Suppose W > 0, C, and satisfy the intertemporal budget constraint (13.38). Define the consumption-reinvested wealth

Suppose W > 0, C, and π satisfy the intertemporal budget constraint

(13.38). Define the consumption-reinvested wealth process W† by (13.43).

(a) Show that W† satisfies the intertemporal budget constraint (13.44).

(b) Show that W†

t − Wt = W†

t t

0 Cs W†

s ds for each t.

Hint: Define Y = W/W† and use Itô’s formula to show that dY = − C W† dt .

Conclude that Wt W†

t

= 1−

t 0

Cs W†

s ds for all t.

(c) Let M be an SDF process and assume MW† is a martingale. Use this assumption and iterated expectations to show that, for any t < T, Et



MTW†

T T

0 Cs W†

s ds



= MtW†

t t

0 Cs W†

s ds+ Et



T t

MsCs ds



.

(d) Let M be an SDF process and assume MW† is a martingale. Use the results of the previous two parts to show that (13.39) is a martingale.