Consider a monopoly where the inverse demand for its product is given by P = 50 − 2Q.Total costs for this monopolist are estimated to be C(Q)= 100 + 2Q + Q².At the profit-maximizing combination of output and price,deadweight loss is: A) $32. B) $64. C) $128. D) cannot be determined with the given information.

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Quizplus · Accepted Answer

Deadweight loss in a monopoly can be calculated by finding the area of the triangle formed by the demand curve, the marginal cost curve, and the quantity produced by the monopolist. First, we need to find the monopolist's profit-maximizing quantity and price. The marginal revenue (MR) and marginal cost (MC) are derived from the given functions. The inverse demand function is P = 50 - 2Q, so the total revenue (TR) is P*Q = (50 - 2Q)Q = 50Q - 2Q^2. The marginal revenue (MR) is the derivative of TR with respect to Q, which is MR = 50 - 4Q. The total cost (C) is given as C(Q) = 100 + 2Q + Q^2, so the marginal cost (MC) is the derivative of C with respect to Q, which is MC = 2 + 2Q. Setting MR = MC to find the profit-maximizing quantity: 50 - 4Q = 2 + 2Q, solving this gives Q = 12. The price can be found by substituting Q into the demand equation: P = 50 - 2*12 = 26. To find the deadweight loss, we need to find where the MC (which is also the supply curve in a competitive market) intersects the demand curve. Setting P = MC: 50 - 2Q = 2 + 2Q, solving for Q gives Q = 12 (which is the competitive quantity, but we need this to find the intersection point). The deadweight loss is the area of the triangle formed by the demand curve, the MC curve, and the monopolist's quantity. The base of the triangle is the difference between the competitive quantity and the monopolist's quantity (which in this case are the same, so we need to find the height of the triangle, which is the difference in prices). However, the calculation provided directly does not lead to the correct interpretation for deadweight loss calculation. The correct approach involves finding the area of the triangle where the base is the difference between the quantities (which in this case, we need to find the competitive quantity correctly) and the height is the difference in prices. The error in the explanation above is in not correctly identifying the competitive quantity and price, and thus not correctly calculating the deadweight loss. The correct calculation involves finding where the marginal cost equals the average cost for the competitive outcome, which was not accurately pursued. Given the mistake in the process described, the deadweight loss calculation involves finding the area of the triangle where the base is the difference between the monopolist's quantity and the competitive quantity (if correctly calculated) and the height is the difference between the prices at these quantities. Without the correct competitive quantity and price, the direct calculation of deadweight loss as described is not feasible with the given information, leading to an incorrect direct answer. The correct approach to finding the deadweight loss involves more detailed steps in identifying the competitive equilibrium and comparing it to the monopolist's outcome.

Consider a Monopoly Where the Inverse Demand for Its Product