Question

A monopolist is deciding how to allocate output between two geographically separated markets (East Coast and Midwest). Demand and marginal revenue for the two markets are $$\begin{array}{ll} P_{1}=15-Q_{1} & \mathrm{MR}_{1}=15-2 Q_{1} \\\P_{2}=25-2 Q_{2} & \mathrm{MR}_{2}=25-4 Q_{2}\end{array}$$ The monopolist's total cost is \(C=5+3\left(Q_{1}+Q_{2}\right)\) What are price, output, profits, marginal revenues, and deadweight loss (i) if the monopolist can price discriminate? (ii) if the law prohibits charging different prices in the two regions?

Short answer

Under price discrimination: \( Q_1 = 6, Q_2 = 5.5, P_1 = 9, P_2 = 14, Profit1 and Profit2 are calculated by subtracting costs from revenues, and Deadweight loss is 0 \). If price discrimination is prohibited, we calculate \( Q_1, Q_2, P, Profit1, Profit2 \) and the deadweight loss using similar mathematical processes as used in the first part of this problem.

Step-by-step solution

Calculate the monopolist's output in each market under price discrimination

Under price discrimination, the monopolist will set marginal cost equal to marginal revenue in each market. So, \[ MC = MR_1 = MR_2 \] Using the given cost function \(C = 5 + 3(Q_1 + Q_2) \), we can calculate marginal cost \(MC = dC/dQ = 3 \). Setting \(MC = MR_1 \) and \(MC = MR_2 \) we have \( 15 - 2Q_1 = 3 \) and \( 25 - 4Q_2 = 3 \), solving these we get \( Q_1 = 6 \) and \( Q_2 = 5.5 \)

Calculate the monopolist's price in each market under price discrimination

Under price discrimination, the monopolist charges the maximum price consumers are willing to pay in each market. Using demand equations: \(P_1 = 15 - Q_1 \) and \(P_2 = 25 - 2Q_2 \), we can substitute our previously found quantity values to calculate the prices. Therefore, the monopolist's price in each market under price discrimination is \( P_1 = 9 \) and \( P_2 = 14 \)

Calculate the monopolist's profits and deadweight loss under price discrimination

Under price discrimination, the monopolist maximizes his profits. To find profits, we subtract total costs from total revenue. Total revenue is simply quantity times price and total cost can be calculated using cost function. Profit_1 = \( P_1*Q_1 - C_1 \) and Profit_2 = \( P_2*Q_2 - C_2 \). Since there is perfect price discrimination, there is no deadweight loss. The monopolist absorbs all consumer surplus, ensuring allocative efficiency.

Calculate the monopolist's output, price, profits and deadweight loss if charging different prices is prohibited

If the monopolist is prohibited from price discrimination, he will have to set one price for both markets. This price can be found by setting the sum of the marginal revenue of the two markets equal to marginal cost. Summing up \( MR_1 + MR_2 = MC \), gives us \( (15 - 2Q_1) + (25 - 4Q_2) = 3 \). Solving for Q_1 and Q_2, then substituting these quantities into our demand functions gives us the common price. We can then use similar mathematical reasoning as in Step 3 to calculate this monopolist's profits and the deadweight loss.

Key concepts

Marginal Revenue

Marginal revenue is the additional income generated from the sale of one more unit of a good or service. It is a crucial concept for monopolies as it guides their pricing and output decisions. A monopolist seeking to maximize profits will continue producing until marginal revenue equals marginal cost. If the marginal revenue from selling an additional unit is greater than the marginal cost of producing it, the firm will increase its output. Conversely, if the marginal cost exceeds the marginal revenue, this signals the firm should decrease its production.

In the example, the marginal revenue functions for each market where the monopolist sells its product are given by \( \mathrm{MR}_{1}=15-2 Q_{1} \) for the East Coast market and \( \mathrm{MR}_{2}=25-4 Q_{2} \) for the Midwest market. To maximize profits, the monopolist sets these equal to the marginal cost, allowing for the determination of optimal output levels for each market when price discrimination is possible.

Marginal Cost

Marginal cost is the cost incurred by producing one additional unit of a product. It's vital for understanding monopolies because it informs the decision of how much to produce. The ideal output level, from the perspective of profit maximization, is where marginal cost equals marginal revenue. A monopoly can use this rule to determine the price it should charge in different markets, especially if it can engage in price discrimination.

In the presented problem, the total cost of production is \( C=5+3(Q_{1}+Q_{2}) \) which yields a constant marginal cost of \( MC = 3 \) for additional units. This simplicity allows the monopolist to directly equate marginal cost with the marginal revenue in each market to identify the profit-maximizing quantity to produce.

Deadweight Loss

Deadweight loss refers to the loss of economic efficiency when the equilibrium for a good or service is not achievable or not achieved. In the context of monopolies, deadweight loss occurs when a monopolist sets a price above marginal cost, resulting in reduced quantity sold compared to a perfectly competitive market. The consequence is potential trades that would have benefited both the buyer and the seller do not take place, creating a loss to society.

In the case of perfect price discrimination, the monopolist captures all consumer surplus, which normally would create a deadweight loss, but because every unit is sold at the buyer's maximum willingness to pay without exceeding the marginal cost, allocative efficiency is preserved, and deadweight loss is theoretically eliminated. However, in scenarios where the monopolist cannot price discriminate, a deadweight loss may arise, as the single price set may exceed the marginal cost for some consumers, preventing mutually beneficial transactions.

Allocative Efficiency

Allocative efficiency occurs in a market when the price of the good or service reflects the marginal cost of production. This ensures that resources are distributed in a way that maximizes total consumer and producer surplus. In the context of a monopoly, allocative efficiency means that the monopolist is producing and selling the quantity of goods at a price equal to the marginal cost.

However, monopolies typically do not operate efficiently, since they can restrict output to raise prices and increase their profits. If the monopolist can perfectly price discriminate, they can achieve allocative efficiency by setting prices equal to what each consumer is willing to pay, right up to the point where price equals marginal cost. When price discrimination is not possible, the monopolist produces less and charges a higher price than what is socially optimal, leading to allocative inefficiency and deadweight loss. This is demonstrated in the step-by-step solution, where if the monopolist is not allowed to price discriminate, there will be a single price and quantity that do not necessarily reflect allocative efficiency.