): ^H(r ; r 2 ) = r r 2 r + r 2 Importantly, we can see tat te sum of certainty-equivalents of a matced pair depends only on representative risk aversion. In oter words, a matced pair (r ; r 2 ) acts as a single individual wit CARA utility and absolute risk aversion ^H(r ; r 2 ).

12 And correspondingly, we can de ne te reciprocal of representative risk aversion, te representative risk tolerance: eh(r ; r 2 ) = ^H(r ; r 2 ) = r + r 2 Tus, te sum of certainty-equivalents written as a function of representative risk tolerance is: CE( e H) = H e i log E e H e R ja = c 0 () c(c 0 ()) Intuitively, te joint expected utility pie of a matced pair is some transformation of te representative individual s expected utility from facing te stream of returns R, minus te cost of e ort, were e ort in te rst-best is independent of any risk types. Furtermore, te transformation of te representative individual s expected utility is a special transformation: K Rja t = eh i = log E e H e R ja = c 0 () were K Rja t = is te cumulant-generating function (log of te moment-generating func- eh tion) of te random variable Rja evaluated at te negative of te representative risk tolerance. How does tis contribute to our understanding of equilibrium matcing in te economy? We know from te Sapley and Subik assortative matcing conditions tat a su cient condition for unique PAM in tis setting is supermodularity of CE(r ; r 2 ) in r ; r 2, and a su cient condition for unique NAM in tis setting is submodularity of CE(r ; r 2 ) in r ; r 2. Moreover, we know tat: d 2 CE(r ; r 2 ) = dce dr dr 2 dh e = r r 2 d 2 H e + d2 CE dr dr 2 dh e 2 2 d 2 CE d e H 2 dh e dh e dr dr 2 Hence, a su cient condition for unique PAM is convexity of CE( e H) in e H, wile concavity of CE( e H) in e H ensures unique NAM. Cecking te second derivative of CE( e H) in e H yields te rst-best matcing result. Proposition 3 In te rst-best model, wen e ort is observable and contractible, te unique equilibrium matcing pattern in risk atittude of principals and agents is negative-assortative. 2

13 Proof. Te second derivative of CE( e H) in e H is straigtforward to nd. CE( e H) = e HK Rja t = eh dce( H) e dh e = K Rja t = eh d 2 CE( e H) d e H 2 = eh 3 K00 Rja since te cumulant-generating function is convex in t. c(c 0 eh K0 Rja ()) t = eh < 0 t = eh Hence, te unique equlibrium matc in te rst-best is negative-assortative. Now tat we understand wat appens in te rst-best, we can investigate te second-best. 3.2 Te Second-Best Now, suppose tat e ort is not observable and not contractible. In tis case, te saring rule of a given principal-agent pair must provide incentives as well as insurance. To gain traction on tis problem, I impose a speci c functional form assumption on te distribution of output. Recall tat output given e ort a is described by Rja = a + ". In te previous subsection, it was assumed tat " f ", a general density function wit support on te real line and well-de ned cumulant-generating function. In tis subsection, assume tat f " is a Laplace distribution wit mean 0 and exogenously-given variance V > 0, were V captures te riskiness of te environment. 4 details on te Laplace distribution.) (See Appendix for more Te key features of tis distribution for te setting of tis paper are symmetry. Te distribution resembles te normal distribution, but as fatter tails, and te density function is non-di erentiable at te mean. Loosely, te fatter tails allow us to avoid te Mirrlees critique of linear contracts a realized return in te tail of a Laplace distribution is not in nitely precise about e ort exerted. Ten, te equilibrium saring rule of a principal-agent pair (r ; r 2 ), were te agent r 2 is ensured expected utility at least e v, solves: suc tat: Z a max s(r) e r (R s(r)) Z 2V e V [R a] dr + a (IR) : Z + a Z a e r 2[s(R) c(a)] 2V e V [R e r (R s(r)) 2V e V [R a] dr a] dr+ e r 2[s(R) c(a)] 2V e V [R a] dr e v 4 Assume V < + max(r ) max(r 2, for te problem to be well-de ned. ) 3

14 Z a (IC) : a 2 arg max a2(0;) Z + a e r 2[s(R) c(a)] 2V e V [R e r 2[s(R) c(a)] 2V e V [R a] dr a] dr+ Te complete equilibrium analysis of tis problem can be found in Appendix, but I will provide a sketc of te solution ere, to provide an understanding of te equilibrium beavior of a matced partnersip. First, we need to address te constraints. Te IR constraint clearly binds in equilibrium. Te IC constraint presents more of a callenge. It would be useful to be able to replace te global IC constraint wit its rst-order condition, but none of te existing su cient conditions for te validity of te rst-order approac apply to tis model. Rogerson (985) sows tat su cient conditions for a standard moral azard model in wic te principal may also be risk-averse are: (a) monotone likeliood ratio, and (b) convexity of te distribution function. Condition (a) olds in my model (te likeliood ratio ere is a piecewise constant, c below te mean and c above te mean), but (b) fails very few standard distribution functions satisfy CDFC. Jewitt (988) identi es su cient conditions tat weaken CDFC, but for a model wit a risk-neutral principal. Additionally, utility is assumed to be additively separable in consumption and e ort, wereas it is multiplicatively separable ere. A variety of more recent contributions identify sets of conditions tat weaken CDFC sligtly, at te cost of strengtening oter conditions, but none weakens CDFC enoug for te Laplace distribution. So, te validity of te rst-order approac must be proved from rst principles 5. Since e ort a is cosen from an open set, te optimum will be interior, if it exists. Hence, te rst-order condition of te global IC constraint is a necessary, toug peraps not su cient, condition for te optimum. Tis means tat te rst-order problem (te problem wit te global IC condition replaced by its rst-order condition) is a relaxed problem, so tat te actual optimum must be a solution of te rst-order problem, if it exists, but a solution of te rst-order problem is not necessarily te optimum. Solving te rst-order problem yields te following wage scedule: s(r p < ba) = r r + r 2 R r r + r 2 ba + c(ba) + r 2 v s(r p > ba) = r r + r 2 R p r r + r 2 ba + c(ba) + r 2 v log r 2 r 2 log r r 2 V r + r 2 + r r 2 V r + r 2 + r 2c 0 (ba)v r 2 c 0 (ba)v were r is te risk attitude of te principal, r 2 is te risk attitude of te agent, and ba is te level of e ort "anticipated" by te principal. In equilibrium, te optimal e ort cosen by te agent in response to te wage scedule s(r p jba) sould be a = ba: te principal as no incentive to pay for a iger level of e ort tan se knows will actually be exerted, and te agent as no incentive 5 Again, a rigorous proof can be found in Wang (202b). 4

15 to exert more e ort tan e is compensated for. Wat is equilibrium e ort given tis compensation sceme? It can be sown tat a = ba is a stationary point of agent r 2 s expected utility from exerting e ort a given wage scedule s(rjba), for every possible ba. However, for ba > ba t, were ba t is some tresold, tere will be asecond stationary point at a < ba because te wage scedule is discontinuous for ba 6= c 0 r, if te principal r +r 2 tries to induce a "too-ig" level of e ort, te agent will pro tably deviate to a discretely lower level of e ort. More concisely, for ba ba t, te unique maximizing level of e ort exerted by te agent is a = ba; for ba > ba t, te unique maximizing level of e ort exerted by te agent is a < ba. by: Terefore, te equilibrium ba set by te principal is ba = ba t, were tis tresold is caracterized c 0 (ba t ) r c 0 (ba t ) + = c 00 (ba t ) > 0 r + r 2 r 2 V r 2 Because te agent s expected utility from exerting e ort a given wage s(rjba ) is strictly concave in a, it must be tat s(rjba ) is infact te optimum. Observe tat setting ba t = c 0 r causes te left-and side of te equation to be 0, wile r +r 2 te rigt-and side is positive. Since te left-and side is strictly increasing in ba t (te cost function c(a) is strictly convex), it must be tat ba t > c 0 r. r +r 2 Terefore, te wage scedule witin a principal-agent pair (r ; r 2 ) is piecewise linear: at te anticipated mean level of output, ba, tere is a jump in te wage realized output levels greater tan te mean ba are rewarded at a discretely iger level tan output levels tat are below te r mean. At output levels away from te anticipated mean, te wage is linear wit slope r +r 2. Hence, te equilibrium wage can be cleanly decomposed into insurance provision and incentive provision. Te jump in te wage at te mean provides incentives (since te likeliood ratio is a piecewise constant, in some sense knowing weter output is above or below te mean is di erentially more informative about e ort exertion), and te linearity away from te mean captures risk-saring. Using tis caracterization of equilibrium saring rule and e ort in a given principal-agent pair (r ; r 2 ), we can solve for conditions for unique assortative matcing, under a functional form assumption on cost of e ort: c(a) = a M, M > for convexity. well. We use te same trick as in te rst-best: expected utility is transferable in te second-best as Proposition 4 Expected utility is transferable in te second-best. Proof. Using te equilibrium saring rule of a given pair and te caracterization of equilibrium 5

16 e ort, we can write te certainty-equivalent of principal r and agent r 2 wen matced: CE r = ba 2 c(ba 2 ) 0 2 v r 2 B 6 4 r 2 + r 2c 0 (ba 2 )V i r r2 i r + + r r 2 r2 r +r 2 V r 2 c 0 (ba 2 )V i r r2 + r r 2 r +r 2 V i + r r2 3 7C 5A CE r2 = v r 2 Hence, it is clear tat tere is a one-to-one tradeo in te certainty-equivalents of r and r 2. Tus, expected utility is transferable in tis model. Terefore, a su cient condition for unique positive-assortative matcing is supermodularity of te pairwise sum of certainty-equivalents, and a su cient condition for unique negative-assortative matcing is submodularity of te pairwise sum: CE(r ; r 2 ) = ba 2 c(ba 2 ) 0 2 B 6 4 r 2 + r 2c 0 (ba 2 )V i r r2 i r + + r r 2 r2 r +r 2 V r 2 c 0 (ba 2 )V i r r2 + r r 2 r +r 2 V i + r r2 3 7C 5A Te rst two terms of tis sum can be tougt of as te part of te expected utility pie coming from productivity (e ort exertion and corresponding expected output), wile te tird term can be tougt of as te part of te expected utility pie coming from risk-saring. Te callenge of identifying conditions for assortative matcing in tis model is te one-sided moral azard. Altoug expected utility is transferable, r and r 2 do not enter symmetrically into te sum of certainty-equivalents of te matced pair. Consequently, te matcing conditions will not be distribution-free as tey are in te absence of moral azard (for example, in te rst-best, or in te case of endogenous matcing under te trade-o of ex ante and ex post risk management, as in Wang (202a)). Finding conditions for te supermodularity and submodularity of CE(r ; r 2 ) in r ; r 2 yields te following matcing results. Proposition 5 Let c(a) = a M, M >.. NAM is te unique eqm matcing pattern for M 2 [; M ] [ [M 4; ), were M M 4. a. M is increasing in (te least risk-averse principal s risk aversion) and in r (te r most risk-averse principal s risk aversion), and decreasing in 2; r 2. Furtermore, M is increasing r in V, and decreasing in. b. M 4 is decreasing in and r, and increasing in 2 and r 2. Furtermore, M 4 is decreasing r r in V, and increasing in. 6

17 2. PAM is te unique eqm matcing pattern for M 2 [M 2 ; M 3 ], were M M 2 and M 3 M 4. (It may be tat te interval [M 2 ; M 3 ] is empty.) a. M 2 is increasing in and r, and decreasing in 2 and r 2. Furtermore, M 2 is increasing r r in V, and decreasing in. b. M 3 is decreasing in and r, and increasing in 2 and r 2. Furtermore, M 3 is decreasing r r in V, and increasing in. (Te proof is relegated to te Appendix.) Tis result igligts te key determinants of te equilibrium matcing pattern: te riskiness of te environment V, te across-group and witin-group eterogeneity in risk attitude, captured by te endpoints of te supports of te risk type distributions of principals and agents, and te marginal impact of e ort on mean output,. In words, te takeaways from te main matcing result are te following. First, positiveassortative matcing (PAM) is more likely for moderately convex cost of e ort functions, wile negative-assortative matcing (NAM) is more likely for cost of e ort functions wic are close to linear or extremely convex. Wat delineates te boundaries of "close to linear", "moderately" convex, and "extremely" convex? Te comparative statics on te bounds tell us te following:. Negative-assortative matcing (NAM) is more likely to arise wen principals are distinctly more risk-averse tan agents, wile positive-assortative matcing (PAM) is more likely to arise wen principals are distinctly less risk-averse tan agents. 2. NAM is more likely to arise wen te environment is very risky (V is large), wile PAM is more likely to arise wen te environment is very safe (V is low). 3. NAM is more likely to arise wen te marginal bene t of e ort (for mean ouptut) is low ( is small), wile PAM is more likely to arise wen te margnal bene t of e ort is ig ( is large). Wat is te intuition beind tese comparative statics? Consider a relatively safe environment were te cost of e ort function is moderately convex and te marginal bene t of e ort is large. E ort exertion across di erent risk attitudes is most eterogeneous wen te cost of e ort function is moderately convex. Moreover, wen te environment is relatively safe (tat is, V is relatively low), te "need" for insurance is small and output is a fairly precise signal of e ort. Hence, rewarding e ort based on realized output is e ective, less risk-averse agents are substantially ceaper to incentivize tan more risk-averse agents, and more risk-averse agents are not willing to pay particularly ig risk premia. All of tese forces pus te incentive provision e ect to outwieg te insurance provision e ect, wic favors PAM over NAM. If in addition principals are less risk-averse tan agents, ten te tradeo between incentive provision and insurance provision is particularly stark (since if a principal were more risk-averse tan te agent, te principal would be appy to provide incentives for te agent, as tis would be a metod of self-insurance). 7

18 Hence, in tis environment, te least risk-averse principal experiences te biggest di erence in utility between being paired wit te least risk-averse agent versus a more risk-averse agent. Tus, te least risk-averse principal will outbid te oter principals for te least risk-averse agent, and once te least risk-averse principal and te least risk-averse agent are removed from te pool of candidates, te least risk-averse principal of tose remaining will outbid te oter principals for te least risk-averse agent remaining, and so on, and te equilibrium matcing pattern will be positive-assortative. On te oter and, wen agents ave close to linear or extremely convex cost of e ort, te di erence between e ort exertion across agents of di erent risk types is small eiter all te agents exert very ig e ort, or all te agents exert very low e ort. If te environment is also risky, tat is, V is ig, ten individuals, especially more risk-averse individuals, will be willing to pay a ig price for insurance. Moreover, output is a noisy signal of actual e ort exertion. Hence, te insurance provision e ect will tend to outweig te incentive provision e ect. If in addition principals are more risk-averse tan agents, ten te incentive provision is aligned wit insurance provision: a more risk-averse principal prefers a less risk-averse agent, because te principal desires insurance for erself, and tis will naturally provide incentives to te agent. Tis means tat te most risk-averse principal experiences te biggest di erence in utility between being paired wit te least risk-averse agent versus a more risk-averse agent. Tus, te most risk-averse principal will outbid te oter principals for te least risk-averse agent, and once tey are matced, te most risk-averse principal remaining will outbid te oters for te most risk-averse agent remaining, and so fort, and te equilibrium matcing pattern will be negative-assortative. An interesting but informal insigt tat emerges from te analysis in tis framework is tat, in contrast wit te standard view, tere is a strong case for te more risk-averse individuals to be principals, and te less risk-averse individuals to be agents: a more risk-averse principal is appy to incentivize a less risk-averse agent, because se wants insurance, and te less risk-averse agent doesn t mind te riskiness of te incentives. Finally, a word on e ciency. Proposition 6 Te equilibrium maximizes te sum of certainty-equivalents, and is Pareto e cient. Te equilibrium maximizes te sum of certainty-equivalents, since te conditions for PAM and NAM were derived by nding conditions for te supermodularity and submodularity of te pairwise sum of certainty-equivalents. Since te sum of certainty-equivalents is a social welfare function, and te equilibrium maximizes tis sum, it must be Pareto e cient. Wile te natural measure of welfare for a partnersip and te individuals witin tat partnersip is te pairwise certainty-equivalent, te unweigted sum of certainty-equivalents across pairs is not te rigt way of tinking about economy-wide welfare. A more risk-averse individual needs to be guaranteed a smaller amount tan a less risk-averse individual to be made indi erent between accepting tat amount wit certainty and partaking in er risky equilibrium, but tere is no reason society sould value er less because of tat. Hence, policy in tis framework will be considered 8