The name is absent



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 rr (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=L0 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




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