Exercise 1 Model of an Airlines MarketAmerican Airlines and United Airlines compete for customers on flights between Chicagoand Los Angeles. The total number of passengers flown by these two firms (per quarter),q, is the sum of the number of passengers flown on American, qA, and those flownon United, qU . Assume no other companies can enter. The flight Chicago–Los Angelescosts each airline $147 per passenger. Let p be the price per passenger in dollars.Suppose that the demand function is q(p) = 339 ? p.(1) If American were a monopoly, what would its optimal quantity and price be?What would its profit be?(2) In the duopoly described above, what is the Cournot equilibrium price for thisindustry? What is each airline’s profits?(3) If American were to move first, and United second, what would be the Stackelbergequilibrium quantities? price? profits?(3) Consider a situation where American and United would form an alliance, calledAUA, which behaves as a monopolist on this market. What would AUA optimalquantity be? What would be the resulting price? What would its profit be? Assumingeach Airline gets half of AUA’s profits, what would each Airline’s profitbe? Are these profits greater than their profits in the duopoly (both Cournot andStackelberg)?Exercise 2Consider the two-player games with the following payoff-matrices (rows and columnsdescribe players’ pure strategies).1. Find all the Nash equilibria of this game (including those in mixed strategies).C DA 0, 1 1, ?1B 3, 1 5, 012. Find all the Nash equilibria of this game (including those in mixed strategies).L C RU 2, 3 3, ?1 4, 1M ?2, 4 2, 2 2, 1D 3, 0 4,127, 2Exercise 3Alice and Betsy are playing a game in which each can play either of two strategies,leave or stay. If both play the strategy leave, then each gets a payoff of $300. If bothplay the strategy stay, then each gets a payoff of $600. If one plays stay and the otherplays leave, then the one who plays stay gets a payoff of $C and the one who playsleave gets a payoff of $D. For which values of C and/or D is the outcome (leave, leave)a Nash equilibrium?Exercise 4Imagine three little girls sitting in a circle, each wearing either a red hat or a whitehat (no girl can see the color of her own hat). Suppose that all the hats are red. Whenthe teacher asks if any student can identify the color of her own hat, the answer isalways negative, since nobody can see her own hat. But the teacher reveals that thereis at least one red hat in the room, a fact which is known to every child (who can seetwo red hats in the room). Then the teacher asks if any student can identify the colorof her own hat, and no one can. The teacher asks a second time if any student canidentify the color of her own hat, and no one can. The teacher asks a third and lasttime if any student can identify the color of her own hat, and each little girl answers“red.”Describe the reasoning that has led each little girl to the answer.Exercise 5Boeing and Airbus are competing to fill an order of jets for Singapore Airlines. Eachfirm can offer a price of $10 million per jet or $5 million per jet. If both firms offerthe same price, the airline will split the order between the two firms, 50-50. If one firmoffers a higher price than the other, the lower-price competitor wins the entire order.Here is the profit that Boeing and Airbus expect they could earn from this transaction:$5m $10m$5m 30, 30 270, 0$10m 0, 270 50, 50(1) What is the Nash equilibrium of this game?(2) Suppose that Boeing and Airbus anticipate that they will be competing for orderslike the one from Singapore Airlines every quarter, from now to the foreseeable future.Each quarter, each firm offers a price, and the payoffs are determined according tothe table above. The prices offered by each airline are public information. Suppose2that Airbus has made the following public statement:“ To shore up profit margins, in the upcoming quarter, we intend to be statesmanlikein the pricing of our aircraft and will not cut price simply to win an order. However,if the competition takes advantage of our statesmanlike policy, we intend to abandonthis policy and will compete all out for orders in every subsequent quarter.”Boeing is considering its pricing strategy for the upcoming quarter (assuming theabove statement is a credible commitment from Airbus). What price would you recommendthat Boeing charge? Present a formal reasoning.(Note: to evaluate payoffs, imagine that each quarter Boeing and Airbus receivetheir payoff right away. That is, if in some quarter, Boeing chooses $5m and Airbuschooses $10, Boeing will immediately receive its profit of $270 in that particularquarter). Furthermore, assume that Boeing and Airbus evaluate future payoffs in thefollowing way: a stream of payoffs of $1 starting next quarter and received in everyquarter thereafter has exactly the same value as a one-time payoff of $40 receivedimmediately this quarter.(3) Suppose that aircraft orders are received once a year rather than once a quarter.That is, Boeing and Airbus will compete with each other for an order this year (withpayoffs given in the table above), but their next competitive encounter will not occurfor another year. In terms of evaluating present and future payoffs, suppose thateach firm views a stream of payoffs of $1 starting next year and received in everyyear thereafter as equivalent to $10 received immediately this year. Again assumingthat Airbus will follow the policy in its public statement, what price would yourecommend that Boeing charge in this year and beyond? Present a formal reasoning.3