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Uncertain lifetime and ordinal impatience
Economics Letters North-Holland
24 (1987) 121-125
UNCERTAIN
LIFETIME
Richard
121
AND ORDINAL
IMPATIENCE
*
M. PECK
University of Illinois at Chrcago. Chicago, IL 60680, USA
Padamanbhan
SRINAGESH
Williams College, Williamstown, Received Accepted
MA 01267, USA
18 February 1987 30 March 1987
We consider state dependent preferences exhibit impatience functions are not required.
preferences which do not exhibit impatience when lifetime is certain and show that these when lifetime is uncertain. Assumptions of risk aversion and additively separable utility
1. Introduction The notion that an uncertain lifetime may help to explain time preference has a long tradition in economics. Irving Fisher (1907) stated ‘[t]he chance of death may be said to be the most important rational factor tending to increase our impatience’. Implicit in Fisher’s statement is the claim that an agent with ‘no taste’ for present over future consumption when lifetime is certain will exhibit impatience when lifetime is uncertain. In this note we demonstrate that this claim is valid in a discrete time framework, using state dependent utility functions. ’ Our notion of impatience, stated in section 1, is closely related to Koopmans’ (1960) definition of ordinal impatience. Koopmans’ definition of impatience is, in turn, a formalization of Fisher’s notion of time preference. The utility functions considered are defined over finite length consumption streams and are state dependent, 2 where the state is the agent’s length of life. Utility functions are assumed to be strictly increasing and symmetric to capture the notion of ‘no taste’ for present over future consumption. By symmetric we mean that the value of a utility function remains unchanged when its arguments are arbitrarily permuted. When lifetime is uncertain, we show, in section 3, that these assumptions imply that the expected utility function exhibits impatience. Though Yaari (1965) does not consider a formal notion of impatience, our results are similar in spirit to his results, in particular, his Case A. Yaari’s Case A considers optimal lifetime consumption paths when lifetime is uncertain and a bequest motive and life insurance markets are absent. * We wish to thank E. Kami for comments on an earlier version of this paper which was presented to the 1985 Winter North American Meeting of the Econometric Society. We also thank William White for his helpful comments. Any errors remain our responsibility. ’ This paper considers the existence of impatience when lifetime is uncertain; Peck and Srinagesh (1985) examine implications of the result of this paper for agent behavior. ’ For an extended discussion of state dependent utility functions, along with references to earlier papers, see Kami (1985).
0165-1765/87/$3.50
0 1987, Elsevier Science
Publishers
B.V. (North-Holland)
R.M. Peck, P. Srinagesh
122
/ Uncertain
lifetime and ordinal impatience
However, Yaari draws his conclusions by considering the first-order condition for an interior solution of the consumer’s optimal lifetime consumption path so that the properties established are local. In contrast, our result that an uncertain lifetime induces impatience pertains to the entire consumption set and obtains whether or not perfect life insurance markets are present. 3 In addition, our result does not require concave, continuously differentiable, additively separable utility functions, and obtains when the forms of state dependent utility functions are allowed to change with length of life.
2. Preferences
and impatience
If an agent lives exactly t periods then his preferences over alternative consumption paths of length t can be represented by a state dependent utility function U,, where the state, t, is the agent’s length of life. A consumption path is a vector c = (c,, c2,. . . , c,), where c,, a non-negative real utility function number, designates the consumption level in period i. Each state dependent considered is strictly increasing in each of its arguments. Impatience is defined as follows. Suppose we have a consumption path c= (Cl ,...,ct),
(2.1)
where for some subscripts
i
(2.2)
ci -c c/.
By interchanging E=
(Cl
)...)
CL )...)
cJ and ci, we form a second c”,,...,
path
q),
(2.3)
where 2, = cI’
(2.4)
zj =
(2.5)
c;.
The path c” increases period i consumption to the level c,, while the lower level of consumption been-postponed to period j. Suppose for any c, described by (2.1) and (2.2) and all corresponding S, formed as indicated (2.3), (2.4) and (2.5) we have u,(c)
has by
(2.6)
’ v,(c),
then the agent is said to exhibit and F as described above
time preference,
that is, impatience.
On the other hand,
if for all c
(2.7) 3 Yaari
(1965)
framework
also considers,
remains
in his Cases B and D, a bequest a topic for future consideration.
motive.
The introduction
of a bequest
motive
into our
R.M. Peck, P. Srinagesh
123
/ Uncertain lifetime and ordinal impatience
then the consumer is indifferent to any rearrangement of the components of a given consumption path, and, in particular, does not exhibit impatience. 4 In this case, U, must be symmetric.
3. Uncertain lifetime and impatience There objective
is a maximum possible lifetime of T but for each t less than positive probability pI of living exactly t periods, that is,
or equal
to T there is an
t=l,...,T,
O
and
When lifetime is uncertain, we assume that the consumer ranks alternative consumption i, . . . , cT) according to the expected utility generated. Expected utility is given by (c
paths
EUtc) = 1
(3.1)
~J&,...,ct),
t=l
where c,,..., c, are the first t components of the vector (ci, . . . , cT). U,, t = 1,. . . , T, is the state dependent utility function representing agent preferences when lifetime is exactly t periods. Theorem. Suppose, when lifetime is of certain length t, t = 1,. . , T, an agent’s preferences can be represented by a symmetric, strictly increasing state dependent utility function U,, and, when lifetime is uncertain, preferences can be represented by the corresponding expected utility function. Then, when lifetime is uncertain, preferences exhibit impatience. Proof.
Consider
c=(c ,,..., ci
)...)
two consumption cj
)...)
paths,
c,),
where for some i
(c”,,...,
4 Of course,
F[
by interchanging
)...)
c, and c,:
F,‘...‘FT),
these definitions
do not completely
(3.2)
partition
the set of possible
preferences
R.M. Peck, P. Srinagesh
124
/ Uncertam lifetime and ordinal impatience
where Fk = 5, =
Ck
T,
k=l,...,
)
k#i,
j,
(3.3)
C,)
(3.4)
F, = c I.
(3.5)
Now to compare
EU(c”)-EU(c)=
EU( F) and EU( c), consider
;p,LI,(P, t=1
,...,
Ft)-u,(c
,,...,
ct)]
i-l =
c
P,[ u,(c,
,...>
6)
-
u,(C,
>...,
CJ]
t=l
j-l +
c P,[u,(c”,,..., Et)-
u,(c, ,...> CO]
tg4[Y(c, ,...> c”,)-
U,(c,
t=1
+
9...9
41.
The first right-hand expression is trivially equal to zero, since for t < i the components of c and c”are identical. In the third right-hand expression, for each t, (F1,. . . , Et) and (c,, . . . , c,) differ only in that the ith and jth have been interchanged. Since U,, for each t, is symmetric, this expression also equals zero. To evaluate the second right-hand expression, j-1 c p,[q(c”, t=i
,...’
Et) -
q(c
,,...,
(3.6)
Cl>]>
note that by construction,
for each t E {i,. . . , j - l},
& =
t+i,
k=l,...,
Ck,
t,
and 5, = c, > c, .
Since U, is strictly
increasing,
u,(F, >...> zz) ’ U,(c, ,...’ so that (3.6) is positive EU(C)
ct),
this implies
t=i ,...> j-
and therefore
> EU(c).
This completes
that
the proof of the theorem.
1,
R.M. Peck, P. Srinagesh
/ Uncertain lifetime and ordinal impalience
125
References Fisher, Irving, 1907, Theory of interest (Macmillan Press, New York). Karni, Edi, 1985, Decision making under uncertainty (Harvard University Press, Cambridge, MA). Koopmans, Tjalling, 1960, Stationary ordinal utility and impatience, Econometrica 28, 287-309. Peck, Richard M. and Padamanbhan Srinagesh, 1985, State dependent utility functions, uncertain lifetime and time preference, Paper presented at the 1985 Winter North American Meeting of the Econometric Society, New York. Yaari, Menahem, 1965, Uncertain lifetime, life insurance and the theory of the consumer, Review of Economic Studies 32, 137-158.
24 (1987) 121-125
UNCERTAIN
LIFETIME
Richard
121
AND ORDINAL
IMPATIENCE
*
M. PECK
University of Illinois at Chrcago. Chicago, IL 60680, USA
Padamanbhan
SRINAGESH
Williams College, Williamstown, Received Accepted
MA 01267, USA
18 February 1987 30 March 1987
We consider state dependent preferences exhibit impatience functions are not required.
preferences which do not exhibit impatience when lifetime is certain and show that these when lifetime is uncertain. Assumptions of risk aversion and additively separable utility
1. Introduction The notion that an uncertain lifetime may help to explain time preference has a long tradition in economics. Irving Fisher (1907) stated ‘[t]he chance of death may be said to be the most important rational factor tending to increase our impatience’. Implicit in Fisher’s statement is the claim that an agent with ‘no taste’ for present over future consumption when lifetime is certain will exhibit impatience when lifetime is uncertain. In this note we demonstrate that this claim is valid in a discrete time framework, using state dependent utility functions. ’ Our notion of impatience, stated in section 1, is closely related to Koopmans’ (1960) definition of ordinal impatience. Koopmans’ definition of impatience is, in turn, a formalization of Fisher’s notion of time preference. The utility functions considered are defined over finite length consumption streams and are state dependent, 2 where the state is the agent’s length of life. Utility functions are assumed to be strictly increasing and symmetric to capture the notion of ‘no taste’ for present over future consumption. By symmetric we mean that the value of a utility function remains unchanged when its arguments are arbitrarily permuted. When lifetime is uncertain, we show, in section 3, that these assumptions imply that the expected utility function exhibits impatience. Though Yaari (1965) does not consider a formal notion of impatience, our results are similar in spirit to his results, in particular, his Case A. Yaari’s Case A considers optimal lifetime consumption paths when lifetime is uncertain and a bequest motive and life insurance markets are absent. * We wish to thank E. Kami for comments on an earlier version of this paper which was presented to the 1985 Winter North American Meeting of the Econometric Society. We also thank William White for his helpful comments. Any errors remain our responsibility. ’ This paper considers the existence of impatience when lifetime is uncertain; Peck and Srinagesh (1985) examine implications of the result of this paper for agent behavior. ’ For an extended discussion of state dependent utility functions, along with references to earlier papers, see Kami (1985).
0165-1765/87/$3.50
0 1987, Elsevier Science
Publishers
B.V. (North-Holland)
R.M. Peck, P. Srinagesh
122
/ Uncertain
lifetime and ordinal impatience
However, Yaari draws his conclusions by considering the first-order condition for an interior solution of the consumer’s optimal lifetime consumption path so that the properties established are local. In contrast, our result that an uncertain lifetime induces impatience pertains to the entire consumption set and obtains whether or not perfect life insurance markets are present. 3 In addition, our result does not require concave, continuously differentiable, additively separable utility functions, and obtains when the forms of state dependent utility functions are allowed to change with length of life.
2. Preferences
and impatience
If an agent lives exactly t periods then his preferences over alternative consumption paths of length t can be represented by a state dependent utility function U,, where the state, t, is the agent’s length of life. A consumption path is a vector c = (c,, c2,. . . , c,), where c,, a non-negative real utility function number, designates the consumption level in period i. Each state dependent considered is strictly increasing in each of its arguments. Impatience is defined as follows. Suppose we have a consumption path c= (Cl ,...,ct),
(2.1)
where for some subscripts
i
(2.2)
ci -c c/.
By interchanging E=
(Cl
)...)
CL )...)
cJ and ci, we form a second c”,,...,
path
q),
(2.3)
where 2, = cI’
(2.4)
zj =
(2.5)
c;.
The path c” increases period i consumption to the level c,, while the lower level of consumption been-postponed to period j. Suppose for any c, described by (2.1) and (2.2) and all corresponding S, formed as indicated (2.3), (2.4) and (2.5) we have u,(c)
has by
(2.6)
’ v,(c),
then the agent is said to exhibit and F as described above
time preference,
that is, impatience.
On the other hand,
if for all c
(2.7) 3 Yaari
(1965)
framework
also considers,
remains
in his Cases B and D, a bequest a topic for future consideration.
motive.
The introduction
of a bequest
motive
into our
R.M. Peck, P. Srinagesh
123
/ Uncertain lifetime and ordinal impatience
then the consumer is indifferent to any rearrangement of the components of a given consumption path, and, in particular, does not exhibit impatience. 4 In this case, U, must be symmetric.
3. Uncertain lifetime and impatience There objective
is a maximum possible lifetime of T but for each t less than positive probability pI of living exactly t periods, that is,
or equal
to T there is an
t=l,...,T,
O
and
When lifetime is uncertain, we assume that the consumer ranks alternative consumption i, . . . , cT) according to the expected utility generated. Expected utility is given by (c
paths
EUtc) = 1
(3.1)
~J&,...,ct),
t=l
where c,,..., c, are the first t components of the vector (ci, . . . , cT). U,, t = 1,. . . , T, is the state dependent utility function representing agent preferences when lifetime is exactly t periods. Theorem. Suppose, when lifetime is of certain length t, t = 1,. . , T, an agent’s preferences can be represented by a symmetric, strictly increasing state dependent utility function U,, and, when lifetime is uncertain, preferences can be represented by the corresponding expected utility function. Then, when lifetime is uncertain, preferences exhibit impatience. Proof.
Consider
c=(c ,,..., ci
)...)
two consumption cj
)...)
paths,
c,),
where for some i
(c”,,...,
4 Of course,
F[
by interchanging
)...)
c, and c,:
F,‘...‘FT),
these definitions
do not completely
(3.2)
partition
the set of possible
preferences
R.M. Peck, P. Srinagesh
124
/ Uncertam lifetime and ordinal impatience
where Fk = 5, =
Ck
T,
k=l,...,
)
k#i,
j,
(3.3)
C,)
(3.4)
F, = c I.
(3.5)
Now to compare
EU(c”)-EU(c)=
EU( F) and EU( c), consider
;p,LI,(P, t=1
,...,
Ft)-u,(c
,,...,
ct)]
i-l =
c
P,[ u,(c,
,...>
6)
-
u,(C,
>...,
CJ]
t=l
j-l +
c P,[u,(c”,,..., Et)-
u,(c, ,...> CO]
tg4[Y(c, ,...> c”,)-
U,(c,
t=1
+
9...9
41.
The first right-hand expression is trivially equal to zero, since for t < i the components of c and c”are identical. In the third right-hand expression, for each t, (F1,. . . , Et) and (c,, . . . , c,) differ only in that the ith and jth have been interchanged. Since U,, for each t, is symmetric, this expression also equals zero. To evaluate the second right-hand expression, j-1 c p,[q(c”, t=i
,...’
Et) -
q(c
,,...,
(3.6)
Cl>]>
note that by construction,
for each t E {i,. . . , j - l},
& =
t+i,
k=l,...,
Ck,
t,
and 5, = c, > c, .
Since U, is strictly
increasing,
u,(F, >...> zz) ’ U,(c, ,...’ so that (3.6) is positive EU(C)
ct),
this implies
t=i ,...> j-
and therefore
> EU(c).
This completes
that
the proof of the theorem.
1,
R.M. Peck, P. Srinagesh
/ Uncertain lifetime and ordinal impalience
125
References Fisher, Irving, 1907, Theory of interest (Macmillan Press, New York). Karni, Edi, 1985, Decision making under uncertainty (Harvard University Press, Cambridge, MA). Koopmans, Tjalling, 1960, Stationary ordinal utility and impatience, Econometrica 28, 287-309. Peck, Richard M. and Padamanbhan Srinagesh, 1985, State dependent utility functions, uncertain lifetime and time preference, Paper presented at the 1985 Winter North American Meeting of the Econometric Society, New York. Yaari, Menahem, 1965, Uncertain lifetime, life insurance and the theory of the consumer, Review of Economic Studies 32, 137-158.