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Internet users in a small Colorado town can access the Web in two ways: via their television cable or via a digital subscriber line (DSL) from their telephone company. The cable and telephone companies are Bertrand competitors, but because changing providers is slightly costly (waiting for the cable repairman can eat up at least small amounts of time!), customers have some slight resistance to switching from one to another. The demand for cable Internet services is given by \(q_{C}=100-3 p_{C}+2 p_{T}\) where \(q_{c}\) is the number of cable Internet subscribers in town, \(p_{C}\) is the monthly price of cable Internet service, and \(p_{T}\) is the price of a DSL line from the telephone company. The demand for DSL Internet service is similarly given by \(q_{T}=100-3 p_{T}+2 p_{C}\). Assume that both sellers can produce broadband service at zero marginal cost. a. Derive the cable company's reaction curve. Your answer should express \(p_{C}\) as a function of \(p_{T}\) b. Derive the telephone company's reaction curve. Your answer should express \(p_{T}\) as a function of \(p_{C}\) c. Combine reaction functions to determine the price each competitor should charge. Then determine each competitor's quantity and profits, assuming that the average total costs are zero. d. Suppose that the cable company begins to offer slightly faster service than the telephone company, which alters demands for the two products. Now \(q_{c}=100-2 p_{C}+3 p_{T}\) and \(q_{T}=100-\) \(4 p_{T}+p_{C}\). Show what effect this increase in service has on the prices and profit of each competitor.

Step by step solution

01

Derive Cable Company's Reaction Curve

To derive the reaction curve for the cable company, we need to maximize its profit. Since the marginal cost is zero, the profit \( \pi_C \) is simply the revenue. The cable company will set its price where demand equals zero:\[ q_C = 100 - 3p_C + 2p_T = 0 \]Solve for \( p_C \) in terms of \( p_T \):\[ 3p_C = 100 + 2p_T \]\[ p_C = \frac{100 + 2p_T}{3} \]This is the cable company's reaction curve.

02

Derive Telephone Company's Reaction Curve

Similarly, derive the reaction curve for the telephone company. Set the demand for DSL services to zero:\[ q_T = 100 - 3p_T + 2p_C = 0 \]Solve for \( p_T \) in terms of \( p_C \):\[ 3p_T = 100 + 2p_C \]\[ p_T = \frac{100 + 2p_C}{3} \]This is the telephone company's reaction curve.

03

Calculate Equilibrium Prices

Use the reaction curves to find the equilibrium prices by solving the system of equations. Substitute the telephone company's reaction curve into the cable company's:\[ p_C = \frac{100 + 2 \left( \frac{100 + 2p_C}{3} \right)}{3} \]Solve for \( p_C \):\[ p_C = 30 \]Then, find \( p_T \) using the cable company's price:\[ p_T = \frac{100 + 2(30)}{3} = 40 \]

04

Calculate Quantities and Profits

Calculate the equilibrium quantities using the demand functions:\[ q_C = 100 - 3(30) + 2(40) = 20 \]\[ q_T = 100 - 3(40) + 2(30) = 10 \]Since marginal costs are zero, profits are:\[ \pi_C = p_C \times q_C = 30 \times 20 = 600 \]\[ \pi_T = p_T \times q_T = 40 \times 10 = 400 \]

05

Consider Impact of Faster Cable Service

With the new demand equations reflecting faster cable service, let's solve again:For cable:\[ q_C = 100 - 2p_C + 3p_T \rightarrow p_C = \frac{100 + 3p_T}{2} \]For telephone:\[ q_T = 100 - 4p_T + p_C \rightarrow p_T = \frac{100 + p_C}{4} \]Solve the system of equations:\[ p_C = \frac{100 + 3 \left( \frac{100 + p_C}{4} \right)}{2} \]\[ p_C \approx 28.57, \quad p_T \approx 32.14 \]Profits with new speeds:\[ q_C = 100 - 2(28.57) + 3(32.14) \approx 46 \]\[ q_T = 100 - 4(32.14) + 28.57 \approx 4 \]New profits:\[ \pi_C = 28.57 \times 46 \approx 1314.22 \]\[ \pi_T = 32.14 \times 4 \approx 128.56 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reaction Function

A Reaction Function is a fundamental concept in the study of competition among firms. It captures the way one competitor adjusts its pricing based on the pricing decisions of the other competitor. In a Bertrand competition setup, two firms decide on the prices of their products simultaneously, aiming to maximize their profits.
For the cable company in our exercise, the reaction function is derived by setting the demand function for the cable service to zero. This means finding the price point where the number of subscribers equals zero, given the prices set by the telephone company. Mathematically, if the demand for the cable service is \[ q_C = 100 - 3p_C + 2p_T = 0 \]solving for \(p_C\) gives the cable company's reaction function:\[ p_C = \frac{100 + 2p_T}{3} \]
Similarly, the telephone company's reaction function is found using its own demand function, represented as\[ q_T = 100 - 3p_T + 2p_C = 0 \]Solving this equation for \(p_T\) provides:\[ p_T = \frac{100 + 2p_C}{3} \]
Reaction functions are crucial as they reflect how each firm's price depends on the competitor’s price, allowing us to understand strategic interactions.

Equilibrium Prices

Equilibrium Prices occur when the prices set by competing firms stabilize such that each firm's chosen price is the best response to its competitor’s price. The firms, in this sense, have no incentive to change prices unilaterally as they are already maximizing their profits given the competitor’s price.
To determine the equilibrium prices in our exercise, consider the derived reaction functions:- Cable Company: \( p_C = \frac{100 + 2p_T}{3} \)- Telephone Company: \( p_T = \frac{100 + 2p_C}{3} \)
By substituting one reaction function into the other and solving the simultaneous equations, we find the equilibrium prices for both services:- Cable Service: \( p_C = 30 \)- Telephone Service: \( p_T = 40 \)
These prices reflect the point at which both competitors are satisfied with their pricing strategy, and any deviation from these prices would result in decreased profits.

Demand Functions

Demand Functions describe the relationship between the price of a product or service and the quantity demanded by consumers. These functions are especially important in competitive markets like the one in this exercise, where two companies vie for the same customer base.
The demand for the cable service is expressed as:\[ q_C = 100 - 3p_C + 2p_T \]
This equation demonstrates that the quantity demanded for cable service decreases when its own price increases but increases when the telephone company's price increases.

• \(100\) is the base demand when both prices are zero.
• \(-3p_C\) indicates the negative relationship between price and demand for the cable service.
• \(+2p_T\) reflects the positive impact of the rival’s price on cable demand.

The same logic applies to the DSL service demand function:\[ q_T = 100 - 3p_T + 2p_C \]
Understanding these demand functions helps businesses anticipate how changes in their own pricing and their competitor's pricing could affect their market share.

Marginal Cost

Marginal Cost in economics represents the cost of producing one additional unit of a good or service. For competitive firms, understanding marginal cost is critical in setting prices strategically.
In this exercise, it is given that both firms have zero marginal costs. This implies that producing additional units of Internet service involves no extra cost for either company.
This simplification means: - Companies can focus on maximizing revenue instead of worrying about production costs. - Each unit sold contributes directly to profit since there are no additional costs incurred.
With zero marginal costs, the primary focus for firms in a Bertrand competition is to set optimal prices that maximize profits, taking into account how price changes affect demand due to competitor pricing.