The Birth of Insurance Contracts - The Ataturk Institute for Modern ...

Yet, the model ignores other important forces shaping contracts. It is therefore complementary
to other capital-structure theories based on agency costs, multiple margins for moral hazard,
adverse selection, signaling, transactions costs, and/or incomplete contracting.
A Proof of proposition 1
Let parameter values ˆ λ = ( ˆ θ, ˆ k1, ˆ ˆp¯x
k, ˆy, ˆpy, ˆx, ˆ¯x, ˆpx+ˆp¯x ) with ˆ θ = 0 and ˆ k1 = 0 be such that the sea
loan contract does not sustain program 2’s optimal allocation of consumption, (ĉ2(y), ĉ2(x)). Then,
ĉ2(y) < ˆw(y), as draw in figure 1for CC ′ , and necessary and sufficient Kuhn-Tucker conditions
lead to
∂£(η,µy,µx,c2(y),c2(x);λ)
∂η = − U 2 + E [U2[c2(s)]] = 0
∂£(η,µy,µx,c2(y),c2(x);λ)
∂c2(y) = −pyU ′ 1[w(y) − c2(y)] + η pyU ′ 2[c2(y)] − µy = 0
∂£(η,µy,µx,c2(y),c2(x);λ)
∂c2(x) = − pxU ′ 1[w(x) − c2(x)] + p¯xU ′ 1[w(¯x) − c2(x)]
+η(1 − py)U ′ 2[c2(x)] − µx = 0
evaluated at ˆη, ˆµ(y) = ˆµ(x) = 0, ĉ2(y), ĉ2(x), and ˆ λ, where η, µy, and µx are the Lagrangian
multipliers associated with restrictions (9) and (10), respectively, and £(.) is the Lagrange function
associated with Program 2. 17
Let λ varies in an open neighborhood of ˆ λ such that the pattern of binding and slack constraints
of program 2 does not change. Totally differentiating (12) and applying the Implicit Function
Theorem, one can calculate the comparative statics effects of each parameter value λi on c2(y)
at a solution point (ĉ2(y), ĉ2(x)) and parameter values ˆ λ with ˆη > 0, ˆµ(y) = ˆµ(x) = 0, and
dµy = dµx = 0:
where
ˆ
Hλi =
∂2 £(η,µy,µx,c2(y),c2(x);λ)
(∂η)2
∂2 £(η,µy,µx,c2(y),c2(x);λ)
∂c2(y)∂η
∂2 £(η,µy,µx,c2(y),c2(x);λ)
∂c2(x)∂η
dc2(y)
= −
dλi
ˆ
Hλi
ˆ ,
H
∂2 £(η,µy,µx,c2(y),c2(x);λ)
∂η∂λi
∂2 £(η,µy,µx,c2(y),c2(x);λ)
∂c2(y)∂λi
∂2 £(η,µy,µx,c2(y),c2(x);λ)
∂c2(x)∂λi
∂2 £(η,µy,µx,c2(y),c2(x);λ)
∂η∂c2(x)
∂2 £(η,µy,µx,c2(y),c2(x);λ)
∂c2(y)∂c2(x)
∂2
£(η,µy,µx,c2(y),c2(x);λ)
(∂c2(x))2
and the (binding-constraints) border hessian ˆ H evaluated at ˆ λ and (ĉ2(y), ĉ2(x)) is negative
definite: ˆ
H
> 0.
17 This assumes that for parameter values ˆ λ the solution of program 2 is such that ĉ2(x) < ˆw(x). The case for
which ĉ2(x) = ˆw(x) can be easily derived by adding restriction ∂£(η,µy,µx,c2(y),c2(x);λ)
∂µx
24
= 0 to (12).
(12)