Numerical Questions Ch11 (solutions)

ECO 2144 Chapter 11. December 6, 2023 1 of 6 Question 1 At the current level of output a firm’s marginal cost equal 16 and marginal revenue equals 10. The firm (a) Is producing the profit-maximizing amount. (b) Should produce more. (c) Should produce less. (d) There is not enough information Solution: Answer b is the correct one Question 2 If the inverse demand curve a monopoly faces is p = 100 − 2 Q , and MC is constant at 16, then maximum profit (a) Equals 1,218. (b) Equals 336. (c) Equals 882. (d) Cannot be determined solely from the information provided Solution: In this case, the marginal revenue equation is MR = 100 − 4 Q . The monopolist maximizes profit if 100 − 4 Q = 16. This means that 4 Q = 84, or Q = 21, and P = 100 − 2(21) = 58. To calculate the profit, note that if the marginal cost is constant, then ı = Q ( p − MC ). Thus, ı = 21(58 − 16) = 882. Question 3 If the inverse demand curve a monopoly faces is p = 100 − 2 Q , and MC is constant at 16, then the deadweight loss from monopoly equals Solution: Here, the easiest approach is to illustrate this graphically, as shown below. Notice that under perfect competition, p = MC: Using this condition, we can clearly see that Q = 42 : To calculate the deadweight loss, one should calculate the area of the yellow triangle shown below: (42 − 21) × (58 − 16) 2 = 441.

ECO 2144 Chapter 11. December 6, 2023 2 of 6 Question 4 The inverse demand curve a monopoly faces is p = 100 − 2 Q . The firm’s cost curve is C ( Q ) = 20 + 6 Q . (a) What is the profit-maximizing solution? Solution: In this case, the marginal revenue function is MR = 100 − 4 Q; and the MC = 6. We know that the monopolist maximizes profits when MC = MR: Thus, 6 = 100 − 4 Q . Therefore, Q = 23 : 5 and p = 100 − 2(23 : 5) = 53 : (b) What is the firm’s economic profit? Solution: To calculate profits we have to use the cost function ı = p × Q − C ( Q ). In this case, ı = 53 × 23 : 5 − 20 − 6 × 23 : 5 = 1084 : 5. (c) How does your answer change if C ( Q ) = 100 + 6 Q ? Solution: he maximization problem is not altered because the marginal cost remains the same. What changes is the fixed cost, which goes from 20 to 100. This means that the profits will be reduced by 80, which implies that the new profits are ı = 1004 : 5. Question 5 If a monopoly’s inverse demand curve is p = 12 − 2 Q and its cost function is C ( Q ) =

ECO 2144 Chapter 11. December 6, 2023 4 of 6 1. Marginal Cost Equality Across Plants : Profit-maximizing firms aim to balance marginal costs across different production sites. For Gillette, this means equating the marginal costs of its L.A. and Mexico City plants. 2. Comparing Marginal Costs : The marginal cost at the Mexican plant MC 2 is lower than that at the L.A. plant MC 1 when 1 + 0 : 5 Q 2 < 8, or Q 2 < 14. Thus, for outputs Q < 14 million blades, it’s more cost-effective for Gillette to produce exclusively at the Mexican plant. 3. Output Allocation for Q > 14: When the total output Q exceeds 14 million blades, the cost-minimizing strategy is to produce 14 million blades at the Mexican plant Q 2 = 14 and the remainder at the L.A. plant Q 1 = Q − 14. 4. Determining the Profit-Maximizing Output : Assuming the profit-maximizing quantity Q is greater than 14 million, the relevant marginal cost is 8 cents (the constant cost at the L.A. plant). Setting the Marginal Revenue MR = 968 − 40 Q equal to the marginal cost, we get: 968 − 40 Q = 8. Simplifying this, we find Q = 24 million blades. 5. Verifying the Output Assumption : It’s important to check that the as- sumption of Q > 14 is valid. If we had initially assumed Q < 14 and set MR = MC 2 = 1 + 0 : 5 Q , we would have found Q > 14. This contradicts our assumption, proving it invalid. 6. Final Allocation and Pricing : With Q 2 = 14 (Mexican plant) and Q 1 = 10 (L.A. plant), the total output is Q = 24 million blades. At this output level, the price is determined by the demand function: p = 968 − 20 × 24 = 488 cents, or 4.88 per blade. (b) What is the profit? Solution: ı = p × Q − MC 1 ( Q 1 ) × Q 1 − MC 2 ( Q 2 ) × Q 2 = 488 × 24 − 8 × 10 − (1 + 0 : 5 × 14) × 14 = 11520 cents or 115.20. Question 7 A monopolist has an inverse demand curve given by p ( Q ) = 12 − Q and a cost curve given by C ( Q ) = Q 2 . (a) What will be its profit-maximizing level of output? Solution: In this case, the marginal revenue function is MR = 12 − 2 Q; and the MC = 2 Q . We know that the monopolist maximizes profits when MC = MR: Thus, 12 − 2 Q = 2 Q .

ECO 2144 Chapter 11. December 6, 2023 5 of 6 Therefore, Q = 3. (b) Suppose the government decides to put a tax on this monopolist so that for each unit it sells it has to pay the government 2. What will be its output under this form of taxation? Solution: Here, the new marginal cost function is MC = 2 Q + 2. This means that in order to maximize profits, the equation 12 − 2 Q = 2 Q + 2 must be satisfied. Solving for Q , we get 10 = 4 Q: Therefore, Q = 2 : 5 (c) Suppose now that the government puts a lump sum tax of 10 on the profits of the monopolist. What will be its output? Solution: This does not change the result from part (a) because the marginal cost is unaffected by the introduction of the tax. In other words, the monopolist will still choose to produce 3 units. However, its profits will be reduced. Question 8 The monopsony’s labor supply is w = 400 + 2 L . (a) What is the firm’s marginal expenditure equation? Solution: This is similar to the case of a monopolist. Here, the total expenditure (assuming that the monopolist only uses labor in the production process) is wL . Thus, the total expenditure is E = 400 L + 2 L 2 . Taking the derivative with respect to L , we get ME L = 400 + 4 L . (b) Assuming the demand for labor is w = 800 − 2 L , what would be the difference between the monopsony wage and the competitive one? Solution: The monopsonist will hire workers according to the condition ME L = w , where the wage is given by the demand function. Therefore, 400 + 4 L = 800 − 2 L . This implies that 6 L = 400 or L = 200 3 . To find the wage, we use the supply curve: w = 400+2 × 200 3 = 1600 3 = 533 : 33. In a competitive market, the wage is determined by the intersection of supply and demand. Thus, 400+2 L = 800 − 2 L . Solving for L , we get 4 L = 400. Therefore, L = 100 and w = 400+2 × 100 = 600. The difference between this wage and the wage under monopsony is 600 − 533 : 33 = 66 : 67.