All My Bloody Notes Bloody Maths Calculus Leibniz Rule

Leibniz Rule

AKA how to differentiate integral limits

What is it?

Differentiating a(x)b(x)f(x,t)dt\int_{a(x)}^{b(x)} f(x,t) dt with respect to xx, you get:

ddxa(x)b(x)f(x,t)dt=f(x,b(x))b(x)f(x,a(x))a(x)+a(x)b(x)xf(x,t)dt\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) dt


In the interdependent value auction setting, we have the bidder whose true type is xx but choose a value xx' to maximise the following expected payoff function:

0xv(x,xj)fXjXi(xjXi=x)dxjFXjXi(xXi=x)m(x,x)\int_{0}^{x'} v(x,x_j) f_{X_j|X_i}(x_j | X_i = x) dx_j - F_{X_j|X_i}(x' | X_i = x)m(x, x')

Where m(x,x)m(x,x') denotes the payment bidder ii has to make when he/she wins.

If we assume the strategy β(x)\beta(x) to be the equilibrium strategy, then revealing the truth, i.e. choosing x=xx' = x, must yield the highest expected payoff.

Therefore it must also satisfy the necessary condition for maximum when evaluating the FOC at x=xx' = x:

ddx0xv(x,xj)fXjXi(xjXi=x)dxj=v(x,x)fXjXi(xXi=x)\frac{d}{dx'} \int_{0}^{x'} v(x,x_j) f_{X_j|X_i}(x_j | X_i = x) dx_j = v(x, x') f_{X_j|X_i}(x' | X_i = x)

Thus we have:

v(x,x)fXjXi(xXi=x)x=x=ddxFXjXi(xXi=x)m(x,x)x=xv(x,x') f_{X_j|X_i}(x' | X_i = x)_{|x' = x} = \frac{d}{dx'} F_{X_j|X_i}(x' | X_i = x)m(x,x')_{|x' = x}

i.e. v(x,x)fXjXi(xXi=x)=ddxFXjXi(xXi=x)m(x,x)v(x,x) f_{X_j|X_i}(x | X_i = x) = \frac{d}{dx'} F_{X_j|X_i}(x | X_i = x)m(x,x)

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