Question: The profit function of a firm is π(x, y) = px + qy - αx2 - βy2, where p and q are the prices per unit and αx2 + βy2 are the costs of producing and selling x units of the first good and y units of the other. The constants are all positive.
(a) Find the values of x and y that maximize profits. Denote them by x∗ and y∗. Verify that the second-order conditions are satisfied.
(b) Define π∗(p, q) = π(x∗, y∗). Verify that ∂π∗(p, q)/∂p = x∗ and ∂π∗(p, q)/∂q = y∗. Give these results economic interpretations.