Optimal contract under double moral hazard and limited liability

Abstract

This paper investigates optimal contracts between risk-neutral parties when both exert efforts and the agent faces limited liability. We identify a sufficient and necessary condition for any contract to implement the second-best outcome, i.e., the best possible outcome in double moral hazard even when the agent faces unlimited liability. It is shown that a simple share-or-nothing with bonus contract (SonBo for short) is optimal and implements the second-best outcome when the condition holds. SonBo contracts have one degree of freedom, which is very useful in dealing with heterogeneous circumstances while still maintaining consistency in contracting. SonBo admits as special cases the option-like and step bonus contracts, which are widely used in dealing with limited liability. Nevertheless, we demonstrate that a step bonus contract is more powerful because an option-like contract can be problematic in some situations. The paper also discusses the performance of SonBo when the principal also faces liability constraint and investigates the optimal contract when the second-best outcome is not achievable.

Appendix

This appendix collects the proofs of Propositions 1 and 2 in the paper. The proof of Proposition 5 is similar with the proof of Proposition 1 and Proposition 2 if we treat the agent’s outside option as ${\overline{U}}_e+R_b$. Its proof is in the Section 6 of the Online Appendix.

1.1 Proof of Proposition 1

Name $\eta ^{*}=B-\gamma ^{*}{\widehat{x}}$. Then $w^{*}(x)=\gamma ^{*}x+\eta ^{*}$ when $x\ge {\widehat{x}}$. Next, we prove the SonBo contract $w^{*}(x)$ can make agent-IC, principal-IC and agent-PC hold at the second-best effort levels $(a^{*},e^{*})$.

Step 1: Agent-IC holds

In this section, we prove the contract $w^{*}(x)$ can make agent-IC hold. Given the principal chooses $a^{*}$, the marginal expected compensation of agent choosing effort $e\in R_{+}$ divided by $h_{e}(a^{*},e)$ is:

$$\begin{aligned}&\int _{{\widehat{x}}}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f_{h}\left( x|h\left( a^{*},e\right) \right) \,dx \\&=\left( \gamma ^{*}x+\eta ^{*}\right) F_{h}(x|h(a^{*},e))\Big |_{ \widehat{x }}^{{\overline{x}}}-\int _{{\widehat{x}}}^{{\overline{x}}}\gamma ^{*}F_{h}\left( x|h\left( a^{*},e\right) \right) \,dx \\&=-B\cdot F_{h}\left( {\widehat{x}}|h\left( a^{*},e\right) \right) -\frac{\frac{c^{\prime }\left( e^{*}\right) }{h_{e}\left( a^{*},e^{*}\right) }+B\cdot F_{h}\left( {\widehat{x}} |h\left( a^{*},e^{*}\right) \right) }{-\int _{{\widehat{x}}}^{{\overline{x}} }F_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx}\int _{{\widehat{x}}}^{{\overline{x}}}F_{h}\left( x|h\left( a^{*},e\right) \right) \,dx. \end{aligned}$$

The above equations hold because $F_{h}({\overline{x}}|h)=0$ and $\gamma ^{*}{\widehat{x}}+\eta ^{*}=B$. When $e=e^{*}$, the above expression is equal to $\frac{c^{\prime }(e^{*})}{h_{e}(a^{*},e^{*})}$, implying agent-IC holds.

Step 2: Principal-IC holds

Given the agent chooses $e^{*}$, the marginal expected compensation of principal choosing $a\in R_{+}$ divided by $h_{a}(a,e^{*})$ is:

$$\begin{aligned}&\int _{{\underline{x}}}^{{\widehat{x}}}xf_{h}\left( x|h\left( a,e^{*}\right) \right) \,dx+\int _{ {\widehat{x}}}^{{\overline{x}}}\left[ \left( 1-\gamma ^{*}\right) x-\eta ^{*}\right] f_{h}\left( x|h\left( a,e^{*}\right) \right) \,dx \nonumber \\&\quad =\int _{{\underline{x}}}^{{\overline{x}}}xf_{h}\left( x|h\left( a,e^{*}\right) \right) \,dx+\frac{ \frac{c^{\prime }\left( e^{*}\right) }{h_{e}\left( a^{*},e^{*}\right) }+B\cdot F_{h}\left( {\widehat{x}}|h\left( a^{*},e^{*}\right) \right) }{\int _{{\widehat{x}}}^{{\overline{x}} }F_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx}\cdot \int _{{\widehat{x}}}^{ {\overline{x}} }xf_{h}\left( x|h\left( a,e^{*}\right) \right) \,dx\nonumber \\&\quad -\int _{{\widehat{x}}}^{{\overline{x}}}\eta ^{*}f_{h}\left( x|h\left( a,e^{*}\right) \right) \,dx. \end{aligned}$$

(20)

To prove $w^{*}(x)$ induces principal to choose $a^{*}$, we need to prove Eq. (20) is equal to $\frac{ v^{\prime }(a^{*})}{h_{a}(a^{*},e^{*})}$ when principal chooses $a=a^{*}$. Since

$$\begin{aligned} \int _{{\underline{x}}}^{{\overline{x}}}xf_{h}\left( x|h\left( a^{*},e^{*}\right) \right) dx=\frac{ v^{\prime }\left( a^{*}\right) }{h_{a}\left( a^{*},e^{*}\right) }+\frac{c^{\prime }\left( e^{*}\right) }{h_{e}\left( a^{*},e^{*}\right) }, \end{aligned}$$

to prove Eq. (20) is equal to $\frac{ v^{\prime }(a^{*})}{h_{a}(a^{*},e^{*})}$, we need to prove:

$$\begin{aligned}&\frac{\frac{c^{\prime }\left( e^{*}\right) }{h_{e}\left( a^{*},e^{*}\right) }+B\cdot F_{h}\left( {\widehat{x}}|h\left( a^{*},e^{*}\right) \right) }{\int _{{\widehat{x}}}^{ {\overline{x}} }F_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx}\cdot {\int _{{\widehat{x}} }^{{\overline{x}} }xf_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx}\nonumber \\&\qquad -\int _{{\widehat{x}}}^{{\overline{x}}}\eta ^{*}f_{h}(x|h(a^{*},e^{*}))\,dx=-\frac{c^{\prime }\left( e^{*}\right) }{ h_{e}\left( a^{*},e^{*}\right) } \nonumber \\&\quad \iff \gamma ^{*}{\int _{{\widehat{x}}}^{{\overline{x}} }xf_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx}+\int _{{\widehat{x}}}^{{\overline{x}}}\eta ^{*}f_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx=\frac{c^{\prime }\left( e^{*}\right) }{ h_{e}\left( a^{*},e^{*}\right) } \end{aligned}$$

(21)

Since $w^{*}(x)$ can induce the agent to choose $e^{*}$ when the principal chooses $a^{*}$, we can easily get:

$$\begin{aligned} \gamma ^{*}\int _{{\widehat{x}}}^{{\overline{x}}}xf_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx+\eta ^{*}\int _{{\widehat{x}}}^{{\overline{x}}}f_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx=\frac{c^{\prime }\left( e^{*}\right) }{ h_{e}\left( a^{*},e^{*}\right) }. \end{aligned}$$

So Eq. (21) is true, which implies that the proposed $w^{*}(x)$ can induce the principal to choose $a^{*}$ given the agent chooses $e^{*}$.

Thus we have proved, given there exists $w^{*}(x)$ to make agent-PC binding, the $w^{*}(x)$ can implement second-best effort choices.

Step 3: Conditions for binding Agent-PC

Under SonBo $w^{*}(x)$, agent’s expected compensation, denoted as $G(B, {\widehat{x}})$, is:

$$\begin{aligned} G(B,{\widehat{x}})=\gamma ^{*}\int _{{\widehat{x}}}^{{\overline{x}}}(x- {\widehat{x}})f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx+B\int _{{\widehat{x}}}^{{\overline{x}} }f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx. \end{aligned}$$

$G(B,{\widehat{x}})$ can be factored into two parts: $G(B,{\widehat{x}})=H( {\widehat{x}})+B\cdot L({\widehat{x}})$.

The function $H({\widehat{x}})$ is strictly decreasing on $[{\underline{x}}, {\overline{x}})$. The function $L({\widehat{x}})$ is the weight on bonus B. The domain of $H({\widehat{x}})$ and $L({\widehat{x}})$ is $[{\underline{x}}, {\overline{x}})$. We extend the domain of these two functions to $[\underline{x },{\overline{x}}]$ by defining: $H({\overline{x}})=\lim _{ {\widehat{x}}\rightarrow {\overline{x}}}H({\widehat{x}})$ and $L({\overline{x}})=\lim _{{\widehat{x}} \rightarrow {\overline{x}}}L({\widehat{x}})$. These extensions make $H(\widehat{x })$ and $L({\widehat{x}})$ continuous on $[{\underline{x}},{\overline{x}}]$.

Step 3.1

Define $\eta ^{\star }=-\gamma ^{\star }{\widehat{x}}^{\star }$. Given a feasible B, $G(B,{\underline{x}})=H({\underline{x}})+B$ because $L({\underline{x}})=1$, and

$$\begin{aligned} H({\underline{x}})&=\gamma ^{\star }\int _{{\underline{x}}}^{{\overline{x}}}(x- {\underline{x}})f\left( x|h(a^{*},e^{*})\right) \,dx \\&=\gamma ^{\star }\int _{{\underline{x}}}^{{\overline{x}}}xf\left( x|h\left( a^{*},e^{*}\right) \right) \,dx+\eta ^{\star }\int _{{\underline{x}}}^{{\overline{x}} }f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx\\&\quad -\eta ^{\star }\int _{{\underline{x}}}^{ {\overline{x}}}f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx\\&\quad -\gamma ^{\star }{\underline{x}} \int _{{\underline{x}}}^{{\overline{x}}}f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx \\&=\int _{{\underline{x}}}^{{\overline{x}}}\left[ \gamma ^{\star }x+\eta ^{\star }\right] f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx\\&\quad -\eta ^{\star }-\gamma ^{\star }{\underline{x}}. \end{aligned}$$

We have proved that without limited liability, the optimal contract $w^{\star }(x)=\gamma ^{\star }x+\eta ^{\star }$ can make the agent-PC binding, that is $\int _{{\underline{x}}}^{{\overline{x}}}[\gamma ^{\star }x+\eta ^{\star }]f(x|h(a^{*},e^{*}))\,dx=c(e^{*})+{\overline{U}}_{e}$. So

$$\begin{aligned} G(B,{\underline{x}})=c\left( e^{*}\right) +{\overline{U}}_{e}-\eta ^{\star }-\gamma ^{\star }{\underline{x}}+B. \end{aligned}$$

Given Assumption 4 that the linear sharing contract is not feasible, i.e. $\gamma ^{\star }{\underline{x}}+\eta ^{\star }<0$,

$$\begin{aligned} G(B,{\underline{x}})-c\left( e^{*}\right) -{\overline{U}}_{e}=B-\left( \gamma ^{\star } {\underline{x}}+\eta ^{\star }\right) >0. \end{aligned}$$

So $G(B,{\underline{x}})>c(e^{*})+{\overline{U}}_{e}$.

Step 3.2

Next, we prove $G(B,{\overline{x}})\le c(e^{*})+{\overline{U}}_{e}$ if and only if $\frac{f_{h}({\overline{x}}|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{f({\overline{x}}|h(a^{*},e^{*}))}\ge \frac{c^{\prime }(e^{*})}{c(e^{*})+{\overline{U}}_{e}}$. Given a feasible B, $G(B, {\overline{x}})=H({\overline{x}})$ because $L({\overline{x}})=0$. We will prove $H( {\overline{x}})\le c(e^{*})+{\overline{U}}_{e}$ is equivalent to $\frac{ f_{h}({\overline{x}}|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{f( {\overline{x}}|h(a^{*},e^{*}))}\ge \frac{c^{\prime }(e^{*})}{ c(e^{*})+{\overline{U}}_{e}}$.

$H({\overline{x}})$ is solved from the following limit problem.

$$\begin{aligned} H({\overline{x}})&=\lim _{{\widehat{x}}\rightarrow {\overline{x}}}H({\widehat{x}}) =\lim _{{\widehat{x}}\rightarrow {\overline{x}}}\frac{\int _{{\widehat{x}} }^{ {\overline{x}}}(x-{\widehat{x}})f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx }{ \int _{\widehat{x }}^{{\overline{x}}}F_{h}\left( x|h\left( a^{*},e^{*}\right) \right) \,dx } \frac{ c^{\prime }\left( e^{*}\right) }{h_{e}\left( a^{*},e^{*}\right) }\\&=\frac{f\left( {\overline{x}} |h\left( a^{*},e^{*}\right) \right) }{f_{h}\left( {\overline{x}} |h\left( a^{*},e^{*}\right) \right) } \frac{ c^{\prime }\left( e^{*}\right) }{h_{e}\left( a^{*},e^{*}\right) }. \end{aligned}$$

The above process uses L’Hospital’s rule and $F_{h}({\overline{x}}|h(a^{*},e^{*}))=0$. It should be noticed that $f_{h}({\overline{x}}|h(a^{*},e^{*}))>0$.¹¹

So

$$\begin{aligned} H({\overline{x}})\le c\left( e^{*}\right) +{\overline{U}}_{e}\iff \frac{f_{h}\left( \overline{ x }|h\left( a^{*},e^{*}\right) \right) h_{e}\left( a^{*},e^{*}\right) }{f({\overline{x}} |h\left( a^{*},e^{*}\right) )}\ge \frac{c^{\prime }\left( e^{*}\right) }{c\left( e^{*}\right) + {\overline{U}}_{e}}. \end{aligned}$$

Combining step 3.1 and step 3.2, there must exist one ${\widehat{x}}\in [{\underline{x}},{\overline{x}}]$ to make agent-PC hold.

Step 3.3

Next, we prove given a feasible B, if there exists one ${\widehat{x}}\in [ {\underline{x}},{\overline{x}}]$ to make $G(B,{\widehat{x}})=c(e^{*})+\overline{ U}_{e}$ hold, the condition $\frac{f_{h}( {\overline{x}}|h(a^{*},e^{ *}))h_{e}(a^{*},e^{*})}{f({\overline{x}} |h(a^{*},e^{*}))} \ge \frac{c^{\prime }(e^{*})}{c(e^{*})+ {\overline{U}}_{e}}$ must hold.

If there exists such ${\widehat{x}}$, SonBo achieves the second-best outcome, so agent-IC holds and agent-PC binds. The binding agent-PC and agent-IC are

$$\begin{aligned}&\int _{{\widehat{x}}}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f_{h}\left( x|h\left( a^{*},e^{*}\right) \right) h_{e}\left( a^{*},e^{*}\right) \,dx=c^{\prime }\left( e^{*}\right) , \\&\int _{{\widehat{x}}}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx=c\left( e^{*}\right) +{\overline{U}}_{e}. \end{aligned}$$

Combing these two equations,

$$\begin{aligned} \frac{\int _{{\widehat{x}}}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f_{h}\left( x|h\left( a^{*},e^{*}\right) \right) h_{e}\left( a^{*},e^{*}\right) \,dx}{ \int _{ {\widehat{x}}}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx}=\frac{c^{\prime }\left( e^{*}\right) }{ c\left( e^{*}\right) +{\overline{U}} _{e}}. \end{aligned}$$

Define

$$\begin{aligned} \psi (z)=\frac{\int _{z}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f_{h}\left( x|h\left( a^{*},e^{*}\right) \right) h_{e}\left( a^{*},e^{*}\right) \,dx}{ \int _{z}^{ {\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx} ,\,z\in [{\widehat{x}},{\overline{x}}]. \end{aligned}$$

Let $h^{*}=h(a^{*},e^{*})$ and $h_{e}^{*}=h_{e}(a^{*},e^{*})$.

$$ \psi ^{\prime }(z)=\frac{-\left( \gamma ^{*}z+\eta ^{*}\right) f_{h}\left( z|h^{*}\right) h_{e}^{*}\cdot \int _{z}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f\left( x|h^{*}\right) \,dx+\int _{z}^{{\overline{x}} }\left( \gamma ^{*}x+\eta ^{*}\right) f_{h}\left( x|h^{*}\right) h_{e}^{*}\,dx\cdot \left( \gamma ^{*}z+\eta ^{*}\right) f\left( z|h^{*}\right) }{ \left[ \int _{z}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f\left( x|h\left( a^{*},e^{*}\right) \right) \,dx\right] ^{2}}. $$

SonBo achieves second-best outcome under agent’s limited liability, so it gives the agent non-negative payment at any possible output. Thus, given a feasible B, when $z\ge {\widehat{x}}$, $\gamma ^{*}z+\eta ^{*}>0$, so $\psi ^{\prime }(z)\ge 0$ is equivalent to

$$\begin{aligned} \frac{\int _{z}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f_{h}\left( x|h^{*}\right) \,dx}{\int _{z}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f\left( x|h^{*}\right) \,dx}\ge \frac{\left( \gamma ^{*}z+\eta ^{*}\right) f_{h}\left( z|h^{*}\right) }{ \left( \gamma ^{*}z+\eta ^{*}\right) f\left( z|h^{*}\right) }. \end{aligned}$$

Based on MLRP, for any $z_{1}\ge z_{2}\ge z$,

$$\begin{aligned}&\frac{\left( \gamma ^{*}z_{1}+\eta ^{*}\right) f_{h}\left( z_{1}|h^{*}\right) }{\left( \gamma ^{*}z_{1}+\eta ^{*}\right) f\left( z_{1}|h^{*}\right) }\ge \frac{\left( \gamma ^{*}z_{2}+\eta ^{*}\right) f_{h}\left( z_{2}|h^{*}\right) }{ \left( \gamma ^{*}z_{2}+\eta ^{*}\right) f\left( z_{2}|h^{*}\right) } \\&\quad \iff \left( \gamma ^{*}z_{1}+\eta ^{*}\right) f_{h}\left( z_{1}|h^{*}\right) \left( \gamma ^{*}z_{2}+\eta ^{*}\right) f\left( z_{2}|h^{*}\right) \\&\quad \ge \left( \gamma ^{*}z_{2}+\eta ^{*}\right) f_{h}\left( z_{2}|h^{*}\right) \left( \gamma ^{*}z_{1}+\eta ^{*}\right) f\left( z_{1}|h^{*}\right) . \end{aligned}$$

Since $z_{1}$ and $z_{2}$ are arbitrary, integrate $z_{1}$ from $z_{2}$ to ${\overline{x}}$ and let $z_{2}=z$:

$$\begin{aligned}&\int _{z}^{{\overline{x}}}\left( \gamma ^{*}z_{1}+\eta ^{*}\right) f_{h}\left( z_{1}|h^{*}\right) \,dz_{1}\cdot \left( \gamma ^{*}z+\eta ^{*}\right) f\left( z|h^{*}\right) \\&\quad \ge \left( \gamma ^{*}z+\eta ^{*}\right) f_{h}\left( z|h^{*}\right) \cdot \int _{z}^{{\overline{x}}}\left( \gamma ^{*}z_{1}+\eta ^{*}\right) f\left( z_{1}|h^{*}\right) \,dz_{1} \\&\quad \iff \frac{\int _{z}^{{\overline{x}}}\left( \gamma ^{*}z_{1}+\eta ^{*}\right) f_{h}\left( z_{1}|h^{*}\right) \,dz_{1}}{\int _{z}^{{\overline{x}}}\left( \gamma ^{*}z_{1}+\eta ^{*}\right) f\left( z_{1}|h^{*}\right) \,dz_{1}}\ge \frac{ \left( \gamma ^{*}z+\eta ^{*}\right) f_{h}\left( z|h^{*}\right) }{\left( \gamma ^{*}z+\eta ^{*}\right) f\left( z|h^{*}\right) } \\&\quad \iff \frac{\int _{z}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f_{h}\left( x|h^{*}\right) \,dx}{\int _{z}^{{\overline{x}}}\left( \gamma ^{*}x+\eta ^{*}\right) f\left( x|h^{*}\right) \,dx}\ge \frac{\left( \gamma ^{*}z+\eta ^{*}\right) f_{h}\left( z|h^{*}\right) }{\left( \gamma ^{*}z+\eta ^{*}\right) f\left( z|h^{*}\right) }. \end{aligned}$$

Thus, we have proved $\psi ^{\prime }(z)\ge 0$ for $z\in [{\widehat{x}}, {\overline{x}}]$. Since the function $\psi (z)$ is increasing, $\lim \limits _{z\rightarrow {\overline{x}}}\psi (z)\ge \psi ({\widehat{x}})=\frac{ c^{\prime }(e^{*})}{c(e^{*})+{\overline{U}}_{e}}$.

Notice that

$$\begin{aligned} \lim \limits _{z\rightarrow {\overline{x}}}\psi (z)=\frac{\left( \gamma ^{*} {\overline{x}}+\eta ^{*}\right) f_{h}\left( {\overline{x}}|h\left( a^{*},e^{*}\right) \right) h_{e}\left( a^{*},e^{*}\right) \,dx}{\left( \gamma ^{*}{\overline{x}}+\eta ^{*}\right) f\left( {\overline{x}}|h\left( a^{*},e^{*}\right) \right) \,dx}=\frac{f_{h}\left( {\overline{x}} |h\left( a^{*},e^{*}\right) \right) h_{e}\left( a^{*},e^{*}\right) \,dx}{f\left( {\overline{x}} |h\left( a^{*},e^{*}\right) \right) \,dx}. \end{aligned}$$

So if there exists any ${\widehat{x}}\in [{\underline{x}},{\overline{x}}]$ to make agent-PC binding, we should have

$$\begin{aligned} \frac{f_{h}({\overline{x}}|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{ f({\overline{x}}|h(a^{*},e^{*}))}\ge \frac{c^{\prime }(e^{*})}{ c(e^{*})+{\overline{U}}_{e}}. \end{aligned}$$

1.2 Proof of Proposition 2

Suppose contract w(x) is an optimal solution to [P(LL)]. Next, we prove this contract must satisfy condition (11). Contract w(x) achieves second-best outcome means that it can make agent-PC binding and agent-IC and principal-IC hold when effort levels are $(a^{*},e^{*})$. Based on the binding agent-PC and agent-IC:

$$\begin{aligned}&\int _{{\underline{x}}}^{{\overline{x}}}w(x)f(x|h(a^{*},e^{*}))\,dx=c(e^{*})+{\overline{U}}_{e} , \\&\int _{{\underline{x}}}^{{\overline{x}}}w(x)f_{h}(x|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})\,dx=c^{\prime }(e^{*}) , \end{aligned}$$

which implies

$$\begin{aligned} \frac{\int _{{\underline{x}}}^{{\overline{x}}}w(x)f_{h}(x|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})\,dx}{\int _{{\underline{x}}}^{{\overline{x}} }w(x)f(x|h(a^{*},e^{*}))\,dx}=\frac{c^{\prime }(e^{*})}{ c(e^{*})+{\overline{U}}_{e}}. \end{aligned}$$

(22)

Step 1

If w(x) is an optimal solution of [P(LL)], (22) should hold. Next we prove if (22) holds, condition (11) should hold.

Define $V(x_{c})=\frac{\int _{x_{c}}^{{\overline{x}} }w(x)f_{h}(x|h(a,e))h_{e}(a,e)\,dx}{\int _{x_{c}}^{{\overline{x}} }w(x)f(x|h(a,e))\,dx}$. The effort choices (a, e) in $V(x_{c})$ can be any feasible effort choices including $(a^{*},e^{*})$. Notice that the left-hand side of (22) is equal to $V({\underline{x}})$ when $(a,e)=(a^{*},e^{*})$. We write f(x|h(a, e)) as f(x|h) and $h_{e}(a,e)$ as $h_{e}$ for short. In this step, we prove $V(x_{c})$ is an increasing function of $x_{c}$ for any pair of (a, e) , i.e., ${\frac{ dV(x_{c})}{dx_{c}}}\ge 0$ for any (a, e). We have

$$\begin{aligned} {\frac{dV(x_{c})}{dx_{c}}}=\frac{w(x_{c})h_{e}\big [f(x_{c}|h)\int _{x_{c}}^{ {\overline{x}}}w(x)f_{h}(x|h)\,dx-f_{h}(x_{c}|h)\int _{x_{c}}^{ {\overline{x}} }w(x)f(x|h)\,dx\big ]}{\big [\int _{x_{c}}^{{\overline{x}} }w(x)f(x|h)\,dx\big ] ^{2}}. \end{aligned}$$

When $w(x_{c})=0$, ${\frac{dV(x_{c})}{dx_{c}}}=0$.

When $w(x_{c})>0$, since $h_{e}>0$, ${\frac{dV(x_{c})}{dx_{c}}}\ge 0$ is equivalent to:

$$\begin{aligned} \frac{\int _{x_{c}}^{{\overline{x}}}w(x)f_{h}(x|h)\,dx}{\int _{x_{c}}^{ {\overline{x}}}w(x)f(x|h)\,dx}\ge \frac{f_{h}(x_{c}|h)}{f(x_{c}|h)}. \end{aligned}$$

(23)

Notice that when $w(x_{c})>0$, $\frac{f_{h}(x_{c}|h)}{f(x_{c}|h)}=\frac{ w(x_{c})f_{h}(x_{c}|h)}{w(x_{c})f(x_{c}|h)}$. Then (23) is equivalent to:

$$\begin{aligned} \frac{\int _{x_{c}}^{{\overline{x}}}w(x)f_{h}(x|h)\,dx}{\int _{x_{c}}^{ {\overline{x}}}w(x)f(x|h)\,dx}\ge \frac{w(x_{c})f_{h}(x_{c}|h)}{ w(x_{c})f(x_{c}|h)}. \end{aligned}$$

(24)

Next, we prove (24) holds. Based on MLRP, the function $\frac{ f_{h}(x|h)}{f(x|h)}$ is increasing with $x\ge x_{c}$ when $w(x)\ne 0$. Pick $x_{1}$ and $x_{2}$ such that $x_{1}\ge x_{2}\ge x_{c}$. If $w(x_{1})\ne 0$ and $w(x_{2})\ne 0$,

$$\begin{aligned}&\frac{w(x_{1})f_{h}(x_{1}|h)}{w(x_{1})f(x_{1}|h)}\ge \frac{ w(x_{2})f_{h}(x_{2}|h)}{w(x_{2})f(x_{2}|h)} \\&\quad \iff w(x_{1})f_{h}(x_{1}|h)w(x_{2})f(x_{2}|h)\ge w(x_{2})f_{h}(x_{2}|h)w(x_{1})f(x_{1}|h). \end{aligned}$$

If $w(x_{1})=0$ or $w(x_{2})=0$ or both,

$$\begin{aligned} w(x_{1})f_{h}(x_{1}|h)w(x_{2})f(x_{2}|h)\ge w(x_{2})f_{h}(x_{2}|h)w(x_{1})f(x_{1}|h). \end{aligned}$$

So for $x_{1}\ge x_{2}\ge x_{c}$,

$$\begin{aligned} w(x_{1})f_{h}(x_{1}|h)w(x_{2})f(x_{2}|h)\ge w(x_{2})f_{h}(x_{2}|h)w(x_{1})f(x_{1}|h). \end{aligned}$$

Since $x_{1}$ and $x_{2}$ are arbitrary, integrate $x_{1}$ from $x_{2}$ to ${\overline{x}}$ and let $x_{2}=x_{c}$. Then

$$\begin{aligned}&\int _{x_{c}}^{{\overline{x}}}w(x_{1})f_{h}(x_{1}|h)\,dx_{1}\cdot w(x_{c})f(x_{c}|h)\ge w(x_{c})f_{h}(x_{c}|h)\cdot \int _{x_{c}}^{{\overline{x}} }w(x_{1})f(x_{1}|h)\,dx_{1} \\&\quad \iff \frac{\int _{x_{c}}^{{\overline{x}}}w(x)f_{h}(x|h)\,dx}{ \int _{x_{c}}^{ {\overline{x}}}w(x)f(x|h)\,dx}\ge \frac{w(x_{c})f_{h}(x_{c}|h)}{ w(x_{c})f(x_{c}|h)}. \end{aligned}$$

Thus (24) is true. So the function ${\frac{dV(x_{c})}{dx_{c}}} \ge 0$ when $w(x_{c})>0$.

So far we have proved that given the contract $w(x)\ge 0$ for any $x\ge x_{c}$, ${\frac{dV(x_{c})}{dx_{c}}}\ge 0$.

Step 2

The (a, e) in $V(x_{c})$ is arbitrary. Now let $(a,e)=(a^{*},e^{*})$. Since $V(x_{c})$ is increasing, $V(x_{c})\ge V({\underline{x}})$ for $x_{c}> {\underline{x}}$. Condition (22) implies that $V(x_{c}) \ge \frac{c^{\prime }(e^{*})}{c(e^{*})+{\overline{U}}_{e}}$ for $x_{c}> {\underline{x}}$. Furthermore, we should have

$$\begin{aligned} \lim _{x_{c}\rightarrow {\overline{x}}}V(x_{c})\ge \frac{c^{\prime }(e^{*}) }{c(e^{*})+{\overline{U}}_{e}}. \end{aligned}$$

Based on L’Hospital’s rule,

$$\begin{aligned} \lim _{x_{c}\rightarrow {\overline{x}}}V(x_{c})= & {} \lim _{x_{c}\rightarrow {\overline{x}}} \frac{w(x_{c})f_{h}(x_{c}|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{w(x_{c})f(x_{c}|h(a^{*},e^{*}))}\\= & {} \lim _{x_{c} \rightarrow {\overline{x}}} \frac{f_{h}(x_{c}|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{f(x_{c}|h(a^{*},e^{*}))}=\frac{f_{h}( {\overline{x}}|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{ f(\overline{ x}|h(a^{*},e^{*}))}. \end{aligned}$$

So given the contract w(x) implements the second-best outcome, i.e. condition (22) holds, $\frac{f_{h}({\overline{x}} |h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{f({\overline{x}}|h(a^{*},e^{*}))}\ge \frac{c^{\prime }(e^{*})}{c(e^{*})+{\overline{U}} _{e}}$ holds.

Next, we prove when condition (11) holds, there must exist contracts to implement the second-best outcome of [P(LL)]. The proof is simple as we have proved when condition (11) holds, SonBo contract exists and it implements the second-best outcome.