STEP 2: PROPERTIES OF SGM PROGRAM

We can now establish properties of the SGM program. Define a cost-minimizing contract that implements effort $$e$$ by $$w_e^*$$, which solves $w_e^* \in \mathop {\arg \min }\limits_{w \in W} w \cdot \phi \left( e \right)$subject to incentive compatibility $e \in \mathop {\arg \max }\limits_{\tilde e \in E} w \cdot \phi \left( {\tilde e} \right) - c\left( {\tilde e} \right)$and individual rationality $w \cdot \phi \left( e \right) - c\left( e \right) \ge 0.$The next lemma shows that all equilibrium aggregate contracts are cost-minimizing contracts.

Lemma 5: Suppose $$\bar w$$ solves $\mathop {\max }\limits_{w \in W + \left( {1 - 1/N} \right)\bar w} \Lambda \left( {w,\bar w} \right).$Then $$\bar w = w_e^*$$ for some $$e \in E$$.

Proof of Lemma 5: Suppose $$\bar w$$ solves $\mathop {\max }\limits_{w \in W + \left( {1 - 1/N} \right)\bar w} \left( {B - Nw + \left( {N - 1} \right)\bar w} \right) \cdot \phi \left( {{e^*}\left( w \right)} \right).$Then $$\bar w$$ implements some effort level $$\bar e$$. Suppose there is some contract $$\hat w \in W$$ that also implements effort level $$\bar e$$ but $\hat w \cdot \phi \left( {\bar e} \right) < \bar w \cdot \phi \left( {\bar e} \right).$Since $$\bar w$$ solves the program, it must be the case that $\left( {B - N\bar w + \left( {N - 1} \right)\bar w} \right) \cdot \phi \left( {\bar e} \right) \ge \left( {B - N\hat w + \left( {N - 1} \right)\bar w} \right) \cdot \phi \left( {\bar e} \right),$or $\hat w \cdot \phi \left( {\bar e} \right) \ge \bar w \cdot \phi \left( {\bar e} \right).$Next, note that if $${e^*}\left( {\bar w} \right) = {e^*}\left( {\hat w} \right) = {e^*}$$, then for any $$\lambda \in \left[ {0,1} \right]$$, $${e^*}\left( {\lambda \bar w + \left( {1 - \lambda } \right)\hat w} \right) = {e^*}$$. Clearly, $$\left( {IR - \bar w} \right)$$ and $$\left( {IR - \hat w} \right)$$ imply $$\left( {IR - \left( {\lambda \bar w + \left( {1 - \lambda } \right)\hat w} \right)} \right)$$:$\hat w \cdot \phi \left( {{e^*}} \right) - c\left( {{e^*}} \right) \ge 0$and$\bar w \cdot \phi \left( {{e^*}} \right) - c\left( {{e^*}} \right) \ge 0$imply$\left( {\lambda \bar w + \left( {1 - \lambda } \right)\hat w} \right) \cdot \phi \left( {{e^*}} \right) - c\left( {{e^*}} \right) \ge 0.$Similarly, if $$\left( {IC - \bar w} \right)$$ and $$\left( {IC - \hat w} \right)$$ hold, so that $${e^*}$$ maximizes the agent's utility under either contract, then since he is risk-neutral, $${e^*}$$ also maximizes his utility under any contract that randomizes between those two contracts, so $$\left( {IC - \left( {\lambda \bar w + \left( {1 - \lambda } \right)\hat w} \right)} \right)$$ also holds:$\hat w \cdot \phi \left( {{e^*}} \right) - c\left( {{e^*}} \right) \ge \hat w \cdot \phi \left( e \right) - c\left( e \right)\,\,\,for\,all\,e \in E$and$\bar w \cdot \phi \left( {{e^*}} \right) - c\left( {{e^*}} \right) \ge \bar w \cdot \phi \left( e \right) - c\left( e \right)\,\,\,for\,all\,e \in E$imply that$\left( {\lambda \bar w + \left( {1 - \lambda } \right)\hat w} \right) \cdot \phi \left( {{e^*}} \right) - c\left( {{e^*}} \right) \ge \left( {\lambda \bar w + \left( {1 - \lambda } \right)\hat w} \right) \cdot \phi \left( e \right) - c\left( e \right)\,\,\,for\,all\,e \in E,$since for each $$e \in E$$, the first two inequalities imply the third inequality holds for that $$e$$.

Finally, there are two cases. Either $$\hat w \in W + \left( {1 - 1/N} \right)\bar w$$ or $$\hat w \notin W + \left( {1 - 1/N} \right)\bar w$$. If the former, then $$\hat w$$ is feasible and yields a higher value for the objective function than $$\bar w$$, so $$\bar w$$ cannot be an equilibrium aggregate contract. Now, suppose  $$\hat w \notin W + \left( {1 - 1/N} \right)\bar w$$. Given $$\lambda \in \left[ {0,1} \right]$$ and $$\tilde w \in W + \left( {1 - 1/N} \right)\bar w$$, define $\hat w\left( {\lambda ,\tilde w} \right) = \frac{1}{{1 - \lambda }}\tilde w - \frac{\lambda }{{1 - \lambda }}\bar w$if $$\lambda < 1$$ and $$\hat w\left( {1,\tilde w} \right) = \bar w$$. Suppose $$w \in W$$. Then $$w = \hat w\left( {\lambda ,\tilde w} \right)$$ for some $$\lambda \in \left[ {0,1} \right]$$ and some $$\tilde w \in W + \left( {1 - 1/N} \right)\bar w$$ (i.e., any line segment connecting any point $$w \in W$$ to $$\bar w$$ has to intersect the set $$W + \left( {1 - 1/N} \right)\bar w$$). Then there exists some $$\tilde w \in W + \left( {1 - 1/N} \right)\bar w$$ such that $$\tilde w$$ implements $${e^*}$$ at a strictly lower cost than $$\bar w$$ does. Since $$\tilde w$$ is feasible, $$\bar w$$ cannot be an equilibrium aggregate contract. Q.E.D.