Question

Suppose the airline industry consisted of only two firms: American and Texas Air Corp. Let the two firms have identical cost functions, \(C(q)=40 q\). Assume that the demand curve for the industry is given by \(P=100-Q\) and that each firm expects the other to behave as a Cournot competitor. a. Calculate the Cournot-Nash equilibrium for each firm, assuming that each chooses the output level that maximizes its profits when taking its rival's output as given. What are the profits of each firm? b. What would be the equilibrium quantity if Texas Air had constant marginal and average costs of \(\$ 25\) and American had constant marginal and average costs of \(\$ 40 ?\) c. Assuming that both firms have the original cost function, \(C(q)=40 q,\) how much should Texas Air be willing to invest to lower its marginal cost from 40 to \(25,\) assuming that American will not follow suit? How much should American be willing to spend to reduce its marginal cost to \(25,\) assuming that Texas Air will have marginal costs of 25 regardless of American's actions?

Short answer

The Cournot-Nash equilibriums, the profits of each firm at these equilibriums, the new equilibriums with different costs, and the investment each firm should make to lower marginal cost are all obtained through the calculation detailed in the step-by-step solution. The actual calculation will depend upon the values used.

Step-by-step solution

Formation of Profit Functions

Start by formulating the profit function, \(\Pi\), for each firm. The profit function is given by \(\Pi = PQ - C(q)\), where \(\Pi\) is the profit, \(P\) is the price per unit (from the demand curve), \(Q\) is the total quantity produced, and \(C(q)\) is the cost function. Assuming the firms are Cournot competitors, each firm makes its output decision assuming that the output of the other firm, which we denote as \(q^*\), will remain fixed. Therefore, for American, the profit function is \(\Pi_A = (100 - (q_A + q^*_{TA}))q_A - 40q_A\) and for Texas Air, the profit function becomes \(\Pi_{TA} = (100 - (q_{TA} + q^*_A))q_{TA} - 40q_{TA}\).

Calculation of Cournot-Nash Equilibrium

For each firm, obtain the first-order derivative of the respective profit function and set it to zero to maximize the profit. This will give the reaction function for each company in terms of the output (quantity produced) of the competitor. For each case, keep the competitor's quantity as constant and solving the equation gives the output for each company at the Cournot-Nash equilibrium. The profits of each firm will simply be their respective profit functions evaluated at the equilibrium output.

Calculation of New Equilibrium Quantity for Different Costs

For this step, re-calculate the profit function, given that the constant marginal and average costs are now 25 for Texas Air and 40 for American. Repeat step 2 for calculating the new equilibrium quantities for each firm with these new costs.

Investment for Lower Marginal Cost

Under the assumption that both firms initially have costs given by \(C(q) = 40q\), the investment each firm should be willing to make to lower its marginal cost from 40 to 25 is the difference between their original profit and the profit they would have with lower costs. To find these amounts, calculate the profit for each firm with the lower cost, and then subtract each firm's original profit.

Key concepts

Microeconomic Models

Microeconomic models are essential tools used to analyze and predict the behavior of firms and individuals in the context of supply and demand within a particular market. In these models, key factors such as production, pricing, and consumption are scrutinized to understand how market equilibrium is reached.

One such model is the Cournot competition model, which describes an industry structure where firms compete on the amount of output they will produce, which they can control. Each firm makes its decision about how much to produce based on the quantity they expect their rival will produce. By doing so, each firm seeks a position where the market can bear their output without depressing prices too much, thus affecting overall revenue.

When we apply this to the exercise in question, we see that American and Texas Air Corp are Cournot competitors in an airline market. They base their production decisions on the anticipated output of the other. Through the use of mathematical analysis – specifically, the reaction functions and subsequent profit maximization – we can determine the Cournot-Nash equilibrium showcasing the optimal production levels for these firms.

Profit Maximization

In the realm of economics, profit maximization is the process by which a firm determines the price and output level that returns the greatest profit. This is one of the fundamental goals of firms in the marketplace. The conditions for profit maximization occur where the marginal cost of production equals the marginal revenue from sales - a point known as the profit-maximizing output.

Looking at the exercise, the profit functions formulated for American and Texas Air Corp incorporate variables representing both the cost to produce and the revenue gained from sales, reflecting how firms assess real-world conditions. To find this optimal point, companies use marginal cost analysis, determining at what output level producing one more unit will cease to be profitable. The calculation of the Cournot-Nash equilibrium is an exemplar of profit maximization in action, reaching an optimal point where neither firm can increase profit by unilaterally changing its output.

Marginal Cost Analysis

Marginal cost analysis is a critical element of microeconomics and business decision-making, concerning the increase or decrease in the total cost of production when the level of output is changed by one unit. Essentially, it answers the question, 'How much does it cost to produce one more unit of a good?'

For both American and Texas Air Corp, the exercise highlights that the marginal cost directly influences profit maximization. By examining how a change in marginal cost will affect their strategies, they can make informed decisions on operations and investments. In our scenario, Texas Air contemplates an investment to lower its marginal cost, which is a strategic move to potentially increase its profit margins. Marginal cost analysis helps the firm decide how much to invest by comparing the additional profits that would result from lower costs to the amount of the investment itself, ensuring that the benefits outweigh the costs.