Question

Two firms produce luxury sheepskin auto seat covers: Western Where (WW) and B.B.B. Sheep (BBBS). Each firm has a cost function given by \\[ C(q)=30 q+1.5 q^{2} \\] The market demand for these seat covers is represented by the inverse demand equation \\[ P=300-3 Q \\] where \(Q=q_{1}+q_{2},\) total output. a. If each firm acts to maximize its profits, taking its rival's output as given (i.e., the firms behave as Cournot oligopolists), what will be the equilibrium quantities selected by each firm? What is total output, and what is the market price? What are the profits for each firm? b. It occurs to the managers of WW and BBBS that they could do a lot better by colluding. If the two firms collude, what will be the profit-maximizing choice of output? The industry price? The output and the profit for each firm in this case? c. The managers of these firms realize that explicit agreements to collude are illegal. Each firm must decide on its own whether to produce the Cournot quantity or the cartel quantity. To aid in making the decision, the manager of \(\mathrm{WW}\) constructs a payoff matrix like the one below. Fill in each box with the profit of \(\mathrm{WW}\) and the profit of BBBS. Given this payoff matrix, what output strategy is each firm likely to pursue? d. Suppose WW can set its output level before BBBS does. How much will WW choose to produce in this case? How much will BBBS produce? What is the market price, and what is the profit for each firm? Is WW better off by choosing its output first? Explain why or why not.

Short answer

The calculated values for equilibrium quantities, total output, market price and profits under Cournot, collusion and Stackelberg competition conditions would be the final answer. Since the specific constants are not given in the exercise, the solution would be represented as formulas and depend on the given functions earlier.

Step-by-step solution

Calculate Cournot Equilibrium output

Start with the profit maximization equation for Cournot competitors and set them for each firm. The profit function for each firm i (i can be 1 or 2) is given by \[ \Pi_i = P \cdot q_i - C(q_i) \] where \( P = 300-3Q = 300 - 3(q_1 + q_2) \), and the cost function \( C(q) = 30q + 1.5q^{2} \). Equate the first-order derivative of the profit function with respect to quantity to zero for each firm, and solve the resulting equations simultaneously for \( q_1 \) and \( q_2 \).

Find total output and market price

Find the total output \( Q \) by adding \( q_1 \) and \( q_2 \) obtained in Step 1. Then calculate the market price \( P \) using the inverse demand equation given by \( P = 300 - 3Q \).

Determine the firm's profits

Substitute the obtained values of total output \( Q \) and quantities \( q_1 \) and \( q_2 \) into the profit function \( \Pi = P \cdot q_i - C(q_i) \) to calculate the profits of each firm.

Find Collusion Output

Calculate the collusion output by treating both firms as a single monopolist and maximizing the joint profits. This means that the total cost function is now double the given cost function, and the first order condition derived from joint profit function (sum of the individual profit functions) is set to zero. Solve for \( Q \) to get the collusion output.

Determine output under Stackelberg competition

This involves considering a sequential game where one firm (WW) moves first. WW's profit-maximizing output is found by considering the reaction function of BBBS, where the reaction function is the quantity BBBS would optimally produce for each possible quantity WW might produce. WW takes this into account and then chooses its own output to maximize profit. Find this output of WW, and then substitute this value into BBBS's reaction function to find BBBS's output.

Calculate Stackelberg market price and profits

Substitute the obtained values of WW and BBBS's output into the inverse demand function to get the market price. Then substitute these values along with market price into each firm's profit function to get their respective profits.

Key concepts

Profit Maximization

In economic terms, profit maximization is the process by which firms determine the best output level to achieve the highest possible profit. For firms like Western Where (WW) and B.B.B. Sheep (BBBS) involved in Cournot competition, this means choosing quantities based on their rivals' output.

To maximize profits, each firm must consider costs and revenues. This involves using the profit function:

• \( \Pi_i = P \cdot q_i - C(q_i) \)
  
• Where \( P = 300 - 3(q_1 + q_2) \) is the price, and \( C(q) = 30q + 1.5q^2 \) is the cost.

By setting the derivative of the profit function to zero, firms identify their optimal production levels, balancing revenue and costs effectively.

This step is critical in locating the Cournot equilibrium, where no firm can increase profit by solely changing their output.

Collusion in Oligopoly

Collusion refers to an agreement between firms in an oligopoly to restrict output and increase prices to maximize collective profits. Although illegal in many jurisdictions, understanding its concept helps in analyzing market behaviors.

In our context, if WW and BBBS were to collude, they'd act as a monopoly. They'd combine their outputs to find the profit-maximizing level for the whole market. This can be done by adding their cost functions together and setting the derivative of the joint profit function to zero. This typically results in:

• Higher market price.
• Lower total output compared to a competitive market.
• Increased profits for both firms.

Collusion in oligopoly showcases the potential benefits for firms to act together but highlights anti-competitive behaviors that regulators closely watch.

Stackelberg Competition

Stackelberg competition deviates from Cournot by introducing a leader-follower dynamic in an oligopoly. This form of competition assumes one firm can set its output first, influencing the second firm's decisions.

For WW and BBBS, if WW sets its output first, it gains a strategic advantage. WW can predict BBBS's reaction based on its output choice, using a reaction function. This way:

• WW selects an output level considering BBBS's expected response.
• BBBS then optimizes its output given WW’s choice.

The leader, in this case, may increase its profits by influencing market conditions, often resulting in a more favorable position than in Cournot competition.

Stackelberg highlights the power dynamics in markets where firms play through strategic timing.

Inverse Demand Function

An inverse demand function showcases the relationship between price and quantity demanded. Unlike a regular demand function, it expresses price as a function of quantity. For WW and BBBS, it's given by:

\( P = 300 - 3Q \)

Here, \( P \) is the price and \( Q \) is the total quantity produced by both firms. It indicates how price decreases with an increase in total production.

This function is essential in determining:

• The market price for given output levels.
• How output decisions affect market dynamics.

Understanding the inverse demand function allows firms to anticipate how changes in production impact their revenues by looking at how price shifts with varying quantities.