8. Appendix: Heterogeneity and majoritarian elections
In districts d ∈ [0, 2 ], J = 1 constitutes a share 1++β of the electorate while J = 2
constitutes only a share 1-β of voters while in districts d ∈ (2, 1], it is the opposite.
Groups J =3, 4 each form a quarter of the electorate in each district.
8.1. Equilibrium policy choices
As in sections 3 and 4, we derive first the equilibrium policy under coalition and
single-party government and then analyze equilibrium party formation.
8.1.1. Coalition government.
We first look at the case where parties P =3, 4 remain split.
Split opposition (4 parties) In districts d ∈ [0,1 ],the competition will be
between P =1 and P =3or 4. The latter are expected to have an equal number
of votes so let us focus on the competition between P =1 and P =3.Thevote
share for P =1 is given by
v1 = 4 ' F (V1 - V1* - δ) +
(A1)
14
8 £ F (V J - V J ∙ - δ)
J=3
Similarly, the vote share of P =3is given by
' (1 - F (V1 - V1*
1 . (i - F (V 2 -
V2*
- δ)) +
- δ))
(A2)
(8.1)
4(1 - F (V3 - V3* - δ))
Note that P =3gets only half of the votes of the disgruntled voters from
J = 1 since P =4gets the other half. We can then calculate
Pr ob(v1 ≥ 2 + 2 φ 4 ■ .+ [(1 + β)2(V1 - V 1∙> (A3)
- β(V2 - V2*) + 3(V3 - V3*) + 1(V4 - V4*)]
2 22
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