Optimal Rent Extraction in Pre-Industrial England and France

Proof of Proposition 2

A higher probability to commit leads to more tax-farming since

∂ff∙ = - s - 1 ,2(1 - f)⅞⅛ a

(35)

∂λ sλ — λ + 1 (1 + δ) φ²

where φ > 0.

The official’s profit is increased because

^dπNPV ^{(1 + δ}) ^(s^-¹) ⁽n+1 ) (¹^- c ) n+1₉ T ∩ ∕o_fi∖

=32π > 0. (36)

∂λ φ³

The CA’s profit is reduced as

^d∏Npv = _- (¹^- c) ^(s^-¹) ⁽¹^- c ^- n+ι ^{(1 + δ}) ^,sλ^-^{λ + 1})) ₂_πτ < о
∂λ (1 + δ)(sλ — λ + 1)² √^{2 π}

(37)

where 1 — ∣ — n-ɪ (1 + δ) (sλ — λ + 1) > 0 for θ > 0.

The debt, finally, is reduced since

∂Debt
∂λ

_ ∂θ* a — 2θ*(1 —

= ∂λ⁽ 2b

^λ+λs) ) — _β∙θ∙(_s

1) < 0.

(38)

CorPapers