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.

Key concepts

Profit Maximization

Profit maximization is the goal of any company operating as a monopoly. A monopolist seeks to find the level of output that maximizes its profits. This involves equating its marginal costs (MC) with its marginal revenue (MR) because at that point, the cost of producing an additional unit is just covered by the revenue from selling it. To understand this, think of profit as the difference between total revenue (TR) and total costs (TC). By maximizing this difference, the firm achieves its best financial outcome.

The process involves two main steps. First, derive the marginal revenue formula from the total revenue, which usually requires knowledge of the demand curve. Second, set MR equal to MC and solve for the quantity (Q). After obtaining the profit-maximizing quantity, substitute it back into the demand equation to find the price (P) that consumers are willing to pay. In summary, the product of this process gives us the price-quantity pair, often referred to as the profit-maximizing price and quantity.

Marginal Cost

Marginal cost is the cost of producing one additional unit of a good. For a monopolist, it is crucial to know this cost to make sure that the resources are being used efficiently, by comparing it to marginal revenue. In this exercise, we see that the marginal cost is constant and equal to 10, which simplifies calculations. Since it doesn’t change regardless of how much is produced, it can be represented as a horizontal line on a graph.

In the context of profit maximization, knowing the marginal cost allows the monopolist to determine the optimal output where the marginal cost intersects with marginal revenue. If the marginal revenue is greater than the marginal cost, it makes sense to produce more. Conversely, if marginal revenue falls below marginal cost, the firm should decrease production. The consistency of a constant marginal cost means that once it intersects with MR, it sets a clear boundary for profit-optimizing production levels.

Demand Curve

The demand curve is a graphical representation of the relationship between the price of a good and the quantity demanded. In a monopoly, this curve indicates how many units consumers are willing to buy at different price points. The shape of the demand curve can greatly affect a monopolist’s price and production strategies.

In the exercise, various demand curves are presented, each with different implications for the monopoly. A steeper demand curve (greater slope) shows that consumers are less sensitive to price changes, while a flatter demand curve implies more sensitivity. This sensitivity impacts how much the monopolist can charge for its product while maintaining a certain level of sales. For a monopolist, understanding how to read and use the demand curve is essential for calculating the equilibrium price-quantity combination that maximizes profit.

Marginal Revenue

Marginal revenue describes the additional income gained by selling one more unit of a product. For monopolies, understanding the marginal revenue is crucial because it helps determine the most profitable level of output. Unlike competitive markets, in a monopoly, selling additional units typically results in lowering the price of remaining units to increase sales, making MR less than the price.

To find the marginal revenue, we need to take the derivative of the total revenue function with respect to quantity. Once the MR is found, it is used to find where it equals the marginal cost. This equality point is where profit is maximized. Each demand curve gives rise to a unique marginal revenue curve, and because of this, monopolists don't have a traditional supply curve. They set production at the level where MR equals MC and adjust the price according to the demand curve to maximize profits.