Problem set ECOS3003 Due date 1300 23 March
Please keep your answers brief and concise. Excessively long and irrelevant answers will be penalised. You can handwrite your answers if you wish.
1. Consider the following game. Two workers A and B simultaneously choose to either work on project 1 (P1) or project 2 (P2). The payoffs are as follow. If both players opt for P1 the payoffs are 10 to A and 20 to B. If both players opt for P2, the payoffs are 8 to A and 16 to B. If the choices are either P1 and P2 or P2 and P1 each player gets 0.
a. What are all of the Nash equilibria?
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b. Now assume that there is a principal who can send a message to both players before they make their choices. What would a potential outcome be? Would the players be willing to pay for the principal to be involved? Interpret your answer in terms of the willingness for people to become employees.
2. Two people, A and B, can simultaneously choose to work on task 1 (T1) or task 2 (T2). There are two ways of organising their work. Firstly, A and B can work in the same business in which they are rewarded by group incentive payments. In this case the payoffs are: 10 to each of them if they both choose T1; 6 to each player if by both opted for T2; and 7 each if one player opted for T2 and the other T1. The alternative way of organising production is to have each person be an independent contractor, in which their payoffs are based on their individual returns (profits). In this case, the payoffs are: 10 each if they both choose T1; 8 each if they both choose T2; 11 to A and 3 to B if A chooses T2 and B chooses T1; and, finally, A will get 3 and B will get 11 if A chooses T1 and B choose T2.
a. What are the equilibria under each organisation structure?
b. Which structure is preferred? Interpret your answer in light of the transactions cost perspective of the firm.
c. What are the shortcomings of this example as a theory of why firms exist?
ECOS3003 Problem set 1 of 4
3. Consider the following delegation versus centralisation model of decision making, loosely based on some of the discussion in class.
A principal has 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 the principal is that she does not know what decision is appropriate given the current state of the economy, but she would like to implement a decision that exactly equals what is 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 the principal’s utility Up (or loss from the maximum possible profit) is given by
UP =− −s d . With such a utility function, maximising utility really means making the loss as small as possible. For simplicity, the two possible levels of s are 0.4 and
0.6, and each occurs with probability 0.5.
There are two division managers A and B who each have their own biases. Manager A always wants a decision of 0.4 to be implemented and incurs a disutility UA that is increasing the further from 0.4 the decision d that is actually implement, specifically, U−= −0.4 d . Similarly, Manager B always wants a decision of 0.6 to be
implement, and incurs a disutility UB that is (linearly) increasing in the distance between 0.6 and the actually decision that is implemented – that is U − = −0.6
d . Each manager is completely informed, so that each of them knows exactly what the
(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 for recommendations from her two division managers. Centralisation means that the principal commits to implement a decision that is the average of the two recommendations she received from her managers. The recommendations are sent simultaneously and cannot be less than 0 or greater than 1.
Assume that the state of the economy s = 0.6. What is the report (or recommendation) that Manager A will send if Manager B always truthfully reports s? Explain your answer.
(b) The principal is going to centralise the decision and will ask for a recommendation from both managers, as in the previous question. Now, however, assume that both managers strategically make their recommendations. What are the recommendations rA and rB made by the Managers A and B, respectively, in a Nash equilibrium? Again, provide some economic intuition for your answer.
(c) Can you design a contract for both of the managers that can help the principal implement their preferred option? Why might this contract be problematic in the real world?
(d) What if the principal instead delegates decision-making entirely to manager A (that is, A can decide on her own what d is without any consultation). Does this make the principal better or worse off than with centralisation and communication (as in part b)? Provide some intuition for your answer.
state of the economy s is.
ECOS3003 Problem set 2 of 4
4. Consider a variant on the Aghion and Tirole (1997) model. Portia, the principal, and Angus, the agent, together can decide on implementing a new project, but both are unsure of which project is good and which is really bad. Given this, if no one is informed they will not do any project and both parties get zero. Both Portia and Angus can, however, put effort into discovering a good project. Portia can put in
effort E; this costs her effort cost 1 E2 , but it gives her a probability of being 2
informed of E. If Portia gets her preferred project she will get a payoff of $1. For all other projects Portia gets zero. Similarly, the agent Angus can put in effort e at a cost
of 1 e2 ; this gives Angus a probability of being informed with probability e. If Angus 2
gets his preferred project he gets $1. For all other projects he gets zero. Note also, that the probability that Portia’s preferred project is also Angus’s preferred project is α (this is the degree of congruence is α). It is also the case that α if Angus chooses his preferred project that it will also be the preferred project of Portia. (Note, in this question, we assume that α = β from the standard model studied in class.)
(a) Assume that Portia has the legal right to decide (P-formal authority). If Portia is uninformed she will ask the agent for a recommendation; if Angus is informed he will recommend a project to implement. First consider the case when both Angus and Portia simultaneously choose their effort costs. Write out the utility or profit function for both Portia and Angus. Solve for the equilibrium level of E and e, and show that Portia becomes perfectly informed (E = 1) and Angus puts in zero effort in equilibrium (e = 0). Explain your result, possibly using a diagram of Portia’s marginal benefit and marginal cost curves. What is Portia’s expected profit?
(b) Now consider the case when the agent Angus has the formal decision making rights (Delegation or A-formal authority). In this case, if Angus is informed he will decide on the project if he is informed; if not he will ask Portia for a recommendation. Again calculate the equilibrium levels of E and e.
(c) Consider now the case when Portia can decide to implement a different timing sequence. Assume now that with sequential efforts first Angus puts in effort e into finding a good project. If he is informed, Angus implements the project he likes. If Angus is uninformed he reveals this to Portia, who can then decide on the level of her effort E. If Portia is informed she then implements her preferred project. If she too is uninformed no project is implemented.
Draw the extensive form of this game and calculate the effort level Portia makes in the subgame when the Agent is uninformed. Now calculate the effort that Angus puts in at the first stage of the game. Calculate the expected profit of Portia in this
sequential game and show that it is equal to (1−α)α + 1α . 2
5. Recent research by Meagher and Wait (2020) found that if workers trust their managers that delegation of decision making is more likely and that workers tend to
ECOS3003 Problem set 3 of 4
trust their managers less the longer the worker has been employed by a particular firm (that is, worker trust in their manager is decreasing the longer the worker’s tenure)
Interpret these results in the context of the infinitely repeated game studied in class. What are some possible empirical issues related to interpreting these results.
ECOS3003 Problem set 4 of 4
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