 Virtual-surplus revenue identity

For an independent private values model, let $X_i(v)$ and $P_i(v)$ be interim allocation and payment. The interim payment identity gives $P_i(v)=v{X}_i(v)-U_i({\underline{v}}_i)-{\int }_{{\underline{v}}_i}^v{X}_i(t)dt$. Applying Fubini's theorem to reverse the order of integration yields $\mathbb{E}{\int }_{{\underline{v}}_i}^{V_i}{X}_i(t)dt=\int X_i(t)(1-F_i(t))dt$. Subtracting this term from $\int v{X}_i(v)f_i(v)dv$ proves the identity. Interim individual rationality bounds the lowest-type utilities below by zero.