Given our distributional assumptions and the concavity of H(g), a unique equilibrium exists. One immediate feature of this equilibrium is that both A and B choose the same policy. Formally, they face the same maximization problems, as pB = (1 — pA) and as qA and qB enter (3.5) symmetrically, but with opposite signs. Intuitively, they have the same selfish preferences and possess the same technology for converting policy promises into expected votes.
To characterize the equilibrium policy, we maximize party A’s objective function (3.1) with regard to qA, taking qB as given. Exploiting (2.1), (2.2), and (3.5), and evaluating the resulting first-order conditions at the point qA = qB, we obtain the conditions which must hold at an equilibrium.
A first result concerns the pattern of redistribution to the voters. The equilibrium involves positive redistribution to group 2 only; that is, b2 > 0, and b1 = b3 = 0. This stark result follows as there are more swing voters in group 2, by our assumption that s2 > s1 ,s3, and as the marginal utility of private consumption is constant. Thus both parties target their redistribution programs towards the middle class, because this group contains the most responsive voters.
The equilibrium supply of public goods then follows from the optimal trade-off between g and b2. The corresponding condition is:
where 1 refers to the marginal utility of private consumption (see 2.1) and superscripts refer to groups. Intuitively, one more unit of redistribution for the middle-class group can be achieved by cutting the supply of the public good by the same amount. This means a gain of votes proportional to s2 • 1 in group 2 (captured by the LHS), but a loss of votes in every group i proportional to s* • Hg (g) (the RHS). It is optimal for the two parties to equate the marginal gain of votes to the marginal loss of votes.
A similar tradeoff, between t and b2, pins down the optimal tax rate. An additional unit of redistribution to the middle-class group can also be achieved by raising the tax rate by one third on all voters. This leads to the complementary slackness condition: