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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 ƒ{ƒÑ ƒÑ ƒy ƒÑ .

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
ƒ{ƒÑ ƒÑ ƒy ƒÑ .

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