Appendix: Proof of Lemma 1
Substitute RbH = RbL + Ψ0(e) throughout and write the first order conditions
for e and RbL :
(1) ∂L = RH - Rl - Ψ0(e) - eΨ00(e) + λ [eΨ00(e)] = 0
(2)∂RL = -1 + λ ≤ 0 and (-1 + λ)RbL = 0
where λ is the Lagrange multiplier for the (IR) constraint.
(i) If (IR) does not bind at the optimum, then λ =0 and eM is determined
by RH - RL = Ψ0(eM) + eMΨ00(eM). From Ψ00(e)>0 and the definition of the
first best effort e*, it follows e < e*. As (IR) is slack, λ = 0 and thus RL = 0.
For this to be the case, the level of effort must satisfy: eMΨ0(eM) -Ψ(eM) ≥ W,
i.e. it must be W ≤ W1.
(ii) Suppose instead that W>W1. Then the contract in (i) cannot be the
solution to the program, as it fails to satisfy (IR). Then, either RbH or RbL must
be raised. As eM < e*, it is optimal to raise only the entrepreneur’s payment
in the high state, so as to raise his effort. Therefore, RbH = Ψ0(eM(W)), where
eM(W) satisfies (IR) as an equality: eM(W)Ψ0(eM(W)) - Ψ(eM (W)) = W.
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