A key condition of the median-voter theorem is that the policy space be onedimensional, i.e. that the policy be representable by a real number. However, in many real-world cases the policy space may be multi-dimensional. Here we study a simple example of redistributive politics. Consider a society consisting of N≥2
groups (e.g. a nation made up of smaller regions), who must divide a resource among themselves. We normalize the value of this resource to 1 . Each group ghas a population of sizeπ g, and we normalize the size of the entire population to 1. The policy is an allocation, assigning to each group a share of the resource. More precisely, each group
g∈{1,…,N}obtains a shareα g ≥0such thatg=1∑Nα g ≤1. The preferences of voteriin group are represented by the functionu ig =α g. The policy must be decided by majority voting. (a) Provide the condition for some allocation αto be (Pareto) efficient. (b) We claim that any efficient policy can be fully characterized by an
N−1−dimensional vector. Is the claim correct? Explain. (c) In what follows, we will always assume that the policy is efficient. Suppose thatN=2, so there are only two groups, denoted 1 and 2 respectively. Suppose thatπ 1 >π 2.
