As is evident from the condition, equilibrium rents r can well be positive. As p is equal to 1 in equilibrium, this is more likely when R (the exogenous rents) are low. The reason why electoral competition does not eliminate rent seeking is that the parties are only perfect substitutes for swing voters, but not for any other voters. This implies that dp is negative, but finite. Equilibrium rents are thus larger, the more imperfect substitutes the two parties are; that is, the smaller is the number of swing voters. Hence, we have a second comparative statics result; a larger number of ideologically neutral voters (a higher s2) reduces equilibrium rents in an interior optimum. Finally, rents are lager, the higher is the variance in electoral outcomes (the lower is d). Higher variance implies that the expected vote share is not very sensitive to policy anyway; given this, the candidates find it optimal to take a greater risk by insisting on larger rents.
What if elections are instead conducted under plurality rule (first past the post) in single-candidate electoral districts? Specifically, assume that there are three electoral districts and consider the following electoral rule: winning the elections (and setting policy) requires winning at least two districts. One can interpret this setting as a parliamentary election, in which two competing parties have candidates in all three districts. The party who wins in two districts has a majority in the assembly and can thus implement its pre-announced policy. Alternatively, one can interpret it as a Presidential election (as in the US), where a candidate only needs a majority of the votes in a majority of the districts (rather than a majority of the population) to be elected President. We continue to talk of parties (rather than candidates) throughout this section, so implicitly, we adopt the first interpretation which also forms the basis for the empirical work to follow.
We start with a simplifying assumption: the three electoral districts coincide with the three groups in the population. We then show that all comparative politics results easily generalize if groups and districts do not completely overlap. Under majoritarian elections, existence of equilibrium is not guaranteed without further assumptions. Basically, we must assume that the ideological bias towards party A in group 1 and towards party B in group 3 are large enough; that is, the group-specific means a1 and a3 are sufficiently distant from zero. If this is the case, an equilibrium exists where A and B announce equal policies and where the entire competition takes place in the ’’marginal district” made up of the middle class (group 2) voters. Party A wins district 1 with large enough a probability and loses district 3 with large enough a probability, so that neither candidate finds it optimal to seek voters outside the marginal district; recall that only two districts are required for winning the election.
Under these assumptions, the relevant expression for party A’s probability of winning can be written as:
the marginal district, district 2. The other districts are neglected, because there party A is either very likely to win or very likely to lose. We may then follow the same steps as in the previous subsection to characterize the policies in a convergent electoral equilibrium. Obviously, only the middle class—the sole group in the marginal district—gets all the redistribution. Furthermore, it is optimal for both candidates to propose more redistribution than under proportional elections. Intuitively, the benefit to the parties of such redistribution is the same as under proportional elections, namely the marginal votes gained from the middle class voters. But the costs are smaller, as the parties now do not internalize the votes lost in the non-marginal districts.