Problem set ECOS3003Due date 14.00 Tuesday 31 MarchPlease keep your answers brief and concise. Excessively long and irrelevant answerswill be penalised.1. Considered the following market. Chrome can choose when launching its newproduct either to do it LARGE or as NICHE. After Chrome has chosen its action,Firefox observes Chrome’s choice and then can choose to ADAPT to RETAIN itsown product. After Firefox has chosen its action, the game ends and the payoffs aremade. The payoffs are as follows. If Chrome chooses LARGE and Firefox ADAPT,the payoffs are 25 and 40 to Chrome and Firefox, respectively. If Chrome goesLARGE and Firefox RETAINS the payoffs are 30 and 50 to Chrome and Firefox. IfChrome plays NICHE and Firefox ADAPTS, the payoffs are (40 Chrome, 30 Firefox)and if Chrome plays NICHE and Firefox RETAINS the payoffs are (20, 20) forChrome and Firefox, respectively.a. What is a Nash equilibrium? Outline the Nash equilibrium or equilibria in thegame, and explain your answer.b. What is a subgame perfect equilibrium and how would you find such anequilibrium? What is outcome in the the subgame perfect equilibrium in this game?Explain your answer.c. Does the subgame perfect equilibrium equilibrium result in the outcome thatmaximises total surplus? If it is, explain why. If not, is there any way the surplusmaximising outcome can be obtained? Interpret your answer in the context of theCoase Theorem. Why might your solution not work?2. Two workers (workers 1 and 2) on a production line both have the choice to cometo work Late or Early. If both come Late their payoffs are 5 and 4 to worker 1 and 2,respectively. If worker 1 comes Late and worker 2 comes Early the payoffs are 2 toeach worker. If worker 1 comes Early and 2 comes Late the payoffs are 1 to each ofthem. Finally, if both workers come Early the payoffs are 3 to each worker.Draw the normal form of this game and determine all of the Nash equilibria. Do youthink this game could represent a true production process?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 whatECOS3003 Problem set1 of 3is 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 theprincipal’s utility Up (or loss from the maximum possible profit) is given byU P s 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,U A 0.4 d . Similarly, Manager B always wants a decision of 0.7 to beimplement, and incurs a disutility UB that is (linearly) increasing in the distancebetween 0.7 and the actually decision that is implemented – that is U B 0.7 d .Each 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 –given she does not know what the state of the economy is – she asks forrecommendations 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 principal’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 being2ECOS3003 Problem set2 of 3informed 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 e 2 ; 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 Poppy’s preferred project is also Aiden’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 Poppy’smarginal benefit and marginal cost curves. What is Poppy’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 willdecide 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 ) .2ECOS3003 Problem set3 of 3