[Economics] Auction Theory: Proving that the found equilibrium is indeed optimal

Solution 1:

$$ G(z) (z-x) = \int_x^z G(z) dy $$ and since $G$ is increasing on $[x,z]$, the right hand side is larger than $\int_x^z G(y) dy$.

Solution 2:

Although there already is an accepted answer, there is another way to see the global optimality - or rather the same way with a different formulation.

By construction, $$\frac{\partial \pi}{\partial b}(b,x) = - G((\beta)^{-1}(b)) + (x-b) \frac{G'((\beta)^{-1}(b))}{(\beta)'((\beta)^{-1}(b))}\Bigg{|}_{b=\beta(x)}= 0,$$ where $\frac{\partial \pi}{\partial b}(b,x)$ is increasing in $x$.

Now consider some bid $\widehat b<\beta(x)$. By continuity of $\beta$, there is a type $\widehat x<x$ such that $\beta(\widehat x)=\widehat b$. Hence, because $\widehat x<x$, $$\frac{\partial \Pi}{\partial b}(\widehat b,x) \geq \frac{\partial \Pi}{\partial b}(\widehat b, \widehat x) = \frac{\partial \Pi}{\partial b} (\beta(\widehat x),\widehat x) = 0. $$ Thus, the expected utility $\Pi( b,x)$ is increasing in $b$ for all $ b<\beta(x)$. Analogously, $\Pi(b,x)$ is decreasing for all $\widehat b'>\beta(x)$.