Question

Suppose a monopoly can produce any level of output it wishes at a constant marginal (and average) cost of \(\$ 5\) per unit. Assume the monopoly sells its goods in two different markets separated by some distance. The demand curve in the first market is given by $$Q_{1}=55-P_{1}$$ and the demand curve in the second market is given by $$ Q_{2}=70-2 P_{2}$$ a. If the monopolist can maintain the separation between the two markets, what level of output should be produced in each market, and what price will prevail in each market? What are total profits in this situation? b. How would your answer change if it only cost demanders \(\$ 5\) to transport goods between the two markets? What would be the monopolist's ncw profit level in this situation? c. How would your answer change if transportation costs were zero and the firm was forced to follow a singlc-pricc policy? d. Suppose the firm could adopt a linear two-part tariff under which marginal prices must be equal in the two markets but lump-sum entry fees might vary. What pricing policy should the firm follow?

Short answer

Answer: To determine optimal levels of output, prices, and profits in each market for different scenarios, follow the steps mentioned in the solution: a) For a monopolist maintaining separation between the two markets, follow the steps 1-3 to calculate equilibrium quantities and prices in each market, and the total profit for the monopolist. b) For a transportation cost of $5 between the two markets, follow step 1 in part (b) to calculate the adjusted demand curve in market 2. Then, follow steps 1-3 in part (a) with the new demand curve and calculate the monopolist's new profit levels in this situation. c) For a single-price policy with zero transportation costs, follow step 1 in part (c) to combine the two demand curves into a single one and then determine the equilibrium output, price, and profit for this single-price policy. d) For a monopolist adopting a linear two-part tariff, follow step 1 in part (d) to determine the two-part tariff, which consists of a marginal price equal in both markets and a lump-sum entry fee. Determine the optimal pricing policy by maximizing the firm's profit.

Step-by-step solution

Calculate marginal revenue for both markets.

Using the given demand curves, \(Q_{1}=55-P_{1}\) and \(Q_{2}=70-2P_{2}\), we can derive the total revenue as expressed in terms of price. For the first market: \(TR_{1}=P_{1}Q_{1}=P_{1}(55-P_{1})\) For the second market: \(TR_{2}=P_{2}Q_{2}=P_{2}(70-2P_{2})\) Next, we find the marginal revenue for both markets by taking the derivative of total revenue with respect to the corresponding quantities: \(MR_{1}=\frac{d(TR_{1})}{dQ_{1}}\) and \(MR_{2}=\frac{d(TR_{2})}{dQ_{2}}\)

Equate marginal cost to marginal revenue in each market.

Since the monopolist can maintain the separation between the two markets, we can apply the profit-maximization rule by equating marginal cost (MC) and marginal revenue (MR) for each market separately: \(MC = 5\) Setting MC equal to MR in both markets and solving for the equilibrium quantities: \(5=MR_{1}\) and \(5=MR_{2}\)

Find the equilibrium quantities and prices in each market.

Plug the equilibrium quantities into their respective demand curves to find the equilibrium prices: \(Q_{1}=55-P_{1}\) and \(Q_{2}=70-2P_{2}\) Now we can calculate the total profit for the monopolist by subtracting total cost from total revenue: \(\Pi=TR_{1}+TR_{2} - TC\) b. Transportation cost of \(\$ 5\) between the two markets.

Calculate the adjusted demand curve in market 2.

Since transportation cost is equal to $5, we will subtract this from the second market's demand curve: \(Q_{2} = 70 - 2(P_{2}+5)\) Now, repeat the process from steps 1-3 in part (a) with the new demand curve, and calculate the monopolist's new profit levels in this situation. c. Transportation costs are zero, and the firm follows a single-price policy.

Combine the two demand curves into a single one.

Since the firm must follow a single-price policy, we must find a single demand curve to represent the total market. Add the equilibrium quantities from both markets to create a combined demand curve: \(Q_{total} = Q_{1} + Q_{2}\) Now, follow steps 1-3 in part (a) to determine the equilibrium output, price, and profit for this single price policy. d. Monopolist adopting a linear two-part tariff.

Determine the two-part tariff.

The two-part tariff consists of a marginal price equal in both markets and a lump-sum entry fee. To find the optimal marginal price, set the MC equal to the average of the two MR's: \(MC = \frac{MR_{1}+MR_{2}}{2}\) Now, find the optimal lump-sum entry fees for each market by charging the difference between the consumer's willingness to pay and the optimal marginal price. This can be obtained from each demand curve: \(Lump\_sum_{1}=P_{1}-MC\) \(Lump\_sum_{2}=P_{2}-MC\) Determine the optimal pricing policy by maximizing the firm's profit.

Key concepts

Marginal Revenue

In economics, marginal revenue represents the additional income that a firm earns from selling one more unit of its product. For monopolies, understanding marginal revenue is crucial for profit maximization. Here, the firm in the exercise is faced with two separate markets, each with its own demand curve. To compute the marginal revenue (MR) for each market, we start by expressing the demand curves for the markets:

• For Market 1: Demand Curve is given as \(Q_{1} = 55 - P_{1}\). The total revenue \(TR_{1}\) can be computed as \(TR_{1} = P_{1} \times Q_{1} = P_{1}(55 - P_{1})\).
• For Market 2: Demand Curve is given as \(Q_{2} = 70 - 2P_{2}\). Here, \(TR_{2} = P_{2} \times Q_{2} = P_{2}(70 - 2P_{2})\).

By differentiating these total revenue functions with respect to the respective quantity, we get the marginal revenue:

• For Market 1: The derivative of \(TR_{1}\) gives us \(MR_{1}\).
• For Market 2: The derivative of \(TR_{2}\) gives us \(MR_{2}\).

Knowing the marginal cost (MC) is constant at \$5, the monopoly equates MC to MR for each market separately to find the profit-maximizing output level. This is important, as it guides the firm to set the correct output levels to maximize profits without overproducing or underutilizing resources.

Two-Part Tariff

A two-part tariff is a pricing strategy often used by monopolies to maximize profits by charging consumers in two parts: a lump-sum fee and a per-unit price. This method can be adopted when the monopolist can sell in both markets at different rates. The lump-sum fee is usually charged for access or entry, while the per-unit charge is applied to each unit purchased.
When implementing a two-part tariff, the firm sets the marginal price across both markets equal to the marginal cost. In this exercise, because the marginal cost is constant at \\(5, the company would ideally set the variable price at \\)5. However, for maximizing revenue, the company could vary the lump-sum fee based on consumer willingness to pay.

• In Market 1, the lump-sum entry fee would be linked to consumer surplus, which is determined by the difference between what consumers are willing to pay according to the demand curve and the marginal price of \$5.
• Similarly, for Market 2, the firm would calculate a different entry fee based on those consumers' willingness to pay.

This pricing policy allows the monopolist to capture consumer surplus as additional profit, and each market's fee can be tailored independently to reflect consumer preferences and demand elasticity.

Demand Curve Analysis

Understanding demand curves is fundamental to setting prices in monopolistic markets. A demand curve shows the relationship between the price of a good and the quantity demanded by consumers. In different markets, these curves can vary, reflecting differing consumer preferences or external factors. In the exercise, we dealt with two different demand curves:

• Market 1's demand curve is \(Q_{1} = 55 - P_{1}\), indicating a linear and downward slope. This suggests that as prices decrease, consumers demand more.
• Market 2's demand curve, \(Q_{2} = 70 - 2P_{2}\), is also linear but less responsive to price changes due to the coefficient of 2 impacting the slope.

By analyzing these curves, the monopoly can determine the impact of price changes on demand in each market. For instance, in the exercise, when transportation costs affect market separation, they impact the slope or demand outright. Similarly, under a single-price policy, understanding the aggregate demand curve created by combining individual market curves helps the monopoly decide on optimum collective pricing.
The demand curve analysis guides decisions on quantity and pricing to ensure that any pricing strategy effectively capitalizes on consumer behavior while maximizing revenue across different scenarios.