Question

Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of \(\$ 10\) per unit. Monopolized marginal costs rise to \(\$ 12\) per unit because \(\$ 2\) per unit must be paid to lobbyists to retain the widget producers' favored position. Suppose the market demand for widgets is given by \\[ Q_{0}=1,000-50 P \\] a. Calculate the perfectly competitive and monopoly outputs and prices. b. Calculate the total loss of consumer surplus from monopolization of widget production. c. Graph your results and explain how they differ from the usual analysis.

Short answer

Answer: The total loss of consumer surplus due to monopolization is -\$632. In a perfectly competitive market, the price and output are \$10 and 500, respectively. In a monopolized market, the price and output are \$11.20 and 440, respectively.

Step-by-step solution

Find the demand curve and inverse demand curve

The given market demand function is \\ \[ Q_{0}=1,000-50 P \\] We need to find the inverse demand curve by solving for P: \\ \[ P = \frac{1000 - Q_0}{50} \\]

Calculate marginal cost for both market structures

Under perfect competition, the marginal cost is constant at \(MC_{PC} = \$10\). For monopolies, the marginal cost is given to be \(MC_{M} = \$12\).

Calculate marginal revenue curves for perfect competition and monopoly

In perfect competition, marginal revenue (MR) is equal to the market price. So, for a perfectly competitive firm, \\ \[ MR_{PC} = P \\] For monopolies, we differentiate the total revenue function (TR = P * Q) with respect to Q to find the marginal revenue function: \\ \[ TR = PQ = (\frac{1000 - Q_0}{50})Q_0 \\] Differentiating with respect to Q_0, we get\\ \[ MR_{M} = \frac{1000 - 2Q_0}{50} \\]

Calculate equilibrium outputs and prices for both market structures

In perfect competition, \\ \[ MR_{PC} = MC_{PC} \\] \[ P = \$10 \\] Substitute P in the inverse demand curve to find Q: \\ \[ Q_{PC} = 1000 - 50*10 = 500 \\] In the monopolized market, \\ \[ MR_{M} = MC_{M} \\] \[ \frac{1000-2Q_{M}}{50} = 12 \\] Solving for Q_M, \\ \[ Q_{M} = \frac{1000 - 50 * 12}{2} = 440 \\] Substitute Q_M in the inverse demand curve to find P: \\ \[ P_{M} = \frac{1000 - 440}{50} = \$11.20 \\]

Calculate the consumer surplus under both market structures

Consumer surplus can be found by calculating the area of the triangle formed by the demand curve, x-axis, and price charged. For perfect competition, \\ \[ CS_{PC} = \frac{1}{2}(1000 - 500)(10) \\] \[ CS_{PC} = \$2500 \\] For monopoly, \\ \[ CS_{M} = \frac{1}{2}(1000 - 440)(11.20) \\] \[ CS_{M} = \$3132 \\]

Calculate the total loss of consumer surplus from monopolization

The total loss of consumer surplus due to monopolization is the difference between consumer surpluses in both market structures. \\ \[ \Delta CS = CS_{PC} - CS_{M} \\] \[ \Delta CS = \$2500 - \$3132 \\] \[ \Delta CS = -\$632 \\]

Graph and analyze the results

Plot the demand curve, inverse demand curve, marginal cost curves, and marginal revenue curves for both market structures on a graph, marking the equilibrium points and consumer surplus areas. The loss of consumer surplus can be seen as the area between the representative equilibrium points. In conclusion, under perfect competition, the price and output are \(P = \$10\) and \(Q=500\). Under a monopolized market, the price and output are \(P = \$11.20\) and \(Q = 440\). The total loss of consumer surplus from monopolization is \(-\$632\).