Question

3. A monopolist chooses output (x) to maximize profit (T) where r(x) = p(x)x-c(x) In this equation, p(x) denotes the price of x and c(x) is the cost of producing x. Since demand curves are downward sloping, we assume that p,(x) <0. In addition, we will assume that marginal cost is positive and non-decreasing; that is, c'(x) > 0 and c"(x) 2 0. Derive a condition on the demand curve such that marginal revenue is downward sloping. Derive the first and second-order conditions for the firm's profit maximizing choice of x. Interpret these two conditions in terms of marginal revenue and marginal cost. Now, suppose that in order to capture some of the monopoly profits, the government imposes a tax on revenue of an amount t so that the monopolist's profit function becomes (x) - p(x)x -c(x)-tp(x)x Suppose that the government keeps the revenue from the tax. Does this tax increase or decrease the monopolist's output? Show your work and explain your logic. How would you answer be different if the tax were levied on profit instead of revenue? Explain.