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.

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.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). $\square$