Simple contracts under observable and hidden actions

Abstract

We consider a general framework for multitask moral hazard problems with observable and hidden actions. Ideally, the principal in our framework can design optimal contracts that depend on both observable (and verifiable) actions and realized outcomes. Given a mild assumption on the existence of a punishment scheme, we identify a general equivalence result, dubbed the “forcing principle,” which states that every optimal contract in our framework is strategically equivalent to a simple forcing contract, which only specifies an outcome-contingent reward scheme and an action profile, and the agent receives the outcome-contingent reward only if he follows the recommended observable actions (and is otherwise punished severely). The forcing principle has useful implications: it confers analytical advantage for the existence and computation of optimal contracts in our setting. It also highlights and makes explicit the importance of the existence of the punishment scheme in characterizing first-best benchmarks in moral hazard problems.

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Notes

  1. 1.

    A precursor of the model in Holmström and Milgrom (1991) can be found in Holmström and Milgrom (1987) on linear contracts as optimal compensation schemes in dynamic environments.

  2. 2.

    This is the so-called linear exponential normal (LEN) model in the literature.

  3. 3.

    We equate multitasks with multi-dimensional actions in our analysis, although there could be literal differences in different contexts.

  4. 4.

    For example, the famous Gantt’s task and bonus plan in industrial management specifies a worker’s reward scheme that varies in both the worker’s output and working hours in a complex way, which are called hybrid reward schemes hereafter. See Example 1.

  5. 5.

    See Mirrlees (1974) and Grossman and Hart (1983), among many others.

  6. 6.

    It is clear though that a typical forcing contract in a first-best benchmark is only a special case of our general forcing principle.

  7. 7.

    While this is perhaps a minor issue in that the existence of such a punishment is realistic and is either implicitly or explicitly recognized in most of the moral hazard literature, we think that it is still sensible and useful to raise the issue explicitly.

  8. 8.

    Technically, it is difficult to establish sequential continuity of the two parties’ expected utility functions via Delbaen’s lemma, crucial in Page’s approach, when the observable action component also enters the reward scheme. We discuss this in detail in Sect. 4.2.

  9. 9.

    Page (1991, 1992a) extend this methodology to general models with both moral hazard and adverse selection. Page (1992b) presents an existence result for Bayesian incentive compatible mechanisms for Stackelberg games with incomplete information.

  10. 10.

    Recently, Kadan et al. (2017) have also established an existence result for principal-agent problems that encompass pure moral hazard, pure adverse selection, and problems with both. However, the existence result in Kadan et al. (2017) is mainly for optimal randomized mechanisms.

  11. 11.

    Here and in the sequel, the observability of a given object implies both verifiability by a third party and measurability of the object with respect to the corresponding distribution function.

  12. 12.

    To be specific, \({\mathscr {K}}\) denotes the set of all practically available pure outcome-contingent reward schemes, given current market/technology conditions and legal customs and the principal’s abilities for computation and accounting, as in Page (1987). For instance, \({\mathscr {K}}\) can be the set of bounded and monotone contracts or a set of contracts that are restricted to be linear in outcomes.

  13. 13.

    In (P1) if \(A_{0}\) is a singleton, then the problem reduces to a pure moral hazard model, while if \(A_{1}\) is a singleton, then (P1) corresponds to the first-best contracting problem with only observable (and verifiable) actions. We use the notation “max” rather than “sup” in the principal’s and agent’s problems in (P1). The existence of solutions in (P1) will be established with further assumptions in Sect. 4.

  14. 14.

    For future comparison, such forcing will take place in (P2 ) in Sect. 3.

  15. 15.

    It is known, however, that the informativeness principle may not hold in a multitask principal-agent models (see, e.g., Holmström and Milgrom 1991).

  16. 16.

    “For Bethlehem Steel, the average monthly output of the shop from March 1, 1900, to March 1, 1901, was 1,173,000 pounds, and from March 1, 1901, to August 1, 1901 (having implemented the task and bonus system) was 2,069,000 pounds. The shop had 700 men in it and we were paying on the bonus plan only about 80 workmen out of the entire 700” (Chapter VII, Gantt 1919).

  17. 17.

    This intuitive approach is perhaps consistent with many business practices and academic research emphasizing the importance of key performance indicators for successes in various organizations.

  18. 18.

    The outcome-contingent portion of the reward scheme, however, is similar to an optimal contract solving a canonical principal-agent model in Holmström (1979), which may not take a simple form.

  19. 19.

    If on the other hand the agent is protected by limited liability or minimum wage legislation from receiving harsh putative wages, then Assumption 1 will be violated. See Sect. 4.1.

  20. 20.

    We thank our referee for bringing the additional implications of \(t\left( \omega \right) \) to our attention.

  21. 21.

    All proofs in the paper are relegated to an “Appendix.”

  22. 22.

    In such simple contracts, the principle can simply state some necessary or minimal requirements on \(a_{0}\) for the agent to get any compensation (for example, minimal working hours, project completion deadlines, exclusive sales territories, etc).

  23. 23.

    See, for instance, the characterization of first-best benchmarks in standard textbooks (Laffont and Martimort 2009; Bolton and Dewatripont 2005).

  24. 24.

    Throughout Example 3, we assume that the principal is always willing to hire the agent given the minimum wage constraint—this can be guaranteed by adjusting the level of the minimum wage without qualitatively affecting the structure of the optimal contracts.

  25. 25.

    The socially optimal action in the setting is identified by comparing the net surpluses generated by \(a_{0}=2\) and \(a_{0}=1\), which are \(\left( \alpha p\left( 2\right) -2\right) \) and \(\left( \alpha p\left( 1\right) -1\right) \) , respectively.

  26. 26.

    A number of studies in the previous literature have studied moral hazard problems where limited liability or a lower/minimum bound on the agent’s payment is imposed (so that Assumption 1 is possibly violated). See, for example, Sappington (1983), Innes (1990), Kim (1997), Dewatripont et al. (2003), etc.

  27. 27.

    Notice that we will consider sequential topological properties for the reward scheme sets. Since such reward scheme sets are sets of functions, they may not always be metrizable from the outset. Consequently, sequential topological properties will be more meaningful for the proof of existence. (For instance, sequential compactness is not equivalent to compactness for sets of functions.)

  28. 28.

    It can be readily verified that fixing x, the agent’s expected utility is strictly concave in effort y, and the classic first-order approach (FOA) can be applied here. When FOA cannot be applied, one can employ a polynomial approach introduced by Renner and Schmedders (2015) which (approximately) transforms the principal’s (bilevel) optimization problem into a simpler nonlinear optimization problem. More recently, Ke and Ryan (2018) also have provided a general approach to solve the standard moral hazard problem via a max–min–max formulation when FOA fails.

  29. 29.

    It can be verified that after substituting the binding (IR) constraint, the objective function (\(r_{s}^{5}(W-r_{s})\)) is strictly quasi-concave in the relevant range, and the unique global maximizer can be derived via the first-order condition as \(r_{s}^{*}=\frac{5W}{6}\).

  30. 30.

    In a specific R&D funding setting with moral hazard, adverse selection, and observable actions, Rietzke and Chen (2018) discuss the characterization of optimal mechanisms by directly adopting forcing contracts.

  31. 31.

    In cases in which \(\int _{\varOmega }v(t(\omega ),a_{0},a_{1})P(d\omega ;a_{0},a_{1})=r\) for some recommended actions \(\left( a_{0},a_{1}\right) \) and the agent is indifferent, we assume that the agent takes the recommended actions \(\left( a_{0},a_{1}\right) \).

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Correspondence to Bo Chen.

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Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Yu Chen passed away unexpectedly in February of 2019, just weeks before the paper was accepted. His friendship and talents will be missed by his coauthors. We thank our referee and editor for very constructive comments. We also gratefully acknowledge helpful comments and discussion from Michael Alexeev, Sascha Baghestanian, Robert Becker, Filomena Garcia, Junichiro Ishida, Jianpei Li, Frank Page, Daniela Puzzello, Eric Rasmussen, Satoru Takahashi, Ning Yang, and Jinghua Yao. Yu Chen was supported by the National Natural Science Foundation of China (Grant No. 71673133). The usual disclaimer applies.

Appendix: Proofs

Appendix: Proofs

Proof of Proposition 1

Let \(k^{*}=(f^{*},a_{0}^{*},a_{1}^{*})\) be a feasible contract. By Assumption 1, there is a Borel measurable schedule, \( t:\varOmega \rightarrow D\) such that for any \((a_{0},a_{1})\in A\), \( \int _{\varOmega }v(t(\omega ),a_{0},a_{1})P(d\omega ;a_{0},a_{1})\le r\). Define \(s:\varOmega \rightarrow D\) such that \(s(\omega )=f^{*}(\omega ,a_{0}^{*})\) for all \(\omega \), and consider the proposed \({\tilde{f}}\) with the simple punishment \(t\left( \omega \right) \):

$$\begin{aligned} {\tilde{f}}(\omega ,a_{0})=\left\{ \begin{array}{l} s(\omega ),\text { if }a_{0}={a}_{0}^{*} \\ t(\omega ),\text { otherwise}. \end{array} \right. \end{aligned}$$

By construction, \(s\in {\mathscr {K}}\), \({\tilde{f}}\in {\mathscr {R}}_{{\mathscr {K}}}\) and s,  and \({\tilde{f}}\) are both Borel measurable. Moreover, \({\tilde{f}}(\omega ,a_{0}^{*})=f^{*}(\omega ,a_{0}^{*})\). We next show that \(\tilde{ k}=({\tilde{f}},a_{0}^{*},a_{1}^{*})\) satisfies both (IR) and (IC).

Under \({\tilde{k}}\), if the agent follows the recommendation \(a_{0}^{*}\), then for all \(\omega \), \({\tilde{f}}(\omega ,a_{0}^{*})=s(\omega )=f^{*}(\omega ,a_{0}^{*})\). Hence, the agent’s expected utility under \({\tilde{k}}\) from choosing \(a_{0}^{*}\) is

$$\begin{aligned} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{*}),a_{0}^{*},a_{1}^{*})P(d\omega ;a_{0}^{*},a_{1}^{*})=\int _{\varOmega }v(f^{*}(\omega ,a_{0}^{*}),a_{0}^{*},a_{1}^{*})P(d\omega ;a_{0}^{*},a_{1}^{*})\ge r, \end{aligned}$$

where the inequality follows from feasibility of \(k^{*}\). Thus, \({\tilde{k}}\) satisfies (IR).

To show (IC), let \((a_{0},a_{1})\) be an arbitrary action profile for the agent, and consider the reward schedule \({\tilde{f}}\). If \(a_{0}\ne a_{0}^{*}\) then \({\tilde{f}}(\omega ,a_{0})=t(\omega )\) for all \(\omega \). Given Assumption 1, the agent indeed has incentive to choose \( a_{0}^{*}\). If \(a_{0}=a_{0}^{*}\) then \({\tilde{f}}(\omega ,a_{0})=f^{*}(\omega ,a_{0}^{*})\) for all \(\omega \). Now if the agent chooses \((a_{0}^{*},a_{1})\), his expected utility is:

$$\begin{aligned} \int _{\varOmega }v(f^{*}(\omega ,a_{0}^{*}),a_{0}^{*},a_{1})P(d\omega ;a_{0}^{*},a_{1})\le & {} \int _{\varOmega }v(f^{*}(\omega ,a_{0}^{*}),a_{0}^{*},a_{1}^{*})P(d\omega ;a_{0}^{*},a_{1}^{*}) \\= & {} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{*}),a_{0}^{*},a_{1}^{*})P(d\omega ;a_{0}^{*},a_{1}^{*}). \end{aligned}$$

The above follows from the optimality of \((a_{0}^{*},a_{1}^{*})\) for the agent under the reward scheme \(f^{*}\) and that \({\tilde{f}}(\omega ,a_{0}^{*})=f^{*}(\omega ,a_{0}^{*})\) for all \(\omega \). Hence, the contract \({\tilde{k}}\) satisfies both (IC) and (IR), and is feasible. Moreover, the fact that \({\tilde{f}}(\omega ,a_{0}^{*})=f^{*}(\omega ,a_{0}^{*})\) for all \(\omega \) implies that the principal and the agent attain the same expected utility under \({\tilde{k}}\) and \(k^{*}\). \(\square \)

Proof of Theorem 1

By Proposition 1, for any feasible contract \((f,a_{0}^{\prime },a_{1}^{\prime })\), there is \(s\in {\mathscr {K}}\) with \(s(\omega )\equiv f(\omega ,a_{0}^{\prime })\) and a corresponding feasible punishment contract \(({\tilde{f}},a_{0}^{\prime },a_{1}^{\prime })\) with

$$\begin{aligned} {\tilde{f}}(\omega ,a_{0})=\left\{ \begin{array}{l} s(\omega ),\text { if }a_{0}=a_{0}^{\prime } \\ t(\omega ),\text { otherwise}, \end{array} \right. \end{aligned}$$
(2)

and the principal obtains the same expected utility under \(({\tilde{f}} ,a_{0}^{\prime },a_{1}^{\prime })\) and \((f,a_{0}^{\prime },a_{1}^{\prime })\) . Since we can always choose a feasible contract \((f,a_{0}^{\prime },a_{1}^{\prime })\) with a punishment contract \(({\tilde{f}},a_{0}^{\prime },a_{1}^{\prime })\), the original problem (P1) is equivalent to the following \(\tilde{(\mathbf P }{} \mathbf 1 )\):

$$\begin{aligned} {({\tilde{\mathbf {P}}}1)} \begin{array}{l} \displaystyle \max \nolimits _{(s,a_{0}^{\prime },a_{1}^{\prime })\in {\mathscr {K}}\times A}\int _{\varOmega }u(\omega ,{\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime }) \\ \displaystyle \text {s.t. } \begin{array}{l} \displaystyle \text {(IR}^{\prime }\text {): }\int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })\ge r\\ \displaystyle \text {(IC}^{\prime }\text {): }(a_{0}^{\prime },a_{1}^{\prime })\in \arg \max \nolimits _{({\widetilde{a}}_{0},{\widetilde{a}}_{1})\in A}\int _{\varOmega }v( {\tilde{f}}(\omega ,{\widetilde{a}}_{0}),{\widetilde{a}}_{0},{\widetilde{a}} _{1})P(d\omega ;{\widetilde{a}}_{0},{\widetilde{a}}_{1}). \end{array} \end{array} \end{aligned}$$

Notice that the reward scheme \({\tilde{f}}\) in \(\tilde{(\mathbf P }{} \mathbf 1 )\) takes a specific form that is determined jointly by \(s(\omega )\) and \( a_{0}^{\prime }\) (see (2)).

Consider a new problem \(\hat{(\mathbf P }{} \mathbf 1 )\) and recall that \(t\left( \omega \right) \) is defined in Assumption 1:

$$\begin{aligned} {(\hat{\mathbf {P}}1)} \begin{array}{c} \displaystyle \max \nolimits _{(s,a_{0}^{\prime },a_{1}^{\prime })\in \mathscr {K}\times A}\int _{\varOmega }u(\omega ,s(\omega ),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })\text { s.t.} \\ \begin{array}{l} \displaystyle \text {(IR): }\int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })\ge r \\ \displaystyle \text {(IC}_{0}):\,\int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })\\ \displaystyle \quad \ge \int _{\varOmega }v(t\left( \omega \right) ,a_{0},a_{1})P(d\omega ;a_{0},a_{1}) \text {, }\forall a_{0}\ne a_{0}^{\prime }, a_{1}\in A_{1} \\ \displaystyle \text {(IC}_{1}):\,a_{1}^{\prime }\in \arg \max \nolimits _{a_{1}\in A_{1}}\int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1})P(d\omega ;a_{0}^{\prime },a_{1}). \end{array} \end{array} \end{aligned}$$

\(\square \)

Lemma 1

Given the definition in (2), the set of constraints (IR\(^{\prime }\)) and (IC\(^{\prime }\)) in \(\tilde{(\mathbf P }{} \mathbf 1 )\) is equivalent to the set of constraints (IR), (IC\(_{0}\)), and (IC\(_{1}\)) in (P̂1 ), i.e., any \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \in {\mathscr {K}} \times A\) satisfying (IR\(^{\prime }\)) and (IC\(^{\prime }\)) also satisfies (IR), (IC\(_{0}\)), and (IC\(_{1}\)), and vice versa.

Proof of Lemma 1

First suppose that \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) satisfies (IR\(^{\prime }\)) and (IC\(^{\prime }\)). Then (IR) holds for \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) given the definition of \({\tilde{f}}(\omega ,a_{0})\) in (2) and (IR\(^{\prime }\) ). Next, since \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) satisfies (IC\(^{\prime }\)), we have that for all \(\left( a_{0},a_{1}\right) \in A\),

$$\begin{aligned} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })\ge \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}),a_{0},a_{1})P(d\omega ;a_{0},a_{1}). \quad \end{aligned}$$
(3)

In particular, by the definition of \({\tilde{f}}(\omega ,a_{0})\), for \( a_{0}\ne a_{0}^{\prime }\), (3) implies

$$\begin{aligned} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })= & {} \int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime }) \\\ge & {} \int _{\varOmega }v(t(\omega ),a_{0},a_{1})P(d\omega ;a_{0},a_{1}), \end{aligned}$$

which is (IC\(_{0}\)) in \(\tilde{(\mathbf P }{} \mathbf 1 )\). Moreover, again by (3), we have for all \(a_{1}\in A_{1}\),

$$\begin{aligned} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })\ge & {} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1})P(d\omega ;a_{0}^{\prime },a_{1}) \\= & {} \int _{\varOmega }v(s\left( \omega \right) ,a_{0}^{\prime },a_{1})P(d\omega ;a_{0}^{\prime },a_{1}), \end{aligned}$$

which is (IC\(_{1}\)) in \(\tilde{(\mathbf P }{} \mathbf 1 )\). Hence, \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) satisfies (IR), (IC\(_{0}\)), and (IC\(_{1}\)).

Now suppose that \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) satisfies (IR), (IC\(_{0}\)), and (IC\(_{1}\)). Then similarly (IR\(^{\prime }\)) holds for \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) given the definition of \({\tilde{f}}(\omega ,a_{0})\) in (2) and (IR). We now show that \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) also satisfies (IC\( ^{\prime }\)). Suppose not, i.e., given s, there is \(\left( a_{0}^{\prime \prime },a_{1}^{\prime \prime }\right) \ne \left( a_{0}^{\prime },a_{1}^{\prime }\right) \) in A such that \(\left( s,a_{0}^{\prime \prime },a_{1}^{\prime \prime }\right) \) satisfies (IC\(_{0}\)) and (IC\(_{1}\)) but

$$\begin{aligned} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime \prime }),a_{0}^{\prime \prime },a_{1}^{\prime \prime })P(d\omega ;a_{0}^{\prime \prime },a_{1}^{\prime \prime })>\int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime }). \nonumber \\ \end{aligned}$$
(4)

Suppose \(a_{0}^{\prime \prime }\ne a_{0}^{\prime }\). Then by definition of \( {\tilde{f}}(\omega ,a_{0})\) and \(t\left( \omega \right) \), we have

$$\begin{aligned} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime \prime }),a_{0}^{\prime \prime },a_{1}^{\prime \prime })P(d\omega ;a_{0}^{\prime \prime },a_{1}^{\prime \prime })=\int _{\varOmega }v(t\left( \omega \right) ,a_{0}^{\prime \prime },a_{1}^{\prime \prime })P(d\omega ;a_{0}^{\prime \prime },a_{1}^{\prime \prime })\le r, \end{aligned}$$

contradicting (IR) and (4). Hence, we must have \(a_{0}^{\prime \prime }=a_{0}^{\prime }\), i.e., (4) becomes

$$\begin{aligned} \int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime \prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime \prime })>\int _{\varOmega }v({\tilde{f}}(\omega ,a_{0}^{\prime }),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime }), \end{aligned}$$

which further implies

$$\begin{aligned} \int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1}^{\prime \prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime \prime })>\int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime }), \end{aligned}$$

which, however, contradicts (IC\(_{1}\)). Hence, \(\left( s,a_{0}^{\prime },a_{1}^{\prime }\right) \) satisfies (IR\(^{\prime }\)) and (IC\(^{\prime }\)). This proves Lemma 1.

Given Lemma 1 and \({\tilde{f}}(\omega ,a_{0})\) defined in (2), the maximization problems of \(\hat{(\mathbf P }{} \mathbf 1 )\) and \(\tilde{(\mathbf P }{} \mathbf 1 )\) are therefore equivalent.Footnote 31 Now since in \(\hat{(\mathbf P }{} \mathbf 1 )\), for any recommended action \(a_{0}^{\prime }\), (IC\(_{0}\)) holds automatically given (IR) and Assumption 1, we can further rewrite (P̂1) as

$$\begin{aligned} \begin{array}{c} \displaystyle \max \nolimits _{(s,a_{0}^{\prime },a_{1}^{\prime })\in {\mathscr {K}}\times A}\int _{\varOmega }u(\omega ,s(\omega ),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime }) \\ \text {s.t. } \begin{array}{l} \displaystyle \text {(IR): }\int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1}^{\prime })P(d\omega ;a_{0}^{\prime },a_{1}^{\prime })\ge r \\ \displaystyle \text {(IC}_{1}):\,a_{1}^{\prime }\in \arg \max \nolimits _{a_{1}\in A_{1}}\int _{\varOmega }v(s(\omega ),a_{0}^{\prime },a_{1})P(d\omega ;a_{0}^{\prime },a_{1}) \end{array} \end{array} \end{aligned}$$

which is exactly (P2). Since (P1) and ( P2) are equivalent optimization problem under Proposition 1, it follows that bullet points 1, 2, and 3 in Theorem 1 hold as desired. \(\square \)

Proof of Proposition 2

This proof follows the proof of existence of optimal contracts in Page (1987) by replacing sequences of \(f_{n}\)’s by sequences of \( (s_{n},a_{0,n})\)’s. Delbaen’s lemma (Delbaen 1974) will imply that U and V are sequentially continuous. Moreover, the constraint set of (P2)\(Gr(A^{*})=\{(s,a_{0},a_{1})\in {\mathscr {K}}\times A:(s,a_{0})\in {\mathscr {L}}(r)\), \(a_{1}\in A^{*}(s,a_{0})\}\) is non-empty and sequentially closed, and \(U^{*}:=\sup _{(s,a_{0},a_{1})\in Gr\left( A^{*}\right) }U(s,a_{0},a_{1})\) is finite. Next, since \(U^{*}\) is a supremum, there is a sequence \(\{(s_{n},a_{0,n},a_{1,n})\}_{n}\) in \(Gr\left( A^{*}\right) \) such that \(U(s_{n},a_{0,n},a_{1,n})\rightarrow U^{*}\) . Given the sequential compactness of \({\mathscr {K}}\) and the compactness of A, there is a subsequence \(\{(s_{n_{k}},a_{0,n_{k}},a_{1,n_{k}})\}_{k}\) in \(Gr\left( A^{*}\right) \) and a triple \((s^{*},a_{0}^{*},a_{1}^{*})\) in \({\mathscr {K}}\times A\) such that \(s_{n_{k}}\rightarrow s^{*}\) pointwise on \(\varOmega \) and \((a_{0,n_{k}},a_{1,n_{k}})\rightarrow (a_{0}^{*},a_{1}^{*})\) under the metric on A. Since \(Gr\left( A^{*}\right) \) is sequentially closed, \((s^{*},a_{0}^{*},a_{1}^{*})\in Gr\left( A^{*}\right) \). Finally, the sequential continuity of U implies that \(U(s^{*},a_{0}^{*},a_{1}^{*})=U^{*}\). Therefore, \((s^{*},a_{0}^{*},a_{1}^{*})\) is the solution to (P2). \(\square \)

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Chen, B., Chen, Y. & Rietzke, D. Simple contracts under observable and hidden actions. Econ Theory 69, 1023–1047 (2020). https://doi.org/10.1007/s00199-019-01187-7

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Keywords

  • First-best benchmark
  • Forcing contract
  • Forcing principle
  • Moral hazard
  • Observable actions

JEL Classification

  • C61
  • C62
  • D86