TIME ALLOWANCE: 2 hours

Answer any THREE questions. All questions carry equal weight.

In cases where a student answers more questions than requested by the examination rubric, the policy of the Economics Department is that the student’s first set of answers up to the required number will be the ones that count (not the best answers) . All remaining answers will be ignored.

1. A politician has to set an income tax rate. She does not know anything about the state of the economy and she believes that the ideal tax rate x is drawn by nature from the uniform distribution on [0, 1].  She seeks an advice from a consultant who observes the ideal tax rate x. After receiving an advice from the consultant, the politician implements some tax rate y .  The preferences of the politician are described by the following utility function

Up (x, y) = 2xy · x2 · y2 .

The preferences of the consultant are given by

Uc (x, y) = 2bx + 2xy · 2by · x2 · y2

where x is the ideal tax rate randomly drawn by nature, y is the tax rate implemented by the politician and b > 0 is the parameter that describes the bias of the consultant.

(a) Show that the politician cannot learn the ideal tax rate from the consultant in

equilibrium if b > 0.

Answer: This is a standard cheap talk game. Assume that the equilibrium strategy of the consultant µ is invertible, i.e. the all the information is transmitted from the consultant to the policymaker.   If the state of the economy is x, the consultant

strictly prefers to send µ(x · b) instead of µ(x) which is a contradiction.             口

(b) Find an equilibrium in which the politician does not pay attention to the consultant’s

recommendation. Is it unique?

Answer: There are many babbling equilibria. One example is when the consultant always sends a message 0, and the policymaker always implements y =  .  In all these equilibria the message is independent of the state of the economy and the

policy implemented is always  .                                                                           口

(c) Find an equilibrium in which the politician learns some useful information about the ideal tax rate from the consultant. Does it always exist?

Answer: The partially separating equilibrium exists if the bias is small. For example the equilibrium with two messages exists if b <  . The sender sends a message m1  if the state is below x_ and sends the message m2 otherwise. In equilibrium y(m1 ) =2(北*) and y(m2 ) = 1北* . The marginal type at x_  is indifferent hence

x_                                         1 + x_

2                               2

or

x_  = 0.5 + 2b

The beliefs of the policymaker are x 1 m1  ~ U [0, x_] and x 1 m2  ~ U [x_ , 1].  This equilibrium exists only when b < 1/4. There are equilibria with finer partitions, but

those impose a more stringent requirement on b.                                                     口

2.  Consider a Rubinstein-St˚ahl bargaining model: Two players are bargaining over a division of a dollar.  In the odd periods, player 1 makes a proposal (i.e.  a possible division, for instance (0.6, 0.4)) and player 2 can either accept or reject it.  In even periods players switch roles: player 2 makes a proposal and player 1 can either accept or reject it. Once a division is accepted by any of the two players, the game stops and they get their shares of a dollar.  Assume that players discount their future payoffs with a common discount factor δ .

(a)  Solve for a Subgame Perfect Nash equilibrium in this game.

(b)  Suppose player 1 has an outside option of r—i.e., when he receives a proposal from

player 2 he can unilateraly execute his outside option and stop the game.  If that happens, player 1 gets a payoff of r and player 2 gets 0. Solve for a Subgame Perfect Nash equilibrium in this modified game.

(c)  Show that equilibria you found in 2a and 2b are unique.

Answer: Two classes of subgames: Player 1 is a proposer and player 2 is a proposer. Subgames within a class are strategically identical to each other.

● Let vi be the maximum (supremum) equilibrium payoff of a player i in a subgame in which he is a proposer and

● let wi  be his maximum (supremum) eq. payoff in a subgame in which his oppo- nent is proposing.

● Let v  i  be the minimum (infinum) equilibrium payoff of a player i in a subgame in which he is a proposer and

● let w  i  be his minimum (infinum) eq. payoff in a subgame in which his opponent is proposing.

Clearly, vi  > v  i  and wi  > w  i .

● The sum of the payoff cannot exceed 1 because the players are dividing 1 dollar: v1 + w  2  < 1                                                 (1)

v  2 + w1  < 1                                                 (2)

● Player 1 can always offer 62v2  and this offer will be accepted:

v  1  > 1 · 62v2                                                                          (3)

● Player 1 can always reject all offers

w  1  > 61v1                                                                              (4)

●  Start with inequality (3) and plug in (1) and then (4):

v  1  > 1 · 62v2  > 1 · 62 + 62  w  1  > 1 · 62 + 62 61  v  1

Solving for v  1  we get

v  1  >                                                 (5)

●  Start with inequality (1) and plug in (4) and then (5):

v1  < 1 · w  2  < 1 · 62  v  2  < 1 · 62

Solving for v1  we get

v1  <                                                 (6)

●  Combine (5) and (6) together to get

 < v  1  < v1  < 

from which we conclude that

v1  = v  1  =

(7)

●  Similarly for agent 2

1 · δ 1  

v2  = v  2  =

● From (4) and (7) :

w  1  >

● From (2) and (8):

w1  < 1 · v  2  =

●  Combine (9) and (10) together to get

δ 1 (1 · δ2 )                         δ 1 (1 · δ2 )

1 · δ 1 δ2                                          1 · δ 1 δ2

from which we conclude that

δ 1 (1 · δ2 )

1 · δ 1 δ2

●  Similarly for agent 2

δ2 (1 · δ 1 )

1 · δ 1 δ2

●  Since w1 + v2  = w2 + v1  = 1, there is no delay and the dollar is divided imme- diately

● When agent i is a proposer, he proposes a division (vi , 1 · vi )

● A receiver j accepts if and only if the offer is weakly above wj .

If the outside option of player 1 is smaller than his equilibrium payoff in the subgame where he responds to offers, it does not matter (i.e., equilibrium remains the same). Otherwise, he is offered his outside option by player 2 (and player 2 gets to keep the rest). In the subgame where player 1 makes a proposal, he offers δ(1 · r) to player 2 and any proposal that is weakly more generous gets accepted by player 2.         口

3. A community that consists of N households is trying to build a road.  Each household independently decides whether to contribute to building the road.  The road is built if and only if each of N households contributes.

The value of the road for household i is vi  and the cost of contribution is c < (N · 1)/N . The value vi  is drawn independently across households from a distribution with c.d.f. F (v) = v (the support of distribution is [0, 1]). The payoff of each household is the value of the road, if the latter is built, net of the cost.

(a)  Suppose that each household follows a threshold strategy, i.e.  a household i con-

tributes if and only if vi  < v_ , where v_ is the threshold. Find the symmetric Bayesian Nash equilibrium in which all households follow the same threshold strategy.

Answer: If household i contributes, its expected payoff is

Ui  = vi Pr{Aj  i contribute} · c

where

Pr{Aj  i contribute} = (1 · v_ )(N – 1) .

Otherwise its payoff is 0.  The marginal type v_  must be indifferent between con- tributing or not, therefore

v_ (1 · v_ )(N – 1)  = c.

The left-hand side of the equation is quasi-concave, and it is equal to zero at zero and one. Therefore this equation has two solutions as long as c < (N · 1)/N       口

(b) Is such an equilibrium unique?

Answer: No it is not. In addition to two equilibria outlined above, there is always an equilibrium in which v_  > 1—i.e. in which nobody contributes no matter what. 口

(c) What would happen if c > (N · 1)/N?

Answer:  There would be a unique symmetric BNE: nobody would contribute be- cause the costs are too high.                                                                                     口

4. Three job candidates (c1 , c2 , c3 ) are looking for a job. There are three employers (e1 , e2 , e3 ) and each employer has only one job opening. Employers’ and candidates’ preferences are given in the following two tables (employers’ preferences are in the left table; for example, employer e1  prefers candidate c2  over candidate c3  and he also prefers candidate c3  over candidate c1 ):

e1      e2      e3                 c1      c2      c3

(a)  Suppose that the allocation of candidates to employers is implemented via serial dic-

tatorship mechanism using the following order of employers: e2 , e1 , e3  (i.e. employers are making choices in this order). Find this allocation. Is it Pareto efficient?

Answer: The allocation is (e1 , c3 ), (e2 , c2 ), (e3 , c1 ). It is Pareto efficient.               口

(b)  Suppose that the candidates are initially allocated to their employers in the following

way:  (e1 , c1 ), (e2 , c2 ) and (e3 , c3 ). To improve the initial allocation, the government runs the top trading cycle mechanism in which only job candidates’ preferences are taken into account. Find the allocation produced by this mechanism.

Answer:   The allocation is (e1 , c2 ), (e2 , c1 ), (e3 , c3 ). It is Pareto efficient.             口

(c) Find a stable allocation that candidate 3 prefers over any other stable allocation.   Answer: In order to find such an allocation, we need to run DAA in which candi- dates are proposing. The resulting allocation is (e1 , c3 ), (e2 , c1 ), (e3 , c2 ). It is Pareto efficient.                                                                                                                     口