Advanced Political Economics
Problem Set 1
Due Date: Tuesday, October 18, 2011
1. A Pie-Splitting Problem
Suppose there are three individuals in the economy, who are interested in receiving a share of a pie of
resources. The size of the pie is exogenously given, and fixed at one. Let xi be the share enjoyed by
individual i, with i ∈ {1, 2, 3}. The utility function of individual i is the following: Ui (x1 , x2 , x3 ) =
(xi )α , with α ∈ (0, 1).
(a) Find the utilitarian optimum, i.e. the allocation of the pie that would maximize a utilitarian
welfare function (Hint: it is useful, if not necessary, to set up the Lagrangean of the constrained
maximization problem).
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(b) Suppose now that the benevolent planner gives a weight wi to citizen i, with i=1 wi = 1. For
simplicity, also assume that α = 21 . Compute the socially optimal shares of the pie, as a function
of the wi s. How can you differently interpret the wi s? (hint: they sum up to one! )
(c) Suppose now that Ms. A and Mr. B are two opportunistic candidates that are competing
for electoral office. In order to be elected, they commit to a policy platform, i.e. a promised
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distribution of the pie among the three citizens. Let xA
i (xi ) be the slice of the pie offered by
candidate A (B) to citizen i. The candidates receive a payoff of one if elected, and zero otherwise.
Show analytically that there is no Nash equilibrium of the voting game (Hint: Think about all
possible configurations of the proposed slices, and show that for each of them there is always a
profitable deviation for at least one of the candidates.)
(d) Now, consider a simple probabilistic voting model with two opportunistic candidates. Differently
from the initial setup, each citizen type i corresponds to a continuum of citizens who enjoy xi
as a local public good. Each citizen belonging to each group has a preference for the public
good (according to the utility function given above) and an ideological preference for one of the
two candidates. As in the lecture notes, this ideological factor is uniformly distributed around
zero with density φi . There is also a common popularity shock that hits the candidates before
the elections, which is assumed to be uniformly distributed with unitary density. For simplicity,
assume again that α = 12 . Finally, assume that the size of group i in the population is wi . Given
these assumptions, calculate the equilibrium policy vector. (Hint: at some point you must bring
the resource constraint back into the picture...)
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2. Bargaining in legislatures
Suppose that a legislature is made of three members, who are interested in receiving a share of a size-one
pie. A proposal passes by simple majority, and there is closed rule, i.e. no possibility of amendments.
Further assume that a member who is indifferent between accepting or rejecting a proposal would
accept it. All members have a common discount factor δ ∈ [0, 1]. Finally, if in the last stage of the
game no proposal is accepted, all members receive a default payoff of zero.
(a) Suppose that there are three sessions in the bargaining game, and that members have equal
probability of recognition. Further assume that a recognized member, if to choose to whom to
promise a share of the pie, would randomize with equal probabilities. What is the subgame-perfect
Nash equilibrium of the game?
(b) What is the ex ante value of the game described above? Let the proposal power of the recognized member be calculated as the difference between her payoff and the expected payoff of not
recognised members. How does this proposal power change as a function of δ?
(c) Now, suppose that there are only two sessions in the bargaining game, and that recognition
probabilities are now different. More precisely, let p1 = 0.5, p2 = 0.3 and p3 = 0.2. What is the
subgame-perfect equilibrium of the game? Does the assumption of equal probabilities of inclusion
in the winning coalition make sense here? Finally, you should also calculate the ex ante value of
the game for the three members.
3. True or False. State clearly whether it is true or false. Briefly explain your answer.
(a) In the citizen-candidate model (Besley and Coate, 1997), the unique equilibrium of the game
features two candidates running for elections.
(b) In the provision of local public goods, political decisions taken under decentralised spending and
centralised financing are socially optimal.
(c) Lobbying models always entail a policy cost for citizens, and no benefits.
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