Appendix : The Continuous - type Model of “ Competitive Non - linear Taxation and Constitutional Choice ” by Massimo

For robustness check, in this section we extend our analysis to the continuous type model, which can be regarded as the limiting case of many finite types. As an overview, with a continuum of types, the tax schedule chosen under each regime is characterized by a second-order differential equation with two boundary values. By focusing on the case where the vertical types are distributed uniformly, we are able to show that under independent taxation, the higher the mobility, the higher the consumption for all but the highest and lowest types; the rich (types sufficiently close to the highest type) pay lower tax, and the poor (types sufficiently close to the lowest type) receive lower subsidy under competition; there exists a cutoff type θ∗ so that all types above θ∗are better off, and all types below θ∗ are worse off with competition. Our computations confirm most of the findings from the three type model regarding the preferences of the median type, who is responsible for the constitutional choice. Specifically, in the vertical dimension worker-consumers are distributed on [θ, θ] with density function f(θ), where f(θ) is continuous, strictly positive everywhere in its support. All the other assumptions are the same as those in the previous discrete type model. As in the discrete type model, citizens can only be sorted in the vertical dimension. Thus, offering a tax schedule T (Q) is equivalent to offering a menu of consumption and production pairs {C(θ),Q(θ)}θ∈[θ,θ]. Define the tax function T (θ) = Q(θ)− C(θ). In the autarkic economy (no tax), a citizen’s optimal consumption is determined by u 0 (c∗) = 1/θ. Again we will consider unified and independent taxation rules. Under either the unified or independent taxation rule, incentive compatibility has to hold for each type of citizen conditional on her State of residence. Define V (θ,bθ) = u(C(bθ))− Q(bθ) θ to be the utility for a citizen with (vertical) type θ who accepts contract {C(bθ), Q(bθ)}. Incentive compatibility requires that V (θ, θ) ≥ V (θ,bθ) ∀(θ,bθ) ∈ [θ, θ].