Problem set ECOS3003NOTE Due date 14.00 Friday 8 AprilPlease keep your answers brief and concise. Excessively long and irrelevant answerswill be penalised.1. Consider the following game, which has been loosely based on the trust modelsstudied in class. For the purposes of this game, focus on pure-strategies only.Each agent moves simultaneously. Agent 1 can take either action T, M or B, whereasAgent 2 can take actions L, C, or R.The payoffs are shown in the following normal form game.a. Outline and explain the Nash equilibria if the game is played once. (Again, focusonly on pure-strategy equilibria.)b. Now consider the case when the game is played twice. That is, in the first period atthe firm, Agents 1 and 2 simultaneously choose their actions. Their choices arerevealed before in the second period, again the agents simultaneously choose theiractions. Then the game ends. There is no discounting of payoffs between periods.ECOS3003 Problem set1 of 4Outline how, as part of a subgame equilibrium that the threat to play either B or R inthe final period rather than M or C can help sustain the cooperative outcome of (T,L)in the first period. Interpret this two-period game as a trust game. Explain why â€˜trustâ€™(or cooperation) can be achieved in the first period of this game without having toresort to an infinite-horizon game.2. Consider a firm with two agents â€“ 1 and 2. Both agents have to choose between twooptions: Client Focus or Cost Focus. If both choose Client the payoffs to 1 are 20 and10 to agent 2. If both agents choose to play Cost the payoffs are 15 to agent 1 and 25to agent 2, respectively. Finally, if any other combination of actions is chosen thepayoffs to each agent are 0.a. Assume that the agent choose their actions simultaneously. Draw the normal formof the game and derive all of the Nash equilibria.b. Now assume that the game is played sequentially: Agent 1 makes her choice ofaction first, this is observed by Agent 2, who then makes his choice. Draw theextensive form of the game and find the subgame perfect equilibria. Briefly interpretthis game in the context of: (i) leadership and corporate culture; and (ii) the BasicValue Maximisation Principle.3. Consider the following delegation versus centralisation model of decision making,loosely based on some of the discussion in class.A principal wishes to implement a decision that has to be a number between 0 and 1;that is, a decision d needs to be implemented where 0 d 1 . The difficulty for theprincipal is that she does not know what decision is appropriate given the current stateof the economy, but she would like to implement a decision that exactly equals whatis required given the state of the economy. In other words, if the economy is in state s(where 0 s 1 ) the principal would like to implement a decision d = s as theprincipalâ€™s utility Up (or loss from the maximum possible profit) is given byUPs d . With such a utility function, maximising utility really means makingthe loss as small as possible. For simplicity, the two possible levels of s are 0.4 and0.7, and each occurs with probability 0.5.There are two division managers A and B who each have their own biases. ManagerA always wants a decision of 0.4 to be implemented, and incurs a disutility UA that isincreasing the further from 0.4 the decision d that is actually implement, specifically,UA0.4 d . Similarly, Manager B always wants a decision of 0.7 to beimplement, and incurs a disutility UB that is (linearly) increasing in the distance0.7 d .between 0.7 and the actually decision that is implemented – that is U BEach manager is completely informed, so that each of them knows exactly what thestate of the economy s is.(a) The principal can opt to centralise the decision but before making her decision â€“given she does not know what the state of the economy is â€“ she asks forECOS3003 Problem set2 of 4recommendations from her two division managers. Centralisation means that theprincipal commits to implement a decision that is the average of the tworecommendations she received from her managers. The recommendations are sentsimultaneously and cannot be less than 0 or greater than 1.Assume that the state of the economy s = 0.7. What is the report (or recommendation)that Manager A will send if Manager B always truthfully reports s?(b) Again the principal is going to centralise the decision and will ask for arecommendation from both managers, as in the previous question. Now, however,assume that both managers strategically make their recommendations. What are therecommendations rA and rB made by the Managers A and B, respectively, in a Nashequilibrium?(c) What is the principalâ€™s expected utility (or loss) under centralised decision making(as in part b)?(d) Can you design a contract for both of the managers that can help the principalimplement their preferred option? Why might this contract be problematic in the realworld?4. Consider a variant on the Aghion and Tirole (1997) model. Poppy, the principal,and Aiden, the agent, together can decide on implementing a new project, but both areunsure of which project is good and which is really bad. Given this, if no one isinformed they will not do any project and both parties get zero. Both Poppy andAiden can, however, put effort into discovering a good project. Poppy can put in1effort E; this costs her effort cost E 2 , but it gives her a probability of being2informed of E. If Poppy gets her preferred project she will get a payoff of $1. For allother projects Poppy gets zero. Similarly, the agent Aiden can put in effort e at a cost1of e2 ; this gives Aiden a probability of being informed with probability e. If Aiden2gets his preferred project he gets $1. For all other projects he gets zero. Note also, thatthe probability that Poppyâ€™s preferred project is also Aidenâ€™s preferred project is α(this is the degree of congruence is α). It is also the case that α if Aiden chooses hispreferred project that it will also be the preferred project of Poppy. (Note, in thisquestion, we assume that α = β from the standard model studied in class.)(a) Assume that Poppy has the legal right to decide (P-formal authority). If Poppy isuninformed she will ask the agent for a recommendation; if Aiden is informed he willrecommend a project to implement. First consider the case when both Aiden andPoppy simultaneously choose their effort costs. Write out the utility or profit functionfor both Poppy and Aiden. Solve for the equilibrium level of E and e, and show thatPoppy becomes perfectly informed (E = 1) and Aiden puts in zero effort inequilibrium (e = 0). Explain your result, possibly using a diagram of Poppyâ€™smarginal benefit and marginal cost curves. What is Poppyâ€™s expected profit?(b) Now consider the case when the agent Aiden has the formal decision makingrights (Delegation or A-formal authority). In this case, if Aiden is informed he willECOS3003 Problem set3 of 4decide on the project if he is informed; if not he will ask Poppy for arecommendation. Again calculate the equilibrium levels of E and e.(c) Consider now the case when Poppy can decide to implement a different timingsequence. Assume now that with sequential efforts first Aiden puts in effort e intofinding a good project. If he is informed, Aiden implements the project he likes. IfAiden is uninformed he reveals this to Poppy, who can then decide on the level of hereffort E. If Poppy is informed she then implements her preferred project. If she too isuninformed no project is implemented.Draw the extensive form of this game and calculate the effort level Poppy makes inthe subgame when the Agent is uninformed. Now calculate the effort that Aiden putsin at the first stage of the game. Calculate the expected profit of Poppy in this1sequential game and show that it is equal to (1 ).25. Bloom et al (2012) â€˜The organization of firms across countries, Quarterly Journalof Economics) has found that delegation is more likely in firms that are located incountries in which the management can trust workers. More recently, Meagher andWait (2015) have found that delegation of decision-making authority is more likelywhen the workers trust the management.In the context of the infinitely repeated game studied in class, briefly discussion bothresults.ECOS3003 Problem set4 of 4
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