STOCHASTIC

CONSUMPTION AND PORTFOLIO RULES 399
For this example, X(t) is calculated as follows:
Solving for X(t) from (87), we have that
n
= - — e~" rm £ ^ e-«e-»*. (87)
L g-J7l'(0>-A(-hAe-'' f i
'J
X(t) = Y(0) + Af(l - c-"')/'?- (88)
The capitalized value of the Certainty-equivalent wage income flow is
/•» (•» r Ad e - '"')
| e-"A-(i) rfj = f Y(fl) er" ds + f ^ —^ jr" ds
_ 7(0) A(l - e—)
r "^ v*
Thus, for this example, 30 the individual, in computing the present value
of future earnings, determines the Certainty-equivalent flow and then
capitalizes this flow at the (certain) market rate of interest.
The third example of a Poisson process differs from the first two because
the occurrence of the event does not involve an explicit change in a state
variable. Consider an individual whose age of death is a random variable.
Further assume that the event of death at each instant of time is an independent
Poisson process with parameter A. Then, the age of death, T, is
the first time that the event (of death) occurs and is an exponentially
distributed random variable with parameter A. The optimality criterion
is to
(89)
max £0 j f U(C, t)dt + B( W(T), T) J (90)
and the associated optimality equation is
0 = U(C*, t) + \[B(1V, t) - J(W, t)] +