Contracts in informed-principal problems with moral hazard

Abstract

In many cases, an employer has private information about the potential productivity of a worker, who in turn has private information about the effort she exerts on the job. Much of the analysis of this environment in the literature restricts the employer to offer contracts that depend only on observable outcomes (e.g., profit). This paper studies the advantages to the employer of offering the worker a set of potential contracts from which the employer will choose after the worker has accepted the offer, so called menu-contracts. Specifically, in a two-state principal-agent problem with moral hazard, I show when the principal can obtain strictly higher expected payoffs than the restricted contracts of the literature by offering a menu-contract.

Appendix: Proofs

Lemma 1

Suppose Assumption 1 holds. The RSW equilibrium is the least-cost separating equilibrium that has type-L principal offering $(\mathbf {w}^{*L}, a^{*L})$ and the type-H principal offering the solution to

$$\begin{aligned} I^{\text {RSW}}: \begin{array}{llll} \max \limits _{\left( \mathbf {w}^H, a^H\right) }&\varvec{\pi }_H(a^H)\cdot \left( \mathbf {q}-\mathbf {w}^H\right)&\text {s.t.}&IR(1,0),\ \text{ AIC }(1,0)\text { and } \text{ PIC }(L,H). \end{array} \end{aligned}$$

The RSW contract always exists.

Proof

Let $\left( \tilde{\mathbf {w}}^H,\tilde{a}^H\right)$ be a solution to $I^\text{ RSW }$. First, I claim that the constraint $\text{ AIC }\left( 1,0\right)$ in problem $I^{\text{ RSW }}$ must bind. Suppose $\text{ AIC }\left( 1,0\right)$ holds with strict inequality and let $\pi ^H_s\ge \pi ^H_f$. Then, decrease $w_s^H$ and increase $w_f^H$ slightly to $(\tilde{w}_s^H-\epsilon _s,\tilde{w}_f^H+\epsilon _f)$ for small $\epsilon _s,\epsilon _f>0$ so that IR$\left( 1,0\right)$ and $\text{ AIC }\left( 1,0\right)$ still hold. Since $\pi ^H_s>\pi ^L_s$, $(\epsilon _s,\epsilon _f)$ can be chosen such that the right hand side of $\text{ PIC }(L,H)$ (possibly weakly) decreases while the objective function strictly increases. If $\pi ^H_s<\pi ^H_f$, increase $w_s^H$ and decrease $w_f^H$ slightly to $(\tilde{w}_s^H+\epsilon _s,\tilde{w}_f^H-\epsilon _f)$ for small $\epsilon _s,\epsilon _f>0$ so that $IR\left( 1,0\right)$ and $\text{ AIC }\left( 1,0\right)$ still hold. Since $\pi ^H_s>\pi ^L_s$, $(\epsilon _s,\epsilon _f)$ can again be chosen such that the right hand side of $\text{ PIC }(L,H)$ (possibly weakly) decreases while the objective function strictly increases.

Second, I claim that $\left( (\tilde{\mathbf {w}}^H,\tilde{a}^H),(\mathbf {w}^{*L},a^{*L})\right)$ is incentive compatible. This is vacuously true for the type-L principal since $\text{ PIC }(L,H)$ is imposed in problem $I^{\text {RSW}}$ and $\mathbf {w}^{*L}$ is incentive compatible for the agent by construction. Further, $\text {AIC}(1,0)$ is imposed in problem $I^\text {RSW}$. It remains to show that

$$\begin{aligned} \varvec{\pi }^H(\tilde{a}^H)\cdot \left( \mathbf {q}- \tilde{\mathbf {w}}^H\right) \ge \varvec{\pi }^H(a^{*L})\cdot \left( \mathbf {q}-\mathbf {w}^{*L} \right) . \end{aligned}$$

(5)

I claim that (5) holds with strict inequality. Note that the curve in $(w_s,w_f)$ space implicitly defined by the agent’s RSW incentive compatibility constraint for the type-H principal,

$$\begin{aligned} \left( \pi ^H_s(\tilde{a}^H)-\pi ^H_s(a_1)\right) \left[ U(\tilde{w}_s^H)-U(\tilde{w}_f^H)\right] = \tilde{a}^H-a_1, \end{aligned}$$

is strictly above that of the type-L principal,

$$\begin{aligned} \left( \pi ^L_S( a^{*L})-\pi ^L_S(a_1)\right) \left[ U(w_s^{*L})-U(w_f^{*L})\right] = a^{*L}-a_1 \end{aligned}$$

due to Assumption 1. Further, the indifference curves of the type-H principal are steeper than the type-L principal’s. Therefore, the indifference curves possess the single crossing property. If $\text{ PIC }(L,H)$ holds with equality, $\tilde{\mathbf {w}}^H$ lies to the north-west of $\mathbf {w}^{*L}$ in $(w_s,w_f)$-space which implies that (5) strictly holds. Otherwise, $\tilde{\mathbf {w}}^H=\mathbf {w}^{*H}$ and (5) strictly holds since $\pi ^H_s(a^{*H})>\pi ^L_s(a^{*L})$.

The RSW problem for the type-H principal is more constrained than $I^\text {RSW}$, but $\left( \tilde{\mathbf {w}}^H,\tilde{a}^H\right)$ solves the latter problem and satisfies all the constraints of the former (with the type-L contract specified as $(\mathbf {w}^{*L},a^{*L})$). Therefore, it solves the RSW problem for the type-H principal: $\left( \hat{\mathbf {w}}^H,\hat{a}^H\right) =\left( \tilde{\mathbf {w}}^H,\tilde{a}^H\right)$. Similarly, the RSW problem for the type-L principal is more constrained than the public information problem, but $\left( \mathbf {w}^{*L},a^{*L}\right)$ solves the latter problem and satisfies all the constraints of the former (with the type-H contract specified as $(\hat{\mathbf {w}}^{*H},a^{*H})$). Therefore, it solves the RSW problem for the type-L principal.

To see that this menu-contract exists I first claim that $\mathbf {w}^{*L}(a)$ exists for any a. For $a=a_1$, $\mathbf {w}^{*L}(a)=(h(\bar{U}+a_1),h(\bar{U}+a_1))$. For $a=a_2$, the constraints $\text{ AIC }(0,1)$ and $IR(a_2;\{0,1\})$ will be satisfied with equality. Since $w_f^L(\cdot ;IR)$ is strictly decreasing and $w_f^L(\cdot ;\text{ AIC })$ is strictly increasing, they must intersect exactly once in $\mathbb {R}^2$. Denote this intersection point $(w_s',w_f')$. If $(w_s',w_f')\in [\underline{w},\infty )^2$ I am done: $\mathbf {w}^{*L}=(w_s',w_f')$. Otherwise, the solution is $\mathbf {w}^{*L}=(w_s'',\underline{w})$ where $w_s''$ satisfies $w^L_f(w_s'',\text{ AIC })=\underline{w}$. If the type-L principal is indifferent between $a_1$ and $a_2$, set $a^{*L}=a_2$.

$I^\text {RSW}$ can be broken down into separate problems of minimizing the cost of implementing each effort then choosing most profitable effort. Note that $a^{*H}=a_1$ implies that $a^{*L}=a_1$ since the expected payoff from the agent’s effort is strictly higher for they type-H principal. Thus, if $a^{*H}=a_1$, $\hat{\mathbf {w}}^H=(h(\bar{U}+a_1),h(\bar{U}+a_1))$ which satisfies all the constraints of $I^\text {RSW}$ given our previous statement.

If $a^{*H}=a_2$ and $\text{ PIC }(L,H)$ does not bind, the solution to $I^\text {RSW}$ is simply $\mathbf {w}^{*H}$, which exists by an argument analogous to the previous one for the existence of $\mathbf {w}^{*L}$. Otherwise, the solution to $I^\text {RSW}$ is defined by

$$\begin{aligned}&\left( \pi ^H_s(a_2)-\pi ^H_s(a_1)\right) \left( U(\hat{w}_s^H)-U(\hat{w}_f^H)\right) = a_2-a_1 \end{aligned}$$

(6)

$$\begin{aligned}&\varvec{\pi }^L\left( a^{*L}\right) \cdot \left( \mathbf {q}-\mathbf {w}^{*L}\right) = \varvec{\pi }^L(a_2)\cdot \left( \mathbf {q}-\hat{\mathbf {w}}^H\right) . \end{aligned}$$

(7)

Equation (6) implicitly defines a strictly increase line in $(w_s,w_f)$-space while equation (7) defines a strictly decreasing line in $(w_s,w_f)$-space. These lines therefore intersect exactly once in $\mathbb {R}^2$. Denote this intersection point $(w_s',w_f')$. If this $(w_s',w_f')\in [\underline{w},\infty )^2$ I am done: $\tilde{\mathbf {w}}^H=(w_s',w_f')$ . Otherwise, the solution is $\mathbf {w}^{*L}=(w_s'',\underline{w})$ where $w_s''$ satisfies equation (6) with $\hat{w}_f^H=\underline{w}$. $\square$