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 stccper) and is given by \\[ Q=45-.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

Answer: The profit-maximizing price-quantity combinations and profits are as follows: 1. Original demand curve: The price-quantity combination is (35, 25) and the profit is 625. 2. Steeper demand curve: The price-quantity combination is (50, 20) and the profit is 800. 3. Flatter demand curve: The price-quantity combination is (30, 40) and the profit is 800.

Step-by-step solution

a. Profit-maximizing price-quantity combination and Profit for original demand curve

To find the profit-maximizing price-quantity combination, we first need to find the marginal revenue (MR) function. Since demand is given by \\(Q = 60 - P\\), we can rewrite this as \\(P = 60 - Q\\). Revenue (R) can be written as \\(R = PQ = Q(60-Q)\\), and then we can differentiate R with respect to Q to find the MR: \\[MR = \frac{dR}{dQ} = 60 - 2Q\\]. Next, set MC equal to MR and solve for Q: \\[10 = 60 - 2Q \Rightarrow Q = 25\\]. Now, plug Q=25 back into the demand curve to find the price: \\[P = 60 - 25 = 35\\]. Therefore, the profit-maximizing price-quantity combination is (35, 25). Finally, calculate the firm's profit by finding the total revenue (TR), total cost (TC), and then calculating the profit (π): \\[TR = PQ = 35 \times 25 = 875\\], \\[TC = AC \times Q = 10 \times 25 = 250\\], \\[\pi = TR - TC = 875 - 250 = 625\\].

b. Profit-maximizing price-quantity combination and Profit for steeper demand curve

For the new demand curve given by \\(Q = 45 - 0.5P\\), we first need to rewrite it in terms of P: \\(P = 90 - 2Q\\). Next, find the revenue function and differentiate it with respect to Q to find the new MR: \\[R = PQ = Q(90 - 2Q)\\], \\[MR = \frac{dR}{dQ} = 90 - 4Q\\]. Set MC equal to MR and solve for Q: \\[10 = 90 - 4Q \Rightarrow Q = 20\\]. Plug Q=20 back into the new demand curve to find the price: \\[P = 90 - 2 \times 20 = 50\\]. Therefore, the profit-maximizing price-quantity combination is (50, 20). Calculate the firm's new profit: \\[TR = PQ = 50 \times 20 = 1000\\], \\[TC = AC \times Q = 10 \times 20 = 200\\], \\[\pi = TR - TC = 1000 - 200 = 800\\].

c. Profit-maximizing price-quantity combination and Profit for flatter demand curve

For the demand curve given by \\(Q = 100 - 2P\\), rewrite it in terms of P: \\(P = 50 - 0.5Q\\). Find the revenue function and differentiate it with respect to Q to find the new MR: \\[R = PQ = Q(50 - 0.5Q)\\], \\[MR = \frac{dR}{dQ} = 50 - Q\\]. Set MC equal to MR and solve for Q: \\[10 = 50 - Q \Rightarrow Q = 40\\]. Plug Q=40 back into the new demand curve to find the price: \\[P = 50 - 0.5 \times 40 = 30\\]. Therefore, the profit-maximizing price-quantity combination is (30, 40). Calculate the firm's new profit: \\[TR = PQ = 30 \times 40 = 1200\\], \\[TC = AC \times Q = 10 \times 40 = 400\\], \\[\pi = TR - TC = 1200 - 400 = 800\\].

d. Graph the three different situations and explain the lack of a supply curve in monopoly

Begin with drawing three graphs with quantity (Q) on the x-axis and price (P) on the y-axis for each demand curve scenario: 1. For the original demand curve \\(Q = 60 - P\\), plot a downward-sloping straight line that goes through (0,60) and (60,0). Mark the profit-maximizing point (35, 25) as A. 2. For the steeper demand curve \\(Q = 45 - 0.5P\\), plot a downward-sloping straight line that goes through (0, 90) and (45,0), with a steeper slope than the first graph. Mark the profit-maximizing point (50, 20) as B. 3. For the flatter demand curve \\(Q = 100 - 2P\\), plot a downward-sloping straight line that goes through (0,50) and (100,0), with a flatter slope than the first graph. Mark the profit-maximizing point (30, 40) as C. Next, plot the MC curve as a horizontal line at P=10 in all three graphs. Notice that MR differs for each demand curve, so the intersection between MR and MC leads to different price-quantity combinations. In a competitive market with a supply curve, price is solely determined by the intersection of the supply and demand curves. However, in a monopoly, there is no fixed relationship between price and quantity supplied. The monopolist chooses the profit-maximizing quantity where MC=MR, and then charges the corresponding price given by the demand curve. Because MR and the corresponding optimal quantity differ for each individual demand curve, there is no real supply curve for a monopolist.