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 equilibrium quantities are Qa = 20 and Qt = 20 with each firm making a profit of $1200 with the original cost function. If both firms have different cost functions with Texas Air having marginal cost of $25 and American having marginal cost of $40, the equilibrium quantities are Qa = 15 and Qt = 30. Texas Air should be willing to spend up to $375 to lower its marginal cost while American should be willing to pay up to $225 to lower its marginal cost, assuming that Texas Air will have marginal costs of 25 regardless of American's actions.

Step-by-step solution

Calculate the Cournot-Nash equilibrium

First start by calculating the Cournot-Nash Equilibrium. This is done by setting up the profit function for each firm, differentiating them with respect to their respective quantities and setting equal to zero (maximization of profits). The profit function is \( \pi = P*Q - C(Q) \) where P is the price, Q is quantity and C(Q) is the cost function. Substitute P and C(Q) with the given equations to get \( \pi = (100 - Qa - Qt)*Qa - 40*Qa \) for American and \( \pi = (100 - Qa - Qt)*Qt - 40*Qt \) for Texas Air. Solve these simultaneously to find the quantity for each firm in equilibrium.

Calculate New Equilibrium Quantity

In step two, redo the profit maximization but with new costs for both firms. For Texas Air, the cost is constant at 25 while for American, it remains at 40. So the profit functions now become \( \pi = (100 - Qa - Qt)*Qa - 40*Qa \) for American and \( \pi = (100 - Qa - Qt)*Qt - 25*Qt \) for Texas Air. Solve these simultaneously to find the new equilibrium quantity.

Calculate investment to lower Marginal Costs

After changing the cost function for Texas Air to the lower marginal cost of 25, calculate how much each firm should be willing to spend to lower their marginal costs to 25. This is done by finding the difference between the profits with the original cost function and the new lower cost function, and this difference equals the amount they should be willing to spend.