Production priorities in dynamic relationships

Forand, Jean Guillaume; Zapal, Jan Article Production priorities in dynamic relationships Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Forand, Jean Guillaume; Zapal, Jan (2020) : Production priorities in dynamic relationships, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 15, Iss. 3, pp. 861-889, https://doi.org/10.3982/TE2963 This Version is available at: https://hdl.handle.net/10419/253449 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 15 (2020), 861–889 1555-7561/20200861 Production priorities in dynamic relationships Jean Guillaume Forand Department of Economics, University of Waterloo Jan Zápal CERGE-EI, a joint workplace of Charles University and the Economics Institute of the Czech Academy of Sciences We characterize optimal contracts in a dynamic principal–agent model of joint production in which project opportunities are heterogenous, utility from these projects is nontransferable, and the agent has the option to quit the relationship at any time. To demand the production of projects that benefit her but not the agent, the principal must commit to produce projects that benefit the agent in the future. Production at all stages of the relationship is ordered by projects’ cost-effectiveness, which is their efficiency in transferring utility between the principal and the agent: cost-effective demands impose relatively low costs on the agent and cost-effective compensation imposes relatively low costs on the principal. Over time, optimal contracts become more generous toward the agent by adding commitments to less cost-effective compensation. In turn, because this new compensation cannot be profitably exchanged against less cost-effective demands, the principal narrows the scope of her demands. Keywords. Dynamic contracts, incentive provision, heterogenous projects. JEL classification. C73, D86, L24. 1. Introduction Productive relationships generate a variety of joint project opportunities over their lifetimes. This raises two related questions: what criteria guide decisions to produce some opportunities and pass up others, and how does project selection evolve over time? In this paper, we address these questions in a dynamic principal–agent model in which (a) heterogenous project opportunities arrive according to an arbitrary stochastic process, (b) utility from these projects is nontransferable (although transferable utility is Jean Guillaume Forand: [email protected] Jan Zápal: [email protected] We thank Georgy Egorov, Paul Klein, Colin Stewart, Jakub Steiner, Xin Zhao, seminar participants at Queen’s, Toronto, and Wilfred Laurier, as well as audiences at the 2015 Annual Conference of the Canadian Economic Association, the 2016 Annual Meeting of the Midwest Political Science Association, the 2016 Canadian Economic Theory Conference, the 2016 Joint Meeting of the European Economic Association and Econometric Society, the Fall 2016 Midwest Economic Theory Conference, and the 2017 North American Summer Meeting of the Econometric Society. Finally, two anonymous referees provided excellent feedback. This paper was previously circulated under the title “The demand and supply of favours in dynamic relationships.” Forand acknowledges support from a SSHRC Insight Development Grant. ©2020 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE2963 862 Forand and Zápal Theoretical Economics 15 (2020) a special case of our model), and (c) the principal is contractually committed to production decisions, but the agent can walk away from the relationship at any time. We characterize optimal contracts in this setting and detail the dynamics of the principal’s demand and supply of projects (i.e., the production of projects that benefit the principal, but are costly for the agent and vice versa). Although we model a canonical principal–agent relationship, for the remainder of the Introduction, we fix ideas by focusing on a manager–worker pair within a larger firm. The simplest model of their interaction features two project opportunities: at each stage, the manager demands effort from the worker and supplies a wage (Mirrlees 1976). We allow for a rich set of productive activities that arise randomly over the course of this relationship. The manager can make demands on the worker that differ in their benefits for the manager and their costs to the worker. For example, the manager may need the worker to deal with an emergency, like a failure in the firm’s server, or she may ask the worker to complete a routine project that is less time-sensitive, like writing a plan for the firm’s information technology infrastructure. Similarly, the manager can supply a number of projects to the worker in the form of both financial and nonmonetary compensation. For example, the manager can recommend the worker for a bonus, offer perks like travel opportunities, accommodations for family issues, and better office space, or can tilt task allocations toward those that benefit the worker’s career (e.g., involving training programs). More broadly, the (stochastic) dynamics of project opportunities within the relationship can be driven by the business cycle, industry trends, or human capital accumulation by both the manager and the worker. In the case in which the manager demands effort and supplies money, it is well known (Lazear 1981) that she benefits from delaying the worker’s compensation: whereas current payments are sunk when the manager makes future demands for effort, committing to pay the worker in the future motivates both current and future effort. Not surprisingly, the optimal contracts in our model also feature backloaded compensation, but our central task is to determine how the manager selects the projects she uses to reward the worker. If, for example, the worker prefers increased flexibility in his schedule to access to a job training program, will the manager prioritize the former type of compensation over the latter? The answer, in general, is “no,” because focusing only on the workers’ preferences neglects the manager’s costs from supplying projects. If job training increases the worker’s productivity, then its net cost for the firm can be small relative to the cost of scheduling flexibility, which offers less countervailing benefits to the firm. If the worker does not value the two types of compensation too differently, then the manager always benefits from substituting job training for schedule flexibility. Therefore, she will commit to sending the worker to all available job training programs before making any promises about future scheduling flexibility. This intuition underlies our main result that characterizes optimal contracts. We show that the manager always prioritizes her supply of projects according to their costeffectiveness, i.e., their benefit for the worker relative to their cost for the manager. At any point in the relationship, the optimal contract identifies a threshold supply project and projects that are more cost-effective than this threshold are supplied whenever they arrive. Projects that are less cost-effective than the threshold are not supplied unless Theoretical Economics 15 (2020) Production priorities 863 (a) the manager makes a new demand and (b) the worker’s participation constraint requires fresh commitments to future compensation. Because the manager adds new supply commitments through cost-effectiveness, the threshold project transitions to less cost-effective projects over time. How does the growth in the scope of the worker’s compensation affect the manager’s demand for projects? We show that the manager demands only those projects that are more cost-effective than the threshold supply project (where cost-effectiveness for demand projects measures the benefit for the manager relative to the cost for the worker). Therefore, the manager’s accumulation of increasingly less cost-effective supply commitments is tied to rationing of her demands on the worker, which become concentrated on the most cost-effective projects. To further illustrate our results, suppose that the manager can demand emergency or routine projects, with the effort cost being the same for both types of projects, but emergency projects being more important for the manager. Therefore, emergency projects are more cost-effective. Suppose also that all demands are more cost-effective than supplying the worker with job training, but that only emergency projects are more costeffective than supplying the worker with schedule flexibility. Early in the relationship, the manager demands both emergency and routine projects, and commits only to supply job training1: because the manager benefits from trading both emergency and routine projects against promises of job training, she will not pass up any demand until all future training opportunities have been promised. Later in the relationship, the manager demands only emergency projects, and supplies both job training and schedule flexibility: because the manager prefers to scale back her demands for routine projects to avoid promising schedule flexibility, she will pass over the former once she must supply the latter to incentivize the worker to take up emergency projects. The inefficiency generated by the worker’s inability to commit to remain in her job is captured by the fact that production decisions for the same project can differ over time: both parties could be made better off ex ante if the manager could use training opportunities that are passed over early in the relationship to incentivize demands for routine projects that are passed over later on. In fact, we show that ex ante Pareto-efficient contracts involve a time-invariant threshold project. Cost-effectiveness pins down project priorities, but not the exact dynamics of production. For example, for how long can the manager keep demanding routine projects? Answering such questions requires determining the worker’s value from the relationship at any point in time, which sets the level of his participation constraint. This value, which is endogenous, incorporates the worker’s utility from producing projects, his time preferences, and the availability of projects in the future: when the process driving project opportunities is arbitrary, the value has little structure. In Section 5, we specialize the model to the case of Markov project processes and construct optimal contracts directly. In doing so, we rank the manager’s demands by how expensive they are for her: more expensive demands require that a broader scope of projects be supplied to the worker. 1This illustrates typical dynamics of optimal contracts, the details of which depend on, among other things, the process driving project opportunities. We revisit this example in Section 4. 864 Forand and Zápal Theoretical Economics 15 (2020) Because we study how future opportunities provide incentives for current production, our work has connections to the literature on informal risk-sharing in the presence of stochastic endowment shocks (Thomas and Worrall 1988,Kocherlakota 1996, Dixit et al. 2000). Important generalizations of this work incorporate hidden information about endowment shocks or utility from production as well as sequential actions. The former literature analyzes chips mechanisms (Möbius 2001,Hauser and Hopenhayn 2008), and dynamic contracts with and without commitment (Guo and Hörner 2015,Li et al. 2017,Lipnowski and Ramos 2020). The latter literature studies holdup situations (Thomas and Worrall 1994,2018,Board 2011) and has close links to the relational contracts literature (Levin 2003). Furthermore, our work is related to the literature on dynamic principal–agent interactions (Lazear 1981,Rogerson 1985,Spear and Srivastava 1987,Sannikov 2008). Our focus on selection from heterogenous project opportunities is the key difference between these contributions and ours. Furthermore, we assume that players are risk-neutral, so that risk-sharing plays no role in our results, we do not rely on transfers, we place no restrictions on the process driving project opportunities, as opposed to the standard independent and identically distributed (i.i.d.) or Markov assumptions,2and we abstract from information asymmetries and holdup problems. Three papers are most closely related to ours. First, Ray (2002) shows that any optimal principal–agent relationship backloads the agent’s compensation: by increasing the agent’s continuation value, the principal relaxes the agent’s no-deviation constraint, so that she can make the agent work harder and improve efficiency. In our model, this logic is one of the forces that drive the backloading of the agent’s utility: by promising to supply the threshold project in the future, the principal gains the ability to demand projects that are more cost-effective than this threshold. The other reason is the rationing in the principal’s demands stemming from her accumulation of increasingly less cost-effective supply commitments. This latter reason has no analog in Ray (2002), which features a repeated stage game and, hence, no heterogeneity in future production opportunities.3 Second, Bird and Frug (2019a) study project production in a closely related dynamic principal–agent model in which project arrivals follow independent Poisson processes and are privately observed by the agent. Like us, they highlight the criterion of costeffectiveness for prioritizing project production. Unlike us, they show that the principal frontloads the agent’s compensation in that she might make less cost-effective supply commitments before exhausting all more cost-effective supply commitments. Informational asymmetries are the key to understanding why our results differ from theirs. In their environment, the principal’s only tool to incentivize the agent to disclose the arrival of a demand project is the growth in the agent’s continuation value. Therefore, frontloading the agent’s compensation allows the principal to free up incentives for future disclosures, and the principal trades off prioritizing cost-effective projects against 2From a technical point of view, this rules out standard recursive approaches to characterizing optimal dynamic contracts (Spear and Srivastava 1987,Thomas and Worrall 1988,Abreu et al. 1990). In contrast, our proofs rely on direct arguments. 3See also Bird and Frug (2019b), who study conditions under which principal–agent relationships increasingly favor the agent over time. Theoretical Economics 15 (2020) Production priorities 865 future flexibility.4In our model with commonly observed project opportunities, it is the level of the principal’s future commitments that underpins the agent’s incentives. Consequently, the agent’s compensation is backloaded because the principal always follows cost-effectiveness when making supply commitments, and the (inefficient) variability in the set of projects that the principal demands and supplies vanishes in the long run. Third, in a contemporaneous paper, Samuelson and Stacchetti (2017) study the role of transfers in a version of our model with two-sided lack of commitment and an i.i.d. process driving project opportunities. They show that the principal uses variation in either continuation values or transfers to generate incentives when transfers are either absent or present, respectively. Their model, however, does not admit a simple description of the relationship’s dynamics. In our model, we can capture transfers to the agent or to the principal through suitably defined supply and demand projects. Because the principal follows cost-effectiveness when committing to supply projects, our results imply that the principal will not start paying the agent until she has exhausted more cost-effective means to reward him. Moreover, the relationship dynamics in our model initially favor the principal and eventually favor the agent. This implies that, when available, transfers flow toward the principal early in the relationship and toward the agent later in the relationship. 2. Model A principal and an agent participate in a long-lived relationship in which a joint project opportunity arises in each period t=12. Specifically, let U⊂R2be a finite set and let u={ut}t≥1be a U∪{(00)}-valued stochastic process that describes the arrival of projects over time, where ut=(00)denotes the absence of a project at t.Let ut=(u1ut)denote a project history at tand let Hdenote the set of all such histories for all times t. Because optimal contracts are indeterminate at histories that occur with zero probability, we assume that P0(ut)>0for all project histories ut. This is the only assumption that we impose on the project process ufor our main results, and we do so mainly to ease the exposition.5 Given a project utat time t, the principal and the agent simultaneously decide whether to participate in the production of the project, and project utis produced if and only if both players agree to produce it. We let ut=(uPtuAt)denote the payoffs to the principal and the agent if project utis produced, and we normalize each player’s payoff from no production to 0. For simplicity, we assume that the players’ stage preferences over the production of projects are strict, that is, that uAt = 0and uPt = 0for all projects ut∈U. Therefore, player i(myopically) prefers to participate in the production of project utif uit >0and prefers not to participate if uit <0. Finally, the players discount future payoffs with common factor δ∈(01). We model projects parsimoniously, 4See also Hopenhayn et al. (2006), in which a planner rations a fixed (expected discounted) stock of rights to future monopoly power to retain the ability to reward a sequence of competing innovators. 5Any process with zero-probability events can be expressed as the limit of a sequence of processes without such events, and the limit of the corresponding sequence of optimal contracts is an optimal contract for the limiting process. 866 Forand and Zápal Theoretical Economics 15 (2020) but we can accommodate projects that are more complicated ventures with uncertain outcomes: in this case, utis interpreted as the expected utilities to the principal and the agent from these richer lotteries. Similarly, production of project utmight generate payoffs in periods beyond t:inthiscase,utis the present value to the principal and the agent of that payoff flow. Project histories and production decisions, and, hence, all players’ payoffs, are publicly observed and verifiable. A contract κ:H→[01]maps project histories into production probabilities. Given a project history utat time t,κ(ut)(henceforth κtfor short, with history utunderstood) is the probability with which contract κspecifies that the project at tis produced. Let Kdenote the set of all contracts. We make the strong assumption that production decisions can be verifiably conditioned on a public randomization device. However, our model also admits an interpretation in which all production decisions are deterministic. Specifically, we can reinterpret κtas specifying the intensity with which project utis produced. In this view, interior production probabilities represent reducing the scale of a project’s implementation.6 Given a contract κand a history utat time t,let Uit =Et ∞  t=t δt−tκtuit denote the associated expected discounted sum of payoffs to player istarting from t. The expectation is taken conditional on the information contained in project history ut, but, as for contracts, we leave the history dependence of payoffs implicit to lighten notation. Notice that the linearity of stage utilities in production probabilities implies that intertemporal smoothing of production decisions due to risk aversion plays no role in our results. We assume that the principal commits to contracts. Meanwhile, the agent has the option to irreversibly quit the relationship at the beginning of every period t, after the arrival of project utbut before the realization of the contract’s production decision (determined by κt). If the agent remains in the relationship, then he is committed to following the outcome of the public randomization device for that period. Quitting yields apayoffof0to both players, which is the payoff they receive when no project is ever produced. It follows that an optimal contract κ∗is a solution to the problem max κ∈K E0UP1 subject to UAt ≥0for all project histories ut.(IR At) In words, an optimal contract maximizes the principal’s ex ante utility from the relationship subject to being individually rational for the agent following all project histories.7 6Allowing the contract to depend on a richer notion of histories, which record past outcomes of the randomization in production, would not change any of our results. By using this randomization, the principal can offer the agent random continuation utility, but this randomization can only (weakly) hurt the principal due to the convexity of the underlying utility possibility set. 7Standard arguments establish the existence of an optimal contract (e.g., Dixit et al. 2000). Theoretical Economics 15 (2020) Production priorities 867 As we show below, production probabilities in optimal contracts are often bang-bang, in which case our restriction to ex ante individual rationality constraints for the agent (i.e., prior to the realization of the production decision) is not constraining. However, following some histories, the agent’s ex post individual rationality constraint could fail if the public randomization device calls for some project to be produced. In such cases, our results exploit the fact that the principal can provide adequate incentives to the agent ex ante by committing to interior production probabilities. In any period and given any project over which the preferences of the principal and the agent are aligned, optimal contracts must specify jointly optimal production decisions.8 Lemma 1. If contract κ∗is optimal, then (i) if uPtuAt >0,thenκ∗ t=1; (ii) if uPtuAt <0,thenκ∗ t=0. Common interest projects contribute to the value of the relationship, but Lemma 1 confirms that optimal contracts can be identified with the production decisions they prescribe for those projects on which the principal and the agent disagree. To this end, define the sets D={u∈U:uP>0>u A}and S={u∈U:uA>0>u P},andassume,to avoid trivialities, that Dand Sare both nonempty. Given a contract κ, we say that the principal demands a project with probability κtat twhenever ut∈Dand, conversely, that the principal supplies a project with probability κtat twhenever ut∈S.Thedecomposition of an optimal contract into the demand and supply of projects turns out to be useful for describing project selection and its dynamics. To simplify notation, we denote a typical element of Dby vandatypicalelementofSby w, and any statement referring to demand project v(respectively, supply project w) should be read as being restricted to projects u∈D(respectively, u∈S). 3. Benchmark:ExantePareto efficiency A useful benchmark is that of ex ante Pareto-efficient contracts in which the agent can commit to production decisions. These contracts maximize the principal’s ex ante utility subject to a lower bound uon the agent’s ex ante utility. An efficient contract κeis a solution to max κ∈K E0UP1subject to E0UA1≥u Efficient contracts resolve many of the same trade-offs as optimal contracts. Therefore, we introduce and discuss these key properties in this simpler setting, and in Section 4, we detail how they are affected when the agent must be continually incentivized to support production. Define an ordering of projects in D∪Ssuch that uuif and only |uP/uA|>|u P/u A|. In words, if vv,thenprojectvis more cost-effective to demand than project vfor 8The proofs of all results are provided in the Appendix. 868 Forand and Zápal Theoretical Economics 15 (2020) Figure 1. Ex ante Pareto-efficient contracts. Here, the threshold Ueis a supply project. The more cost-effective projects vand ware always produced, and the less cost effective vand ware never produced. the principal: in this case, the ratio vP/|vA|—the principal’s benefit per util cost to the agent—measures the productivity of project vas a tool for extracting utility from the agent. Conversely, if ww,thenprojectwis more cost-effective to supply than project wfor the principal: in this case, the ratio |wP|/wA—the principal’s cost per util benefit to the agent—measures the productivity of project was a tool for providing utility to the agent. Notice that more cost-effective demands are ranked higher by , while more cost-effective supplies are ranked lower by . This is illustrated in Figure 1,where points in the plane represent projects, projects in the northwestern quadrant can be demanded by the principal, projects in the southeastern quadrant can be supplied, and more cost-effective projects are represented by larger dots. For simplicity, we assume that the ordering is complete on D∪S, i.e., that all project pairs are ranked strictly by cost-effectiveness. Our first result shows that the principal’s demand and supply of projects in efficient contracts are determined by cost-effectiveness. Proposition 1. Fix any ex ante Pareto-efficient contract κeand any time t.Theprincipal demands and supplies projects that are more cost-effective than some threshold: there exists a project Uesuch that κe t=1if vtUe 0if Uevt and κe t=1if Uewt 0if wtUe9(1) 9Recall that, by our notational convention, statements like vtUeshould be read as applying only to histories with ut∈D, and statements like Uewtshould be read as applying only to histories utwith ut∈S. Theoretical Economics 15 (2020) Production priorities 875 history ut−1, we assume that ut=wwith probability γt,whereγt+1>γ t. Alternatively, if training opportunities are declining, then given any time t>1and any history ut−1, we assume that ut=wwith probability βt,whereβt+1<β t.17 In this case, we also assume that training opportunities become exceedingly rare over time: limt→∞ βt=0.To isolate the effect of how job training opportunities are distributed over time, we fix their (expected discounted) quantity across both scenarios: ∞ t=2δt−1γt=∞ t=2δt−1βt.This means that when training opportunities are growing, the manager has few of them to offer the worker initially relative to the case of decline, but that the opposite is true later in the relationship. We make further assumptions to tighten the link between optimal and efficient contracts in this example. First, because optimal contracts are conditioned on the arrival of a first demand by the manager and ex ante Pareto-efficient contracts depend on the workers’ individual rationality constraint evaluated prior to the realization of an initial project, we assume that the project at time t=1is v. Second, because optimal contracts deliver all ex ante surplus to the manager, we focus on those efficient contracts that yield expected utility u=0to the worker. Notice that given our assumption that the ex ante quantities of all projects are fixed whether training opportunities are growing or declining, efficient contracts are identical in both scenarios. Finally, suppose that the efficient contract is such that the manager always demands both emergency and routine projects, and always supplies job training but never supplies schedule flexibility. If training opportunities are growing, the manager’s ability to reward the worker increases over time and her potential demands are time-invariant: for any time t> 2,∞ t=tδt−1γt>∞ t=2δt−1γtand ∞ t=tδt−1pu=∞ t=2δt−1pufor all u∈{vvw}. Therefore, in this case the efficient contract satisfies the worker’s individual rationality constraint at all times t≥1and is optimal. In contrast, optimal contracts in the declining scenario cannot be efficient. First, because vw, the manager must eventually supply schedule flexibility in exchange for continued demands for emergency projects when her commitments to job training no longer provide meaningful incentives to the worker. Second, because wv, the manager must stop demanding routine projects when she starts rewarding the worker through commitments to schedule flexibility. In the long run, production in the declining scenario is essentially reduced to exchanging emergency projects against scheduling flexibility (although job training is provided in the rare cases when it is available). After observing a manager extend both job training and schedule flexibility to a worker, it would be natural to interpret this as a sign that the firm has a plentiful supply of rewards to offer its workforce. Our results suggest the opposite interpretation: an increase in the scope of the worker’s nonmonetary compensation points to scarcity in those rewards that are most effective from the firm’s perspective. Because the manager benefits from substituting job training for schedule flexibility, observing the latter means that the bound on the availability of training opportunities is binding. Similarly, it would be natural to conjecture that a worker receiving nonmonetary compensation 17Recall that project processes take values in U∪{(00)}, so that the growth and decline scenarios will have different probabilities of having no project arrive at any given time. For simplicity we leave these and other feasibility constraints on the two processes implicit. 876 Forand and Zápal Theoretical Economics 15 (2020) from a variety of sources would produce more for the firm. Again, our results predict the opposite: diversified rewards are tied to rationing in the worker’s tasks. Because the manager benefits from substituting decreased schedule flexibility for routine projects, a worker who is rewarded by job flexibility cannot be profitably asked to work on routine projects. ♦ 5. Markov project processes From Section 4, we know that the optimal supply threshold can only transition to less cost-effective supply projects if the principal makes additional demands. However, in general, we cannot identify which project histories, and, in particular, which demands, lead to changes in the threshold supply project. In this section, we sharpen our results by assuming that the project process uis Markov. Proposition 3. Suppose that the project process is Markov and fix any optimal contract κ∗. For all demands v, there exists a supply project Wvsuch that, given any history utwith ut=v, the optimal threshold project W∗ tis the least cost-effective of projects W∗ t−1 and Wv: W∗ t=max W∗ t−1Wv In the Markov case, the updating rule for the optimal supply threshold has a simple form: following any history (ut−1v), the supply threshold is updated to Wvif this project is less cost-effective than W∗ t−1, while it remains at W∗ t−1otherwise. We provide more details about the relationship between the threshold Wvand its associated demand v below. Here we note that Wvcaptures the minimal level of future supply commitments (and, correspondingly, the maximal level of future demands) that provide incentives for a demand for v. Therefore, if W∗ t−1is more cost-effective than Wv, then the agent’s individual rationality constraint at tbinds as a result of the principal’s demand for vand the optimal contract must become more generous toward the agent by adding commitments to less cost-effective projects (and, correspondingly, dropping less cost-effective demands). Furthermore, because the project process is Markov and the threshold Wvis history-independent, the continuation contracts following all histories at which a binding individual rationality constraint is met at demand vare identical. This stationary updating rule and corresponding “amnesia property” for optimal contracts are analogous to well known results in related models, notably those of Thomas and Worrall (1988)and Kocherlakota (1996) for the i.i.d. case and of Ligon et al. (2002) for the Markov case. We can use Proposition 3 to order demands by the scale of the incentives that must be supplied to the agent so as to produce them. If demand projects vand vare such that Wvis less cost-effective than Wv, then we say that demand vis more expensive for the principal than demand v:inreturnforv, the principal must commit to supply more projects to (and demand less from) the agent in the future. An important note is that the expensiveness of a demand vis different from its cost-effectiveness: the latter is the ratio of the principal’s benefit from vto the agents’ cost, while the former is a measure of the stringency of the agent’s individual rationality constraint following v.Intuitively, Theoretical Economics 15 (2020) Production priorities 877 expensiveness depends on two potentially countervailing factors: the agent’s stage cost from producing v(given by |vA|) and the value to the agent of future project opportunities conditional on having reached project v, which depends on both the discount factor δand the project process. However, if the project process is i.i.d., then the relationship’s future production opportunities are history-independent. In that case, a demand’s expensiveness is determined solely by its cost to the agent. Corollary 1. Suppose that the project process is i.i.d. If demands vand vare such that |vA|≥|vA|,thenvcannot be more expensive for the principal than v. This is analogous to a result from Thomas and Worrall (1988). In their model, the agent’s opportunity cost from outside offers (which are i.i.d.) stands for the cost of the principal’s demand in that state, and the level of the corresponding optimal wage stands for the expensiveness of this demand for the principal. In line with Corollary 1,their Proposition 3 shows that optimal wages are nondecreasing in outside market wages. For the general Markov case, we derive the ranking of demands by expensiveness through our construction of optimal contracts in the proof of Proposition 3, and it tracks the solutions to a recursive sequence of reduced problems: v1, the most expensive demand for the principal, has the least cost-effective threshold Wv1,overallv∈D, associated to the problem of finding an optimal contract subject only to (a) the initial project being v(i.e., u1=v) and (b) the individual rationality constraint for the agent at time 1 (i.e., UA1≥0). Then v2, the second most expensive demand for the principal, has the least cost-effective threshold Wv2,overallv∈D\{v1}, to the problem of finding an optimal contract for the principal subject only to (a) u1=v,(b)UA1≥0, and (c) the fact that the threshold transitions to Wv1whenever v1arrives, and so on. As opposed to the i.i.d. case, the ranking of demands by expensiveness depends on the project process. For example, the most expensive demand v1might impose a low cost on the agent if the continuation process following v1provides few opportunities to reward him. Our final goal in this section is to derive sufficient conditions for ranking demands by their expensiveness in the non-i.i.d. case. To this end, fix two demands vand v. First, Condition 1 is that for vto be more expensive than v, it is more costly for the agent: |vA|≥|vA|. Second, note that the thresholds associated to both demands vand v must take into account future transitions to thresholds associated to demands that are more expensive than both of them (as in constraint (c) of the reduced problem defining v2in the previous paragraph). Correspondingly, Condition 2 is that the distributions over these future transitions are identical following vand v: for all t>1,Pv[ut=u]= Pv[ut=u]for all u/∈S,wherePvstands for the distribution of process uconditional on u1=v.18 Third, the expensiveness of demand vshould capture the idea that the principal has worse opportunities to supply projects following v. Because production decisions are ordered by cost-effectiveness, this requires that the project process puts less weight on more cost-effective supply projects following v. However, such a condition would not be sufficient on its own, as less cost-effective supplies may yield high 18For related reasons, Condition 2 also requires that the distribution of projects that are neither supply nor demand projects (i.e., projects in (U∪{(00)})\(D∪S)) are also identical following vand v. 878 Forand and Zápal Theoretical Economics 15 (2020) stage benefits to the agent, so that the effect on the agent’s utility following vwould be ambiguous. Correspondingly, Condition 3 is a joint restriction on the cost-effectiveness and the stage benefit of supply opportunities following vand v:forallt>1and all c≥0, Pv|wPt|/wAt ≤cEvwAt||wPt|/wAt ≤c ≤Pv|wPt|/wAt ≤cEvwAt||wPt|/wAt ≤c(4) Condition 3 says that, given any fixed supply threshold, the agent’s expected rewards are higher following vthan following v. Corollary 2. Suppose that the project process is Markov. If demands vand vsatisfy Conditions 1–3, then vcannotbemoreexpensivefortheprincipalthanv. Conditions 1–3 are clearly stringent, but our preceding remarks highlight that this is to some degree by necessity. Notice that Conditions 2 and 3 are satisfied if the project process is i.i.d. Returning to the Example, what would be needed to conclude that routine projects cannot be more expensive for the manager than emergency projects? We have assumed that the agent is indifferent between producing both types of projects, so that Condition 1 is satisfied. Ignoring Condition 2, Condition 3 can be satisfied if job training programs are more likely to arrive following a routine project, so that the worker can expect to be compensated more often with the manager’s preferred supply project. But because the agent prefers schedule flexibility to job training, the former cannot arrive too rarely following a routine project. Otherwise, the agent’s expected compensation could be lower following routine projects when the principal rewards the agent with both job training and schedule flexibility. 6. Conclusion We recap our main results by discussing their relationship to two key assumptions of our model: that no transfers are available to support production and that the principal can commit production decisions. Transfers. While we have not explicitly allowed for monetary payments between the principal and the agent, models with transfers are special cases of our model. Indeed, a transfer of kdollars to the agent can be represented by a supply project mS=(−kk) ∈ S, while a transfer of kdollars to the principal can be represented by a demand project mD=(k−k) ∈D,wheremSand mDare equally cost-effective. The flexibility of the project process allows for different specifications of transfer opportunities. On the one hand, if all nonmonetary projects are followed by transitions to both mSand mD,and if kis large, then transfers are always available and essentially unrestricted in size. On the other hand, if mSarrives at fixed intervals, then the principal has infrequent but regular opportunities to pay a bonus to the agent. While our results apply to all models with transfers, they provide specific implications for the use of money in the dynamic relationships captured by our environment. First, the principal’s ability to use transfers to reward the agent does not crowd out supply through production: the principal Theoretical Economics 15 (2020) Production priorities 879 does not start paying the agent until she has committed to supply projects that are more cost-effective than money in all their future occurrences. Furthermore, in an optimal contract the principal may even supply projects that are less cost-effective than money if the availability of future transfer opportunities is sufficiently constrained. However, if kis large and transfer opportunities are frequent, then the principal would always use money instead of less cost-effective projects. Second, the direction of the flow of money between the principal and the agent varies over the relationship’s lifetime: the principal demands transfers from the agent early in the relationship and supplies transfers to the agent later in the relationship. No commitment for the principal. If the principal cannot commit to production decisions, then the model must be augmented with history-dependent individual rationality constraints for the principal that cap her supply of projects. Cost-effectiveness still drives project selection decisions, but with an important qualification: if the principal supplies a less cost-effective project, then she must also supply more cost-effective projects in all succeeding histories in which none of her individual rationality constraints has been binding. This implies that some characterization of the optimal contract in terms of threshold supply projects would still be possible without commitment by the principal, but that pinning down general properties of the optimal contract dynamics would be difficult. Recall that if both sides can commit to production decisions, then the threshold supply project is fixed over time, and if only the principal has commitment power, then the threshold becomes more favorable to the agent over time to incentivize demands. If the principal cannot commit either, then she must have incentives to supply projects, which would imply a threshold that becomes less favorable to the agent following some histories. Therefore, in contrast to our results, the optimal contract typically does not stabilize in the long run. Absence of commitment power for the principal would generate an inefficiency closely related to that discussed in Section 4: the principal and the agent would be better off if past demands could incentivize the principal’s current supply of projects, but without commitment, these can only be supported by future demands. Appendix Proof of Lemma 1. Suppose, toward a contradiction, that κ∗is optimal and that, for some project history utsuch that uPtuAt >0,wehavethatκ∗ t<1.Fixacontract˜κthat is identical to κ∗except that ˜κt=1at ut. It follows that ˜κis individually rational because κ∗is individually rational. Furthermore, ˜ UPt >U∗ Pt, yielding the desired contradiction. The proof for the case of utsuch that uPtuAt <0is similar and is omitted. We prove Proposition 2 before proving Proposition 1 to avoid repeating several arguments that simplify in the context of Proposition 1. Proof of Proposition 2. We proceed in a number of steps. 880 Forand and Zápal Theoretical Economics 15 (2020) Step 1. Fix an optimal contract κ∗, project history ut, its superhistories utand ut , and projects ww. Suppose that (i) ut=wand (ii) ut =wand t−1  s=t+1 κ∗ sIus∈DIU∗ As=0=019 We show that if κ∗ t<1then κ∗ t =0 To see this suppose, toward a contradiction, that κ∗ t<1at utand that κ∗ t >0at ut . Now consider an alternative contract ˜κ, identical to κ∗except that (i) κ∗ t<˜κt≤1at ut, (ii) 0≤˜κt <κ ∗ t at ut , (iii) ˜ UAt −U∗ At =δt−tPtut˜κt−κ∗ twA−δt−tPtut κ∗ t −˜κt wA=0(5) and (iv) U∗ Ar +δt−rPr(ut )[˜κt −κ∗ t ]wA≥0for any history urthat is a proper superhistory of utand a proper subhistory of ut (i.e., with t+1≤r≤t −1),andsuchthatκ∗ r>0 and ur∈D. Because U∗ Ar >0for any history urin (iv), such a contract always exists. Furthermore, ˜κis individually rational for the agent. To see this, first note that because ˜ UAt =U∗ At ≥0,wehavethat˜κsatisfies (IRAr) for all times r≤t. Second, because ˜ UAt>U∗ At≥0, it follows that given any time r>tand history urthat is not a subhistory of ut ,wehavethat ˜ UAr ≥U∗ Ar ≥0. Third, even though we have that ˜ UAt <U∗ At , because ˜κt uAt =˜κt wA≥0it also follows that ˜ UAt ≥δEt U∗ At+1 ≥0 Finally, consider history urthat is a proper superhistory of utand a proper subhistory of ut (i.e., with t+1≤r≤t −1). Suppose ˜κsatisfies (IRAr+1)forur+1that is a subhistory of ut .If˜κruAr ≥0, then it follows that ˜ UAr ≥δEr˜ UAr+1 ≥0 If ˜κruAr <0, then ˜κr=κ∗ r>0and ur∈D, and from (iv) it follows that ˜ UAr =U∗ Ar +δt−rPrut ˜κt −κ∗ t wA≥0 19Throughout, t−1 s=t+1κ∗ sIus∈DIU∗ As=0=0denotes that, given history utand its superhistory ut , for any history usthat is a proper superhistory of utand a proper subhistory of ut (i.e., with t+1≤s≤t −1), either κ∗ s=0or us/∈Dor U∗ As >0. Theoretical Economics 15 (2020) Production priorities 881 It thus follows recursively that ˜κsatisfies (IRAr) for all times t+1≤r≤t −1. It remains only to note that, by (5), we have ˜ UPt −U∗ Pt =−δt−tPtut˜κt−κ∗ t|wP|+δt−tPtut κ∗ t −˜κt |wP| =δt−tPtut κ∗ t −˜κt |wP|1−|wP|/wA |wP|/wA >0 where the inequality follows because ww, contradicting the optimality of κ∗. Step 2. Step 1 implies that to any optimal contract κ∗there corresponds a historydependent threshold project mapping W∗:H→Ssuch that, for all times tand histories ut, κ∗ t=1if W∗ twt 0if wtW∗ t where for simplicity we denote W∗(ut)by W∗ t, with the project history understood. For any history ut=(ut−1ut),thethresholdisgivenby W∗ut=max W∗ut−1∪w:Ptκ∗ t>0ut=w t−1  s=t+1 κ∗ sIus∈DIU∗ As=0=0>0 where we set W∗(u0)=minS.20 By construction, W∗ tis nondecreasing with respect to and is such that, given any history utand its superhistory ut,W∗ t=W∗ tif t s=t+1κ∗ sIus∈DIU∗ As=0=0.Thisprovesparts(i) and (iii) of Proposition 2. Step 3. Fix an optimal contract κ∗and project history ut=(ut−1vt),wherevt min{wW∗ t−1}. We show that κ∗ t>0. To see this suppose, toward a contradiction, that κ∗ t=0. Because κ∗ t=0,wehaveW∗ t−1=W∗ t. Hence, because vtmin{wW∗ t−1},there exists ut+1=(ut−1vtwt+1)such that vtwt+1W∗ t,wherewt+1W∗ tand part (i) of Proposition 2 imply κ∗ t+1=0. Now consider an alternative contract ˜κ, identical to κ∗ except that (i) κ∗ t<˜κt≤1at ut, (ii) κ∗ t+1<˜κt+1≤1at ut+1, and (iii) ˜ UAt −U∗ At =˜κt−κ∗ tvAt +δPtut+1˜κt+1−κ∗ t+1wAt+1=0(6) Such a contract always exists. Furthermore, ˜κis individually rational for the agent: because ˜ UAt =U∗ At ≥0,˜κsatisfies (IRAr) for all times r≤t,andbecause ˜ UAt+1≥ U∗ At+1≥0,˜κsatisfies (IRAr) for all times r≥t+1. It remains only to note that, by (6), 20Throughout, Pt(κ∗ t>0ut=wt−1 s=t+1κ∗ sIus∈DIU∗ As=0=0)denotes Pt(ut)of a superhistory utof ut with κ∗ t>0,ut=w, and t−1 s=t+1κ∗ sIus∈DIU∗ As=0=0. 882 Forand and Zápal Theoretical Economics 15 (2020) we have ˜ UPt −U∗ Pt =˜κt−κ∗ tvPt +δPtut+1˜κt+1−κ∗ t+1wPt+1 =˜κt−κ∗ tvPt1−|wPt+1|/wAt+1 vPt/|vAt| >0 where the inequality follows because vtwt+1, contradicting the optimality of κ∗. Step 4. Fix an optimal contract κ∗and project history ut=(ut−1vt),where max{W∗ t−1w}vt. We show that κ∗ t=0. To see this suppose, toward a contradiction, that κ∗ t>0.WehaveeitherW∗ t=W∗ t−1or W∗ tW∗ t−1. Hence, because max{W∗ t−1 w}vt,thereexistsut+1=(ut−1vtwt+1)such that W∗ twt+1vt,whereW∗ twt+1 and part (i) of Proposition 2 imply κ∗ t+1=1. Now consider an alternative contract ˜κ, identical to κ∗except that (i) 0≤˜κt<κ ∗ tat ut, (ii) 0≤˜κt+1<κ ∗ t+1at ut+1, and (iii) U∗ At −˜ UAt =κ∗ t−˜κtvAt +δPtut+1κ∗ t+1−˜κt+1wAt+1=0(7) Such a contract always exists. Furthermore, ˜κis individually rational for the agent. To see this, first note that because ˜ UAt =U∗ At ≥0,˜κsatisfies (IRAr) for all times r≤t.Second, even though we have that ˜ UAt+1<U∗ At+1, because ˜κt+1wAt+1≥0, it also follows that ˜ UAt+1≥δEt+1U∗ At+2 ≥0 and, hence, ˜κsatisfies (IRAr) for all times r≥t+1. It remains only to note that, by (7), we have U∗ Pt −˜ UPt =κ∗ t−˜κtvPt +δPtut+1κ∗ t+1−˜κt+1wPt+1 =κ∗ t−˜κtvPt1−|wPt+1|/wAt+1 vPt/|vAt| <0 where the inequality follows because wt+1vt, contradicting the optimality of κ∗.Steps 3and 4jointly imply part (ii) of Proposition 2. Proof of Proposition 1. We specialize the results of Proposition 2 for optimal contracts to establish our results for efficient contracts. Recall that efficient contracts are solutions to max κ∈K E0UP1subject to E0UA1≥u Fix any efficient contract κeand consider histories utand ut. First, arguments closely mirroring those of Step 1 in the proof of Proposition 2 show that if κe t<1then κe t=0if either wtwtor vtvt Theoretical Economics 15 (2020) Production priorities 883 Second, arguments mirroring those of Steps 3and 4show that if κe t<1then κe t=1if vtw if κe t>0then κe t=0if wtvt In fact, all the arguments from the previous steps are simplified because the only individual rationality constraint for the agent is the ex ante one. Third, define the following sets of projects: Se 1=w:κe t=1for any utwith ut=wDe 1=v:κe t=1for any utwith ut=v Se 0=w:κe t=0for any utwith ut=wDe 0=v:κe t=0for any utwith ut=v Se i=S\Se 1∪Se 0De i=D\De 1∪De 0 Note that from our preceding results we have that given any u∈De 0∪De i∪Se 1∪Se iuvimplies v∈De 0and uwimplies w∈Se 1 given any u∈De i∪De 1∪Se i∪Se 0vuimplies v∈De 1and wuimplies w∈Se 0 (8) Therefore, De iand Se ieach include at most one project and at most one of these sets is nonempty so that even De i∪Se iincludes at most one project. Therefore, we can set Ue⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∈De i∪Se iif De i∪Se i= ∅ =max  De 0∪Se 1if De i∪Se i=∅and De 0∪Se 1= ∅ =min  D∪Sif De i∪Se i=∅and De 0∪Se 1=∅ Proposition 1 then follows from (8). Notice that our results above do not pin down production probabilities at projects in De i∪Se i. However, a simple selection from the set of efficient contracts allows a complete characterization of κe. Specifically, given the linearity of payoffs in production probabilities, it is immediate that it is without of loss of generality for optimal payoffs to assume that κeprescribes equal production probability at all histories utwith ut∈De i∪Se i: we can restrict attention to contracts such that κe t=k∗ D∈[01]for any history utwith ut∈De iand κe t=k∗ S∈[01]for any history utwith ut∈Se i. Now define threshold project u∗=Ue∈D∪Salong with threshold production probability k∗∈[01]as k∗=⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 1−k∗ Dif u∗∈De i k∗ Sif u∗∈Se i 1if De i∪Se i=∅and De 0∪Se 1= ∅ 0if De i∪Se i=∅and De 0∪Se 1=∅ The reason for expressing k∗=1−k∗ Dwhen u∗∈De iwill become clear in the proof of Proposition 3 below, where we apply our results on efficient contracts to characterize optimal contracts when the project process is Markov. There we order demands v∈D 884 Forand and Zápal Theoretical Economics 15 (2020) by their expensiveness to the principal, so that whether the threshold u∗is a demand or a supply project, our formulation of the threshold production probability k∗always identifies the scale of the principal’s costs. Finally, by (8), it follows that, given any history ut,wehavethat κe t=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1if vtu∗ 1−k∗if vt=u∗ 0if u∗vt (9) and that κe t=⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1if u∗wt k∗if wt=u∗ 0if wtu∗ (10) Proof of Proposition 3. Our characterization of optimal contracts with Markov project processes in Proposition 3 shows how to define the cutoff supply project from Proposition 2 through a recursive rule involving fixed threshold {Wv}v∈Dassociated to all demand projects. Our proof of this result follows from the construction of an optimal contract. We proceed in a number of steps. Step 1. Fix project v∈Dand suppose that u1=v. We define the reduced problem max κ∈KUP1subject to UA1≥0. (11) Notice that problem (11) is a special case of the problem solved by efficient contracts. Therefore, as in the proof of Proposition 1, we can conclude that the solution κ∗to (11) can be characterized by threshold project u∗and production probability k∗, as described in (9)and(10). Step 2. We can rank the solutions to (11) for various v∈Dfor which u1=vin terms of how expensive they are to the principal. Specifically, fix vv ∈Dand consider the associated solutions κ∗and κ∗to the problem (11)withu1=vand u1=v, respectively. If either u∗u∗or u∗=u∗and k∗>k ∗, then we say that the contract κ∗is more expensive for the principal than contract κ∗. In words, when these conditions are met, then κ∗demands less of every project v∈Dand supplies more of every project w∈Sthan does κ∗. Formally, this is a different definition of expensiveness for demand projects as that in the text, and all references to expensiveness in the remainder of the proof refer to this definition. The notion of expensiveness in the text, which is based on the thresholds Wvdefined in this proof, is easily seen to be a consequence of the notion defined here. Fix any project uand some history utsuch that ut=u,andletUidenote the payoff to ifrom contract κ∗starting from utand let Uidenote the payoff to ifrom contract κ∗ starting from ut: these payoffs are history-independent because uis Markov and both contracts κ∗and κ∗are stationary. Furthermore, it follows that if κ∗is more expensive for the principal than κ∗, so then we have that UA≥UA. An implication is that, for any t, contract κ∗must still satisfy (IRAt)ifut=v, but that contract κ∗does not in general satisfy (IRAt)ifut=v.