ANTI-COMPETITIVE FINANCIAL CONTRACTING: THE DESIGN OF FINANCIAL CLAIMS.

Appendix: Proof of Lemma 1

Substitute R_b^H = R_b^L + Ψ⁰(e) throughout and write the first order conditions
for e and R_b^L :

(1) ∂L = RH - R^l - Ψ⁰(e) - eΨ⁰⁰(e) + λ [eΨ⁰⁰(e)] = 0

(2)∂RL = -1 + λ ≤ 0 and (-1 + λ)R_bL = 0

where λ is the Lagrange multiplier for the (IR) constraint.

(i) If (IR) does not bind at the optimum, then λ =0 and e^M is determined
by R^H - R^L = Ψ⁰(e^M) + e^MΨ⁰⁰(e^M). 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: e^MΨ⁰(e^M) -Ψ(e^M) ≥ 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 R_b^H or R_b^L 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, R_b^H = Ψ⁰(e^M(W)), where
e^M(W) satisfies (IR) as an equality: e^M(W)Ψ⁰(e^M(W)) - Ψ(e^M (W)) = W.

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