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))
—Rab*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
b*0 (r)
c*0 (r)
(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
c*0 (r) =
1
Ra (1 — p — r)
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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