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.