Uncertainty and Risk - DARP

Microeconomics
So now view the situation from the position of the beginning of the lifetime.
The person gets utility
u (x 1 ) + [u (x 2 ) + u (x 3 )] (8.13)
if he makes it through to period 2, where the expression in square brackets
in (8.13) is just the rest-of-lifetime expected utility if you get to period 2,
taken from (8.12); of course if the person does not survive period 1 he gets
just u (x 1 ). So, using the same reasoning as before, from the standpoint
of period 1 lifetime expected utility is now
Rearranging this we have
[u (x 1 ) + [u (x 2 ) + u (x 3 )]]
+ [1 ] u (x 1 ) :
u (x 1 ) + u (x 2 ) + 2 2 u (x 2 ) : (8.14)
It is clear that the same argument could be applied to T > 2 periods and
that the resulting utility function would be of the form
u (x 1 ) + u (x 2 ) + 2 2 u (x 2 ) + ::: + T T u (x 2 ) : (8.15)
In other words we have the standard intertemporal utility function with
the pure rate of time preference replaced by the modi…ed rate of time
preference 0 := .
cFrank Cowell 2006 125