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1. The following table shows the demand curve facing a monopolist who produces at a constant marginal cost of $10: Price Quantity 18 0 16 4 14 8 12 12 10 16 8 20 6 24 4 28 2 32 0 36 a. Calculate the firm’s marginal revenue curve. To find the marginal revenue curve, we first derive the inverse demand curve. The intercept of the inverse demand curve on the price axis is 18. The slope of the inverse demand curve is the change in price divided by the change in quantity. For example, a decrease in price from 18 to 16 yields an increase in quantity from 0 to 4. Therefore, the slope of the inverse demand is ∆ P ∆ Q
= -0.5, and the demand curve is therefore
P 18 0.5Q.
The marginal revenue curve corresponding to a linear demand curve is a line with the same intercept as the inverse demand curve and a slope that is twice as steep. Therefore, the marginal revenue curve is MR 18 Q. b. What are the firm’s profit-maximizing output and price? What is its profit? The monopolist’s profit-maximizing output occurs where marginal revenue equals marginal cost. Marginal cost is a constant $10. Setting MR equal to MC to determine the profit-maximizing quantity: 18 Q 10, or Q 8. To find the profit-maximizing price, substitute this quantity into the demand equation:
P 18 (0.5) (8) = $14.
Total revenue is price time’s quantity: TR (14) (8) = $112. The profit of the firm is total revenue minus total cost, and total cost is equal to average cost times the level of output produced. Since marginal cost is constant, average variable cost is equal to marginal cost. Ignoring any fixed costs, total cost is 10Q or 80, and profit is 112 80 $
2. Suppose that an industry is characterized as follows: C = 100 + 2q^2 each firm’s total cost function MC = 4q firm’s marginal cost function P = 90 – 2Q industry demand curve MR = 90 – 4Q industry marginal revenue curve a. If there is only one firm in the industry, find the monopoly price, quantity, and level of profit. If there is only one firm in the industry, then the firm will act like a monopolist and produce at the point where marginal revenue is equal to marginal cost: 90 4Q 4Q Q 11.25. For a quantity of 11.25, the firm will charge a price P 90 2(11.25) $67.50. PQ C $67.50(11.25) [100 2(11.25)^2 ] $406.25. b. Graphically illustrate the demand curve, marginal revenue curve, marginal cost curve, and average cost curve. The graph below illustrates the demand curve, marginal revenue curve, and marginal cost curve. The average cost curve is not shown because it makes the diagram too cluttered. AC reaches its minimum value of $28.28 and intersects the marginal cost curve at a quantity of 7.07. The profit that is lost by having the firm produce at the competitive solution as compared to the monopoly solution is the difference of the two profit levels as calculated in parts a and b: $406.25 350 $56.25. On the graph below, this difference is represented by the lost profit area, which is the triangle below the marginal cost curve and above the marginal revenue curve, between the quantities of 11.25 and 15. This is lost profit because for each of these 3.75 units, extra revenue earned was less than extra cost incurred. This area is (0.5) (3.75) (6030) $56.25. Another way
