Further Practice Questions

# Further Practice Questions - Further Practice Questions...

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Unformatted text preview: Further Practice Questions: Chap 12 Post-Mid (With Solutions) Question 8 Suppose the airline industry consisted of only two firms: American and Texas Air Corp. Let the two firms have identical cost functions, C(q) = 40q. Assume 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? To determine the Cournot-Nash equilibrium, we first calculate the reaction function for each firm, then solve for price, quantity, and profit. Profit for Texas Air, π 1 , is equal to total revenue minus total cost: π 1 = (100 - Q 1 - Q 2 ) Q 1 - 40 Q 1 , or π π 1 1 1 2 1 2 1 1 1 1 2 1 2 100 40 60 =--- =-- Q Q Q Q Q Q Q Q Q , . or The change in π 1 with respect to Q 1 is ∂ ∂ =-- 1 1 1 2 60 2 π Q Q Q . Setting the derivative to zero and solving for Q 1 in terms of Q 2 will give Texas Air’s reaction function: Q 1 = 30 - 0.5 Q 2 . Because American has the same cost structure, American’s reaction function is Q 2 = 30 - 0.5 Q 1 . Substituting for Q 2 in the reaction function for Texas Air, Q 1 = 30 - 0.5(30 - 0.5 Q 1 ) = 20. By symmetry, Q 2 = 20. Industry output, Q T , is Q 1 plus Q 2 , or Q T = 20 + 20 = 40. Substituting industry output into the demand equation, we find P = 60. Substituting Q 1 , Q 2 , and P into the profit function, we find π 1 = π 2 = 60(20) -20 2 - (20)(20) = \$400 for both firms in Cournot-Nash equilibrium. 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? By solving for the reaction functions under this new cost structure, we find that profit for Texas Air is equal to π 1 1 1 2 1 2 1 1 1 2 1 2 100 25 75 =--- =-- Q Q Q Q Q Q Q Q Q . The change in profit with respect to Q 1 is ∂ ∂ =-- π 1 1 1 2 75 2 Q Q Q . Set the derivative to zero, and solving for Q 1 in terms of Q 2 , Q 1 = 37.5 - 0.5 Q 2 . This is Texas Air’s reaction function. Since American has the same cost structure as in 8 .a ., American’s reaction function is the same as before: Q 2 = 30 - 0.5 Q 1 ....
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Further Practice Questions - Further Practice Questions...

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