Uncertainty and Risk - DARP
Uncertainty and Risk - DARP Uncertainty and Risk - DARP
Microeconomics CHAPTER 8. UNCERTAINTY AND RISK Exercise 8.9 A person lives for 1 or 2 periods. If he lives for both periods he has a utility function given by U (x 1 ; x 2 ) = u (x 1 ) + u (x 2 ) (8.7) where the parameter is the pure rate of time preference. The probability of survival to period 2 is , and the person’s utility in period 2 if he does not survive is 0. 1. Show that if the person’s preferences in the face of uncertainty are represented by the expected-utility functional form X ! u (x ! ) (8.8) !2 then the person’s utility can be written as What is the value of the parameter 0 ? u (x 1 ) + 0 u (x 2 ) : (8.9) 2. What is the appropriate form of the utility function if the person could live for an inde…nite number of periods, the rate of time preference is the same for any adjacent pair of periods, and the probability of survival to the next period given survival to the current period remains constant? Outline Answer: 1. Consider the person’s lifetime utility with the consumption x 1 and x 2 in the two periods. If the person survives into the second period utility is given by u (x 1 ) + u (x 2 ) otherwise it is just u (x 1 ). Given that the probability of the event “survive to second period”is expected lifetime utility is [u (x 1 ) + u (x 2 )] + [1 ] u (x 1 ) : On rearranging we get in other words the form (8.9) with 0 = . u (x 1 ) + u (x 2 ) ; (8.10) 2. Apply the argument to one more period. Now there is consumption x 1 ,x 2 ; x 3 in the three periods and the probability of surviving into period t + 1 given that you have made it to period t is still . Consider the situation of someone who survives to period 2. The person gets utility u (x 2 ) + u (x 3 ) (8.11) if he survives to period 3 and u (x 2 ) otherwise. His expected utility for the rest of his lifetime, contingent on having reached period 2 is therefore cFrank Cowell 2006 124 [u (x 2 ) + u (x 3 )] + [1 ] u (x 2 ) = u (x 2 ) + u (x 3 ) (8.12)
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
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Microeconomics CHAPTER 8. UNCERTAINTY AND RISK<br />
Exercise 8.9 A person lives for 1 or 2 periods. If he lives for both periods he<br />
has a utility function given by<br />
U (x 1 ; x 2 ) = u (x 1 ) + u (x 2 ) (8.7)<br />
where the parameter is the pure rate of time preference. The probability of<br />
survival to period 2 is , <strong>and</strong> the person’s utility in period 2 if he does not<br />
survive is 0.<br />
1. Show that if the person’s preferences in the face of uncertainty are represented<br />
by the expected-utility functional form<br />
X<br />
! u (x ! ) (8.8)<br />
!2<br />
then the person’s utility can be written as<br />
What is the value of the parameter 0 ?<br />
u (x 1 ) + 0 u (x 2 ) : (8.9)<br />
2. What is the appropriate form of the utility function if the person could live<br />
for an inde…nite number of periods, the rate of time preference is the same<br />
for any adjacent pair of periods, <strong>and</strong> the probability of survival to the next<br />
period given survival to the current period remains constant?<br />
Outline Answer:<br />
1. Consider the person’s lifetime utility with the consumption x 1 <strong>and</strong> x 2<br />
in the two periods. If the person survives into the second period utility<br />
is given by u (x 1 ) + u (x 2 ) otherwise it is just u (x 1 ). Given that the<br />
probability of the event “survive to second period”is expected lifetime<br />
utility is<br />
[u (x 1 ) + u (x 2 )] + [1 ] u (x 1 ) :<br />
On rearranging we get<br />
in other words the form (8.9) with 0 = .<br />
u (x 1 ) + u (x 2 ) ; (8.10)<br />
2. Apply the argument to one more period. Now there is consumption<br />
x 1 ,x 2 ; x 3 in the three periods <strong>and</strong> the probability of surviving into period<br />
t + 1 given that you have made it to period t is still . Consider the<br />
situation of someone who survives to period 2. The person gets utility<br />
u (x 2 ) + u (x 3 ) (8.11)<br />
if he survives to period 3 <strong>and</strong> u (x 2 ) otherwise. His expected utility for<br />
the rest of his lifetime, contingent on having reached period 2 is therefore<br />
cFrank Cowell 2006 124<br />
[u (x 2 ) + u (x 3 )] + [1 ] u (x 2 )<br />
= u (x 2 ) + u (x 3 ) (8.12)
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