Question

Consider two firms facing the demand curve \(P=50-5 Q\), where \(Q=Q_{1}+Q_{2}\). The firms' cost functions are \(C_{1}\left(Q_{1}\right)=20+10 Q_{1}\) and \(C_{2}\left(Q_{2}\right)=10+12 Q_{2}\) a. Suppose both firms have entered the industry. What is the joint profit- maximizing level of output? How much will each firm produce? How would your answer change if the firms have not yet entered the industry? b. What is each firm's equilibrium output and profit if they behave noncooperatively? Use the Cournot model. Draw the firms' reaction curves and show the equilibrium. c. How much should Firm 1 be willing to pay to purchase Firm 2 if collusion is illegal but a takeover is not?

Short answer

The joint profit-maximizing level of output, each firm's equilibrium output, profit in a non-cooperative environment and the takeover price for Firm 1 to purchase Firm 2 are all significant in this problem. These values will vary depending on the specifics of the demand and cost functions provided.

Step-by-step solution

Identify the Functions & Variables

Given, demand function \(P=50-5Q\) and cost functions \(C_{1}=20+10Q_{1}\) and \(C_{2}=10+12Q_{2}\). We also know \(Q=Q_{1}+Q_{2}\). Also, the profit for each firm can be given as \(\pi = PQ - C_{i}\) where \(i=1,2\).

Compute Joint Profit Maximizing Output

We need to find \(Q_{1}\) and \(Q_{2}\) such that joint profits are maximized. Let’s write the profit functions for each firm first:\(\pi_{1} = (50-5(Q_{1}+Q_{2}))Q_{1} - (20+10Q_{1})\) and \(\pi_{2} = (50-5(Q_{1}+Q_{2}))Q_{2} - (10+12Q_{2})\).To find the profit-maximizing quantity for each firm, we take derivative of each profit function with respect to its own quantity and set it to 0. Solve the equations to get the quantities.

Compute Cournot Equilibrium

Under Cournot model, the firms choose their quantity taking into account the other firm's quantity. The reaction function of a firm shows the quantity it will produce at each potential level of output of the other firm. Reacting functions are derived by solving each firm individual profit maximization problem which is similar to what we did earlier. We get equilibrium by solving the two reaction function simultaneously.

Computing Takeover Price

Firm 1 would be willing to pay up to the additional profits that it would receive from acquiring Firm 2. This additional profit is the difference between the joint profit when both firms are under one ownership (which we calculated earlier) and the total profit of two firms when they are operating separately (which is sum of profits calculated in above step under Cournot model).

Key concepts

Profit Maximization

In economics, profit maximization is the process by which firms determine the best output level and pricing to achieve maximum profitability. In the scenario presented, we have two firms sharing a demand curve given by \( P = 50 - 5Q \), where \( Q = Q_1 + Q_2 \). Firm 1 and Firm 2 have their cost functions defined as \( C_1(Q_1) = 20 + 10Q_1 \) and \( C_2(Q_2) = 10 + 12Q_2 \), respectively.

To find the joint profit-maximizing output, we compute the profit functions for both firms and take derivatives with respect to their respective quantities. This helps to determine \( Q_1 \) and \( Q_2 \) when they make joint decisions to maximize profits together. In essence, the firms need to choose outputs that maximize their collective profits by balancing revenue from sales and production costs.

This strategy depends on cooperating to equate the marginal revenue (MR) with marginal cost (MC) for both firms. By solving these equations, firms can find their combined optimal output and maintain joint profitability equilibrium while avoiding the competitive pressures when acting separately.

Reaction Curves

In the Cournot competition framework, firms react to the expected output decisions of each other. This is where the concept of reaction curves comes into play. Reaction curves illustrate how one firm adjusts its output in response to the output set by another firm.

With this numeric problem, we derive Firm 1's reaction curve by maximizing its profit while assuming Firm 2's output is constant. A similar approach is used to derive Firm 2's reaction curve. These curves are then used to determine the Cournot equilibrium, where both firms choose quantities such that neither has an incentive to unilaterally change its output.

• The intersection of reaction curves represents equilibrium in outputs where each firm’s strategy is optimal given the strategy of the other.
• Reaction functions can be expressed as mathematical equations, and solving these equations gives the equilibrium output levels for both firms.

Understanding reaction curves is vital, as they demonstrate how strategic interdependence affects firms' decisions in a noncooperative, competitive setting.

Takeover Strategy

A takeover strategy involves one firm seeking to acquire another to consolidate market position or enhance profitability. In scenarios where collusion is illegal, a takeover could be a legal alternative to achieving similar outcomes.

Here, Firm 1 may consider taking over Firm 2 to internalize competition and maximize profits that come from a unified strategy rather than competing. The maximum price Firm 1 should be willing to pay for Firm 2 is the added profit realized from the acquisition, which is the difference between the joint profits as one entity and the combined profits when operating separately.

• Calculating this takeover price involves understanding both joint profits and those under Cournot competition when firms operate individually.
• It's crucial to factor in the cost savings or additional synergies gained after merging when evaluating the worth of a takeover.

This strategy enables the firm to potentially achieve a monopoly-like status without violating anti-trust laws that ban explicit collusion.

Noncooperative Behavior

Noncooperative behavior in economic games like the Cournot model means firms choose their output levels independently to maximize individual profits without collaboration or agreements. Such behavior reflects real-world business scenarios where firms act in self-interest.

In our exercise, the firms are assumed to use noncooperative behavior under the Cournot framework. Each firm independently decides its output to maximize its profit, considering the other's production as given. This situation complicates profit maximization as neither firm can fully control the market via cooperation.

• Firms expect the other to act similarly, resulting in a Nash equilibrium where neither can improve profit by unilaterally altering its output.
• Noncooperative equilibrium tends to result in lower joint profits compared to cooperative strategies, yet it is stable since no firm can benefit by changing its own output alone.

Understanding noncooperative behavior is crucial in competitive markets, as it shapes strategies and outcomes of firm interactions.