Question: A firm produces and sells two commodities. By selling x tons of the first commodity the firm gets a price per ton given by p = 96 - 4x. By selling y tons of the other commodity the price per ton is given by q = 84 - 2y. The total cost of producing and selling x tons of the first commodity and y tons of the second is given by C(x, y) = 2x2 + 2xy + y2.
(a) Show that the firm's profit function is P(x, y) = -6x2 - 3y2 - 2xy + 96x + 84y.
(b) Compute the first-order partial derivatives of P, and find its only stationary point.
(c) Suppose that the firm's production activity causes so much pollution that the authorities limit its output to 11 tons in total. Solve the firm's maximization problem in this case. Verify that the production restrictions to reduce the maximum possible value of P(x, y).