Auction Theory: Proving that the found equilibrium is indeed optimal I have been looking at auction theory and in the book Auction Theory by Krishna, there is one (seemingly simple) inequality that I just cannot follow.
Context: given a private valuation $x$, the optimal bidding strategy has been found $\beta(x)$. Now, the author wants to show that behaving and bidding as if you were of type $z$, $\beta(z)$ does not increase profits.
Then, calculating the difference between the profit in the optimum and the profit if you would behave as if you were of type $z$ leads to the following inequality.
$G(x)$ being a probability distriubtion:
$$\pi(\beta(x),x) - \pi( \beta(z),x) = G(z)(z-x) - \int_x^zG(y)dy \geq 0$$
The profit functions were calculated from a first-price auction in case it helps anyone.
My question is why the inequality holds. Why is $G(z)(z-x) - \int_x^zG(y)dy$ larger than 0?
I hope you can help me :)
$$
G(z) (z-x) =
\int_x^z G(z) dy
$$
and since $G$ is increasing on $[x,z]$, the right hand side is larger than
$\int_x^z G(y) dy$.