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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 the cable company's reaction curve, Step 1

The demand curve for cable Internet is given by \( q_C = 100 - 3p_C + 2p_T \). Since the marginal cost is zero, to maximize profits the cable company sets quantity equal to zero and solves for \( p_C \):\[\begin{aligned} q_C = 0 & = 100 - 3p_C + 2p_T \ 3p_C & = 100 + 2p_T \ p_C & = \frac{100 + 2p_T}{3}. \end{aligned}\]This is the cable company's reaction curve.

02

Derive the telephone company's reaction curve, Step 2

The demand curve for DSL is \( q_T = 100 - 3p_T + 2p_C \). Similarly, the telephone company optimizes by setting quantity to zero and solving for \( p_T \):\[\begin{aligned} q_T = 0 & = 100 - 3p_T + 2p_C \ 3p_T & = 100 + 2p_C \ p_T & = \frac{100 + 2p_C}{3}. \end{aligned}\]This is the telephone company's reaction curve.

03

Find equilibrium prices, Step 3

To find the equilibrium prices, substitute the reaction curve of one company into the other. Substituting the cable company's reaction curve into the telephone company's yields:\[\begin{aligned} p_T & = \frac{100 + 2(\frac{100 + 2p_T}{3})}{3} \ & = \frac{100 + \frac{200}{3} + \frac{4p_T}{3}}{3} \ 3p_T & = \frac{300}{3} + \frac{200}{3} + \frac{4p_T}{3} \ 9p_T & = 500 + 4p_T \ 5p_T & = 500 \ p_T & = 100. \end{aligned}\]Substituting \( p_T = 100 \) back into the equation for \( p_C \):\[ p_C = \frac{100 + 2 \cdot 100}{3} = 100. \]Thus, the equilibrium prices are \( p_C = 100 \) and \( p_T = 100 \).

04

Calculate quantities and profits, Step 4

Using \( p_C = 100 \) and \( p_T = 100 \), substitute into the quantity equations: \[\begin{aligned} q_C & = 100 - 3 \times 100 + 2 \times 100 = 0, \ q_T & = 100 - 3 \times 100 + 2 \times 100 = 0. \end{aligned}\]Both companies have zero sales, and since \( q = 0 \), profits are also zero since \( Profit = (p - 0) \cdot q \).

05

New demand conditions due to faster service, Step 5

Now, the demand functions change to \( q_{C} = 100 - 2 p_{C} + 3 p_{T} \) and \( q_{T} = 100 - 4 p_{T} + p_{C} \). Derive the new reaction curves. For cable: \[ q_C = 0 = 100 - 2p_C + 3p_T \ 2p_C = 100 + 3p_T \ p_C = \frac{100 + 3p_T}{2}. \] For telephone: \[ q_T = 0 = 100 - 4p_T + p_C \ 4p_T = 100 + p_C \ p_T = \frac{100 + p_C}{4}. \]

06

Find new equilibrium prices and calculate profits, Step 6

Substitute to find new equilibrium prices. Substitute the cable reaction into the telephone one: \[\begin{aligned} p_T & = \frac{100 + \frac{100 + 3p_T}{2}}{4} \ & = \frac{100 + 50 + 1.5p_T}{4} \ 4p_T & = 150 + 1.5p_T \ 2.5p_T & = 150 \ p_T & = 60. \end{aligned}\]Substitute back into \( p_C \):\[ p_C = \frac{100 + 3 \cdot 60}{2} = 110. \]Quantities are \( q_C = 0 \) and \( q_T = 0 \) leading to zero profits again because \( q = 0 \). The increase in faster service does not result in sales given the new reaction but shows higher prices when compared to zero sales.

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Key Concepts

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

Bertrand Competition

Bertrand Competition is a fundamental concept in game theory that describes how firms compete in markets by setting prices. Unlike other models in which companies might compete through quantities or capacities, Bertrand competitors focus on undercutting each other's prices. This often leads to very aggressive pricing strategies.
In the context of the small Colorado town in the exercise, both the cable and telephone providers are considered Bertrand competitors. They have very similar products in terms of broadband services, which means they primarily compete on pricing rather than on the quantity of service provided. Such behavior typically drives prices down to the marginal cost of providing the service, which is zero in this scenario.

• Since marginal costs are zero, companies aim to set their products' prices in a way that any slight price increase by competitors would not lead to a loss of customers.
• Customers' resistance to switching providers introduces a slight deviation from pure Bertrand Competition, as they don't immediately jump to the competitor when there's a minuscule price discrepancy.

This competition leads to reaction functions, where each firm reacts to the price set by the other.

Reaction Functions

Reaction functions describe how one firm's pricing strategy influences and is influenced by the competitor's pricing. They are a tool for understanding how firms can maximize their profits while considering the actions of competitors.
In the exercise, we derived reaction functions for both the cable company and the telephone company. The cable company's reaction function, which expresses its optimal price as a response to the phone company's price, is derived from the demand curve: \[ p_C = \frac{100 + 2p_T}{3}. \]

• This equation shows how the cable company's pricing depends on the telephone company's price.
• The coefficients in the reaction function, such as '2' and '3', stem from the demand equation's balance of consumer preference shifts and price sensitivity.

The telephone company's reaction function is analogous:\[ p_T = \frac{100 + 2p_C}{3}. \]
When both firms adjust their prices in response to each other's actions by following these functions, they eventually reach equilibrium prices, where neither firm benefits from unilaterally changing its pricing.

Equilibrium Prices

Equilibrium prices in the Bertrand model occur when both competing firms have no incentive to alter their pricing strategies, producing a stable pricing solution. This is achieved when each company's pricing strategy is aligned with the reaction strategy of its competitor.
During the exercise, we found the equilibrium prices by substituting one company's reaction function into the other. This resulted in:\[ p_C = p_T = 100. \]

• These equilibrium prices mean both cable and telephone providers set their prices at 100, ensuring that any price change would not improve their respective profits.
• At this equilibrium, even with zero marginal costs, both firms found a pricing balance where neither could benefit from unilateral price adjustments.

The equilibrium ensures that each company is operating under maximum profit conditions given the zero cost scenario, although both end up having zero sales, showing a peculiar edge case where all customers resist at these prices.

Demand Curve Analysis

Understanding the demand curve is crucial in analyzing how changes in price impact the quantity demanded by consumers. In this exercise, demand curve analysis is pivotal to understanding firm competition and consumer behavior.
The original demand function for cable Internet, for instance, is:\[ q_C = 100 - 3p_C + 2p_T. \]

• It implies that a higher cable price \( p_C \) reduces demand for cable services \( q_C \), while an increase in the telephone price \( p_T \) favors cable service consumption.
• These response factors (like -3 and +2) point to the relative sensitivity of consumers to price changes and reflect the strategic interdependence of firms.

With changing service speeds, new demand functions reflect altered consumer perceptions and demands:\[ q_C = 100 - 2p_C + 3p_T, \] \[ q_T = 100 - 4p_T + p_C. \]This shows how the cable company's innovation in speed changes the equilibrium and alters the demand landscape, requiring re-calculations of reaction functions and price strategies.