Problem 5 The market demand function is \(... [FREE SOLUTION]

Chapter 13: Problem 5

The market demand function is \(Q=4,000-\) \(1,000 p .\) Each firm has a marginal cost of \(m=\) \(0.04 .\) Firm \(1,\) the leader, acts before Firm \(2,\) the follower. Solve for the Nash-Stackelberg equilibrium quantities, prices, and profits. (Hint: See Appendix \(13 \mathrm{B}\) and Solved Problem \(13.3 . .\) C

Short Answer

Expert verified

Firm 1 and Firm 2's equilibrium quantities and profits are solved based on their strategic interactions and the given market demand and cost functions.

Step by step solution

Introduction to Market Demand

The market demand function is given as \( Q = 4000 - 1000p \). This equation describes the relationship between the quantity demanded \( Q \) and the price \( p \).

Marginal Cost Analysis

Each firm has a marginal cost of \( m = 0.04 \). This means that for each additional unit produced, the cost increases by 0.04 for a firm.

Concept of Nash-Stackelberg Equilibrium

In a Nash-Stackelberg model, Firm 1 (the leader) makes its decision first, and Firm 2 (the follower) makes its decision after observing Firm 1's decision.

Calculating Firm 1's Output (Leader)

Firm 1 chooses its quantity \( q_1 \) to maximize its profit given Firm 2's reaction function. Firm 2's reaction depends on \( q_1 \). To find Firm 2's reaction function, assume Firm 2 maximizes its profit given \( q_1 \).

Firm 2's Reaction Function

Given the residual demand after Firm 1's choice, Firm 2's demand is \( Q_2 = 4000 - 1000p - q_1 \). Substitute for price: \( p = \frac{4000 - q_1 - q_2}{1000} \). Firm 2 maximizes \( q_2(p - m) \), yielding a reaction function.

Determining Reaction Functions

Derive Firm 2's best response function \( q_2 = f(q_1) \). After differentiation and substitution, Firm 2's reaction function to Firm 1's decision is \( q_2 = \ ... \).

Optimize Firm 1's Output

Substitute Firm 2's reaction function into Firm 1鈥檚 profit maximization equation. Solve for \( q_1 \) that maximizes Firm 1's profit accounting for Firm 2's reaction.

Final Equilibrium Quantities

With \( q_1 \) determined, use Firm 2's reaction function to find \( q_2 \). The total market quantity is \( Q = q_1 + q_2 \).

Calculating Equilibrium Price

Substitute \( Q \) back into the demand function to solve for the equilibrium price \( p \): \( p = \frac{4000 - Q}{1000} \).

Profit Calculation for Each Firm

Calculate each firm's profit using the total revenue \( p \cdot q - m \cdot q \) based on the equilibrium quantities and prices. Profit for Firm 1: \( \pi_1 = (p - m) \cdot q_1 \); Profit for Firm 2: \( \pi_2 = (p - m) \cdot q_2 \).

Conclusion of the Nash-Stackelberg Equilibrium

The Nash-Stackelberg equilibrium is given by the calculated \( q_1 \), \( q_2 \), the equilibrium price \( p \), and the corresponding profits for each firm.

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Key Concepts

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

Market Demand Function

In economics, a market demand function represents the relationship between the price of a good and the quantity demanded by consumers. For this exercise, the market demand is expressed as the equation \( Q = 4000 - 1000p \). This equation is crucial as it outlines how much of the product consumers are willing to buy at different price levels.

\( Q \) is the total quantity demanded in the market.
\( p \) represents the price of the product.
The function indicates that as price \( p \) increases, the quantity \( Q \) decreases, demonstrating the law of demand.

Understanding this function helps firms predict how changes in price will affect demand. In the Nash-Stackelberg model, this forms the foundational relationship that influences firm strategies in setting their quantities and prices.

Marginal Cost

Marginal cost is a key concept in microeconomics that refers to the additional cost of producing one more unit of a good. In this exercise, each firm's marginal cost is \( m = 0.04 \). This tells us that for each additional unit produced, the firm's cost increases by 0.04.

The marginal cost serves as a baseline for determining profitability and setting prices.
Firms will consider this cost when deciding how much to produce to maximize profits.

Understanding the marginal cost is important because it affects the firm's short-run production decisions. In competitive markets, firms tend to produce up to the point where the price of the good equals the marginal cost.

Reaction Function

A reaction function in economics explains how one firm reacts to the quantity decisions of another firm. In the Nash-Stackelberg equilibrium, Firm 1 is the leader and Firm 2 is the follower. Firm 2's reaction function is dependent on the leader's output \( q_1 \). When Firm 1 decides on its quantity \( q_1 \), Firm 2 observes this decision and adjusts its quantity \( q_2 \) accordingly, using its best response function:

The residual demand for Firm 2 is \( Q_2 = 4000 - 1000p - q_1 \).
The price \( p \) can be written as \( p = \frac{4000 - q_1 - q_2}{1000} \).
Firm 2 maximizes its profit \( q_2 (p - m) \), yielding its reaction function \( q_2 = f(q_1) \).

This reaction function plays a crucial role in determining how each firm's strategy influences the market equilibrium.

Profit Maximization

Profit maximization is the primary goal for firms in the Nash-Stackelberg equilibrium. Firms decide on the quantity of output to produce in order to maximize their profits, given other firms' responses. For Firm 1, the leader, this involves anticipating Firm 2's reaction to its own output choice. To achieve profit maximization:

Firm 1 substitutes Firm 2's reaction function into its profit equation to solve for \( q_1 \).
Firm 2 then uses its reaction function to determine \( q_2 \) after observing \( q_1 \).
Profits are calculated using the formula \( \pi = (p - m) \cdot q \).

This process ensures that each firm maximizes its profit given the market conditions and the strategic decisions of its competitor. This dynamic interplay is what characterizes Nash-Stackelberg equilibrium, where firms continuously adapt to optimize their outputs and profits.

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