Relational enforcement
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Achim, Peter; Knoepfle, Jan Article Relational enforcement Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Achim, Peter; Knoepfle, Jan (2024) : Relational enforcement, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 19, Iss. 2, pp. 823-863, https://doi.org/10.3982/TE5183 This Version is available at: https://hdl.handle.net/10419/320254 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. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. 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 19 (2024), 823–863 1555-7561/20240823 Relational enforcement Peter Achim Department of Economics, University of York Jan Knoepfle School of Economics and Finance, Queen Mary University of London A principal incentivizes an agent to maintain compliance and to truthfully announce any breaches of compliance. Compliance is imperfectly controlled by the agent’s private effort choices, is partially persistent, and is verifiable by the principal only through costly inspections. We show that in principal-optimal equilibria, the principal enforces maximum compliance using deterministic inspections. Periodic inspection cycles are suspended during periods of self-reported noncompliance, during which the agent is fined. We show how commitment to random inspections would benefit the principal, and discuss possible ways for the principal to overcome her commitment problem. Keywords. Relational contracts, dynamic enforcement, persistence, costly inspections. JEL classification. C73, D83. 1. Introduction In 2018 and 2019 two plane crashes killed 346 people and led to a worldwide grounding of the Boeing 737 MAX.1An investigation by the U.S. Congress concluded that the accidents were to a large extent due to “grossly insufficient oversight by the FAA” (Federal Aviation Administration).2Starting from the early 2000s, the FAA had increasingly trusted manufacturers to certify their own planes to save costs. By 2018, Boeing had self-certified nearly all of its work (Kitroeff, Gelles, and Nicas (2019)). Boeing rushed the development of the 737 MAX at the expense of safety. This case illustrates the risk in relying on self-reported quality assurances without sufficient oversight. Peter Achim: [email protected] Jan Knoepfle: [email protected] We are thankful to Francesc Dilmé, Daniel Hauser, Florian Hoffmann, Martin Pollrich, Sven Rady, and Alex Smolin for valuable comments, and thank participants at the Canadian Economic Theory Meeting in Vancouver and the Econometric Society Summer Meeting in St Louis in 2017, as well as seminar participants at Bonn and HECER. Achim is grateful to Chris Shannon and the Economics Department at UC Berkeley for their hospitality during a productive visit in the fall of 2016 and to the Hausdorff Center for Mathematics in Bonn for providing financial support. Knoepfle acknowledges financial support by the German Research Foundation through CRC TR224-B04 and from the Academy of Finland (Project 325218). 1As a result, Boeing suffered an operational loss of over $20 billion. The estimated impact on the U.S. economy as a whole was a 0.4 percentage points loss in gross domestic product (GDP) growth (di Giovanni et al. (2020)). 2See U.S. House of Representatives (2020). ©2024 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE5183
824 Achim and Knoepfle Theoretical Economics 19 (2024) In this paper, we study enforcement relationships in which the agent privately controls and observes the state of compliance and makes reports to a principal without commitment power. Compliance is partially persistent over time and can be observed by the principal only through costly inspections. The principal schedules inspections and imposes fines to incentivize the agent to exert effort and to self-report instances of noncompliance. We show that the principal can induce the agent to exert full effort and report truthfully at all times through relational incentives. The principal carries out inspections despite knowing the result beforehand. Our analysis highlights the importance of the persistent effect of effort. Further, the principal cannot gain from randomized inspections when she lacks commitment, but random inspections would be optimal with commitment. Public-sector applications of our model include banking supervision to ensure that banks maintain functioning internal risk assessments3and environmental protection where the corresponding government agency ensures the enforcement of regulation by firms.4Similarly, private-sector organizations must ensure internally that employees follow regulations.5 We consider principal-optimal equilibria in which the agent truthfully discloses all instances of noncompliance and exerts maximum effort throughout. The principaloptimal equilibrium we derive in our main result (Theorem 1) entails two phases: a monitoring phase and a penalty phase. The agent is in the monitoring phase when he reports compliance. During the monitoring phase, the agent is not fined, but is subject to periodic inspections that would result in the maximal possible fine in the off-path event that the inspection revealed misreporting. The agent is in the penalty phase when he reports noncompliance. He pays a constant flow fine, but is never inspected. He also pays a lump sum fine each time the state transitions from compliance to noncompliance. Crucially, this transition fine features penalty reductions for early disclosures of noncompliance, an aspect that is consistent with voluntary disclosure schemes commonly used in practice. The penalty reduction prevents the agent from delaying a report of an incidence of noncompliance in the hope that he can regain compliance before the next inspection.6 Notably, inspection times in this equilibrium are entirely predictable for the agent, which implies that the principal cannot gain from randomized inspections. Intuitively, 3See Section 5for a brief discussion of banking supervision practices in Germany. 4For the United States, Blundell, Gowrisankaran, and Langer (2020) measure the benefits of dynamic procedures used by the EPA. 5For instance, the European Commission (2019) supports exporting firms in elaborating internal compliance programs(ICP) to “mitigate risks associated with dual-use trade controls and to ensure compliance” internally. Dual-use goods have civil and military applications and fall under special regulation to promote international security, e.g., by “countering risks associated with the proliferation of Weapons of Mass Destruction” (European Commission (2019, p. 17)). 6Blundell, Gowrisankaran, and Langer (2020) point out that when determining the gravity of fines, the EPA takes into account whether a violation was self-reported or not. See also Kapon (2022), who studies optimal design of fine reductions (amnesties) granted for self-reports of illegal activity when detections arrive at an exogenous rate. Focusing on deterministic fine reduction paths, Kapon (2022)alsofindsa cyclical structure of the optimal mechanism. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 825 the principal”s motive to inspect is derived from her desire to maintain a reputation for vigilance.7Predictable inspections provide the strongest incentive for the principal to inspect. As long as the principal inspects as prescribed by her equilibrium strategy, the agent continues to expect to be monitored and, thus, has an incentive to exert effort and report truthfully. However, when the principal delays inspections in a way that is detectable by the agent, then the agent will infer that the principal has become nonvigilant. This in turn induces the agent to shirk, which ultimately leads to a breakdown of the relationship that is costly for the principal. If the principal uses a random strategy and mixtures are unobservable for the agent, deviations by the principal are harder to detect for the agent. This destroys any potential benefit for the principal in equilibrium. We exploit the optimality of predictable inspection schedules for the construction of the principal-optimal equilibrium in Theorem 1: her equilibrium payoffs coincide with the value of an auxiliary mechanism-design problem in which the principal is restricted to nonrandom inspections. We then transform this optimization into a dynamic programming problem that uses the agent’s promised utility as a state variable. Comparative statics reveal the importance of persistence for relational enforcement. In equilibrium, the persistent effect of effort on compliance allows the principal to deter the agent from deviating through isolated inspections. As the state’s persistence vanishes, the inspection costs necessary to enforce compliance grow arbitrarily large. We then contrast the relational enforcement equilibrium with stochastic inspection mechanisms. The ability to commit to random inspections decreases the principal’s inspection costs relative to the deterministic inspections that are required in the noncommitment case. Deterministic inspections are more costly because of delay and noise in the compliance process, and due to the transition penalties that are needed to generate incentives for voluntary disclosure. Comparative statics highlight the contrast between relational enforcement and the commitment case with random inspections. As the persistence of the state of compliance vanishes, the random inspection costs decrease monotonically. We also discuss ways to overcome the principal’s commitment problem, including institutional separation of planning and execution of oversight and inspection sampling combined with publicly accessible and verifiable records. The rest of the paper is organized as follows. After discussing related literature, the model setup is presented in Section 2.Section3characterizes the agent’s incentive constraints, shows that the principal-optimal equilibrium can be determined by solving an auxiliary mechanism-design problem, and outlines how to solve the auxiliary problem. We present the principal-optimal equilibrium in Section 4, followed by comparative statics. Section 5discusses random inspections. All proofs are contained in the Appendix. 7Here, “maintaining a reputation” means following equilibrium actions because deviating leads to a less favorable continuation value (see Chapter 22 in Ljungqvist and Sargent (2018)). This “history-dependence” notion of reputation is distinct from the “adverse-selection” approach to reputation (Mailath and Samuelson (2006, p. 459)), in which incentives stem from the desire to convince the opponent that you are of a specific type. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
826 Achim and Knoepfle Theoretical Economics 19 (2024) Related literature Our paper is closely related to the literature on costly state verification (CSV). Early papers, including Townsend (1979), Gale and Hellwig (1985), Mookherjee and Png (1989), and Border and Sobel (1987), focus on one-shot interactions. One of the main findings in this literature is the optimality of cutoff verification protocols, an insight that has been influential in explaining the use of debt contracts and the role of financial intermediaries. A number of papers consider dynamic extensions. In many of these, the principal’s observation reveals the agent”s current private information with no intertemporal link to past actions or states.8By contrast, the state in our model is partially persistent, so inspections reveal information about past behavior. Inspections of a persistent state are analyzed in Ravikumar and Zhang (2012)and Kim (2015). These papers study pure adverse-selection problems with exogenous private information when the principal has commitment. In Ravikumar and Zhang (2012), the contracting friction is driven by risk-sharing concerns. They find that random inspections are optimal, and, after each inspection, there is a grace period without inspections. In Kim (2015), the contracting friction is driven by the agent’s limited liability. They find that random inspections are optimal for incentive provision when truthful disclosure is attainable, but periodic inspections are optimal to guide environmental protection activities when the fines are insufficient to attain truth-telling. Our setting features an adverse-selection and moral-hazard problem, the principal lacks commitment power, and the agent is risk-neutral so that the contracting friction stems from limited liability. We find deterministic inspections are optimal when the principals lacks commitment. Our result for the commitment case is in line with their findings that random inspections provide incentives more effectively. Most closely related is the paper by Varas, Marinovic, and Skrzypacz (2020), which studies a pure moral-hazard model with full commitment and without fines.9In their model, the agent is incentivized by the desire to maintain a good reputation and inspections make the agent’s type public. Additionally, inspections serve an informationacquisition purpose for the principal. The authors find random inspections are optimal for incentive provision, but deterministic inspections are optimal for information acquisition. In contrast, in our model, the agent discloses the state of compliance, so that inspections do not reduce the uncertainty about the state. Ball and Knoepfle (2023)study 8For dynamic moral-hazard problems in which monitoring reveals the current action, see Antinolfi and Carli (2015), Piskorski and Westerfield (2016), Dilmé and Garrett (2019), Chen, Sun, and Xiao (2020), Li and Yang (2020), Dai, Wang, and Yang (2022), Wong (2022). For dynamic adverse-selection problems in which verification reveals the agent’s current information that is independent and identically distributed (i.i.d.) across periods, see Chang (1990), Webb (1992), Monnet and Quintin (2005), Wang (2005), Popov (2016), Malenko (2019). 9In both papers, state transitions are based on the reputation for quality model (Board and Meyer-terVehn (2013)). In Board and Meyer-ter-Vehn (2013) quality becomes publicly observable at random times. In the present paper and in Varas, Marinovic, and Skrzypacz (2020), the principal chooses the times at which the state becomes publicly observable at a cost. In Halac and Prat (2016) and Dilmé and Garrett (2019), the principal invests in building her persistent monitoring capabilities, and monitoring reveals information about current actions of the agents. In contrast to the setup in the present paper, the principal’s actions are private and she cannot perfectly control the time at which she signals vigilance. In Halac and Prat (2016) this leads to a breakdown of the relationship with positive probability after the agent’s effort remains unrecognized for too long. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 827 optimal inspections with commitment and show that random inspections are optimal for incentives when the agent must avoid a breakdown and deterministic inspections are optimal when the agent must achieve a breakthrough. The driver of nonrandom inspections in our paper is the principal’s lack of commitment. The persistent effect of effort is important for relational incentives. This is also highlighted for a collaboration problem without commitment in Ramos and Sadzik (2023). Similar to our comparative statics in Section 4.2, the authors show that relational incentives vanish without persistence. Without persistence, commitment is crucial for enforcement with costly inspections: when the agent is expected to comply, the principal has no incentive to pay the inspection cost to reveal information she already knows. Indeed, Reinganum and Wilde (1985) confirm for a nonrepeated setting that compliance is not achievable without full commitment. With repeated interactions, continuation play can provide punishment for insufficient inspection. Ben-Porath and Kahneman (2003) prove a folk theorem, showing that full compliance can be obtained without commitment in the undiscounted limit. In our game, full compliance is attainable even with discounting. This difference stems from the persistence of the state and the observability of inspections by the agent in our model. 2. Model Players, actions, and state dynamics There are an agent and a principal. Time t∈[0, ∞) is continuous. The agent, at each instant t, privately chooses effort ηt∈[0, 1]to comply with exogenously given regulation as best he can. The state of compliance at time t is θt∈{0, 1}, where we refer to state 0 as noncompliant and to state 1 as compliant. Effort affects the transitions of the process {θt}t≥0: there are parameters λ>0andα∈ (0, 1)such that the state changes from 0 to 1 at Poisson rate ηtλα and from 1 to 0 at rate λ(1−ηtα). We may interpret λand αas follows. There is a Poisson process of shocks arriving at rate λ. Whenever there is a shock at time t, the resulting state is θt=1 with probability ηtαand it is θt=0 with probability 1 −ηtα; between shocks the state remains unchanged. Thus, λmeasures the variability of compliance and αmeasures the responsiveness to the agent’s effort conditional on a shock; α<1 implies that the agent cannot always maintain compliance despite his best efforts. The agent observes θtat all times and sends report ˆ θt∈{0, 1}to the principal. The agent can exit the relationship unilaterally at any time. The principal chooses inspections and fines to incentivize the agent. We denote by NI tthe cumulative number of inspections and by Ftthe cumulative fines up to and including time t.Thatis,dNI t≡Nt−limstNI s∈{0, 1}is equal to 1 if and only if there is an inspection at time tand dFt≥0 is the fine paid by the agent at time t. Information and timing The agent observes the history of all paths ht=ηs,θs,ˆ θs,NI s,Fss∈[0,t]. The principal never observes the agent’s effort and is able to observe the state θtonly by performing an inspection at time t. To allow for randomized inspections, we equip 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
828 Achim and Knoepfle Theoretical Economics 19 (2024) the principal with a private random signal π, defined on a sufficiently rich sample space . The principal observes histories of the form hP t={π,ˆ θs,NI s,Fs,θs:NI s= 1}s∈[0,t]. Heuristically, we can describe the timing of events within each instant [t,t+dt) as follows.10 First, the agent chooses effort ηt. Subsequently, nature determines whether a shock arrives and, conditional on the arrival of a shock and the effort, draws a new state θt. The agent then observes the realized θtand sends a report ˆ θt∈{0, 1}to the principal. The principal chooses whether to inspect, dNI t∈{0, 1}, and sets a fine dFtincurred immediately by the agent, where the fine can be contingent on the true state θtif and only if the principal chose to inspect. Payoffs and equilibrium The principal and the agent are risk-neutral and discount future payoffs at a common rate r>0. The principal is tasked with ensuring that the agent complies with the regulation. She incurs a lump-sum cost κ>0 from each inspection. For a realized history h={ηt,θt,ˆ θt,NI t,Ft}t∈[0,∞), the discounted net present cost of the principal at time tis kt=∞ t e−r(s−t)κdNI s.(1) The principal does not benefit directly from compliance or from fining the agent. Fines are interpreted as remedial actions that negatively impact the agent.11 To ensure that the principal is willing to bear the inspection costs, assume that when the relationship breaks down, i.e., the agent exits or ceases to exert effort, the principal suffers cost ¯ K.We assume throughout that the bound ¯ Kis large enough such that it exceeds the expected inspection costs necessary to incentivize the agent.12 The agent incurs effort cost of cηtdtwith c>0 and disutility dFtfrom fines. His discounted net present payoff at time tis given by ut=∞ t e−r(s−t)(−cηsds−dFs).(2) The agent is protected by limited liability. If he chooses to exit, the relationship ends permanently, which results in a continuation payoff of −B. This implies a constraint on the severity of fines the principal can impose. We assume that the exogenously given 10We outline the sequentiality at a given instant to give an intuition about the order of moves. Formally, the order is captured by continuity properties of the respective action and state paths. It is well known that in continuous-time games with observable actions, strategies may not produce well defined action paths. To focus the exposition in the main text on the main economic forces, we defer a more formal treatment to Supplemental Appendix A (available at http://econtheory.org/supp/5183/supplement.pdf), where we adopt an approach by Kamada and Rao (2023) to impose restrictions on strategies that guarantee well defined action paths. 11Our results do not rely on this assumption. When the principal benefits from fines, her preferred equilibrium differs from the one we present only in an initial fine paid by the agent (see Section 6). 12This serves as a concise way to deliver incentives for inspection for the principal when analyzing the equilibrium problem without commitment. Alternatively, we could explicitly incorporate an (unobserved) flow reward θtRor ηtRin our model so that, for R>0 large enough, the principal’s expected payoff from inducing effort by the agent exceeds the necessary inspection costs. Our results would be unaffected. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 829 bound Bis large enough: B> ¯ B≡c(r+λ)/(λαr). Otherwise, the maximal punishment is insufficient to incentivize effort even if θtwere public at all times. Given a strategy profile, the principal and the agent form expectations about history hbased on their past observations. For strategies that induce measurable action processes on path, we denote the expected cost of the principal and payoff of the agent at time tby Kt=EP t−[kt]and Ut=EA t−[ut]. The expectation is with respect to the process {θs}s∈[0,∞)and the randomization device π, and it is conditional on the information that is available to the principal and the agent, respectively.13 In continuous-time games with observable actions and stochastic environments, players’ behavior may be nonmeasurable. We do not impose restrictions on strategies that rule out nonmeasurable behavior. Instead, our equilibrium definition below requires that strategies lead to measurable actions on path. Histories away from the equilibrium path may lead to nonmeasurability. Payoffs at such histories can be assigned freely within the feasible bounds. In our game, the lower bounds on payoffs can be reached by either player unilaterally through exit or by imposing the maximal fine. Therefore, potential nonmeasurabilities off path and the assigned payoffs cannot be used as a threat to enlarge the equilibrium set (see also the discussion of this approach in Kamada and Rao (2023)). We define a strategy profile, together with processes {Kt,Ut}t≥0,tobeaperfect Bayesian equilibrium if the following statements hold. •The strategies of the principal and the agent are sequentially rational. •Along the equilibrium path, Ktand Utare equal to the conditional expectations given above. Away from the equilibrium path, Ktand Utare equal to the conditional expectations whenever these are well defined. •At all histories and all times, Kt∈[0, ¯ K]and Ut∈[−B,0 ]. We say that the agent’s strategy is truthful if ˆ θt=θtat all histories along the equilibrium path. Further, we call the agent’s strategy maximally compliant if ηt=1atall histories along the equilibrium path. Note that with full effort by the agent, the probability of compliance at any given time is maximized. We refer to an equilibrium as truthful or maximally compliant if the agent’s strategy in this equilibrium has the respective property. Henceforth, we restrict attention to such equilibria (see the discussion in Section 6). 13For the principal, the expectation is with respect to the natural filtration generated by the process {NI s,Fs,θs:dNs=1}s∈[0,t)∪{ˆ θs}s∈[0,t]when taking her inspection decision, and with respect to the natural filtration generated by {ˆ θs,NI s,Fs−,θs:dNs=1}s∈[0,t]for the fine. For the agent, the expectation is with respect to the natural filtration generated by the process {ηs,θs,ˆ θs,NI s,Fs}s∈[0,t)for his effort choice, and with respect to the natural filtration generated by {ˆ θs,NI s,Fs}s∈[0,t)∪{ηs,θs}s∈[0,t]for his report. As mentioned above, see Supplemental Appendix A for a formal treatment of permissible strategies. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
830 Achim and Knoepfle Theoretical Economics 19 (2024) We say that inspections are predictable for the agent if he knows for certain whether or not his current report will lead to an inspection at any history.14 Henceforth, we refer to inspections as random whenever they are nonpredictable for the agent. 3. Agent’s and principal’sproblem 3.1 Agent: Incentive compatibility Fix an arbitrary principal strategy of fines and inspections, and let Utbe the agent’s associated expected discounted continuation payoff at time tunder the assumption that he exerts full effort and reports truthfully. We characterize recursively the conditions under which truthful reporting and maximal compliance are a best response for the agent in terms of the evolution of his promised utility at all times. Due to the persistence in the agent’s private information, the recursive characterization of incentive compatibility requires tracking two state variables:15 the agent’s expected continuation utility given that θt=0andgiventhatθt=1. Formally, fix a principal strategy and define for any strict history at time t, U0 t=EA t−[Ut|θt=0]and U1 t=EA t−[Ut|θt=1].(3) These are the agent’s expected continuation utilities when history ht−is followed by the realization of θt=0orθt=1. Here, EA t−represents the expectation conditional on all available information before time t. Following Zhang (2009), we call U1 tthe persistent payoff if θt−=1andthetransitional payoff in case θt−=0, and vice versa for U0 t. Our first result provides a complete characterization of the agent’s incentivecompatibility constraints in terms of the evolutions of U0 tand U1 t. The construction is based on the martingale representation for marked point processes (Last and Brandt (1995)), which is presented in detail in Appendix A. We exploit the fact that the agent’s time-texpectation of his total discounted lifetime utility is a martingale. For the inspection counting process NI,thecompensator is a predictable process νI={νI t}t≥0such that the compensated process NI t−νI tis a martingale. The compensator exists under very general conditions and can be interpreted as the predictable drift of the underlying (nonpredictable) stochastic process. We can think of the compensator as a generalization of the cumulative hazard function, and, consequently, think of dνI/dtas the hazard rate of inspections (whenever it exists). Furthermore, let the predictable process I={I t}t≥0measure the jump in the persistent payoff if an inspection is performed at time t.16 Lemma 1. A principal’s strategy induces maximal compliance and truthful reporting if and only if it generates the processes {U1 t,U0 t}t≥0of promised utilities satisfying for i=θt− and j=1−θt−, and at all twith dNI t=0and θt−=θt, 14Formally, predictability means that the process NIis measurable with respect to the information available to the agent (see Davis (1993, p. 67, for a definition in the context of jump processes)). 15This is based on Fernandes and Phelan (2000), who introduce a recursive approach with serially correlated states in discrete time. See Zhang (2009) for a treatment in continuous time. 16That is, given that an inspection occurs at time t(and θt−=1), then I t=U1 t−U1 t−, supposing that the inspection confirms that the agent reported truthfully, which is the case along the equilibrium path. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 837 Figure 3. The evolution of an example path realization starting in the compliant state. Solid curves depict the agent’s persistent payoff in the current state; dashed curves depict the transitional payoff, to which the agent’s payoff jumps when the state changes. agent of the principal’s continued oversight. Demonstrated vigilance shapes the agent’s perception that he will eventually be detected if he were to deviate. While random inspections may be supported in a relational contract, such arrangements require strong deterrents for the principal to ensure her adherence to the equilibrium strategy. This requirement ultimately renders randomization nonbeneficial for the principal (Lemma 2). The equilibrium in Theorem 1naturally features penalty reductions for early disclosures of noncompliance. This is consistent with voluntary disclosure schemes that are commonly used in practice. The U.S. environmental protection agency (EPA) employs a self-reporting program called “Incentives for Self-Policing” that requires firms voluntarily disclose any violations that are detected internally. Similar to the way the agent is incentivized in the above equilibrium, firms that disclose violations early are rewarded by a reduction in penalties and a suspension of inspections until compliance is restored. Theorem 1provides insights into how enforcement agencies can benefit from offering regulated firms incentives for voluntary disclosure. Voluntary disclosure allows the principal to limit inspection to periods of compliance and, thus, lowers the overall inspection costs. The EPA points out that the advantage of these incentives lies in “making formal EPA investigations and enforcement actions unnecessary.”22 In the theoretical literature, the observation that voluntary disclosure reduces monitoring costs dates back to Kaplow and Shavell (1994), who introduced self-reporting into the enforcement model by Becker (1968). Without the agent’s disclosure, the principal in our model would not be able to consistently avoid inspections during phases of noncompliance.23 22https://www.epa.gov/compliance/how-we-monitor-compliance. 23See Varas, Marinovic, and Skrzypacz (2020) for a model without reports. Our results confirm the conjecture in that paper that voluntary disclosure can avoid unnecessary inspections (see Varas, Marinovic, and Skrzypacz (2020, p. 2921)). 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
838 Achim and Knoepfle Theoretical Economics 19 (2024) 4.2 Comparative statics How do variations in the parameters affect the length of inspection cycles and the inspection costs? As one would expect, if the penalty bound Bincreases or the effort cost cdecreases, the agency problem becomes less severe: the inspection cycle T∗becomes larger and the expected costs decrease.24 The effect of a change in the arrival rate of shocks λis more intricate. An increase in λdecreases the state’s persistence and has a non-monotone effect on the length of inspection cycles and the overall costs. The following result makes these statements precise. To ensure that the equilibrium in Theorem 1always exists, we require that λ>λ≡cr/(Brα −c)>0, fixing all other parameters.25 Lemma 3. Asthearrivalrateofshocksλincreases, the following statements hold. •The inspection cycle length increases for low λand decreases for high λ,with limλ↓λT∗(λ)=limλ↑∞ T∗(λ)=0. •The discounted inspection costs decrease for low λand increase for high λ,with limλ↓λK∗ 0(λ)=limλ↑∞ K∗ 0(λ)=∞. For the cycle length T∗, there are two opposing effects if λincreases. First, at any given instance, the state is more likely to change in response to current effort. The marginal benefit from effort is higher and it is easier to incentivize the agent, allowing for an increase in T∗. Second, the state becomes more fragile: the link between current effort and future compliance weakens. Delayed inspections have less incentive power, forcing the principal to shorten inspection cycles. Lemma 3shows that the first effect dominates for low λand the second effect dominates for high λ. For any fixed T∗, the total inspection costs decrease in λas any cycle of fixed length is more likely to be interrupted by a breach of compliance, so the inspection is less likely to be carried out. Thus, for low λ, this effect and the increase in T∗work in the same direction. Inspection costs decrease in λ.Forhighλ, the two effects work in opposite directions. Lemma 3shows that the decrease in T∗is fast enough to outdo the second effect; the inspection costs increase in λ. Both inspection intensity and inspection costs grow arbitrarily large at both extremes. As λgoes to infinity and state persistence vanishes, inspections must be immediate to deter deviations. This highlights a key disadvantage of nonrandom inspections and the absence of commitment. Intuitively, a shirking agent faces an “effective” discount rate of r+λwhen considering the impact of the next inspection. This is because the state today determines the state at the next inspection only with probability e−λT .To ensure inspection effectiveness, T∗(λ)must approach zero fast enough so as to keep limλ→∞ λe−(r+λ)T∗(λ)strictly positive. For the principle, in contrast, the effective discount rate is r+λ(1−α), which is smaller than that for the agent. The limit of the 24A formal proof for the changes in Band c, as well as comparative statics with respect to α,canbefound in a working paper version, which is available upon request. 25Observe that the lower bound on Brequired for feasibility of effort, ¯ B=c(r+λ) λαr , grows arbitrarily large as λgoes to 0. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 839 principal’s cost is proportional to limλ→∞ λe−(r+(1−α)λ)T∗(λ). It is then easy to see that this cost must be infinite for α<1whenlimλ→∞ λe−(r+λ)T∗(λ)>0. The high compliance cost for large λarises from the agent’s opportunity to regain compliance with high probability unless the next inspection is imminent. Imminent inspections (T∗near 0) inflate costs. Random inspections may be valuable, allowing the principal to threaten immediate inspections without performing them constantly. We now confirm that random inspection schedules outperform predictable ones when feasible. With randomization, the principal strictly prefers higher arrival rates λ. 5. Commitment Our results show that without commitment, the principal cannot benefit from randomization. In this section, we confirm that if the principal could commit to follow through with random inspection schedules, this would decrease inspection costs. One way to enhance commitment to a profitable random procedure in arm’s-length enforcement is to separate planning and execution of inspections, as seen in German banking supervision. The European Central Bank or the supervisory agency at the Finance Ministry (BaFin) schedules audits, while the German Bundesbank executes them (BaFin (2016)). The inspection cost is not incurred by the party making the inspection decision, eliminating the temptation to delay or skip inspections. This separation differs significantly from two seemingly similar alternatives: outsourcing all oversight or compensating the principal for inspection costs. Outsourcing only shifts the problem one layer further; compensation requires precise knowledge of the cost to avoid inefficient inspections.26 Alternatively, the lack of detectability, which hinders profitable randomization, can be overcome if the principal is responsible for overseeing a large pool of independent agents and there are public records. The principal can then regularly inspect a fixed proportion of agents and make the results publicly available to create a verifiable signal of continued vigilance. For example, the EPA’s database “Enforcement and Compliance History Online” collects over 44,000 inspected facilities within the 12 months up to April 2021;27 the Public Company Accounting Oversight Board (PCAOB) publicizes approximately 100–300 inspection reports per year.28 To confirm the benefit of randomization, consider the following mechanism, which is optimal in the class of stationary random mechanisms.29 •Inspections are performed only during compliance with constant Poisson arrival rate m∗ R=r ¯ B B−¯ B. 26If the compensation falls short of the actual cost and effort required for an inspection, the incentive to skip it persists. If the compensation exceeds the cost, this creates an incentive to inspect inefficiently often. 27https://echo.epa.gov. 28https://pcaobus.org/oversight/inspections/firm-inspection-reports. 29For a proof of this claim, see Appendix C. Note that we do not claim that the mechanism presented here is the optimal commitment mechanism. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
840 Achim and Knoepfle Theoretical Economics 19 (2024) •Fines are levied only during noncompliance with a constant flow fine f∗ R=r¯ B. •If an inspection reveals noncompliance, then the agent pays the maximal fine. Similar to the equilibrium with predictable inspections, there are two phases. Inspections but no fines during compliance, and fines but no inspections during noncompliance. The differences are that inspections arrive at random and there is no lump-sum fine at transitions to noncompliance. Inserting into Lemma 1the values dFt=0anddνI t=m∗ Rdtin case i=1, and the values dFt=f∗ Rdtand dνI t=0incasei=0, it is straightforward to verify that the payoffs of the agent are constant at U1 R=− c rα,U0 R=− c rα −c λα.(8) Here, U1 Ris the persistent payoff and U0 Ris the transitional payoff when the agent reports compliance, and vice versa when the agent reports noncompliance. It is straightforward that all constraints are satisfied at all times, with (H) binding in i=1and(O) binding in both states. The next result shows that the principal’s inspection costs with predictable inspections are generally higher than with random inspections. In contrast to the predictable inspection schedule, a high arrival rate λbenefits the principal in this random mechanism. Theorem 2. The inspection costs in the stationary random mechanism defined above are strictly lower than in the principal-optimal equilibrium in Theorem 1.Furthermore,the costs in this random mechanism are decreasing in λfor all λ,with lim λ↓λKR(λ)=∞ and lim λ↑∞ KR(λ)=cα Brα −cκ. Random inspections dominate predictable inspection procedures for two reasons. First, by the argument at the end of Section 4.2, noise and delay make periodic inspections less effective. The threat of an imminent inspection at all times is more effective in our setting, even when holding the payoff impact of each inspection fixed.30 That is, if the agent’s initial utility is fixed at some level u, the costs from the random mechanism implementing this utility level are strictly below the costs in the predictable equilibrium implementing the same level. Second, in our setting with fines and self-reported compliance, random inspections allow for a greater payoff impact of an inspection on the deviating agent: with predictable inspections, self-reporting requires a transition fine whenever a breach of compliance occurs. The risk of the transition fine reduces the agent’s overall equilibrium payoff. Since the lower bound on payoffs is fixed at B,thisreduction decreases the maximum loss that the principal can impose when an inspection 30See also Varas, Marinovic, and Skrzypacz (2020), who show that a constant inspection rate provides incentives most effectively under commitment when the payoff consequence of each inspection is fixed. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 841 reveals a misreport. Thus, predictable inspections have a smaller payoff impact, making them overall less powerful. Our finding that random inspections provide incentives more effectively is consistent with Varas, Marinovic, and Skrzypacz (2020), who study a setting without voluntary disclosure. They show that (partially) predictable inspections can be optimal when the principal derives direct value from information, i.e., when her flow payoff is convex in the posterior belief. In our model, the principal induces honest self-disclosure. Along the equilibrium path, the principal always knows the true state. Therefore, introducing convexity in the principal’s value as a function of her belief would not affect our results; her belief is always 0 or 1. Varas, Marinovic, and Skrzypacz (2020) identify a trade-off according to which incentive provision recommends randomization while information acquisition makes predictable inspections more profitable. Our analysis suggests that self-reporting can resolve this trade-off in favor of randomization when the current state is known to the agent and monetary incentives are feasible. 6. Conclusion We study enforcement through inspections and fines. In relational enforcement, maximum compliance and truthful disclosure are attained through nonrandom inspections. A fully committed principal would benefit from random inspections. The persistent effect of the agent’s effort on compliance makes it possible to create incentives through isolated inspections. An intermediate level of persistence is optimal in the case of relational enforcement. If the principal can commit to random inspections, inspection costs are increasing in the level of persistence as compliance becomes less responsive to effort. This highlights the importance of persistence in relational enforcement. Throughout the analysis, we assume that the principal does not benefit from the fines imposed on the agent. This assumption is innocuous. Given that the agent exerts full effort at all times, when his initial promised utility is u, the expected discounted sum of fines paid by the agent is equal to −u−c/r. If the principal were to benefit from fines at rate β∈(0, 1], her objective would be to maximize −K(u)+β(−u−c/r)instead of maximizing −K(u)in the baseline model. Denoting the maximizer of −K(u)by u∗,it is easy to see that the optimal equilibrium consists of an initial fine B+u∗paid to the principal, followed by the equilibrium of Theorem 1.31 A possible variation to our model is to allow the principal to pay subsidies to the agent when successfully passing inspections. If the principal could reward the agent for passed inspections, the upper bound on the agent’s continuation utility would increase. The principal could then decrease the inspection frequency as the maximal punishment increases. With commitment to random inspections, the principal could essentially avoid all inspection costs if rewards were unbounded. She could offer an arbitrarily large reward after inspecting with vanishing probability. 31The agent’s initial utility (before paying the fine) is at his outside option −Band then jumps to u∗.The principal’s payoff −K(u∗)+β(B−c/r)is clearly an upper bound for −K(u)+β(−u−c/r)among u≥−B. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
842 Achim and Knoepfle Theoretical Economics 19 (2024) The assumptions that the principal implements full effort is natural in many situations, for example, when the principal, tasked with monitoring compliance, is not the same institution as the one designing the regulation. The assumption is also important for tractability. To let the principal choose effort, the model would need to account explicitly for the principal’s benefit from compliance.32 More importantly, the optimization problem would become substantially more complex. Maximizing over the effort level would add a continual control at all times.33 Similarly, we focus on equilibria with truthful self-reports. This focus is natural in many applications in which it is essential for regulators to accurately identify compliance violations. In the auxiliary mechanism design problem with principal commitment, having only one agent ensures that the revelation principle applies. With commitment, the principal can replicate the outcome of any combination of mechanism and reporting strategy with the corresponding direct mechanism and a truthful reporting strategy. However, in the equilibrium problem without commitment, we do not rule out potential benefits from nontruthful behavior. Since we find the optimal predictable equilibrium via the auxiliary mechanism design problem, the only remaining concern is whether the principal could exploit nontruthful reporting to benefit from random inspections in equilibrium. We suspect this is not the case, but verifying the conjecture is beyond this article’s scope. Appendix A: Proofs for Section 3 Proof of Lemma 1. The proof of Lemma 1consists of two intermediate results. Lemma Aprovides a martingale representation for the agent’s lifetime expected utility, and Lemma Bprovides necessary and sufficient conditions for the path of expected payoffs such that full effort and truthful reporting are a best response for the agent. Define Wtto be the agent’s lifetime expected utility, with expectations taken with respect to the information that is available at time t: Wt=t 0 e−rs(−dFs−cηsds)+e−rtUt. By construction, the process {Wt}t≥0is a martingale (Davis (1993, p. 20)). There are three types of events: changes in the state, changes in reports, and inspections. Inspections are governed by the process NIgiven by the principal’s strategy. For consistency, we introduce the counting processes Nθ={Nθ t}t≥0and Nˆ θ={Nˆ θ t}t≥0that count the number of changes in the state of compliance and in the reports, respectively. For each process 32In some cases, when the principal’s benefit is large enough, implementing full effort is optimal and the analysis would be unaffected. 33Indeed, for predictable inspections, we sidestep the problem of continual controls by showing that it is without loss to levy no fines between inspections. This difficulty is also the reason why we do not claim that the random mechanism in Section 5is optimal among all inspection mechanisms. While it is optimal among any Markovian procedure, confirming that it is optimal generally would require a verification argument that deals with continual controls that can change both continuously or impulsively. We are not aware of existing dynamic programming results to verify that the recursive approach we employ remains valid in this class of optimization problems. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 843 Nawith a∈{θ,ˆ θ,I}, define the compensator to be a predictable process νa={νa t}t≥0 such that the compensated process Na t−νa tis a martingale. The compensator exists under very general conditions and can be interpreted as the predictable drift of the underlying (nonpredictable) stochastic process. For the hazard rate of transitions in compliance, we shall write qt(ηt):=dνθ t/dtor, more explicitly, qt(ηt)=θt−λ(1−αηt)+(1−θt−)λαηt.(9) The martingale representation theorem for marked point processes (Last and Brandt (1995)) implies the following result.34 Lemma A. There exist predictable processes θ,ˆ θ,andIsuch that the evolution of the agent’s expected utility is given by dUt=rUtdt+dFt+cηtdt+ a∈{θ,ˆ θ,I} a tdNa t−dνa t. (10) The processes θ,ˆ θ,andIhave an intuitive interpretation: They represent the jump in utility at time tthat results from a change in compliance, a change in reported compliance, or an inspection. The following lemma will complete the proof of Lemma 1. Lemma B. A mechanism that induces the payoffs {Ut}t≥0is incentive compatible with full effort and truthful reporting if and only if for all t≥0, (i) (r+qt(1))ˆ θ t−dνI t(I t−ˆ θ t)≥dˆ θ twhen θt= ˆ θt (ii) (1−2θt−)λα(θ t+ˆ θ t)≥cwhen θt=ˆ θt (iii) Ut∈[−B,0 ]. Proof. Define Wt=t 0 e−rs(−dFs−cηsds)+e−rt ˜ Ut to be the agent’s expected payoff from choosing effort {˜ηs}and report {ˆ θs}up to time t with maximum effort and truthful reporting thereafter. Here ˜ Utis the expected continuation payoff. We may have ˜ Ut= Utif the agent has reported nontruthfully, i.e., ˆ θt−= θt−. Consider first the case in which the agent’s report regarding his type at time tis truthful, so that ˜ Ut=Ut. Differentiating with respect to tyields dWt=e−rt(−dFt−cηtdt)−re−rtUtdt+e−rt dUt. 34A formal proof for our setting, which is a straightforward adaptation of the proof of Theorem 1.13.15 in Last and Brandt (1995) p. 25, is provided in Supplemental Appendix B. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
844 Achim and Knoepfle Theoretical Economics 19 (2024) Using Lemma Ato replace dUtyields dWt=e−rt(1−ηt)cdt+ a∈{θ,ˆ θ} a tdNa t−qt(1)dt+I tdNI t−dνI t. If the agent deviates for an additional instant (but still reports truthfully), then dNθ t=dNˆ θ t=1 with probability qt(˜ηt)dt 0 with probability 1 −qt(˜ηt)dt. Taking expectations therefore yields EA t[dWt]=e−rtEA(1−ηt)cdt+θ t+ˆ θ tqt(˜ηt)−qt(1)dt. It follows from condition (ii) that θ t+ˆ θ tq(˜ηt)−cηt≤θ t+ˆ θ tqt(1)−c. Thus EA t[dWt]≤0. We thus obtain the chain of inequalities EA 0[Wt]=EA 0t 0 dWs+W0=t 0 EA 0[dWs]+EA 0[W0] =t 0 EA 0EA s[dWs]+W0≤W0. (11) Now consider the case in which the agent’s most recent report at time tis false, that is, θt−= ˆ θt−, and he continues the nontruthful strategy for an additional moment at time t. If no change in the state occurs at the additional moment, then the agent must correct his report immediately thereafter. If a change occurs, then the previously false statement becomes truthful, and thus his report does not change. Therefore, we have d˜ Ut=˜ Ut−˜ Ut−dt =dNθ tUt−Ut−dt−ˆ θ t−dt+dNI tUt+I t−Ut−dt−ˆ θ t−dt +1−dNθ t−dNI tUt+ˆ θ t−Ut−dt−ˆ θ t−dt =dNθ tdUt+dˆ θ t−ˆ θ t+dNI tdUt+dˆ θ t−ˆ θ t+I t +1−dNθ t−dNI tdUt+dˆ θ t =dUt+dˆ θ t−dNθ tˆ θ t+dNI tI t−ˆ θ t. Again using Lemma Ato replace dUt,weobtain dWt=e−rt(−dFt−cηtdt)−re−rtUt+ˆ θ t +e−rtrUtdt+dFt+cdt+θ tdNθ t−q∗ t+dˆ θ t−dNθ tˆ θ t+dNI tI t−ˆ θ t. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 845 It follows from the honesty constraint (i) that, in expectation, dˆ θ t≤(r+qt(1))ˆ θ t− dνt(I t−ˆ θ t). Substituting it into dWtand simplifying, again using ˜ Ut=Ut+ˆ θ t,gives EA t[dWt]=e−rt(1−ηt)cdt+θ t−ˆ θ tq(˜ηt)−qt(1)θ t−ˆ θ t. Now θ t−ˆ θ t=(θ t+Ut)−(ˆ θ t+Ut)is the payoff difference from a change in the state without a change in report and a change in report without a change in the state. Since θt−= ˆ θt−by hypothesis, this is identical to ˜ θ t+˜ ˆ θ tafter the history in which the true state was identical to his report. Thus (ii) implies that ηt=1 maximizes the right-hand side, so that EA t[dWt]≤0.Bythesameargumentasin(11), we have EA 0[Wt]≤W0=U0, so that the agent cannot profit from deviating. Taking the limit, we find that lim t→∞ EA 0[Wt]≤U0, which implies that the agent cannot gain from deviating from maximum effort and truthful reporting. Conversely, if the incentive constraint (i) is violated, then the above inequalities are inverted, so that the agent has a strict incentive to be dishonest. Likewise, if (ii) is violated, the agent has a strict incentive to exert no effort, and a violation of (iii) leads to exit by the agent. To complete the proof of Lemma 1, we show that condition (Pk) follows from Lemma A,and(H),(O),and(P) are equivalent to conditions (i), (ii), and (iii) in Lemma B. Consider a mechanism and a strategy for the agent that jointly generate the payoff process {Ut}t≥0for the agent, and denote by {U1 t,U0 t}t≥0the associated pair of promised utilities defined in (3). Step 1. By the definition of U0 t,U1 t,wehave θ t+ˆ θ t=U1 t−U0 tif θt−=ˆ θt−=0 U0 t−U1 tif θt−=ˆ θt−=1, qt(1)=qt(1)=λα if θt−=0 (1−α)λif θt−=1. (12) Combining these two expressions, we can write more succinctly qt(1)θ t+ˆ θ t=λ(θt−−α)U1 t−U0 t. Lemma Athen implies that, conditioning on the event that dNθ t=dNˆ θ t=dNI t=0, we get exactly condition (Pk) in Lemma 1. Step 2. Next, suppose that the agent is not truthful after some history at time t.Let i=θtbe the true state and suppose the agent reports j=1−θt.ThenUi t=Ut+ˆ θ tand dUi t=Ut+dt+ˆ θ t+dt−Ut+ˆ θ t=rUtdt+dFt+cdt−qt(1)θ t+dˆ θ t ≤rUtdt+dFt+cdt−qt(1)θ t+r+qt(1)ˆ θ t−dνtI t−ˆ θ t =rUi t+λ(i−α)U1 t−U0 t−dνtI t−ˆ θ t+dFt+cdt. (13) 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
846 Achim and Knoepfle Theoretical Economics 19 (2024) The second line follows from Lemma A, and the inequality in the third line follows from condition (i) in Lemma B, where we take expectations conditional on the event that dNθ t=dNˆ θ t=0. The last equality in (13) holds since qt(1)θ t−ˆ θ t=qt(1)Ut+θ t−Ut+ˆ θ t=qt(1)Uj t−Ui t=λ(i−α)U1 t−U0 t. Punishment is without cost for the principal and, therefore, it is optimal to impose the most severe punishment after an inspection reveals a dishonest report. The severity of punishments is restricted by the limits of enforcement that require the agent’s continuation value not to fall below the lower bound −B<0. Therefore, we have I t−ˆ θ t=Ut+I t =−B −Ut+ˆ θ t =Ui t=− B+Ui t. Substituting this last equation into (13) yields dUi t=rUi t+λ(i−α)U1 t−U0 tdt+dνtB+Ui t+dFt+cdt, which is equal to condition (H) in Lemma 1. Conversely, if (i) does not hold at some t, then using the same steps as above, the inequality is reversed, so that (H) is violated. Step 3. Substituting (12) into the obedience constraint (ii), we obtain, for each θt−, θ t+ˆ θ t(1−2θt−)λα =λαU1 t−U0 t≥c. The last inequality is identical to (O) in Lemma 1. Conversely, if (ii) is violated at some t, then the inequality is reversed, so that (O) is violated. Proof of Lemma 2. The proof of Lemma 2consists of two results, stated and proven formally below. Lemma C. For any truthful and maximally compliant equilibrium, there exists a principal strategy such that truthful reporting and maximal compliance are a best response for the agent and (i) inspections are predictable for the agent whenever he reports compliance (ii) it generates weakly lower inspection costs for the principal. Proof. Take any truthful maximally compliant equilibrium. Let U0 tand U1 tbe the continuation payoffs of the agent in this equilibrium. The following steps present a modified inspection schedule satisfying the properties stated in Lemma C. As the original equilibrium is truthful and maximally compliant, U0 tand U1 tsatisfy the constraints from Lemma 1. First, we argue that for any inspection following a high report, it is without loss to assume that truthful reports are never punished more than misreports to the low state at the time of an inspection. That is, if the persistent payoff U1 tjumps downward on the path after an inspection, it will do so by less than the distance from the transitional utility to the lower bound −B. Formally, let ¯ U1 tbe the agent’s persistent payoff right after 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 853 Proof. Using Claim 5, there is no loss in generality in assuming that ˆ θ0=1. Consider the optimal initial utilities (u0,u1), where we assume to the contrary that u1−u0> c/(λα).Denotebyt∗the minimizer of U1 t.LetTbe the first inspection time conditional on no transition, and let the promised utilities at that time be ˆ u1and ˆ u0.Nowfix >0 sufficiently small, and consider an alternative mechanism identical to the original mechanism, except that the first time of inspection is (T+), and with initial utilities (˜ u1,˜ u0).IfT<t ∗,thenlet ˜ u0=˜ u1−u1+u0and let ˜ u1solve ˆ u1=er(T+)˜ u1+(1−α)eλ(T+)−1u1−u0−c1−er(T+)/r. Thus, by shifting the initial promised utilities up, the first inspection date is postponed, while maintaining incentive compatibility and keeping the terminal values constant. Consequently, the initial utilities could not have been optimal. If T≥t∗,thenlet ˜ u1=u1 and let ˜ u0solve ˆ u1=er(T+)˜ u1+(1−α)eλ(T+)−1˜ u1−˜ u0−c1−er(T+)/r. Thus, by shifting u0up while keeping u1constant, the first inspection date can be postponed while maintaining incentive compatibility and keeping the terminal values constant. In either case, a pair of initial utilities with u1−u0>c/ (λα)cannot be optimal. Without loss, we can now restrict attention to initial pairs of utility (u0,u1)such that u1−u0=c/(λα).Letu=u1denote the initial utility for the agent in the high state. The paths of promised utilities are then described by φ0(t,u)and φ1(t,u). Define K1 n(u)=min 0≤t≤T(u) u≥φ1(t,u)t 0 e−(r+λ(1−α))sλ(1−α)K0 nds+e−(r+λ(1−α))tK1 n−1u+κ(19) to be the maximum payoff for the principal at initial utility ufor the agent, where the principal maximizes over stopping times and the post inspection utility uresulting from the terminal promised utility φ1(t,u)and a potential fine at the time of an inspection. Let u∗ nbe a minimizer of K1 nand denote by t∗ nthe associated first inspection date. Claim 7. Let u∗ n−1be a minimizer of K1 n−1(u)and suppose K1 n−1 (u)>0for all u>u ∗ n−1. Then t∗ n=T0(u∗ n)and φ1(t∗ n,u∗ n)>u ∗ n−1. Proof. First we show that t∗ n=T0(u∗ n). Suppose, to the contrary, that t∗ n<T(u∗ n).If φ1(t∗ n,u∗ n)>u ∗ n−1, then because φ1is strictly increasing in its second argument, we can find a lower initial utility u<u ∗ nsuch that φ1(t∗ n,u)<φ 1(t∗ n,u∗ n). Since K1 n−1 (˜ u)>0for ˜ u>u ∗ n−1,wehaveK1 n(u)<K 1 n(u∗ n), contradicting optimality of u∗ n.Ifφ1(t∗ n,u∗ n)≤u∗ n−1, then the optimal initial utility in step n−1isu=−u∗ n−1. We can thus find t>t ∗ nsuch that φ1(t,u∗ n)<u ∗ n−1. Thus, the first inspection was delayed, while the continuation utility for the agent remains constant, contradicting optimality of u∗ n. Thus, we have t∗ n=T0(u∗ n). Now suppose φ1(T0(u∗ n),u∗ n)<u ∗ n−1. Then we can find a new initial utility u>u ∗ nsuch that φ1(T0(u),u)=u∗ n−1. Since T0(·)is increasing we have T0(u)>T0(u∗ n), contradicting the optimality of u∗ n. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
854 Achim and Knoepfle Theoretical Economics 19 (2024) In light of the result of Claim 7, there will be no loss in limiting our attention to the case t=T(u)and u=φ1(t,u). The principal’s expected costs for given utility uare, therefore, K1 n(u)=T(u) 0 e−(r+λ(1−α))sλ(1−α)K0 nds+e−(r+λ(1−α))T(u)K1 n−1φ1(t,u)+κ. Define β0=λα/(r+λα)and β1=λ(1−α)/(r+λ(1−α)). Solving the integrals and rearranging, the principal’s payoff can be expressed more succinctly as K1 n(u)=a(u)+b(u)K1 n−1φ1T(u),u, where a(u)=e−(r+λ(1−α))T(u) 1−β0β1+β0β1e−(r+λ(1−α))T(u)κ b(u)=e−(r+λ(1−α))T(u) 1−β0β1+β0β1e−(r+λ(1−α))T(u). Simple calculus reveals a(u)=−e(r+λ−αλ)T(u)(r+λ−αλ)2(r+αλ)r(r+λ) (1−α)αλe(r+λ−αλ)T(u)r(r+λ)2κT(u)and b(u)=−e(r+λ−αλ)T(u)r(r+λ)(r+λ−αλ)2(r+αλ) (1−α)αλ2+e(r+λ−αλ)r(r+λ)2T(u), so that sign a(u)=sign b(u)=sign T(u).From(17), it follows that a(u)<0ifu<u ∗ 1 >0ifu>u ∗ 1 ,b(u)<0ifu<u ∗ 1 >0ifu>u ∗ 1. Step 0 Consider the case n=0, so the principal cannot perform any inspections. The principal has no way to incentivize the agent so that her value function is equal to the lower bound Kθ 0=¯ K. Step 1 Suppose the principal can inspect at most once, so that n=1.Let tbe the first inspection if no transition occurs; let ube the initial utility for the agent. The expected costs for the principal when inspecting at time tare K1 1(u)=a(u)+b(u)¯ K. The marginal cost increase in utility uis K1 1 (u)=a(u)+b(u)¯ K. We have K1 1 (u)<0foru<u 1∗and K1 1 (u)>0foru>u ∗ 1,withu∗ 1>¯ u.Thusu1∗minimizes K1 1. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 855 Step 2 Suppose there are two inspections left to be performed. The principal’s payoff can be written as K1 2(u)=a(u)+b(u)K1 1φ1T(u),u. When u>u ∗ 1,then(K1 2)(u)>0 and, therefore, the optimizer does not exceed u∗ 1.Because K1 2(u)is maximal when ulies at the participation boundary and is continuous in between, there must be a minimizer u∗ 2. The marginal cost increase is K1 2 (u)=a(u)+b(u)K1φ1T(u),u+b(u)Duφ1T(u),u)K1 1 φ1T(u),u. Here Duφ1(T(u),u)is the total derivative of φ1(T(u),u)with respect to u, which can be shown to be Duφ1T(u),u=erT(u)1+T(u)ceλT (u)−11−α α r λ+ru+ceλT(u)1−α α+1>0. Thus, for u>u ∗ 1(>¯ u), K1 2 (u)=a(u)+b(u)K1φ1T(u),u+b(u)Duφ1T(u),u)K 1φ1T(u),u >a (u)+b(u)K1φ1T(u),u>a (u)+b(u)¯ K=K1 1 (u). In particular, this means u∗ 2<u ∗ 1. Step nK 1 n(u)=a(u)+b(u)K1 n(φ1(T(u),u)) has a minimum at u∗ n. The marginal cost at u>u ∗ n−1(>¯ u)is K1 n (u)=a(u)+b(u)Kn−1φ1T(u),u+b(u)Duφ1T(u),u)K n−1φ1T(u),u <a (u)+b(u)K1 n−1(u)+b(u)Duφ1T(u),u)K n−2φ1T(u),u <a (u)+b(u)K1 n−2(u)+b(u)Duφ1T(u),u)K n−2φ1T(u),u, where the first line follows from our induction hypothesis. Therefore, u≥u∗ n−1implies K1 n (u)<K 1 n−1 (u)<0. The induction shows that u∗ n<u ∗ n−1for all n≥0. It follows immediately from the definition of ¯ uthat u∗ n>¯ ufor all n.Hence,{u∗ n}is a decreasing and bounded sequence, so that by the monotone convergence theorem, the sequence converges to a limit ˆ u≥¯ u. Since {u∗ n}is convergent, it is a Cauchy sequence, so that by Claim 3, lim n→∞u∗ n−u∗ n−1=lim n→∞u∗ k−φ1Tu∗ n,u∗ n=0⇒ˆ u=¯ u. This establishes the mechanism characterized in Theorem 1as the optimum in the auxiliary problem. We now verify that it is also optimal in the original problem. B.2 Verification for the proof of Theorem 1 B.2.1 No fines between inspections We now show that the mechanism described in the previous section remains optimal when we remove assumption (A). To this end, we show 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
856 Achim and Knoepfle Theoretical Economics 19 (2024) that when performing the iteration over the number of available inspections n, the principal cannot gain from imposing fines between inspections when ninspections are left. Consider again Step nof the iteration in the previous section. By the same argument as before, we have u1−u0=c/(λα)and the first time of inspection is at the first time tat which U0 t=−B. The evolution of the paths of promised utilities are given by dU1 t=rU1 tdt−λ(1−α)U1 t−U0 tdt+cdt+dFt dU0 t=rU0 tdt−λαU1 t−U0 tdt+cdt+dFt−dμt, where we let dμt≥0 denote the slacking in the honesty constraint. The evolution of the difference in utilities is dU1 t−U0 t=(r+λ)u1−u0+dμt, which implies that the utility paths diverge at least exponentially, and are independent of any fines and increasing in threats. If the first inspection takes place at t, conditional on no transition before t, this means that U0 t=−Band U1 t=−B+e(r+λ)tc λα +t 0 e(r+λ)sdμs. The last term has to be zero because otherwise we could find a pair of initial promised utilities with ˆ u1<u 1and set dμs=0foralls∈(0, t), and a time t>tsuch that the promised utilities at time tunder the new initial conditions are as with the original pair at time t, thus increasing the principal’s payoff. Therefore, at the first time of inspection, U1 t=−B+e(r+λ)tc λα. Given that Ut 1is independent of any fines in step n, and there no fines in step n−1 onward, we must have U1 t=φ1(T(u∗ n),u∗ n). This means that the policy of the previous section with initial promised utilities (u∗ n,u∗ n−c/(λα)) remains optimal even when fines between inspections are available. B.2.2 General mechanisms in the relaxed problem Parts B.1 and B.2.1 demonstrate that the mechanism described in the theorem is an optimal Markovian mechanism under the relaxing of assumption (B). It remains to verify that no (non-Markovian) mechanism can do better. Let Kθt(U)denote the expected costs for the principal in our mechanism that delivers the agent with promised payoffs of U=(U0,U1). We show that the expected value in state θtfrom any incentive-compatible mechanism that delivers the initial promised payoff U0=(U0 0,U1 0)to the agent cannot exceed Kθt(U0). Since both the inspection cost and the set of feasible continuation utilities do not depend on their values prior to inspection, we can apply Proposition 54.18 and Theorem 54.28 in Davis (1993, pp. 235 & 242) to conclude that Kn, the value function with no more than ninspections, converges to value function Kof the problem without bound on the number 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 857 of inspections, and that Kis the unique bounded and continuous function that solves the quasi-variational inequality UKθ(u)−rKθ(u)≥0 WK θ(u)−Kθ(u)≥0 UKθ(u)−rKθ(u)WK θ(u)−Kθ(u)=0 on the state space {(θ,u0,u1)|θ∈{0, 1},(u0,u1)∈[−B,0 ]2,u1−u0≥c/(λα)}.Here, U denotes the extended generator of the piecewise deterministic Markov process that is defined by the relationship35 EP 0Kθt(Ut)=Kθ0(u)+EP 0t 0 UKθs(us)ds in case no inspection occurs before t,andWis the expected cost at an inspection time: WK θ=min u0,u1Kθ(u0,u1)+κ. Consider an arbitrary incentive-compatible mechanism with inspection process {dNI t}t and define the expected value at time tby Gt=t 0 e−rsκdNI s+e−rtKθt(Ut). For t=0, we have G0=Kθ0(U0).Fort>0, we can represent Gtby the differential formula (see Theorem 31.3 in Davis (1993, p. 83)) as Es[Gt]−Gs=t s e−r(z−s)UKθz(Uz)−rKθz(Uz)dz +Est s e−r(z−s)WK θz(Uz)−Kθz(Uz)dNI z. By the variation inequality above, both integrals are positive so that the process (Gt)t≥ 0 is a submartingale bounded by 0. This implies that E0[Gt]≥G0for any t≥0. In particular, taking the limit as tapproaches infinity, we get E0[∞ 0e−rs(κdNI s)] = E0[limt→∞ Gt]≥G0=Kθ0(U0). Hence, any incentive-compatible maximal-compliance mechanism leads to weakly higher inspection costs. B.2.3 Optimality in the original problem We now consider the original model, in which we remove assumption (A) so that the honesty constraint must hold in both states. We show that during noncompliance, the honesty constraint does not bind and, therefore, the solution of the relaxed problem is also a solution to our original problem. The proof is constructive. In the optimal mechanism of the relaxed problem, the pair of promised utilities at the outset and during noncompliance is (u0,u1):=(¯ u,¯ u−c/(λα)). 35See Davis (1993, pp. 27–33). 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
858 Achim and Knoepfle Theoretical Economics 19 (2024) Since dU0 t≤0, we have U0 t≤u0.SetdFt=u0−U0 tand dμt=u1−U1 t+u0−U0 t.Next, while θ=0, set dFt=−ru0+λα(u1−u0)and dμt=c+(r+λ)(u1−u0). Substituting into the promise-keeping and truth-telling constraints, it follows that dUi t=0for each i=0, 1 and dNI t=0 while θt=0, which is identical to the solution in the relaxed problem. Proof of Lemma 3 Define (T)≡(B−c/r)1−e−rT −c/(λα)eλT erT −α+c/(λα)(1−α). (20) By Theorem 1,wehaveT∗=inf{T>0:(T)=0}. This exists and is unique whenever B> ¯ B(is increasing from 0 at T=0 and crosses 0 from above exactly once). The function is continuously differentiable in λand Ton a neighborhood of T∗.Bythe implicit function theorem, we have ∂T∗ ∂λ =−λ TT=T∗ , where xdenotes the partial derivative of with respect to x. As mentioned above, (T)crosses 0 from above at T=T∗so that T|T=T∗<0. Hence, for all parameters, we have sign∂T∗ ∂λ =sign(λ|T=T∗). Consider in (20)asλλ=cr/(Brα −c),whichisthelowerboundonλsuch that the feasibility assumption B> ¯ B=c(r+λ) rλα is fulfilled. Then is equal to B−c r1−e−rT−B−c rαeλT erT −α−(1−α). This can be equal to 0 only if T=0 because it is concave in Tand the Tderivative is 0atT=0. Hence, limλ↓λT∗(λ)=0andT∗is initially increasing in λ. Finally, to show that T∗(λ)λ→∞ −→ 0, consider (20)andobservethat(T∗)=0implies lim λ→∞ e(r+λ)T∗(λ) λ=0. This implies that λT∗(λ)is either finite or grows at lower than logarithmic rate as λbecomes arbitrarily large. In particular, T∗(λ)must go to 0. Considering the costs, let Let K0 EQ and K1 EQ denote the expected discounted inspection cost when starting in state 0 or 1, respectively. For fixed inspection cycle length T, they follow the nested equations K0 EQ =∞ 0 e−(r+λα)tλαK1 EQ dt=λα r+λαK1 EQ 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
Theoretical Economics 19 (2024) Relational enforcement 859 and K1 EQ =T 0 e−(r+λ(1−α))tλ(1−α)K0 EQ dt+e−(r+λ(1−α))Tκ+K1 EQ =1−e(r+λ(1−α))Tλ(1−α) r+λ(1−α)K0 EQ +e−(r+λ(1−α))Tκ+K1 EQ. Inserting K0 EQ and solving for K1 EQ gives K1 EQ =r+λα r(r+λ)·r+λ(1−α)e−(r+λ(1−α))T 1−e−(r+λ(1−α))T·κ. Note that K1 EQ is decreasing in Tand decreasing in λfor fixed T. Thus, given that T∗is increasing in λfor low λ, it follows immediately that the costs decrease for low λ. Further, K1 EQ approaches ∞as Tgoes to zero for any positive and finite λ. Since lim λλT∗(λ)=0, it follows that lim λλKEQ(λ)=0. For the limit as λgrows arbitrarily large, note that the total cost in the limit is given by lim λ→∞ K1 EQ =(1−α)α rlim λ→∞ λ e(1−α)λT∗(λ). Recall from above that λT ∗(λ)grows to ∞at lower than logarithmic rate so that the term above must be ∞. Appendix C: Proofs for Section 5 Proof of Theorem 2. We first argue that as the mechanism described in the main text is optimal among the class of stationary mechanisms, consider an alternative stationary stochastic mechanism that delivers some given promised utility u.From the promise-keeping constraint (Pk) and the honesty constraint (H) for i=1, it is straightforward to obtain that the constant rate mR(u)of inspection that keeps the promised utility in state 1 stationary at the level u∈[−B+c/(λα),−c/(rα)] is mR(u)= r(c−αλu)/(αλ(B+u)−c). The principal’s expected monitoring costs in the stationary random mechanism that provides promised utility U1 t=uthroughout can be determined recursively. Denoting by K1 R(u)the expected costs while in compliance, we have K1 R(u)=r+λα r mR(u) r+λ=r+λα r r(c−αλu) (r+λ)αλ(B+u)−c. (21) 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
860 Achim and Knoepfle Theoretical Economics 19 (2024) It is easy to see that KR(u)is decreasing in u. Given that −c rα is an upper bound on the promised utility for the agent during compliance (it is the maximum payoff for the agent subject to satisfying the obedience constraint) and it is the promised utility delivered by the mechanism characterized above, it follows that this mechanism is indeed the optimal stationary mechanism. To show that the costs of the random mechanism are strictly below the equilibrium costs with predictable inspections, express the latter as K0 EQ =∞ 0 e−(r+λ)tλαK1 EQ +(1−α)K0 EQdt K1 EQ =∞ 0 e−(r+λ)tλα˜ K1(τt)+(1−α)K0 EQdt+ ∞ n=1 e−(r+λ)kT∗κ, where K0 EQ denotes the expected costs while in noncompliance and ˜ K1(τ)=e−(r+λ(1−α))(T∗−τ)κ+K1 EQ+T∗ τ e−(r+λ(1−α))(s−τ)λ(1−α)K0 EQ ds denotes the expected costs while in compliance and time τ∈[0, T∗]has passed since the last inspection or transition. Note that ˜ K1(τ)is increasing in τwith ˜ K1(0)=K1 EQ and ˜ K1(T∗)=κ+K1 EQ. Thus, replacing ˜ K1(τt)by K1 EQ in the recursive expression above, and solving the system gives an upper bound on the equilibrium costs K1 EG: K1 EQ ≤˜ K=r+λα r e−(r+λ)T∗ 1−e−(r+λ)T∗κ. (22) To see that K1 Rin (21)islower,use(7)towriteT∗in (22) as a function of u1∗: ˜ Ku1∗=r+λα r e−(r+λ)T∗ 1−e−(r+λ)T∗κ=r+λα r c αλB+u1∗−cκ. Now it is immediate to check that ˜ K(−c rα )=KR(−c rα )and ˜ K(u)<K R(u)for u<−c rα and B> ¯ B. Since −c/(rα)is an upper bound on u, it follows that KR(u1∗)<˜ K(u1∗).It follows from (22)thatK1 EQ >K 1 R. For the comparative statics in λ,consider(21)andobservethatK1 Ris decreasing in λfor fixed mRand decreasing in mR(u). The optimal promised utility −c/(rα)does not change with λand mR(u)is decreasing in λfor any u. For the limit, observe that lim λ→∞ mR−c rα=cr Brα −c and, thus, lim λ→∞ K∗ R=α r cr Brα −cκ=cα Brα −cκ. 15557561, 2024, 2, Downloaded from https://onlinelibrary.wiley.com/doi/10.3982/TE5183 by ZBW Kiel - Hamburg (German National Library of Economics), Wiley Online Library on [04/07/2025]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons License
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