Reference no: EM131240845

The monopoly producer of a single good has a constant unit cost c(ω(t)) at time t, where ω(t) is the firm’s “experience” at that date. (Assume c > 0, c < 0, and limt→∞ c(t) > 0.) Time is continuous and runs from zero to infinity. Experience accumulates with production: dω(t)/dt = q(t), where q(t) is production at date t. (Those who have done the empirical work have assumed, as we do, that production exhibits constant instantaneous returns to scale and that the appropriate measure of experience is cumulative output.) Let R(q) denote the revenue function as a function of quantity (supposing demand is invariant). Assume R > 0 and R < 0. Let r denote the interest rate. The monopolist’s objective function is ∞ 0 [R(q(t)) − c(ω(t))q(t)]e−rtdt. (i) Show that at each instant the monopolist sets marginal revenue equal to the average (discounted) unit cost in the future: A(t) = ∞ t c(ω(s))re−r(s−t) ds. Hint: Consider the current cost and the future savings from changing q(t) slightly. (ii) Show that output increases over time.