Question

A single firm monopolizes the entire market for widgets and can produce at constant average and marginal costs of \\[A C=M C=10.\\] Originally, the firm faces a market demand curve given by \\[Q=60-P.\\] a. Calculate the profit-maximizing price-quantity combination for the firm. What are the firm's profits? b. Now assume that the market demand curve shifts outward (becoming steeper) and is given by \\[Q=45-0.5 P.\\] What is the firm's profit-maximizing price-quantity combination now? What are the firm's profits? c. Instead of the assumptions of part (b), assume that the market demand curve shifts outward (becoming flatter) and is given by \\[Q=100-2 P.\\] What is the firm's profit-maximizing price-quantity combination now? What are the firm's profits? d. Graph the three different situations of parts (a), (b), and (c). Using your results, explain why there is no real supply curve for a monopoly.

Short answer

2. Calculate the firm's profits for each of the three demand curves. 3. Explain why there is no real supply curve for a monopoly.

Step-by-step solution

1a: Finding the profit-maximizing price-quantity combination for the first demand curve

The original demand curve is given by: \\[Q = 60 - P\\] First, we need to find the total revenue (TR) function. To do this, we can rewrite the demand curve as: \\[P = 60-Q\\] Then, multiply it by Q to find the total revenue: \\[TR = P \times Q = (60 - Q)Q\\] To find the marginal revenue (MR), we can take the derivative of TR with respect to Q: \\[MR = \dfrac{d(TR)}{dQ} = 60 - 2Q\\] Now, we can set the MR equal to the MC to find the profit-maximizing quantity: \\[60 - 2Q = 10\\] \\[Q = 25\\] Now that we have the profit-maximizing quantity, we can plug it into the demand curve to find the price: \\[P = 60 - 25 = 35\\] So, the profit-maximizing price-quantity combination for the first demand curve is P = 35 and Q = 25.

1b: Calculating the firm's profits for the first demand curve

To find the firm's profits, we need to calculate the difference between the total revenue and total cost. The total revenue (TR) can be found as: \\[TR = P \times Q = 35 \times 25 = 875\\] The total cost (TC) can be found using the average cost (AC) and the quantity produced: \\[TC = AC \times Q = 10 \times 25 = 250\\] Thus, the firm's profits are: \\[\text{Profit} = TR - TC = 875 - 250 = 625\\]

2a: Finding the profit-maximizing price-quantity combination for the second_demand curve

The second_demand curve is given by: \\[Q = 45 - 0.5P\\] First, we need to find the total revenue (TR) function. To do this, we can rewrite the demand curve as: \\[P = 90 - 2Q\\] Then, multiply it by Q to find the total revenue: \\[TR = P \times Q = (90 - 2Q)Q\\] To find the marginal revenue (MR), take the derivative of TR with respect to Q: \\[MR = \dfrac{d(TR)}{dQ} = 90 - 4Q\\] Now, set the MR equal to the MC to find the profit-maximizing quantity: \\[90 - 4Q = 10\\] \\[Q = 20\\] Now that we have the profit-maximizing quantity, we can plug it into the demand curve to find the price: \\[P = 90 - 40 = 50\\] So, the profit-maximizing price-quantity combination for the second_demand curve is P = 50 and Q = 20.

2b: Calculating the firm's profits for the second_demand curve

To find the firm's profits, we need to calculate the difference between the total revenue and total cost. The total revenue (TR) can be found as: \\[TR = P \times Q = 50 \times 20 = 1000\\] The total cost (TC) can be found using the average cost (AC) and the quantity produced: \\[TC = AC \times Q = 10 \times 20 = 200\\] Thus, the firm's profits are: \\[\text{Profit} = TR - TC = 1000 - 200 = 800\\]

3a: Finding the profit-maximizing price-quantity combination for the third_demand curve

The third_demand curve is given by: \\[Q = 100 - 2P\\] First, we need to find the total revenue (TR) function. To do this, we can rewrite the demand curve as: \\[P = 50 - 0.5Q\\] Then, multiply it by Q to find the total revenue: \\[TR = P \times Q = (50 - 0.5Q)Q\\] To find the marginal revenue (MR), take the derivative of TR with respect to Q: \\[MR = \dfrac{d(TR)}{dQ} = 50 - Q\\] Now, set the MR equal to the MC to find the profit-maximizing quantity: \\[50 - Q = 10\\] \\[Q = 40\\] Now that we have the profit-maximizing quantity, we can plug it into the demand curve to find the price: \\[P = 50 - 20 = 30\\] So, the profit-maximizing price-quantity combination for the third_demand curve is P = 30 and Q = 40.

3b: Calculating the firm's profits for the third_demand curve

To find the firm's profits, we need to calculate the difference between the total revenue and total cost. The total revenue (TR) can be found as: \\[TR = P \times Q = 30 \times 40 = 1200\\] The total cost (TC) can be found using the average cost (AC) and the quantity produced: \\[TC = AC \times Q = 10 * 40 = 400\\] Thus, the firm's profits are: \\[\text{Profit} = TR - TC = 1200 - 400 = 800\\]

4: Graph the three different situations and explain why there is no real supply curve for a monopoly

A supply curve shows the relationship between the quantity supplied and the price in the market. However, in a monopoly, the firm chooses both the quantity and the price that maximize its profit. As a result, there is no unique relationship between quantity and price, as each demand curve results in a different price-quantity combination. When the monopolist faces a different demand curve, the profit-maximizing price and output may change. In other words, there is no single, upward-sloping supply curve for a monopolist because the monopolist's supply decision is tied directly to the market demand curve, which can have many different shapes. Thus, a monopoly doesn't have a real supply curve that can be represented as a simple functional relationship between quantity and price.