[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 $$\widehat x such that $$\beta(\widehat x)=\widehat b$$. Hence, because $$\widehat 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 $$\widehat b'>\beta(x)$$.