The name is absent

Under the constant coefficient of absolute risk aversion Ra and using the FOCs we derive

RaC*' (r) pq (p (1 - q) + r) u⁰ (wo - P (r) - L + c* (r))

—R_ab*0 (r) p (1 — q) (1 — p (1 — q) — r) u⁰ (w₀ — P (r) — L + b* (r)) — u⁰ (w₀ — P (r) — L + c* (r))

and

b^*0 (r)p (1 — q) u⁰ (w0 — P (r) — L + b^* (r)) = c^*0 (r)(1—pq) u⁰ (w0 — P (r) — L + c^* (r))

The last equation implies

(1 — pq) u0 (w₀ — P (r) — L + c* (r))
p (1 — q) u⁰ (w₀ — P (r) — L + b* (r))

and

sign (b^*0 (r)) = sign (c^*0 (r)) .

Substitution into the first equation yields

We thus deduce that

sign (b^*0 (r)) = sign (c^*0 (r)) < 0 for all r<1 —p

sign (b^*0 (r)) = sign (c^*0 (r)) > 0 for all r>1 —p.

As b*⁰ (f) < 0, we have r < 1 — p and thus sign (b*⁰ (r)) = sign (c*⁰ (r)) < 0 for all r < r.
The implicit rate of interest in the market with the broker

r = (1 + γ) p (1 — q) r

is lower than the one in the bilateral case which completes the proof.

References

[1] Abraham, K.S., 2001, The Insurance Effects of Regulation by Litigation, Washington DC, Brookings
Institution.

[2] Anderlini, L., L. Felli and A. Postlewaite, 2003a, “Courts of Law and Unforeseen Contingencies”, working
paper, University of Pennsylvania, Department of Economics.

[3] Anderlini, L., L. Felli and A. Postlewaite, 2003b, “Should Courts Always Enforce What Contracting
Parties Write”, working paper, University of Pennsylvania, Department of Economics.

[4] Berliner, B., 1982, Limits of Insurability of Risks, Prentice-Hall.

[5] Biglaiser, G., 1993, “Middlemen as Experts”, RAND Journal of Economics 24, 212-223.

[6] Bond, E.W. and K.J. Crocker, 1997, “Hardball and the Soft Touch: the Economics of Optimal Insurance
Contracts with Costly State Verification and Endogenous Monitoring Costs”, Journal of Public
Economics 63, 239-264.

[7] Crocker, K.J. and J. Morgan, 1998, “Is Honesty the Best Policy? Curtailing Insurance Fraud through
Optimal Incentive Contracts”, Journal of Political Economy 106, 355-375.

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