Characterize the agent who is indifferent between voting

Question

Lobbying

Consider a society where the population consists of three kinds of voters J = R, M, P with intrinsic parameters α^J. The proportion of agents in group J is denoted by λ^J, and Σ_J=1³λ^J = 1. Besides, Σ_J=1³α^Jλ^J = α. Once more, the voting strategy of voter i in group J is affected by (1) the economic policy that is implemented q, (2) his individual ideological bias σ_iJ toward candidate B, and (3) the popularity δ of politician B. We assume that σ^i,J is uniformly distributed on: [- 1/2Φ^J ; 1/2Φ^J ], where Φ_J is group specific. The preferences of i over the policy implemented by A are summarized by W (q^A, α^J ) whereas his preferences over the policy implemented by politician B are given by W (q^B, α^J)+σ^iJ + δ. Assume that each group can decide to contribute to the campaign of politician P. Let OJ be the indicator variable that takes the value of 1 if group J wants to finance a party and the value of 0 if group J does not contribute to any campaign.

Suppose that the willingness to contribute is exogenous. Let C_P^J denote the contribution per member in group J to politician P. The total contribution of each member in group J is thus C_A^J + C_B^J. For any agent in group J, the cost of contributing to the campaign is D(C_A^J + C_B^J) = 1/2 [(C_A^J)² + (C_B^J)²]. Whenever a candidate receives a contribution, he spends the money and his popularity is affected. For simplicity, party B's popularity is given by:

δ = δ˜ + h(C_B - C_A)

where C_P represents the sum of the contributions received by party P, namely C_P = Σ_JO^Jλ^JC^J_P, and h is a parameter measuring the campaign's effectiveness. Assume that δ˜ is uniformly distributed with density on [- 1/2ψ ; 1/2ψ ]. The timing is as follows. First, each voter observes σi and politicians announce their platforms. Second, groups fix their contributions simultaneously. Third, the popularity parameter is realized and the election takes place. Last, the winner's platform is implemented.

(a) In each group, characterize the agent who is indifferent between voting for politician A and voting for politician B for given levels of contributions. Determine politician A's vote share as well as his probability of winning the election.

(b) From an ex ante perspective, what is the objective function of each member in the group if the latter wants to finance politicians? Determine the optimal contribution per member in each group. Discuss.

(c) Determine the platforms politicians select if all groups are willing to con- tribute (i.e., O^J = 1 for all J ) or no group contributes (i.e., O^J = 0 for all J ). What are the contributions in equilibrium? What happens if σi has the same distribution in all groups, namely Φ^J = Φ for all J . Discuss.

(d) ) Suppose now that some groups prefer not to finance politicians (i.e., there exists J such that O^J = 0). What are the politicians'platforms and which policy is finally implemented? Discuss. Which groups have the strongest incentives to become organized?