for all c ∈ [0, L], i.e. b* (c) > 0 for r = 0. Differentiating dEMw)] ∣b=o with respect to r yields
—--—— ∣b=o = (- (1 - p) u' (wo - P) - pqu' (wo - P - L + c) - p (1 - q) u' (wo - P - L)) < 0.
∂r ∂b
This implies that for each c there exists a unique r(c) > 0 such that b* (c) > 0 for all r < r (c) and b* (c) = 0
for all r ≥ r (c).
The first derivative of expected utility with respect to c is
---L ( )] = pq (- (1 - p) u' (wo - P) + (1 - pq) u' (wo - P - L + c) - p (1 - q) u' (wo - P - L + b)) .
∂c
The second derivative is negative and expected utility therefore globally concave in c for all levels of b. The
FOC thus determines the unique global maximum c* = c* (b). Evaluating the first derivative at c = L yields
∂E [u (w)] l = pqp (1 - q) (u' (wo _ p) _ u' (wo - P - L + b)) .
∂c
As 0 ≤ b* < L, we have dEu(w)] ∣c=L < 0 and thus c* < L.
Evaluating the first derivative at c = b yields
P - L + b) - u' (wo - P)).
Finally, define r = r (c*).
dE [u (w)] = pq (1 - p) (u' (wo
∂c
As 0 ≤ b* < L, we have ',Eyi't |c=b > 0 and thus c* > b*.
A.2 Proof of Proposition 3
The FOCs for b* (r) and c* (r) are
(1 - p) (p (1 - q) + r) u' (wo - P (r)) - pq (p (1 - q) + r) u' (wo
+p (1 - q) (1 - p (1 - q) - r) u' (wo - P (r) - L + b*
P (r)
(r))
L + c* (r))
and
(1 - p) u' (w0 - P (r)) + (1 - pq) u' (w0 - P (r) - L + c* (r)) - p (1 - q) u' (w0
P (r) - L + b* (r)) = 0
where
P (r) = pqc* (r) + p (1 - q) b* (r) + rb* (r) .
Implicitly differentiating both FOCs with respect to r yields
p'm - - (1 - p)(p(1 - q)+r)
P (r)t +p (1 -<
u'' (wo - P (r)) - pq (p (1 - q) + r) u'' (wo
q) (1 - p (1 - q) - r) u'' (wo - P (r)
and
-c*' (r) pq (p (1 - q) + r) u'' (wo - P (r) - L + c* (r))
+b*' (r) p (1 - q)(1 - p (1 - q)
r) u00 (wo - P (r)
L + b* (r))
P (r) - L + c* (r))
pqu0 (wo - P (r) - L + c* (r))
- L + b* (r))
(1 - p) u' (w0 - P (r)) - p (1 - q) u' (w0 - P (r) - L + b* (r))
P0
( ) ʃ - (1 - p) u'' (wo - P (r)) + (1 - pq) u'' (wo - P (r) - L + c* (r)) ɪ
r ɪ -p (1 - q) u'' (wo - P (r) - L + b* (r)) J
+c*' (r) (1 - pq) u'' (w0 - P (r)
- L + c* (r)) -
b*' (r) p (1 - q) u'' (wo - P (r) - L + b* (r))
20
Il