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:<math>\therefore u_e(x^*) + u_g^*(x^*) + u_h^*(x^*) \ge u_e(x) + u_g^*(x) + u_h^*(x) \quad \forall x \in \mathbb{R}</math>
And so the we have both that (as <math>\widehat{U_e}^\hat(x) = u_e(x) + u_g(x) + u_h(x) \quad</math>) the principal's problem is to maximize the joint surplus and that the contribution schedules are truthful.
===Solving For Linear Contribution Terms===
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