Question

Sal's satellite company broadcasts TV to subscribers in Los Angeles and New York. The demand functions for each of these two groups are $$\begin{array}{l}{Q}_{NY}=60-0.25P_{NY}\\ {\text{Q}}_{LA}=100-0.50P_{LA}\end{array}$$ where $Q$ is in thousands of subscriptions per year and $P$ is the subscription price per year. The cost of providing $Q$ units of service is given by $$C=1000+40Q$$ where $Q=Q_{\mathrm{NY}}+Q_{\mathrm{LA}}$ a. What are the profit-maximizing prices and quantities for the New York and Los Angeles markets? b. As a consequence of a new satellite that the Pentagon recently deployed, people in Los Angeles receive Sal's New York broadcasts and people in New York receive Sal's Los Angeles broadcasts. As a result, anyone in New York or Los Angeles can receive Sal's broadcasts by subscribing in either city. Thus Sal can charge only a single price. What price should he charge, and what quantities will he sell in New York and Los Angeles? c. In which of the above situations, (a) or (b), is Sal better off? In terms of consumer surplus, which situation do people in New York prefer and which do people in Los Angeles prefer? Why?

Short answer

Profit-maximizing prices and quantities under separate pricing are $P_{NY}=40$, $Q_{NY}=5$, $P_{LA}=80$, $Q_{LA}=10$. Under single-price setting, they are $P=60$, $Q_{NY}=30$, $Q_{LA}=50$. Sal is better off under single-price setting. New Yorkers prefer separate pricing and Los Angelenos prefer single pricing due to differences in consumer surplus.

Step-by-step solution

Compute Profit-Maximizing Prices and Quantities for each Market

First obtain the inverse demand function for each market and subtract the marginal cost to obtain the marginal revenue. Equate marginal revenue to marginal cost to find profit-maximizing quantity. Then substitute this quantity into the demand function to find the associated price. Equations are: Inverse demand for NY: $$P_{NY}=60-4Q_{NY}$$Inverse demand for LA: $$P_{LA}=100-2Q_{LA}$$Marginal cost: $$MC=40+\frac{dC}{dQ}$$Solving gives $Q_{NY}=5$, $P_{NY}=40$, $Q_{LA}=10$ and $P_{LA}=80$

Compute Outcome for Single-Price Setting

In this scenario, let's say the price in both markets is $P$. Combine the demand functions for the markets into one equation and solve for quantity and price similarly to step 1. We get $Q_{total}=Q_{NY}+Q_{LA}=160-0.75P$. Solving using marginal cost gives $Q_{total}=80$, $P=60$, $Q_{NY}=30$, $Q_{LA}=50$

Compare Sal's Profits in Two Scenarios

Using the price and quantity solutions from both scenarios, compute the profit: $$Profit=P\ast Q-C(Q)$$. This gives profits for scenario (a) as 2000and for scenario (b) as 4000. Thus Sal is better off in scenario (b).

Compute Consumer Surplus

Consumer surplus is given by: $$CS={\int }^Q(P_{max}-P)dQ$$. Compute the consumer surplus for NY and LA in both scenarios. Consumer surplus increases with lower prices and larger quantities. It turns out that NY prefers scenario (a) because their surplus is higher due to the lower price, while LA prefers scenario (b) because their surplus increases with the higher quantity.

Key concepts

Microeconomic Demand Functions

When studying microeconomics, understanding demand functions is crucial. A demand function describes how the quantity of a good demanded by consumers changes in response to its price. In our example, Sal's satellite company has separate demand functions for New York and Los Angeles markets, represented as

$$Q_{NY}=60-0.25P_{NY}$$ $$Q_{LA}=100-0.50P_{LA}$$
where $Q$ denotes thousands of subscriptions, and $P$ represents the annual subscription price. These equations suggest that as the price increases, the quantity demanded decreases, which is a fundamental principle known as the law of demand. To maximize profits in both markets, Sal needs to assess how changes in prices can affect the quantities demanded and subsequently how those quantities can meet Sal's production costs and desired profits.

The inverse demand function is obtained by solving the demand function for price. This form is particularly useful for Sal to determine the prices he should charge for a given quantity of subscriptions sold. Simply put, the inverse demand functions for New York and Los Angeles tell us the highest price that consumers are willing to pay for each level of quantity provided.

Consumer Surplus

Consumer surplus measures the difference between the maximum price that consumers are willing to pay for a good and the market price they actually pay. It reflects the benefit that consumers receive from purchasing goods at lower prices than they are prepared to pay.

In our scenario, the concept of consumer surplus helps to assess which market conditions are better for the consumers in New York and Los Angeles. Consumer surplus is computed as the area below the demand curve but above the price level, mathematically represented by the integral

$$CS={\int }^Q(P_{max}-P)dQ$$
where $P_{max}$ is the highest price consumers are willing to pay, and $Q$ is the quantity.

With Sal's satellite services, when he adjusts prices or the manner in which he sells his packages (one price for both cities vs. different prices for each city), the consumer surplus will shift. For example, people in New York might benefit from a situation wherein Sal charges different prices in each city because the optimal price in New York (as per Step 1 of the solution) might be lower than the unified price he would set if he were charging a single price for both cities.

Marginal Cost Analysis

Marginal cost is the additional cost incurred from producing one more unit of a good or service. It plays a significant role in determining the profit-maximizing output level for a business. In the exercise, we use marginal cost analysis to identify the profit-maximizing quantities and prices for Sal's satellite company.

The marginal cost function provided in the exercise is

$$MC=40+\frac{dC}{dQ}$$
where $dC/dQ$ represents the change in total cost from producing one more unit. In this case, since there are no additional variables affecting cost, the marginal cost is constant at $40 for each additional thousand subscriptions.

By setting the marginal cost equal to the marginal revenue (which is derived from the inverse demand function), we can determine the exact quantity at which profit is maximized. This analysis is used to solve for both the dual-pricing scenario and the single-price scenario in our exercise. Understanding how to equate marginal revenue with marginal cost is a fundamental step in optimizing production and pricing strategies in microeconomics.