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.
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 \(f_{h}({\overline{x}}|h(a^{*},e^{*}))>0\).Footnote 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
$$\begin{aligned} \frac{f_{h}({\overline{x}}|h(a^{*},e^{*}))h_{e}(a^{*},e^{*})}{ f({\overline{x}}|h(a^{*},e^{*}))}\ge \frac{c^{\prime }(e^{*})}{ c(e^{*})+{\overline{U}}_{e}}. \end{aligned}$$
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.