08 Aug ECON 4P34: Assignment 3Due: March 9, 2020 (In Clas
ECON 4P34: Assignment 3Due: March 9, 2020 (In Class)1 To In nity, and Beyond! 10 pointsFigure 1: Modi ed Prisonners’ Dilemmac dc 2, 1 -1, 4d 3, -3 0, 0Player BPlayer ANumber left (right) of comma refers to A’s (B’s) payo .Consider the above game, played in nitely. Both Player A and Player B have the samediscount factor, .(a) Suppose that both players follow a grim trigger strategy: play c as long as theother player plays c if the other player plays d, then subsequently play d forever.Find the minimum such that this is a SPNE. (5 points)(b) Assume that both A and B are playing some SPNE of the in nitely repeatedgame. Let V A and V B be A and B’s discounted values. What are thelowest values that V A and V B could have? (5 points)(Hint: no calculations are needed. Apply the Folk Theorem.)2 From Here to Eternity 12 pointsConsider the following stage game, assuming the discount rate is = 0:99. Checkwhether the strategy pro les are SPNE in the in nitely repeated game.1Figure 2: Prisonners’ Dilemmac dc 5, 5 0, 6d 6, 0 1, 1Player BPlayer ANumber left (right) of comma refers to A’s (B’s) payo .(a) Each player always plays c. (3 points)(b) Each player always plays d. (3 points)(c) Player A alternates between c and d, while player 2 always plays c; if any playerdeviates from this scenario, then each player plays d thereafter. (3 points)(d) Each player plays c, and if a player deviates, then he plays c, and the other playerplays d thereafter. (3 points)3 Sharing Is Weak 20 pointsLet us extend the Tragedy of the Commons problem we discussed during lectures. Con-sider N players, each of whom has the following utility function each period: log(x),where x is the amount consumed. Suppose there is a common property resource of sizey > 0. The game has two stages. In stage one, player, i = 1; :::N can withdraw ci, suchthatNPi=1ci y. In stage two, each player consumes of the remaining quantity, y NPi=1ci.In the case that they attempt to consume in excess of the available amount, then eachplayer splits the resource equally. For the sake of simplicity, letNPi=1ci = c.(a) Write down each player’s intertemporal utility function.1 (2 points)(b) What is the Nash equilibrium in Stage Two? (3 points)(c) What is this game’s SPNE? (6 points)(d) What is the socially optimal level of consumption in Stage One? (4 points)(e) How does the di erence between the SPNE consumption in Stage One, and thesocially optimal consumption in Stage One, change as N becomes very large? Whatis the intuition for this, and does this explain why ecologist Garrett Hardin suggestedsterilization as a solution to this problem? [Hint: take the limit as N approachesin nity.] (5 points)1The term `intertemporal’ means over time. That is, across stages one and two, what is each player’sutility function?2
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