Question

I only need answers for the Bertrand Nash Equilibrium section.

please provide answers with as much details as possible. Thank you

Oligopoly There are two firms competing in the market for Airplanes - Boeing and Airbus. The market demand is given by Q = 120 - P. Boeing has lower Marginal Costs of production than Airbus. Thus MCB = $20, MCA = $40. Assume that TFC = $0 for both firms. (Think of price being in thousands.) Boeing a) Derive Boeing's residual demand curve, assuming that Airbus produces a units. b) What is Boeing's Marginal Revenue? c) Derive Boeing's Reaction Function by setting its MRB = MC. (Make sure you express q as a function of q.) Airbus: a) Derive Airbus' residual demand curve, assuming that Boeing produces q units. b) What is Airbus' Marginal Revenue? c) Derive Airbus' Reaction Function by setting its MRA = MCA (Make sure you express q as a function of q.) Cournot Nash Equilibrium a) Solve the two reaction functions to obtain the duopoly quantities sold by the two firms. b) In the examples we did in class, with identical costs, the two firms produced the same quantity in equilibrium. This is no longer the case. c) Who produces the greater output and why? d) What is the total quantity of airplanes sold in the market? What is the equilibrium price? e) Calculate the profit for each firm. Bertrand Nash Equilibrium: Suppose Boeing and Airbus compete with each in prices instead. Assume that if both firms charge the same price, Boeing wins the entire market. Which of the following price pairs for Boeing and Airbus respectively qualifies as a Nash Equilibrium? In each case explain why the price pair is or isn't a Nash Equilibrium. (1) p = $20, PA = $40. (ii) pB = $20, PA = $20. (iii) pB = $40, PA = $40.

Answer 1

1. the boeing would want to earn the maximum revenue possible. hence given the mc = 20$ and Pa = 40$, Boeing would charge a price of 40$. in this way he would still capture the entire market and earn a larger revenue at the same time. hence this is not a nash equilibrium

2. here Pa = 20$ which is less than the marginal cost of the airbus = 40$. however if Pa increases its price to 40$ or greater, it wouldn't matter as the entire market is captured by the Boeing, and the airbus is not producing anything. hence airbus is not bettter of if changes its price. similarly if it decreases its price then it would make a loss as it would now be supplying to the market and we know it makes a loss on every product since price<marginal cost. hence airbus has no incentive to change.

at the same time if boeing decreases the price then it would be making loss. whereas if it increases the price then it would lose the entire market. hence it has no incentive to move either.

therefore this is a nash equilibrium

3. this is also a nash equilibrium. if boeing decreases the price then it would be making a smaller profit, whereas if it increases the price then it would lose the entire market, hence it has no incentive to move. similarly if airbus lowers its prices themn it would capture the market but would make a loss on every product it produces, thus it would not do that. similarly if airbus increases the price then it wouldn't affect anything as it will not be producing anything in that case either. hence it would not be better off. therefore we can see that it is a nash equilibrium