The name is absent

Appendix 3. Proofs of propositions

Proof of Proposition 1: From (3.1) it follows that for n =1a positive value of δ
will drive ug ≡ u' (W + rk} below up ≡ u' (w + rk}. Since u" < 0, this requires
W>w. Moreover, according to equations (A.16) and (A.17) in Appendix 2 we
have

— > 0,    — = 0 for n = 1 and δ = 0 initially. ¥

∂δ         ∂δ

Proof of Proposition 2: Condition (3.3) in Proposition 2 is equivalent to

Consider equation (3.1) which was derived from the politician’s first-order condi-
tions on the assumption that the recruitment constraint W ≥ w is not binding.
According to (3.1) the inequality in (A.23) would imply ug ≡ u' (W + rk} > u'p ≡
u' (w + rk}, but since this would require W < w, it would violate the recruitment
constraint. Hence this constraint must be binding when (3.3) holds, implying the
absence of rents. ¥

Proof of Proposition 3: The proposition considers a case with many small juris-
dictions (n →∞) where tax competition has eliminated rents so that W = w,
u_g = u_p and u^'_g = u^'_p = u^'.Thefirst-order conditions (A.11) through (A.13) in
Appendix 2 then simplify to

p_o (α + δ) u^' - αλ + η =0,                     (A.24)

po(1 +δ)g^' - λ(τk+ W)=0,                 (A.25)

λ (1 - α)(k + τk^')+ηk - (1 - α) kpou^' =0,            (A.26)

where we have used the definition δ ≡ α_i (p_i - p_o) /p_o. Inserting (A.24) into (A.26)
and noting from (3.1) that ε = -τ k^'/k when n →∞,weget

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