Contracting with moral hazard, adverse selection and risk neutrality: when does one size fit all?

Abstract

This paper studies a principal-agent relationship when both are risk-neutral and in the presence of adverse selection and moral hazard. Contracts must satisfy the limited-liability and monotonicity conditions. We provide sufficient conditions under which the optimal contract is simple, in the sense that each type is offered the same contract. These are: the action and the agent’s type are complements, and the output’s cumulative distribution function is such that the marginal rate of substitution between the action and the agent’s type is the same for each possible output realization. Furthermore, under the average monotone likelihood ratio property, the optimal contract is a call-option contract as in Innes (J Econ Theory 52(1):45–67, 1990). The results shed light on the fact that sometimes contracts are not highly dependent on individual characteristics as predicted in most pure moral hazard and pure adverse selection settings.

Appendices

Appendix

Proofs of moral hazard, adverse selection and limited liability

Observe that, after summation by parts, a type-$\theta$ agent’s utility rewrites as follows:

$$\begin{aligned} U(\theta ,w,a)=w_{0}+\sum \nolimits _{i=0}^{n-1}\triangle w^{i}P^{i}(\theta , a)-c(a) \end{aligned}$$

where $\triangle w^i\equiv w^{i+1}-w^{i}\ge 0$ and $P^{i}(\theta , a)\equiv \sum \nolimits _{j\ge i+1}^{n}p^{j}(\theta , a)$ and the Principal’s payoff from type $\theta$ is given by:

$$\begin{aligned} V(\theta ,w,a)=y_{0}-w_{0}+\sum \nolimits _{i=0}^{n-1}(\triangle y^{i}-\triangle w^{i})P^{i}(\theta , a) \end{aligned}$$

where $\triangle y^{i}\equiv y^{i+1}-y^{i}$.

Observe that FOSD with respect to a implies that, for any $a'\ge a$, $V(\theta ,w,a')\ge V(\theta ,w,a)$, since $\triangle y^{i}\ge \triangle w^{i}$ for all $i\in I/\{n\}$ by MONP and $U(\theta ,w,a')\ge U(\theta ,w,a)$ since $\triangle w^{i}\ge$ for all $i\in I/\{n\}$.

Lemma 1

$U(\theta ,w,a)$ is supermodular in $(\theta ,\triangle w,a)$.

Proof

Recall that $P(\theta ,a)\equiv (P_{1}(\theta ,a),\ldots ,P_{I}(\theta ,a))$ and $P^{i}(\theta ,a)\equiv \sum \nolimits _{h'\ge h}p^{i}(\theta ,a)$. Suppose that $a^{'}>a$, then lets define h(i) as the lowest h such that $P^{i}(\theta , a^{'}, a_{-i})-P^{i}(\theta , a, a_{-i})\ge 0$; the existence of h(i) is ensured by the fact that condition SC implies $P^{i}(\theta , a')-P^{i}(\theta , a)$ rises with h whenever $a'\ge a$. Suppose that $\theta ^{'}>\theta$, then lets define $h(\theta )$ as the lowest h such that $P^{i}(\theta ', a)-P^{i}(\theta , a)\ge 0$; the existence of $h(\theta )$ is ensured by the fact that condition SC implies that $P^{i}(\theta ', a)-P^{i}(\theta , a)$ rises with h whenever $a'\ge a$.

Suppose that $\theta '>\theta$ and $a'\ge a$, then supermodularity requires that $U(\theta ',a',\triangle w)-U(\theta ',a,\triangle w)\ge U(\theta ,a',\triangle w)-U(\theta ,a,\triangle w)$. Observe that:

$$\begin{aligned}&U(\theta ',a',\triangle w)-U(\theta ',a,\triangle w)\ge U(\theta ,a',\triangle w)-U(\theta ,a,\triangle w)\\&\quad =\triangle wP(\theta ',a')-c(a')-(\triangle wP(\theta ',a)-c(a))- (\triangle wP(\theta ,a')-c(a')\\&\qquad -(\triangle wP(\theta ,a)-c(a))) \\&\quad =\triangle w(P(\theta ',a')-P(\theta ',a))-\triangle w(P(\theta ,a')-P(\theta ,a))\\&\quad =\sum \nolimits _{i}\triangle w^{i}\big (P^{i}(\theta ',a')-P^{i}(\theta ',a)-(P^{i}(\theta ,a')-P^{i}(\theta ,a))\big )\\&\quad \ge 0, \end{aligned}$$

where the inequality follows from the fact that condition SC ensures that $P^{i}(\theta ,a')-P^{i}(\theta ,a)$ rises with $\theta$ for all h, and the fact that the monotonicity constraint MONA implies that $\triangle w^{i}\ge 0$.

Suppose that $\theta '>\theta$ and $\triangle w^{i'}>\triangle w^{i}$. Then supermodularity requires that $U(\theta ',a,\triangle w^{i'}, \triangle w_{-h})-U(\theta ',a,\triangle w^{i},\triangle w_{-h})\ge U(\theta ,a,\triangle w^{i'},\triangle w_{-h})-U(\theta ,a,\triangle w^{i},\triangle w_{-h})$. Observe that this inequality is equal to:

$$\begin{aligned} (\triangle w^{i'}-\triangle w^{i})P^{i}(\theta ',a)\ge (\triangle w^{i'}-\triangle w^{i})P^{i}(\theta ,a). \end{aligned}$$

Because $P^{i}(\theta ',a)\ge P^{i}(\theta ,a)$ and $\triangle w^{i'}>\triangle w^{i}$, this holds for all $i\in I$. This proves the claim.

The proof for a is identical to that for $\theta$ and therefore omitted for the sake of brevity.

Suppose that $\triangle w_{h'}^{'}>\triangle w_{h'}$ and $\triangle w^{i'}>\triangle w^{i}$. Then supermodularity requires:

$$\begin{aligned}&U(\theta ,a,\triangle w_{h'}^{'},\triangle w^{i'}, \triangle w_{-hh'})-U(\theta ',a,\triangle w_{h'}^{'},\triangle w^{i},\triangle w_{-hh'})\ge \\&U(\theta ,a,\triangle w_{h'},\triangle w^{i'},\triangle w_{-hh'})-U(\theta ,a,\triangle w_{h'},\triangle w^{i},\triangle w_{-hh'}). \end{aligned}$$

Observe that this inequality is equal to:

$$\begin{aligned} (\triangle w_{h'}^{'}-\triangle w_{h'})P^{i}(\theta ,a)\ge (\triangle w_{h'}^{'}-\triangle w_{h'})P^{i}(\theta ,a). \end{aligned}$$

Because $P^{i}(\theta ,a)>0$ and $\triangle w^{i'}>\triangle w^{i}$, this holds for all $i\in I$. This proves the claim. $\square$

Proof of Proposition 1

We restrict attention to actions and contracts that are piecewise continuously differentiable. The agent’s problem in terms of his revelation of type $\theta '$ and choice of action profile $a'$, or the equivalent in terms of his choice from the menu of contracts offered by the principal, is:

$$\begin{aligned} \max _{\theta '\in \Theta ,a'\in A}U( \theta ,w(\theta '),a'). \end{aligned}$$

For any reported type $\theta '\in \Theta$, the first-order condition for a for any type-$\theta$ is given by:

$$\begin{aligned}&\sum \nolimits _{i}p^{i}_{a}(\theta ,a')w^{i}(\theta ')- c_{a}(a')\le 0,\; a'\ge 0\;\forall \theta \in \Theta \;\mathrm {and}\;\nonumber \\&\left( \sum \nolimits _{i}p^{i}_{a}(\theta ,a')w^{i}(\theta ')- c_{a}(a')\right) a=0. \end{aligned}$$

(A.1)

Let the solution to this problem be $\alpha (\theta ,\theta ')$, and when $\theta '=\theta$ the solution is denoted by $a(\theta )$. Supermodularity of $U(\theta ,w,a)$ in $(\theta ,w,a)$ ensures its existence. If $\alpha (\theta ,\theta ')>0$, because the objective function is jointly continuous in $(\theta ,a)$, this implies that $\alpha (\theta ,\theta ')$ is continuous in $(\theta ,\theta ')$. Conditions MONA and CP ensure that, for any contract for which $\triangle w^{i}(\theta ')>0$ for some i, $U(\theta ,w,a)$ is strictly concave in a and $\alpha (\theta ,\theta ')>0$ for all $\theta ,\theta '\in \Theta$. Thus, local conditions are necessary and sufficient for $\alpha (\theta ,\theta ')$ to be the optimal effort for all $\theta ,\theta '\in \Theta$. To see this result, observe that:

$$\begin{aligned} U(\theta ,w(\theta '),a)=w_{0}+\sum \nolimits _{i=0}^{n-1}\triangle w^{i}(\theta ')P^{i}(\theta , a)-c(a) \end{aligned}$$

Thus,

$$\begin{aligned} U_{a}(\theta ,w(\theta '),a)=\sum \nolimits _{i=0}^{n-1}\triangle w^{i}(\theta ')P_{a}^{i}(\theta , a)-c_{a}(a) \end{aligned}$$

and

$$\begin{aligned} U_{aa}(\theta ,w(\theta '),a)=\sum \nolimits _{i=0}^{n-1}\triangle w^{i}(\theta ')P_{aa}^{i}(\theta , a)-c_{aa}(a) \end{aligned}$$

Because assumption CP ensures that $P^{i}(\theta , a)$ is strictly concave in a and c(a) is convex in a, this, together with MONA, implies that $U(\theta ,w(\theta '),a)$ is strictly concave in a for all $\theta ,\theta '\in \Theta$. Together with the Inada’s type conditions assumed in assumption CP, this ensures that $\alpha (\theta ,\theta ')>0$ for all $\theta ,\theta '\in \Theta$.

Equation (A.1) is a necessary condition for optimality of $\alpha (\theta ,\theta ')$. We also need the following to hold for all $\theta \in \Theta$,

$$\begin{aligned} U_{aa}(\theta , w(\theta '),\alpha (\theta ,\theta '))|_{\theta '=\theta }\le 0. \end{aligned}$$

(A.2)

It follows from the above that:

$$\begin{aligned} U_{aa}(\theta , w(\theta '),\alpha (\theta ,\theta '))\frac{ \partial \alpha (\theta ,\theta ')}{\partial \theta }+\sum \nolimits _{i}p^{i}_{a\theta }(\theta ,\alpha (\theta ,\theta '))w^{i}(\theta ')=0 \end{aligned}$$

(A.3)

and

$$\begin{aligned} U_{aa}(\theta , w(\theta '),\alpha (\theta ,\theta '))\frac{ \partial \alpha (\theta ,\theta ')}{\partial \theta '}+\sum \nolimits _{i}p^{i}_{a}(\theta ,\alpha (\theta ,\theta '))\frac{dw^{i}(\theta ')}{d\theta '}=0 \end{aligned}$$

(A.4)

Maximizing at a point of differentiability yields the first-order condition with respect to $\theta '$; for almost all $\theta \in \Theta$,

$$\begin{aligned}&\sum \nolimits _{i}p^{i}(\theta ,\alpha (\theta ,\theta '))\frac{d w^{i}(\theta ')}{d \theta '}\nonumber \\&\quad + \Big (\sum \nolimits _{i}p^{i}_{a}(\theta ,\alpha (\theta ,\theta '))w^{i}(\theta ')- c_{a}(\alpha (\theta ,\theta '))\Big )\frac{\partial \alpha (\theta ,\theta ')}{\partial \theta '}\Big |_{\theta '=\theta }=0 \end{aligned}$$

(A.5)

Using the first-order condition for the action profile, and assuming that $\alpha (\theta ,\theta ')>0$, this can be re-written as follows: for almost all $\theta \in \Theta$,

$$\begin{aligned} \sum \nolimits _{i}p^{i}(\theta ,\alpha (\theta ,\theta '))\frac{d w^{i}(\theta ')}{d \theta '}\Big |_{\theta '=\theta }=0. \end{aligned}$$

(A.6)

This shows that Eq. (A.6) is necessary for implementability.

Next, we need to show that the indirect utility is concave in $\theta '$ when evaluated at $\theta '=\theta$. That is,

$$\begin{aligned} \frac{\partial ^{2}}{\partial \theta '\partial \theta '}U(\theta , w(\theta '),\alpha (\theta ,\theta ') )\Big |_{\theta '=\theta }\le 0. \end{aligned}$$

Because Eq. (A.6) holds as an identity in $\theta$ when evaluated at the optimal action $\alpha (\theta ,\theta ')$, one can total differentiate (A.6) to find:

$$\begin{aligned} \frac{\partial ^{2}}{\partial \theta '\partial \theta '}U(\theta , w(\theta '),\alpha (\theta ,\theta '))+\frac{\partial ^{2}}{\partial \theta '\partial \theta }U(\theta , w(\theta '),\alpha (\theta ,\theta '))\Big |_{\theta '=\theta }=0. \end{aligned}$$

(A.7)

This implies that, for almost all $\theta \in \Theta$, the local second-order condition can be written as:

$$\begin{aligned} \frac{\partial ^{2}}{\partial \theta '\partial \theta }U(\theta , w(\theta '),\alpha (\theta ,\theta '))\Big |_{\theta '=\theta }\ge 0 \end{aligned}$$

(A.8)

Its derivative is given by:

$$\begin{aligned}&\Bigg (\sum \nolimits _{i}p^{i}_{\theta }(\theta ,\alpha (\theta ,\theta '))\frac{d w^{i}(\theta ')}{d \theta '}+ \sum \nolimits _{i}p^{i}_{a\theta }(\theta ,\alpha (\theta ,\theta '))w^{i}(\theta ')\frac{\partial \alpha (\theta ,\theta ')}{\partial \theta '}\\&\quad +\Big (U_{aa}(\theta , w(\theta '),\alpha (\theta ,\theta '))\frac{ \partial \alpha (\theta ,\theta ')}{\partial \theta '}+\sum \nolimits _{i}p^{i}_{a}(\theta ,a')\frac{d w^{i}(\theta ')}{d \theta '}\Big )\frac{ \partial \alpha (\theta ,\theta ')}{\partial \theta } \\&\quad +U_{a}(\theta , w(\theta '),\alpha (\theta ,\theta '))\frac{ \partial ^{2} \alpha (\theta ,\theta ')}{\partial \theta '\partial \theta }\Bigg )\Big |_{\theta '=\theta }\ge 0. \end{aligned}$$

Noticing that $U_{a}(\theta , w(\theta '),\alpha (\theta ,\theta '))=0$, and that the second term multiplying $\frac{ \partial \alpha (\theta ,\theta ')}{\partial \theta }$ is zero due to Eq. (A.4) and substituting Eq. (A.3) into this, we deduce that the local second-order condition is as follows:

$$\begin{aligned}&\frac{1}{U_{aa}(\theta , w(\theta '),\alpha (\theta ,\theta '))}\Bigg (\sum \nolimits _{i}p^{i}_{\theta } (\theta ,\alpha (\theta ,\theta '))\frac{d w^{i}(\theta ')}{d \theta '}U_{aa}(\theta , w(\theta '),\alpha (\theta ,\theta '))\nonumber \\&-\sum \nolimits _{i}p^{i}_{a\theta }(\theta ,\alpha (\theta ,\theta ')) w^{i}(\theta ')\sum \nolimits _{i}p^{i}_{a}(\theta ,\alpha (\theta ,\theta ')) \frac{dw^{i}(\theta ')}{d\theta '}\Bigg )\Big |_{\theta '=\theta }\ge 0. \end{aligned}$$

(A.9)

Because $U_{aa}(\theta , w(\theta '),\alpha (\theta ,\theta '))\le 0$, a necessary condition for implementability is that the term inside the parenthesis in Eq. (A.9) is negative. This gives rise to the equation in the main text.

Global Incentive Constraints: Suppose next that (A.1), (A.2) (A.6) and (A.9) hold. Then, it must be the case that all agent’s incentive-compatibility conditions hold. To see this result, suppose by contradiction that for at least one type $\theta$ the agent’s incentive compatibility is violated. Then there exists a type $\theta '\ne \theta$, such that the following holds:

$$\begin{aligned} \sum \nolimits _{i}p^{i}(\theta ,\alpha (\theta ,\theta '))w^{i}(\theta ')- c(\alpha (\theta ,\theta '))>\sum \nolimits _{i}p^{i}(\theta ,a(\theta ))w^{i}(\theta )-c(a(\theta )). \end{aligned}$$

Integrating this re-writes as follows:

$$\begin{aligned} \int _{\theta '}^{\theta }\left( \sum \nolimits _{i}p^{i}(\theta ,a(s))\frac{d w^{i}(s)}{ds}{+}\left( \sum \nolimits _{i}p^{i}_{a}(\theta ,a(s))w^{i}(s)-c_{a}(a(s))\right) \frac{d a(s)}{d s}\right) ds<0. \end{aligned}$$

The local incentive constraints, together with (A.9), implies that, if $\theta >\theta '$:

$$\begin{aligned} \int _{\theta '}^{\theta }\Big (\sum \nolimits _{i}p^{i}(\theta ,a(s))\frac{d w^{i}(s)}{ds}+\Big (\sum \nolimits _{i}p^{i}_{a}(\theta ,a(s))w^{i}(s)-c_{a}(a(s))\Big )\frac{d a(s)}{d s}\Big )ds\ge 0, \end{aligned}$$

which leads to a contradiction. If $\theta <\theta '$, the same reasoning leads us to a similar contradiction. This shows that Eqs. (A.1), (A.6), (A.9) are sufficient conditions for incentive compatibility. $\square$

Proof of Proposition 2

It readily follows from Theorem 2 in Milgrom and Segal (2002) that:

$$\begin{aligned} {\dot{U}}(\theta )=U(\underline{\theta })+ \int _{\underline{\theta }}^{\theta }P^{i}_{\theta }(x,a(x))\triangle w^{i}(x)f(x)dx, \end{aligned}$$

where $P^{i}_{\theta }(\theta , a)\equiv \sum \nolimits _{j\ge i+1}p_{\theta }^{j}(\theta , a)\ge 0,\;\forall i\in I\setminus \{n\}$ and the inequality is due to FOSD.

After summation by parts the virtual surplus can be re-written as follows:

$$\begin{aligned} \Phi (\theta ,w(\theta ),a(\theta ))=(S(\theta ,a(\theta ))-H(\theta ) \sum \nolimits _{i\ne n}P^{i}_{\theta }(\theta ,a(\theta ))\triangle w^{i}(\theta ))f(\theta )-U(\underline{\theta }). \end{aligned}$$

Observe also that the agent’s first-order condition for the action re-writes as follows:

$$\begin{aligned} \sum \nolimits _{i\ne n}P^{i}_{a}(\theta ,a(\theta ))\triangle w^{i}(\theta )-c_{a}(a(\theta ))=0, \end{aligned}$$

where $P^{i}_{a}(\theta , a)\equiv \sum \nolimits _{j\ge i+1}p_{a}^{j}(\theta , a)\ge 0,\;\forall i\in I\setminus \{n\}$ due to FOSD.

Let $\gamma (\theta )$ be the Lagrange multiplier for the first-order condition for the action, $\delta ^{i}(\theta )$ be the multiplier corresponding to MONA, and $\eta ^{i}(\theta )$ be the multiplier corresponding to MONP. Then, for each $\theta \in \Theta$, the first-order condition with respect to $\triangle w^{i}$ evaluated at the optimal mechanism, denoted by $({\hat{w}}(\theta ),{\hat{a}}(\theta ))$, is as follows:

$$\begin{aligned}&-H(\theta )P^{i}_{\theta }(\theta ,{\hat{a}}(\theta ))f(\theta )+\gamma (\theta ) P^{i}_{a}(\theta ,{\hat{a}}(\theta ))+ \delta ^{i}(\theta )-\eta ^{i}(\theta )=0, \end{aligned}$$

(A.10)

$$\begin{aligned}&\delta ^{i}(\theta )\ge 0,\; \delta ^{i}(\theta )\triangle {\hat{w}}^{i}(\theta )=0,\;\triangle {\hat{w}}^{i}(\theta )\ge 0,\forall \;i\in I\setminus \{n\}, \end{aligned}$$

(A.11)

$$\begin{aligned}&\eta ^{i}(\theta )\ge 0, \; \eta ^{i}(\theta )(\triangle y^{i}-\triangle {\hat{w}}^{i}(\theta ))=0,\;\triangle {\hat{w}}^{i}(\theta )\le \triangle y^{i}, \forall \;i\in I\setminus \{n\}, \end{aligned}$$

(A.12)

and the one with respect to a is given by:

$$\begin{aligned}&\Big (S_{a}(\theta ,{\hat{a}}(\theta ))-H(\theta ) \sum \nolimits _{i\ne n}P^{i}_{\theta a}(\theta ,{\hat{a}}(\theta ))\triangle {\hat{w}}^{i}(\theta )\Big )f(\theta )\nonumber \\&\quad +\gamma (\theta )\Big (\sum \nolimits _{i=0}^{n-1}P^{i}_{aa}(\theta ,{\hat{a}}(\theta ))\triangle {\hat{w}}^{i}(\theta )-c_{aa}({\hat{a}}(\theta ))\Big )=0. \end{aligned}$$

(A.13)

It readily follows from Eq. (A.10) that, for all $\theta \in \Theta$, $\triangle {\hat{w}}^{i}(\theta )=\triangle y^{i}$ if

$$\begin{aligned} \gamma (\theta )\frac{P^{i}_{a}(\theta ,{\hat{a}}(\theta ))}{P^{i}_{\theta }(\theta ,{\hat{a}}(\theta ))}\ge H(\theta )f(\theta ), \end{aligned}$$

and $\triangle {\hat{w}}^{i}(\theta )=0$ otherwise. To save on notation, define $\xi ^{i}(\theta ,a)\equiv \gamma (\theta )\frac{P^{i}_{a}(\theta ,a)}{P^{i}_{\theta }(\theta ,a)}$ and $\xi (\theta )=H(\theta )f(\theta )$. Then, $\triangle {\hat{w}}^{i}(\theta )=\triangle y^{i}$ if $\xi ^{i}(\theta ,a)\ge \xi (\theta )$ and $\triangle {\hat{w}}^{i}(\theta )=0$ otherwise.

It readily follows from (A.13) that:

$$\begin{aligned} \gamma (\theta )= -\frac{S_{a}(\theta ,{\hat{a}}(\theta ))- H(\theta )\sum \nolimits _{i\ne n}P^{i}_{\theta a}(\theta ,{\hat{a}}(\theta ))\triangle {\hat{w}}^{i}(\theta )}{\sum \nolimits _{i\ne n}P^{i}_{aa}(\theta ,{\hat{a}}(\theta ))\triangle {\hat{w}}^{i}(\theta )-c_{aa}({\hat{a}}(\theta ))}f(\theta ). \end{aligned}$$

(A.14)

Multiplying the first-order condition in Eq. (A.10) by $\triangle {\hat{w}}^{i}(\theta )$, summing over all i and substituting into the first-order condition for the action, one gets that:

$$\begin{aligned} \gamma (\theta ) c_{a}({\hat{a}}(\theta ))=H(\theta ) \sum \nolimits _{i=0}^{n-1}P^{i}_{\theta }(\theta ,{\hat{a}}(\theta ))\triangle {\hat{w}}^{i}(\theta )f(\theta )+\sum \nolimits _{i=0}^{n-1}\eta ^{i}(\theta )\triangle {\hat{w}}^{i}(\theta )\ge 0. \end{aligned}$$

It readily follows from this that $\gamma (\theta )> 0$ for all $\theta \in \Theta \setminus {{\bar{\theta }}}$, since there exists an $i\in I\setminus \{n\}$, such that $\triangle {\hat{w}}^{i}(\theta )>0$; otherwise ${\hat{a}}(\theta )=0$. We deduce from the FOC in Eq. (A.13), that $S_{a}(\theta ,a(\theta ))>0$ for all $\theta \in \Theta \setminus {{\bar{\theta }}}$ and, therefore, ${\hat{a}}(\theta )<a^{**}(\theta )$ for all $\theta \in \Theta \setminus {{\bar{\theta }}}$.

Setting $H({\bar{\theta }})=0$ in Eq. (A.10), we get that: $\gamma ({\bar{\theta }})P^{i}_{a}({\bar{\theta }},{\hat{a}}({\bar{\theta }}))+ \delta ^{i}({\bar{\theta }})=\eta ^{i}({\bar{\theta }})$. Suppose that $\gamma ({\bar{\theta }})>0$, then because $P^{i}_{a}({\bar{\theta }},{\hat{a}}({\bar{\theta }}))>0$, $\eta ^{i}({\bar{\theta }})>0$ for all $i\in I\setminus \{n\}$ and therefore $\triangle {\hat{w}}^{i}({\bar{\theta }})=\triangle y^{i}$ for all $i\in I\setminus \{n\}$. We deduce from the FOC in Eq. (A.13) that ${\hat{a}}({\bar{\theta }})=a^{**}({\bar{\theta }})$ and, therefore, $S_{a}({\bar{\theta }},a({\bar{\theta }}))=0$. It follows from Eq. (A.14), that this implies that $\gamma ({\bar{\theta }})=0$, which is a contradiction. Suppose that $\gamma ({\bar{\theta }})=0$, then $\delta ^{i}({\bar{\theta }})=\eta ^{i}({\bar{\theta }})=0$ for all $i\in I\setminus \{n\}$. It follows from Eq. (A.14) that $S_{a}({\bar{\theta }},a({\bar{\theta }}))=0$, and therefore ${\hat{a}}({\bar{\theta }})=a^{**}({\bar{\theta }})$.

Observe that condition CP, together with the following assumption that $\frac{P^{i}_{a\theta }(\theta ,a(\theta ))}{P^{i}_{a}(\theta ,a(\theta ))}$ is non-decreasing with a, ensures concavity of $\Phi (\cdot )$ with respect to a. A sufficient condition for this is that $P^{i}_{aa\theta }(\theta ,a)\ge 0$ for all $i\in I\setminus \{n\}$. In addition, notice that the optimal action increases with $\theta$ only if conditions SC and CP, together with the following assumption that $\frac{P^{i}_{a\theta }(\theta ,a(\theta ))}{P^{i}_{a}(\theta ,a(\theta ))}$ is non-increasing with $\theta$ hold. A sufficient condition for this is that $P^{i}_{a\theta \theta }(\theta ,a)\le 0$ for all $i\in I\setminus \{n\}$, since $H(\theta )$ is non-increasing with $\theta$. Also, observe that the determinant of the Hessian regarding $\Phi (\theta ,w,a)$ is equal to zero due to the linearity of the objective function with respect to $\triangle w^{i}$. $\square$

Proof of Proposition 3

Assume that $p(\theta ,a)$ satisfies WSEP, and define $Q^{i}=\sum \nolimits _{j\ge i+1}(q^{j}-r^{j})$. Then, it is easy to see that the principal’s problem can be written as follows:

$$\begin{aligned}&\max _{(w,a):\Theta \rightarrow \mathcal {C}\times A} \int _{\theta \in \Theta } \Big (S(\theta ,a(\theta )){-}H(\theta )g_{\theta }(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}(\theta )\Big )f(\theta )d\theta {-}U({\underline{\theta }})\\&\text { subject to }\\&\sum \nolimits _{i\ne n}Q^{i}\triangle {\dot{w}}^{i}(\theta )\ge 0,\\&g_{a}(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}(\theta )-c_{a}(a(\theta ))=0,\\&\triangle w^{i}(\theta )\in [0,\triangle y^{i}],\;\forall \;i\in I\setminus \{n\},\\&U(\underline{\theta })\ge L. \end{aligned}$$

Suppose that there exists an incentive-compatible mechanism $(w(\theta ),a(\theta ))$; we will then show that there is another mechanism that pays the same to everyone, denoted by $w^{*}$, and asks each agent to exert the effort $a^{*}(\theta )\equiv \mathop {\text {argmax}}\nolimits _{a\in A}U(\theta ,w^{*},a)$ which increases the principal’s payoff pointwise and is incentive compatible.

Let $(w(\theta ),a(\theta ))$ be an incentive-compatible mechanism and let contract $w^{*}$ be the contract with $w^{0*}=\inf \{w({\hat{\theta }}): {\hat{\theta }}\in \Theta \}$ and $\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i*}=\sup \big \{\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i*}({\hat{\theta }}): {\hat{\theta }}\in \Theta \big \}$. If $w^{0*}>L$, reducing all payments uniformly will keep all constraints satisfied and will increase the principal’s expected payoff. Thus, we can assume that $w^{0*}=L$. In addition, either there is a contract in the menu such that the $\sup$ and $\inf$ are attained, or $w^{*}$ is a limit point of $\{w({\hat{\theta }}): {\hat{\theta }}\in \Theta \}$ due to the piecewise continuity of $w({\hat{\theta }})$.

Because $w(\theta )$ is incentive compatible, holding the chosen effort constant, the following holds:

$$\begin{aligned}&w_{0}(\theta )+\sum \nolimits _{i=0}^{n-1}\triangle w^{i}(\theta )g(\theta , a(\theta )))Q^{i}-c(a(\theta ))\\&\quad \ge L+\sum \nolimits _{i=0}^{n-1}\triangle w^{i*}g(\theta , a(\theta ))Q^{i}-c(a(\theta )),\;\forall \theta \in \Theta . \end{aligned}$$

It readily follows from this that, for all $\theta \in \Theta$,

$$\begin{aligned}&w_{0}(\theta )-L+\sum \nolimits _{i=0}^{n-1}(\triangle w^{i}(\theta )-\triangle w^{i*})g(\theta , a(\theta )))Q^{i}\ge 0 \end{aligned}$$

(A.15)

We deduce from the above that expected compensation is higher under the incentive-compatible mechanism that under the alternative menu $w^{*}$ whenever the agent chooses the incentive-compatible action $a(\theta )$ . When the agent chooses the action $a^{*}(\theta )\in \mathop {\text {argmax}}\nolimits _{a\in A} U(\theta , w^{*},a)$, we have the following: for all $\theta \in \Theta$,

$$\begin{aligned}&y^{0}-w^{0}(\theta )+\sum \nolimits _{i=0}^{n-1}(\triangle y-\triangle w^{i}(\theta ))g(\theta , a(\theta )))Q^{i}\\&\qquad -\Big ( y^{0}-L+\sum \nolimits _{i=0}^{n-1}(\triangle y-\triangle w^{i*})g(\theta , a^{*}(\theta ))Q^{i}\Big )\\&\quad = L-w^{0}(\theta )+ \sum \nolimits _{i=0}^{n-1}(\triangle y-\triangle w^{i*})(g(\theta , a(\theta ))-g(\theta , a^{*}(\theta )))Q^{i} \\&\qquad +g(\theta ,a(\theta ))\sum \nolimits _{i=0}^{n-1}(\triangle w^{i*}-\triangle w^{i}(\theta ))Q^{i}\\&\quad \le 0, \end{aligned}$$

where the inequality follows from Eq. (A.15), the fact that FOSD, CP and SC imply that $a^{*}(\theta )$ exists and $a^{*}(\theta )\ge a(\theta )$ and MONP implies that $\triangle y-\triangle w^{i*}\ge 0$ for all $i\in I$. Thus, the principal’s payoff is higher when menu $w^{*}$ is offered than when any other incentive-compatible menu is offered.

Taking into account the fact that the incentive-compatible contract that yields the largest payoff to the principal is type-independent, the principal’s problem becomes the following:

$$\begin{aligned}&\max _{(w,a):\Theta \rightarrow \mathcal {C}\times A} \int _{\theta \in \Theta } \Big (S(\theta ,a(\theta ))-H(\theta )g_{\theta }(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}\Big )f(\theta )d\theta -U({\underline{\theta }})\\&\text { subject to }\\&g_{a}(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}-c_{a}(a(\theta ))=0, \\&\triangle w^{i}\in [0,\triangle y^{i}],\;\forall \;i\in I\setminus \{n\},\\&U(\underline{\theta })\ge L, \end{aligned}$$

since for a type-independent contract trivially satisfies the ignored constraint: $\sum \nolimits _{i\ne n}Q^{i}\triangle {\dot{w}}^{i}(\theta )\ge 0$.

It is easy to see that it is optimal to set $U({\underline{\theta }})= L$.

The first-order condition for $w^{i}$ is given by:

$$\begin{aligned} \big (-H(\theta )g_{\theta }(\theta ,a(\theta ))+ \gamma (\theta )g_{a}(\theta ,a(\theta ))\big )Q^{i}+ \delta ^{i}(\theta )-\eta ^{i}(\theta )=0 \end{aligned}$$

(A.16)

and that for a is given by

$$\begin{aligned}&S_{a}(\theta ,a)f(\theta )- H(\theta )g_{a\theta }(\theta ,a(\theta ))f(\theta )\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}\nonumber \\&\quad +\gamma (\theta )\big (g_{aa}(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}-c_{aa}(a(\theta ))\big )=0. \end{aligned}$$

(A.17)

It readily follows from Eq. (A.16) that, if $H(\theta )g_{\theta }(\theta ,a(\theta ))f(\theta )- \gamma (\theta )g_{a}(\theta ,a(\theta ))>0$, then $\delta ^{i}(\theta )>0$ for all $i\in I\setminus \{n\}$. Therefore, $\triangle w^{i}=0$ for all $i\in I\setminus \{n\}$, which implies that $a(\theta )=0$ for all $\theta \in \Theta$. This can never be optimal. Thus, $H(\theta )g_{\theta }(\theta ,a(\theta ))f(\theta )- \gamma (\theta )g_{a}(\theta ,a(\theta ))\le 0$. This requires that $\gamma (\theta )\ge 0$ for all $\theta \in \Theta$.

It readily follows from this and Eq. (A.17) that:

$$\begin{aligned} \gamma (\theta )= -\frac{S_{a}(\theta ,a(\theta ))- H(\theta )g_{a\theta }(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}}{g_{aa}(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}-c_{aa}(a(\theta ))}f(\theta ), \;\forall \theta \in \Theta ,\nonumber \\ \end{aligned}$$

(A.18)

since $g_{aa}(\theta ,a(\theta ))\sum \nolimits _{i\ne n}Q^{i}\triangle w^{i}-c_{aa}(a(\theta ))<0$.

This implies that $S_{a}(\theta ,a(\theta ))> 0,\;\forall \theta \in \Theta \setminus \{{\bar{\theta }}\}$. Hence, the optimal action, denoted by $a^{*}(\theta )$, satisfies the following: $a^{*}(\theta )<a^{**}(\theta ), \;\forall \theta \in \Theta \setminus \{{\bar{\theta }}\}$.

Suppose that $H(\theta )g_{\theta }(\theta ,a(\theta ))f(\theta )- \gamma (\theta )g_{a}(\theta ,a(\theta ))< 0$, then it readily follows from Eq. (A.16) that $\eta ^{i}(\theta )>0$ for all $i\in \setminus \{n\}$, which implies that $\triangle w^{i}=\triangle y^{i}$ for all $i\in I\setminus \{n\}$. This, together with the agent’s first-order condition in Eq. (A.17), implies that $a^{*}(\theta )=a^{**}(\theta ), \;\forall \theta \in \Theta$. This contradicts the fact that if $\gamma (\theta )>0$, then $a^{*}(\theta )<a^{**}(\theta )$.

Thus, $H(\theta )g_{\theta }(\theta ,a(\theta ))f(\theta )-\gamma (\theta )g_{a}(\theta ,a(\theta ))=0$, then $\eta ^{i}(\theta )=\delta ^{i}(\theta )=0$ and $\triangle w^{i}\in [0,\triangle y^{i}]$. This implies that $\gamma (\theta )>0$ for all $\theta \in \Theta \setminus \{{\bar{\theta }}\}$ and $\gamma ({\bar{\theta }})=0$. Thus, $a^{*}(\theta )<a^{**}(\theta ), \;\forall \theta \in \Theta \setminus \{{\bar{\theta }}\}$ and $a^{*}({\bar{\theta }})=a^{**}({\bar{\theta }})$.

Observe that condition CP, together with the following assumption that $\frac{g_{a\theta }(\theta ,a(\theta ))}{g_{a}(\theta ,a(\theta ))}$ is non-decreasing with a, ensures concavity of $\Phi (\cdot )$ with respect to a. A sufficient condition for this is that $g_{aa\theta }(\theta ,a)\ge 0$ for all $i\in I\setminus \{n\}$. In addition, notice that the optimal action increases with $\theta$ only if conditions SC and CP, together with the following assumption that $\frac{P^{i}_{a\theta }(\theta ,a(\theta ))}{P^{i}_{a}(\theta ,a(\theta ))}$ is non-increasing with $\theta$ hold.

A sufficient condition for this is that $P^{i}_{a\theta \theta }(\theta ,a)\le 0$ for all $i\in I\setminus \{n\}$, since $H(\theta )$ is non-increasing with $\theta$. Also, observe that the determinant of the Hessian regarding $\Phi (\theta ,w,a)$ is equal to zero due to the linearity of the objective function with respect to $\triangle w^{i}$. $\square$

Proof of Proposition 4

Let $a(\theta )$ be the incentive-compatible action and w be a type-independent mechanism that implements $a(\theta )$. Because under WSEP the agent’s incentive-compatible action satisfies the following:

$$\begin{aligned} -g_{a}(\theta ,a(\theta ))\sum \nolimits _{i=0}^{n-1}\sum \nolimits _{j=0}^{i}(q^{j}-r^{j})\triangle w^{i}-c_{a}(a(\theta ))=0, \end{aligned}$$

then, any other type-independent mechanism ${\tilde{w}}$ that satisfies the following:

$$\begin{aligned} \sum \nolimits _{i=0}^{n-1}\sum \nolimits _{j=0}^{i}(q^{j}-r^{j})\triangle {\tilde{w}}^{i}=W\equiv \sum \nolimits _{i=0}^{n-1}\sum \nolimits _{j=0}^{i}(q^{j}-r^{j})\triangle w^{i} \end{aligned}$$

(A.19)

is also incentive-compatible. Hence, any incentive-compatible, type-independent mechanism must solve the following:

$$\begin{aligned}&\mathop {\text {max}}\limits _{w\in \mathcal {C}}\int _{\theta \in \Theta }V(\theta ,a(\theta ),w)d\theta \text { subject to } (A.19). \end{aligned}$$

Let $\lambda$ be the Lagrange multiplier for constraint (A.19), $\delta ^{i}$ the Lagrange multiplier for MONA, and $\eta ^{i}$ the Lagrange multiplier for MONP. Then, the first-order condition is given by:

$$\begin{aligned} -\int _{\theta \in \Theta }(1-P^{i}(\theta ,a(\theta ))dF(\theta )+\lambda \sum \nolimits _{j=0}^{i}(q^{j}-r^{j})+\delta ^{i}-\eta ^{i}=0 \end{aligned}$$

(A.20)

Because $\lambda$ is independent of i, and the first-order condition is independent of w, the solution is bang-bang. Observe that, if $\delta ^{i}\ge 0$, then $\eta ^{i}=0$ and vice versa. It is easy to see that FOSD implies that:

$$\begin{aligned} \delta ^{i}\ge 0 \Leftrightarrow \frac{\int _{\theta \in \Theta }(1-P^{i}(\theta ,a(\theta ))dF(\theta )}{\sum \nolimits _{j=0}^{i}(q^{j}-r^{j})}-\lambda \le 0, \end{aligned}$$

and $\eta ^{i}>0$ otherwise. Hence, if the ratio on the RHS is increasing in i, the optimal contract is option-like. Observe that, for any h,

$$\begin{aligned} \frac{\int _{\theta \in \Theta }(1-P^{i+1}(\theta ,a(\theta ))dF(\theta )}{\sum \nolimits _{j=0}^{i+1}(q^{j}-r^{j})}\ge \frac{\int _{\theta \in \Theta }(1-P^{i}(\theta ,a(\theta ))dF(\theta )}{\sum \nolimits _{j=0}^{i}(q^{j}-r^{j})}. \end{aligned}$$

After a few steps of simple algebra, it can be easily shown that this is equivalent to:

$$\begin{aligned} (q_{i+1}-r_{i+1})(1-\sum \nolimits _{j=0}^{i}r^{j})+ r_{i+1}\sum \nolimits _{j=0}^{i}(q^{j}-r^{j})\le 0. \end{aligned}$$

Define $l^{i}\equiv q^{i}/r^{i}$. Notice that EMLRP implies that, for any $i\in I\setminus \{n\}$, $l^{i+1}\ge l^{i}$. Using this notation, the last equation re-writes as follows:

$$\begin{aligned}&r^{i+1}\big ((l^{i+1}-1)(1-\sum \nolimits _{j=0}^{i}r^{j})+\sum \nolimits _{j=0}^{i}r^{j}(l^{j}-1)\big )\le 0\\&\quad \Leftrightarrow l^{i+1}\le 1+\sum \nolimits _{j=0}^{i}r^{j}(l^{i+1}-l^{j}) \end{aligned}$$

Because of EMLRP, the RHS and the LHS increase with i. In addition, it is easy to check that, at $i=n-1$, the RHS is equal to the LHS and is lower than the RHS at $i=0$, and the LHS increases at a rate $l^{i+2}-l^{i+1}$, while the RHS does so at a rate $r^{i+1}(l^{i+2}-l^{i+1})$. Hence, putting this together we get that the inequality above holds for all $i\in I\setminus \{n\}$. $\square$

Proof of Proposition 5

The agent’s problem in terms of his revelation of type $\theta '$, or the equivalent in terms of his choice from the menu of contracts offered by the principal, is:

$$\begin{aligned} \max _{\theta '\in \Theta }U( w(\theta '),a(\theta '),\theta ). \end{aligned}$$

The first-order condition is given by:

$$\begin{aligned}&\Big (\sum \nolimits _{i}p^{i}(\theta ,a(\theta '))\frac{d w^{i}(\theta ')}{d \theta '}+\Big (\sum \nolimits _{i}p^{i}_{a}(\theta ,a(\theta '))w^{i}(\theta ')\nonumber \\&\quad - c_{a}(a(\theta '))\Big )\frac{d a(\theta ')}{d \theta '}\Big )\Big |_{\theta '=\theta }=0 ,\;\forall \theta \in \Theta . \end{aligned}$$

(A.21)

Because this holds as an identity in $\theta$, we can do a total diferentiation to find:

$$\begin{aligned} \Big (\frac{\partial U( \theta ,w(\theta '),a(\theta '))}{\partial \theta '\partial \theta '}+ \frac{\partial U(\theta , w(\theta '),a(\theta '))}{\partial \theta '\partial \theta }\Big )\Big |_{\theta '=\theta }=0 \end{aligned}$$

(A.22)

Because the second-order condition requires the first term to be non-positive, this implies that:

$$\begin{aligned}&\Big (\sum \nolimits _{i}p^{i}_{\theta }(\theta ,a(\theta '))\frac{d w^{i}(\theta ')}{d \theta '}+\sum \nolimits _{i}p^{i}_{a\theta }(\theta ,e(\theta '))w^{i}(\theta ')\frac{d a(\theta ')}{d \theta '}\Big )\Big |_{\theta '=\theta }\ge 0. \end{aligned}$$

(A.23)

The missing details of the proof resemble that of proposition 1 and, for the sake of brevity, are omitted. $\square$

Pure moral hazard case: not to be published

In this subsection, we analyze the case in which the agent’s ability is known to both the principal and the agent before signing the contract, but the action is not observed by the principal.¹⁷

Let the likelihood ratio be $\frac{p^{i}(\theta ,a')}{p^{i}(\theta ,a)}$ for any $a'>a$. In this section we assume the following:

[MLRP]:

$\frac{p^{i}(\theta ,a')}{p^{i}(\theta ,a)}$ increases with i for all $\theta \in \Theta$.

This ensures that any principal who has a payoff function increasing in output prefers the stochastic distribution of returns induced by higher actions. We will show here that the optimal contract under this assumption is option-like, as in Innes (1990) and Matthews (2001).

Suppose that an agent of type $\theta \in \Theta$ is faced with contract $w\in \mathcal {C}$ and chooses action $a\in A$; his expected utility is then given by:

$$\begin{aligned} U( \theta ,w,a) \equiv p(\theta ,a)w-c(a). \end{aligned}$$

(B.1)

Observe that the linearity of the agent’s payoff function in Eq. (B.1) with respect to payments implies that if the principal could offer a contract in which payments are unbounded, the first-best action could be implemented by offering a contract that pays the output minus a face value after each output realization. However, this is prevented by the limited-liability constraint.

Contract w induces an agent of type $\theta$ to choose action $a\in A$ if and only if the following incentive-compatibility constraint holds:

$$\begin{aligned} U( \theta ,w,a)\ge U(\theta ,w,a'),\;\; \forall a'\in A, \end{aligned}$$

(B.2)

and induces him to participate if the following individually-rationality constraint holds:

$$\begin{aligned} U( \theta ,w,a)\ge 0. \end{aligned}$$

(B.3)

Let the principal’s expected utility be $V(\theta ,w,a)\equiv p(\theta ,a)(y-w)$. Then, the principal’s problem when faced with an agent of type $\theta \in \Theta$ is the following:

$$\begin{aligned} \max _{(w,a)\in \mathcal {C}\times A}&V(\theta ,w,a)\;\text {subject to}\;(B.2)\; \mathrm {and} \;(B.3) \end{aligned}$$

Let $a(\theta ,w)$ be largest element satisfying $a(\theta ,w)\in \mathop {\text {argmax}}\nolimits _{a\in A}U(\theta ,w,a)$. Then, the principal’s problem is given by:

$$\begin{aligned} \max _{w \in \mathcal {C}}&V(\theta ,w,a(\theta ,w)) \; \text { subject to } \; U(\theta ,w,a(\theta ,w))\ge 0. \qquad \qquad \qquad \text {P-MH} \end{aligned}$$

Proposition 6

(Pure Moral Hazard) Suppose CP and MLRP hold. Then,

• (i) The optimal contract is a call-option contract of the form $w^{i}(\theta )=\max \{0,y^{i}-{\bar{y}}(\theta )\}$, and the optimal action satisfies $a^{m}(\theta )\le a^{*}(\theta )$; and

• (ii) If SC holds, $a^{m}(\theta )$ rises with $\theta$ and ${\bar{y}}(\theta )$ falls with $\theta$.

Proof

Define the call-option contract with face value ${\bar{y}}$ by $o^{i}({\bar{y}})=\max \{y^{i}-{\bar{y}},0\},\;\forall i\in I$. Under this contract, the principal’s payoff is given by $d^{i}({\bar{y}})=y^{i}-o^{i}({\bar{y}})=\min \{y^{i},{\bar{y}}\}$. In addition, consider any other contract w inducing a return to the principal equal to $z^{i}=y^{i}-w^{i}$. Then, the following lemma, which is closely related to Lemma 1 in Matthews (2001), shows that $p(\theta ,a)d'$ and $p(\theta ,a)z'$ single-cross at the action a satisfying $dp(\theta ,a)=zp(\theta ,a)$, because the payments specified by d are at a maximum for low-output realizations and at a minimum for high-output realizations, given monotonicity constraints.

Lemma 2

Suppose that MLRP holds. Then, for any $(\theta ,a)\in \Theta \times A$ and $w\in \mathcal {C}$, and a call-option contract $o\ne w$, if $p(\theta ,a)o=p(\theta ,a)w$, then:

$$\begin{aligned} p(\theta ,a')o\lessgtr p(\theta ,a')w\text { for } a'\gtrless a \end{aligned}$$

Proof

Let $v=z-d$. Routine algebra proves the result if, for all $a'\in A$, the equality $vp(\theta ,a)=0$ implies that $v(p(\theta ,a')-p(\theta ,a))>0,\forall a'>a$, and $v(p(\theta ,a')-p(\theta ,a))<0,\forall a'<a$.¹⁸ Let $vp(\theta ,a)=0$. Since $v\ne 0$ and $p(\theta ,a)$ has full support, v takes positive and negative values. Let k be the largest i such that $v_i< 0$. Because o is a call-option contract and w satisfies LL, MONA and MONP, $v_i\le 0$ for all $i\le k$ and $v_i>0$ for all $i>k$. Thus, because v has a positive component (since $vp(\theta ,a)=0$), MLRP and $vp(\theta ,a)=0$ implies the following for all $a'>a$:

$$\begin{aligned} v(p(\theta ,a')-p(\theta ,a))&=\sum \nolimits _{i=0}^{n}\Big (\frac{p^{i} (\theta ,a')}{p^{i}(\theta ,a)}-1\Big )p^{i}(\theta ,a)v^{i}\\&>\sum \nolimits _{i=0}^{k}\Big (\frac{p^{k}(\theta ,a')}{p^{k}(\theta ,a)}-1\Big )p^{i}(\theta ,a)v^{i}\\&\quad +\sum \nolimits _{i=k+1}^{n}\Big (\frac{p^{k}(\theta ,a')}{p^{k}(\theta ,a)}-1\Big )p^{i}(\theta ,a)v^{i}\\&= \Big (\frac{p^{k}(\theta ,a')}{p^{k}(\theta ,a)}-1\Big )\sum \nolimits _{i=0}^{n}p^{i}(\theta ,a)v^{i}\\&=\Big (\frac{p^{k}(\theta ,a')}{p^{k}(\theta ,a)}-1\Big )vp(\theta ,a)=0. \end{aligned}$$

The argument is identical for all $a'<a$. $\square$

Lemma 3

Suppose that $a(\theta ,w)\in \mathop {\text {argmax}}\nolimits _{a\in A}U(\theta ,w,a)$, and that $a(\theta ',w)\in \mathop {\text {argmax}}\nolimits _{a\in A}U(\theta ',w,a)$, with $a(\theta ,w)$ in the interior of A. Then, (i) $a(\theta ', w)>a(\theta ,w)$ if $\theta '>\theta$, and $a(\theta ',w)<a(\theta ,w)$ if $\theta '<\theta$; and (ii) $a(\theta ,w)>a(\theta ,w)$ if $\triangle w^{i'}>\triangle w^{i}$, and $a(\theta ,w)<a(\theta ,w)$ if $\triangle w^{i'}<\triangle w^{i}$.

Proof

The proof follows directly from Lemma 1 and theorem 3 in Edlin and Shannon (1998). $\square$

This lemma establishes that the agent’s chosen action rises with the agent’s type as well as with any bonus $\triangle w^{i}$.

Lemma 4

A non-trivial optimal contract exists.

Proof

The proof here follows closely the one in Poblete and Spulber (2012). Let $a(o)\in \mathop {\text {argmax}}\nolimits _{a\in A}U(\theta ,a,o)$, $a(w)\in \mathop {\text {argmax}}\nolimits _{a\in A}U(\theta ,w,a)$, and the call-option contract o be such that $U(\theta ,a(w),o)=U(\theta ,a(w),w)$. Then, the single-crossing property of $U(\theta ,w,a)$ in Lemma 2 implies that: (i) $a(o)>a(w)$; (ii) Letting $U(\theta ,a(w),w)=\max _{a\in A}U(\theta ,w,a')$, single-crossing at a(w) implies that $U(\theta ,a(o),o)\ge U(\theta ,a(w),w)$; and (iii) $V(\theta ,a(o),o)>V(\theta ,a(w),w)$. This follows from the fact that $a(o)>a(w)$ implies that $V(\theta ,a(w),w)=V(\theta ,a(w),o)<V(\theta ,a(o),o)$, where the last inequality follows from MLRP and the fact that $y^{i}-o^{i}$ is increasing with h. From (ii) and (iii) contract o yields greater net benefits to the principal than contract w, and it satisfies the agent’s participation constraint. Hence, if an optimal contract exists, it must be a call-option contract.

Observe that the agent’s utility under a call-option contract $U(\theta ,a({\bar{y}}),o({\bar{y}}))$ is continuous and decreasing in ${\bar{y}}$ and supermodular in $(\theta , a,-{\bar{y}})$, as shown in Lemma 1. Let $A({\bar{y}})=\{a\in A|a\in \mathop {\text {argmax}}\nolimits _{x\in A}U(\theta ,x,o({\bar{y}}))\}$. It is easy to check that the set $A({\bar{y}})$ is closed and bounded. Furthermore, let $a({\bar{y}})$ be the largest element in $A({\bar{y}})$. Then monotone comparative statics show that $a({\bar{y}})$ is decreasing with ${\bar{y}}$, because boundedness and monotonicity $a({\bar{y}})$ is continuous except at countable number points.

The expected payoff of the principal is given by $V(\theta ,a,o({\bar{y}}))=\sum \nolimits _{i}p^{i}(\theta ,a)\min \{y^{i},{\bar{y}}\}$. Observe that $V(\theta ,a({\bar{y}}),o({\bar{y}}))$ is discontinuous only if $a({\bar{y}})$ is discontinuous. Because $a({\bar{y}})$ is decreasing, $V(\theta ,a({\bar{y}}),o({\bar{y}}))$ is decreasing at any discontinuity point. Let ${\tilde{V}}(\theta ,a({\bar{y}}),o({\bar{y}}))=\sup _{{\bar{y}}'\le {\bar{y}}}V(\theta ,a({\bar{y}}'),o({\bar{y}}))$. Hence, ${\tilde{V}}(\theta ,a({\bar{y}}),o({\bar{y}}))$ is non-decreasing and continuous in ${\bar{y}}$ because $V(\theta ,a({\bar{y}}),o({\bar{y}}))$ is decreasing at any discontinuity point. Consider now the following problem:

$$\begin{aligned} \max _{{\bar{y}}}{\tilde{V}}(\theta ,a({\bar{y}}),o({\bar{y}}))\text { subject to }U(\theta ,a({\bar{y}}),o({\bar{y}}))\ge 0 \end{aligned}$$

Clearly, this has a solution, since we are maximizing a continuous function on a closed set and we can restrict ${\bar{y}}$ to be lower than a finite number which makes the set compact, since $U(\theta ,a({\bar{y}}),o({\bar{y}}))$ converges to 0 as ${\bar{y}}$ grows. Let the solution to this problem be ${\tilde{V}}^{*}$, and ${\bar{y}}^{*}$ the lowest value ${\bar{y}}$ such that ${\tilde{V}}(\theta ,a({\bar{y}}),o({\bar{y}}))={\tilde{V}}^{*}(\theta ,a({\bar{y}}),o({\bar{y}}))$. Notice that ${\bar{y}}^{*}$ is well defined since ${\tilde{V}}(\theta ,a({\bar{y}}),o({\bar{y}}))$ is continuous in ${\bar{y}}$.

The claim to be proven is that a call-option contract with face value ${\bar{y}}^{*}$ implements ${\tilde{V}}^{*}$. This means that there exists a sequence $a^{n}\in A({\bar{y}}^{n})$ with $\lim _{n\rightarrow \infty }{\bar{y}}^{n}\rightarrow y^{*}$ and $\lim _{n\rightarrow \infty }a^{n}\rightarrow a^{*}$, such that $\lim _{n\rightarrow \infty }{\tilde{V}}(\theta ,a^{n},o)\rightarrow {\tilde{V}}^{*}$. If ${\bar{y}}^{*}$ cannot implement ${\tilde{V}}^{*}$, then $a^{*}\notin A({\bar{y}}^{*})$. This means that there exists ${\tilde{a}}\in A({\bar{y}}^{*})$, such that $U(\theta ,{\tilde{a}},o({\bar{y}}^{*}))>U(\theta ,a^{*},o({\bar{y}}^{*}))$. Since $U(\theta ,a,o({\bar{y}}))$ is continuous in both arguments $(a,{\bar{y}})$; by picking n sufficiently large this also means that $U(\theta ,{\tilde{a}},o({\bar{y}}^{n}))> U(\theta ,a^{n},o({\bar{y}}^{n}))$, which is contradiction. Finally, suppose that the contract with face value ${\bar{y}}^{*}$ is not optimal. Then, there exists ${\bar{y}}$ such that $V(\theta ,a({\bar{y}}),o({\bar{y}}))>{\tilde{V}}^{*}$ and $U(\theta ,a({\bar{y}}),o({\bar{y}}))\ge 0$. But this implies that ${\tilde{V}}(\theta ,a({\bar{y}}),o({\bar{y}}))\ge V(\theta ,a({\bar{y}}),o({\bar{y}}))> {\tilde{V}}(\theta ,a({\bar{y}}^{*}),o({\bar{y}}^{*}))$, which contradicts the optimality of ${\bar{y}}^{*}$ in the modified problem. $\square$

$\square$

The optimal contract is a call-option contract that makes the principal the full residual claimant for output realizations lower than the face value ${\bar{y}}(\theta )$ by paying the agent nothing, and makes the agent the full residual claimant for high output realizations by paying him the output minus the face value equal to ${\bar{y}}(\theta )$.¹⁹ This yields the result in Innes (1990).

Because the agent is risk neutral, a call-option contract solves the standard principal-agent problem in equation (P-MH). The reason has to do with the fact that a risk-neutral principal and agent only care about the expected payment. Because conditional on a given action a, a call-option contract always yields to the principal a lower expected cost than a non-call-option contract for action a, a call-option contract either yields a greater expected payoff conditional on action a being chosen, or it induces an action $a'\ge a$ at the same expected cost for the principal as a, which rises his expected payoff due to that MLRP implies FOSD. Hence, a call-option contract is optimal for the principal.

The optimal contract is different for each type and better types are offered more powerful incentives and therefore choose higher actions. However, the optimal action is downward distorted with respect to the first-best, since the principal wants to lower the agent’s limited-liability rent by means of making an efficient use of the information or monitoring system.

It is interesting to study how the limited-liability rent changes with the agent’s ability type. Using the Envelope theorem one can show that the limited-liability rent varies with $\theta$ as follows:

$$\begin{aligned} \sum \nolimits _{i}p^{i}_{\theta }(\theta ,a^{m}(\theta ))\max \{y^{i}-{\bar{y}}(\theta ),0\}-\sum \nolimits _{i}p^{i}(\theta ,a^{m}(\theta )) \frac{\partial \max \{y^{i}-{\bar{y}}(\theta ),0\}}{\partial \theta }. \end{aligned}$$

The first term comprises the increase in the limited-liability rent due to an increase in the agent’s type when the contract is held constant. This is due to the fact that the return distribution satisfies MLRP with respect to the agent’s ability parameter $\theta$, and the contract satisfies MONA. The second term consists of the increase in the limited-liability rent due to the effect that a change in $\theta$ has on the optimal face value of the call-option contract. Supermodularity in $(\theta ,a)$ implies that ${\bar{y}}(\theta )$ decreases with $\theta$. The reason stems from the following: on the one hand, an increase in $\theta$ improves the output distribution in the sense of FOSD, which, ceteris paribus, rises the principal’s payoff; on the other hand, an increase in the face value ${\bar{y}}$ reduces the optimal action and, since the marginal return to the action increases with the agent’s type, the principal chooses a lower ${\bar{y}}$ as $\theta$ rises. Hence, the limited-liability rent rises with $\theta$. This implies that if types were unobservable, then every type will claim to be the highest type. Thus, the optimal contracts under pure moral hazard are not implementable under moral hazard and adverse selection.