Econ 460: Problem Set 10Collusion and Extensive Form GamesNovember 15, 2016 Submit the problem set i

Econ 460: Problem Set 10Collusion and Extensive Form GamesNovember 15, 2016 Submit the problem set in class or on ELMS by Wed Nov 30 Q1. There are two …rms A and B in a homogenous product industry. Inverse demand is P = 120 Qwhere Q is the combined output of the …rms. Firm A has a marginal cost of 0 and …rm B has a marginalcost of 10. There is an in…nite sequence of periods in which …rms simultaneously set prices.In this question we will consider whether the following collusive strategies with trigger strategypunishments are a subgame perfect Nash equilibrium: both …rms set a price of 60 (and split outputequally) unless one of the …rms has deviated, in which case they play the static Nash equilibriumforever. The discount factor is .(a) Assume that the static Nash equilibrium involves the lower marginal cost …rm producing allthe output at a price equal to the marginal cost of the higher marginal cost …rm (so A would take theentire market at a price of 10). What is A’s per-period pro…t in the punishment phase? What is B’spro…t?(b) What is A’s per-period pro…t when the …rms are setting the collusive price of 60? What is B’spro…t?(c) Find the critical discount factor for …rm A not to want to deviate from the collusive equilibrium?Clearly show your working.(d) B’s critical discount factor is 0.5. Give a simple verbal explanation for why one of the …rm’shas a higher critical discount factor in this problem.(e) Suppose that the true discount factor lies between the critical discount factors of the two…rms. How might the …rms change the collusive equilibrium to make it possible to sustain collusion?[VERBAL ANSWER ONLY] 1 (f) In general what economic environments are associated with a higher discount factor (i.e., with…rms placing more value on future payo¤s)? (i.e., list at least 2 factors associated with a higher discountfactor) 2. Consider a homogenous goods duopoly. Market demand is Q(p) = 80 p. Firm 1’s capacity is k1 = 30; …rm 2’s capacity is k2 = 20: Both …rms have zero marginal cost and compete in prices.Assume that any rationing is e¢ cient which means that if one …rm has a lower price but is capacityconstrained then the consumers with the highest valuations buy from that …rm.(a) What is the joint-pro…t maximizing outcome (prices, output, pro…t)?(b) Show that the non-collusive Nash equilibrium outcome has each …rm setting the competitiveprice of 30 and producing at capacity.(c) Suppose that the 2 …rms play an in…nitely repeated game, with the stage game describedas above, with discount factor . Assume that they use “grim strategies” to sustain a joint-pro…t maximizing outcome i.e., once anyone deviates they revert forever to the non-collusive Nahs equilibriumoutcome.share of 1 They don’t share collusive pro…ts equally. Instead, …rm 1 gets share and …rm 2 gets . (i) What is each …rm’s discounted sum of current and future pro…ts along the collusive equilibriumpath?(ii) What is the maximum discounted sum of current and future pro…ts that each …rm can get fromcheating on the collusive equilibrium?(iii) Write down …rm 1’s “no cheating constraint”. What is the critical discount factor (as a function of ) above which …rm 1 will not cheat?(iv) Repeat for …rm 2.(v) For what value ofbetween this value of are …rms’ critical discout factors the same? Explain the relationship and …rms’relative capacities. For this value of , what is the critical discout factor? 3. Do some research on the internet to …nd out what is meant by the Department of Justice’s“leniency program†for collusion cases.Write short answers/bullet points (e.g., 3-4 sentences or bullets for each one) for the followingquestions:(a) What is the leniency program, and how long has it been in e¤ect?2 (b) What is the logic behind the program?(c) The program is generally believed to have been successful in the sense that the number ofprosecuted collusion cases has risen and many …rms have applied for leniency.potential disadvantages with the program? But what are the [If you google search on the program you’ll …nd some short reviews by law …rms that discuss some problems]. 4. What are the subgame perfect equilibria in the following three-player extensive form game?What are the Nash equilbria?Player 1L R (6,0,6) Player 2 (8,6,8) L Payoffs listed as(Player 1, Ply 2, Ply 3) Player 3L R Player 1U(0,0,0) R Player 1D U D (7,10,7) (7,10,7) (0,0,0) If you are unfamiliar with game theory you may need to look at a textbook to understand theformulation. 3