14.122 Problem Set #4 Suppose two companies, one American and one Japanese, can each produce a ho-mogeneous good for sale in the U.S. market at marginal cost c,0≤ c<1. However, theJapanese firm must also pay a tariff of t per unit sold. Assume that the inverse demand isgiven by P (Q)=1− Q where Q is the total quantity produced by the two firms.(a) Find the Nash equilibrium of the Cournot-like game where the firms simultaneouslychoose quantities qAand qJ(with t fixed). How high must the tariff be to eliminate allJapanese imports?(b) Suppose the U.S. government chooses t before the firms choose quantities, and thatthe objective of the govermment is to maximize the sum of consumer surplus, the profitof the domestic firm and tax revenues. What tariff level is chosen in a subgame perfectequilibrium?(c) Find a Nash equilibrium of the game above in which the U.S. government achievesa lower utility level than it does in the subgame perfect equilibrium.Consider the following model of monetary policy. In the first stage, the firms inthe economy form an expectation πeof what the inflation rate will be and sign wagecontracts, make investments, etc. on the basis of these expectations. Next, the FederalReserve takes actions which determine the actual inflation rate π. Unexpectedly highinflation stimulates the economy and increases aggregate output Y , but any departuresof inflation from expectations cause losses to the firms. In particular, assume that Y =Y0+ a log(1 + π − πe) and that the firms’ profits are −(π − πe)2. Suppose also that theutility of the Federal Reserve Chairman is given by Ufed= Y − cπ2.Find the inflation level in the subgame perfect equilibrium of this game. Does thesubgame perfect equilibrium involve rational expectations (i.e. is πe= π)? Comment onhow the equilibrium inflation rate varies with a and c?1 4.3.1. For each of the extensive forms below say whether the set of successors of theindicated nodes are or are not subgames. 2. Find all of the subgame perfect equilibria of the following extensive form game andgive the payoffs obtained by the players in each of them. \LR LR0, 02, 2U−10,xy, 00, 0D8, 4U0, 04, 8D4. Consider the one player infinite horizon game in which player 1 chooses an action at∈{0, 1} at t =1, 2, 3,..., and receives a payoff of 1 if limt→∞at>otherwise.12and a payoff of zero(a) Show that the payoffs in this game are not continuous at infinity.(b) Find a strategy profile which is not a subgame perfect equilibrium despite the factthat the player can not change his strategy at a single information set h and improve hispayoff conditional on that information set being reached.2 Suppose that two players play each other for two periods. In the first period theyplay the game at left below, and in the second period they play the game at right. Thereis no discounting between periods. Players observe the action their opponent took in thefirst period before choosing their second period actions.(a) For x ≤ 2 and y ≤ 6, find a subgame perfect equilibrium in which player 1 receivesa payoff of 10.(b) For x = 5 and y = 3, find a subgame perfect equilibrium in which player 2 receivesa payoff of 10.(c) For x = y = 4 show that there is no subgame perfect equilibrium in which (U, L)isplayed in the first