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
(.+.)(.-2>
(A.23)
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,
ug = up and u'g = u'p = u'.Thefirst-order conditions (A.11) through (A.13) in
Appendix 2 then simplify to
po (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 ≡ αi (pi - po) /po. Inserting (A.24) into (A.26)
and noting from (3.1) that ε = -τ k'/k when n →∞,weget
η ≡ pou' (1
ε (α + δ) - δ
α) U - ε (1 - .) ) '
(A.27)
36