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

Proof of Lemma 4: The Implicit Function Theorem also gives the first-order comparative
statics of any parameter on ϕ(c2(y)), for any c2(y) at a solution. In particular,
d ϕ(c2(y), λ)
d k1
= −
∂Υ(c2(y),c2(x);λ)
∂k1
∂Υ(c2(y),c2(x);λ)
∂c2(x)
< 0 ∀c2(y) in the restricted set, (29)
since U ′ (.) > 0, U ′′ (.) < 0 and the utility functions exhibit decreasing absolute risk-aversion
(DARA). QED
Therefore, the optimal allocation establishes c ∗ 2 (y) = k2 − k + y + ɛ with ɛ > 0.
Step 2: Show that c∗ 2 (y) < k2.
This follows from imposing the optimal sharing rule c∗ 2 (y) < c∗2 (x) to the ex-ante participation
constraint of agent 1 holding with equality and noting that all individually rational allocations of
agent 2’s consumption are upper-bounded by the ex- ante participation constraint of agent 1.
Let c1(y) = w(y) − c2(y) ≤ k1−k+y, and, consequently, c2(y) ≥ (k1+k2−k+y)−k1+k−y = k2.
Then, for E [U1[w(s) − c2(s)]] = E [U1[k1 − k + s]] to be satisfied, c1(¯x) = w(¯x) − c2(x) >
k1 − k + ¯x. These, in turn, imply that c2(x) ≤ k2, which is in contradiction with the optimal
sharing rule, c ∗ 2 (y) < c∗ 2 (x).
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