Reference no: EM131582683

Question: If the profit function is differentiable, the envelope theorem implies that Dπ(q) = z(q), that is, the derivative of the profit function at a point is simply the optimal production plan (this is Hotelling's lemma). We will show that the profit function is differentiable whenever ƒ( ) is strictly concave.

Fix a price vector q, and consider the behavior of the profit function as we move away from this point. Using the fact that π(q) = qz(q), show that for any change h in the price vector,

Π(q + h) ≥ Π(q) + hz(q)

Π(q) ≥ Π(q + h) - hz(q + h)

Using (1) and (2), show that

h[z(q + h) - z(q)] ≥ Π(q + h) - Π(q) - hz(q) ≥ 0

Using this expression, show that n is differentiable at q and Dπ(q) = Z(q).

The next problem illustrates how the envelope theorem and the properties of the value function can sometimes be used to obtain comparativestatics results.