Suppose a perfectly competitive industry can produce widgets at a constant marginal cost of \(\$ 10\) per unit. Monopolized marginal costs increase 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_{D}=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.

In a perfectly competitive market, the price is equal to the marginal cost. The marginal cost in this case is given as \(\$10\) per unit. The demand function is given by: \\[Q_{D} = 1000 - 50P,\\] To calculate the perfectly competitive output and price, set the marginal cost equal to the price in the demand function: \\[10 = P.\\] Now, substitute the price back into the demand function to find the quantity: \\[Q_{D} = 1000 - 50(10) = 1000 - 500 = 500.\\] Therefore, the perfectly competitive output is 500 units and the price is \(\$10\).

In a monopoly, the monopolized marginal cost is given as \(\$12\) per unit. To find the output and price in a monopolized market, we need to set the monopolized marginal cost equal to the marginal revenue derived from the demand function. First, we need the total revenue: \\[R = PQ = (1,000 - 50P)P.\\] Now, calculate the marginal revenue by differentiating the total revenue with respect to quantity: \\[\frac{dR}{dQ} = \frac{d}{dQ}(20,000 - 100P) = -100 + 2P.\\] Now, set the monopolized marginal cost (\(12\)) equal to the marginal revenue and solve for the price: \\[12 = -100 + 2P.\\] This gives: \\[P = \frac{112}{2} = \$56.\\] Now, substitute the price back into the demand function to find the quantity: \\[Q_{D} = 1000 - 50(56) = 1000 - 2800 = -1800.\\] Therefore, the monopoly output is 1800 less units compared to the perfectly competitive scenario and the price is \(\$56\).

Consumer surplus is the area under the demand curve and above the market price. In a perfectly competitive market, the consumer surplus is the area of the triangle formed by the demand curve, the price, and the quantity. In the monopolized scenario, the consumer surplus is the area under the demand curve and above the monopolized price up to the monopolized output. Loss of consumer surplus can be calculated as the difference in the areas under the curve: Loss of consumer surplus = (Area under the curve in a perfectly competitive market) - (Area under the curve in a monopolized market) In this case: Loss of consumer surplus = (0.5 × base × height) - (0.5 × base × height) Base and height refer to the respective dimensions of the triangular areas of consumer surplus under the demand curve. Loss of consumer surplus = (0.5 × 500 × 10) - (0.5 × 500 × 46) Loss of consumer surplus = 11,500 Therefore, the total loss of consumer surplus from monopolization of widget production is $11,500.

A graph comparing the perfectly competitive market and monopolized market can be drawn, with the x-axis representing the quantity and the y-axis representing the price. The demand curve will be a downward-sloping linear function intersecting the y-axis at P = 20 and the x-axis at Q = 400. The perfectly competitive equilibrium (price = \(10, quantity = 500) will be a point on the demand curve, while the monopolized equilibrium (price = \)56, quantity = -1800) will be below and to the left of the perfectly competitive equilibrium on the demand curve. The key differences between the perfectly competitive and monopolized scenarios are the prices and quantities produced. In the perfectly competitive market, the quantity produced is higher, and the price is lower, resulting in a larger consumer surplus. In the monopolized market, the quantity produced is lower, and the price is higher, resulting in a smaller consumer surplus and a total loss in consumer surplus due to monopolization.