Economics 8004, Spring 2015.

Instructor: David Rahman, University of Minnesota.

Homework 1—Due April 16 1. In an environment with only two social alternatives, show that majority voting is strategy-proof and non-dictatorial. 2. Given a finite set A of alternatives, consider a social choice function f whose range is some subset B (not necessarily equal to A) such that |B| ≥ 3, and whose domain contains all strict preference profiles over A. Show that f is strategy-proof if and only if it is dictatorial on B, that is, for some individual i, the SCF f selects i’s top-ranked alternative from B. Therefore, any strategy-proof SCF that accommodates at least three alternatives consists of restricting the set of choices and delegating the choice to someone. 3. Consider a quasi-linear exchange economy with n individuals. The space of net trades in non-money commodities is R` . Given a profile v = (v1 , . . . , vn ) of utility functions over non-money commodities, the gains from trade g(v) are given by ( n ) n X X g(v) = max vi (zi ) : zi = 0 . z

i=1

i=1

Write g(v−i ) for the gains from trade for everyone but i, that is, ) ( X X zj = 0 . g(v−i ) = max vj (zj ) : z

j6=i

j6=i

Assume that (i) g(v) and g(v−i ) are finite and well defined throughout, and (ii) the core of g is nonempty at v. The marginal product of individual i is defined as MP i (v) = g(v) − g(v−i ). P

(a) Show that i MP i (v) ≥ g(v) for all v. Identify conditions on v such that P i MP i (v) = g(v). (b) Suppose that n = 2. Let Vi be a convex subset of possible utility functions for Q individual i, and V = i Vi the space of profiles of utility functions. Assuming that |V | > 1, find conditions on V such that (i) there exist functions h1 and h2 such that for every v ∈ V , g(v1 , v2 ) = h1 (v2 ) + h2 (v1 ), and (ii) such functions do not exist. Argue each case. 4. Exercises 23.C.10, 23.D.2, 23.E.1, 23.E.5 and 23.E.6 from MWG. 1