Optimal Rent Extraction in Pre-Industrial England and France – Default Risk and Monitoring Costs

Proof of Proposition 2

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

∂ff∙ = - s - 1   ,2(1 - f)⅞⅛ a

∂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})) _2πτ < 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

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