ECOS3035: Economics of Political Institutions
Homework I
1. Consider a society of three individuals whose preferences over the four possible alternatives are:
Person 1: d ≻ c ≻ b ≻ a
Person 2: d ≻ c ≻ a ≻ b
Person 3: b ≻ a ≻ c ≻ d
Consider the pairwise majority rule and assume that individuals vote sincerely.
(a) For the profile of preferences above, do social preferences satisfy unanimity?
(b) For the profile of preferences above, are social preferences transitive?
(c) Now consider changing person 2’s preferences to a ≻ d ≻ c ≻ b. Are social preferences transitive for the new preference profile?
(d) Relate your answers in parts (b) and (c) to Arrow’s theorem? Be brief here.
2. Consider a society of three individuals and three alternatives a, b and c. Consider the pairwise majority rule and assume that individuals vote sincerely. Consider only strict preferences over alternatives.
(a) How many profiles of preferences are there in this society?
(b) How many profiles of preferences are there in this society where each individual places alternative a at the top of their ranking?
(c) List all the possible profiles of preferences for which the rule above violates transitivity.
3. Consider a society of three individuals A, B, and C, who have to choose over three trade policies, more free trade (F), status qou levels of trade (S), and more protectionist trade (P). Suppose society has the following preferences with individuals voting sincerely.
A: S ≻ F ≻ P
B: F ≻ S ≻ P
C: P ≻ S ≻ F
(a) Suggest an intuitive way to order the policies above from left to right.
(b) For this order of policies, can individuals be ordered in terms of their preferences from left to right? Do this by checking the definitions that we had in class.
(c) Who is the median voter? What policy do they prefer?
(d) Specify a non-dictatorial aggregation rule where societies preferences are the same as this median voter’s preferences?
(e) Why does this rule satisfy all of Arrow’s axioms (in particular, transitivity)?
4. Consider the Hotelling/Downs model of political competition with a unit mass of consumers uniformly distributed over the interval [0, 2]. There are two candidates (candidate 1 and candidate 2) who simultaneously choose a policy on the interval [0, 2], and then voters vote for one of the two candidates. Let sj denote candidate j
′
s policy, j = 1, 2. A voter votes for the candidate whose policy is closest to the voter’s location. A candidate gets a utility of 1 from winning and -1 from losing. Ties are broken with a fair coin toss.
(a) On a graph, with s1 on the horizontal axis and s2 on the vertical axis, plot candidate 2’s best response set (that is, specify all the s2’s which are optimal for candidate 2, given s1).
(b) On a graph, with s1 on the horizontal axis and s2 on the vertical axis, plot candidate 1’s best response set (that is, specify all the s1’s which are optimal for candidate 1, given s2).
(c) Depict the Nash equilibrium on this graph where both players are playing a best response to each other. Provide some intuition for why this is a Nash Equilibrium.
5. Consider the model of vote buying that we did in class (Groseclose and Snyder). In class we considered an example with 7 legislators. Redo the same example, but this time with 9 legislators. What is the optimal number of votes for party A to buy and specify its total cost from doing so?