The name is absent

for all c ∈ [0, L], i.e. b* (c) > 0 for r = 0. Differentiating ^dEM^w)] ∣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.

^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

and

(1 - p) u' (w₀ - P (r)) + (1 - pq) u' (w₀ - P (r) - L + c* (r)) - p (1 - q) u' (w₀

where

P (r) = pqc* (r) + p (1 - q) b* (r) + rb* (r) .

Implicitly differentiating both FOCs with respect to r yields

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