ECOS3003 Problem set 1 of 3

Problem set ECOS3003

Due date 14.00 Tuesday 1 April

Please keep your answers brief and concise. Excessively long and irrelevant answers

will be penalised.

1. Considered the following market. Chrome can choose when launching its new

product 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 its

own product. After Firefox has chosen its action, the game ends and the payoffs are

made. 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 goes

LARGE and Firefox RETAINS the payoffs are 30 and 50 to Chrome and Firefox. If

Chrome 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) for

Chrome and Firefox, respectively.

a. What is a Nash equilibrium? Outline the Nash equilibrium or equilibria in the

game, and explain your answer.

b. What is a subgame perfect equilibrium and how would you find such an

equilibrium? 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 that

maximises 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 the

Coase Theorem. Why might your solution not work?

2. Two workers (workers 1 and 2) on a production line both have the choice to come

to 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 to

each worker. If worker 1 comes Early and 2 comes Late the payoffs are 1 to each of

them. 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 you

think 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 „T d „T1. 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

ECOS3003 Problem set 2 of 3

is required given the state of the economy. In other words, if the economy is in state s

(where 0 „T s „T1) 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

P U ƒ ƒ{ s ƒ{ 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

0.7, 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,

0.4 A U ƒ ƒ{ ƒ{ d . Similarly, Manager B always wants a decision of 0.7 to be

implement, and incurs a disutility UB that is (linearly) increasing in the distance

between 0.7 and the actually decision that is implemented – that is 0.7 B U ƒ ƒ{ ƒ{ d .

Each manager is completely informed, so that each of them knows exactly what the

state of the economy s is.

(a) The principal can opt to centralise the decision but before making her decision ¡V

given she does not know what the state of the economy is ¡V 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 = 0.7. 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 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?

(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 principal

implement their preferred option? Why might this contract be problematic in the real

world?

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 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 Poppy and

Aiden can, however, put effort into discovering a good project. Poppy can put in

effort E; this costs her effort cost 1 2

2

E , but it gives her a probability of being

ECOS3003 Problem set 3 of 3

informed of E. If Poppy gets her preferred project she will get a payoff of $1. For all

other projects Poppy gets zero. Similarly, the agent Aiden can put in effort e at a cost

of 1 2

2

e ; this gives Aiden a probability of being informed with probability e. If Aiden

gets his preferred project he gets $1. For all other projects he gets zero. Note also, that

the probability that Poppy¡¦s preferred project is also Aiden¡¦s preferred project is £\

(this is the degree of congruence is £\). It is also the case that £\ if Aiden chooses his

preferred project that it will also be the preferred project of Poppy. (Note, in this

question, we assume that £\ = £] from the standard model studied in class.)

(a) Assume that Poppy has the legal right to decide (P-formal authority). If Poppy is

uninformed she will ask the agent for a recommendation; if Aiden is informed he will

recommend a project to implement. First consider the case when both Aiden and

Poppy simultaneously choose their effort costs. Write out the utility or profit function

for both Poppy and Aiden. Solve for the equilibrium level of E and e, and show that

Poppy becomes perfectly informed (E = 1) and Aiden puts in zero effort in

equilibrium (e = 0). Explain your result, possibly using a diagram of Poppy¡¦s

marginal benefit and marginal cost curves. What is Poppy¡¦s expected profit?

(b) Now consider the case when the agent Aiden has the formal decision making

rights (Delegation or A-formal authority). In this case, if Aiden is informed he will

decide on the project if he is informed; if not he will ask Poppy for a

recommendation. Again calculate the equilibrium levels of E and e.

(c) Consider now the case when Poppy can decide to implement a different timing

sequence. Assume now that with sequential efforts first Aiden puts in effort e into

finding a good project. If he is informed, Aiden implements the project he likes. If

Aiden is uninformed he reveals this to Poppy, who can then decide on the level of her

effort E. If Poppy 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 Poppy makes in

the subgame when the Agent is uninformed. Now calculate the effort that Aiden puts

in at the first stage of the game. Calculate the expected profit of Poppy in this

sequential game and show that it is equal to (1 ) 1

2

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