An explanation of hyperbolic marginal utility from money
*
Arad Nir
November 2000
Abstract
"Hyperbolic discount functions are characterized by a relatively high discount rate over
short horizons and a relatively low discount rate over long horizons" (Laibson 1997, p.
445). In this theoretical note, we show that individuals hyperbolically discount marginal
utility from money when they follow a cognitive procedure in which they believe that
their wealth might increase or decrease in each future period under the constraint of a
small-perceived probability that wealth will deteriorate below its current level.
JEL classification: D90
Keywords: Hyperbolic discounting, cognitive procedure.
*
School of Economics, Tel Aviv University, Tel Aviv 69970, Israel, E-mail: [email protected]
I thank my supervisor, Chaim Fershtman, for many helpful comments and suggestions. I
benefited from discussions with Avner Shaked, Elchanan Ben-Porath, Oren Bar-Gill, Yoram
Hamo and Workshop participants at Tel Aviv University. Financial support from the David
Horowitz research institute is greatly acknowledged.
2
1. Introduction
A consistent psychological finding on individual time preferences is that discount
functions are hyperbolic, suggesting that people are impatient at present, but claim to be
patient in the future. Phelps and Pollak (1968) introduced hyperbolic discount functions
into economic theory in the context of consumption and savings across generations.
Laibson (1994) utilized hyperbolic discount functions in the context of intertemporal
one-person decisions to study consumption and saving patterns. Thaler (1981) offered
experimental evidence that used hypothetical questions and found that the discount rate
declines sharply with the length of time to be waited1. This note illustrates a cognitive
procedure that would lead people to act as if they were hyperbolic discounters.
Two economic papers have already offered explanations of such inconsistent
behavior. Azfar (1999) showed that hyperbolic discounting could arise when agents are
uncertain about their discount rate or hazard rate, as well as the probability of not
receiving payment. Azfar explains the intuition:
The apparent discount rate at time t depends on the true discount rate and the expected
hazard rate at t. The hazard rate at date t is the weighted average of the initial hazard
rates, where the weights are the probability of survival till t. These weights decline more
rapidly for higher hazard rates, and thus as t rises, the hazard rate converges to the lower
end of the initial distribution. (p. 247)
Rubinstein (2000) claims that the inconsistent behavior is better explained by a
decision-making procedure that is based on similarity relations than by hyperbolic
discounting.
1
Benzion, Rapoport, and Yagil (1989) replicated these findings for undergraduate and graduate
students of economics and finance at two Israeli universities. Kirby and Herrnstein (1995)
demonstrated the same results using both monetary and non-monetary real rewards. Kirby (1997)
showed that subjects are hyperbolic discounters using real rewards when subjects were induced to
convey their true value using second bid auctions.
3
Experiments in hyperbolic discounting generally ask subjects whether they prefer
a small, earlier monetary reward (SER) or a larger, later monetary reward (LLR). [Thaler
(1981), Benzion, Rapoport and Yagil (1989), Kirby and Herrnstein (1995), Kirby
(1997)]. In this note, we will investigate how the perceived marginal utility from money
alters as a function of time. That is because, in the experiments, individuals were asked
about their preferences over a sum of money that was marginal to their total wealth. Thus
we will investigate changes in utility from additional sums of money that the individual
may receive.
Impulsiveness might be seen as a negative attribute and patience as positive.
Some scientists have made moral statements concerning hyperbolic discounting. Kirby
(1997) stated that "[a]fter all, behavior consistent with normative discounting should be
our goal, even if it is not our norm" (p. 68). Kirby and Herrnstein (1995) wrote: "One
may stably prefer a smaller benefit to a larger one that is more remote in time. But if
preference reverses simply because of changes in one's temporal vantage point . . . then
the violation of stationarity meets the ordinary criterion of impulsiveness" (p. 83).
Contrary to these statements, Azfar (1999) and this note suggest that the hyperbolic
discounting phenomenon might be rational. We must stress, however, that perhaps
hyperbolic discounting does not need an explanation. People might simply be impulsive.
Nevertheless, this note offers one possible explanation for the hyperbolic discounting
phenomenon.
We assume that individuals perceive the probability that their wealth will
deteriorate below its current level as relatively small. For simplicity, in the formal model
we assume that probability to be zero. This assumption can be justified by the following:
4
The individual is loss-averse and has a status-quo bias [see Rabin (1998) and Kahneman,
Knetsch and Thaler (1992) for a detailed analysis of these phenomena]. Thus, she is
averse to thinking about a situation in which her financial situation will deteriorate. She
might tell herself that if, in the future, her financial situation deteriorates close to the
current level, she will work hard to try to ensure that it does not deteriorate further.
Also, note that the individual herself is the one who assigns the probabilities to
what might happen to her wealth. We assume that she assigns probabilities congruent
with the above assumptions. Consider the following way of contemplating future
financial situations from the present perspective: The individual understands that future
wealth can fluctuate between periods due to life's changing circumstances, but that the
probability that future wealth will deteriorate beyond the current level is small. This kind
of reasoning generates a hyperbolic marginal utility function from money because, in the
near future, it is highly probable that the individual's financial situation will improve and
that the expected marginal utility from money will decrease. But in the distant future, the
probability that the financial situation will deteriorate is higher than in the near future,
causing the expected marginal utility from money to decrease by a lesser amount than in
the near future. This possible increase of marginal utility in the distant future generates
the hyperbolic marginal utility phenomenon2.
This note aims to highlight the discount rate generated by the fact that individuals
contemplate the amount of wealth they will possess in the future. Thus, we will assume
that there is no discount rate per se. If such a discount rate did exist, the overall discount
2
This is an alternative explanation to the one provided by Azfar (1999). The explanations do not
conflict. When both prevail, marginal utility will be even more hyperbolic.
5
rate would be a function of the real discount rate and the discount rate generated by the
fact that individuals contemplate the amount of wealth they will possess in the future.
2. A three-period example
Consider a three-period example where period 0 stands for the present, period 1
for the near future, and period 2 for the distant future. The hyperbolic discounting theory
would prescribe that an individual will prefer an SER (smaller earlier reward) to an LLR
(larger later reward) in the near future, but an LLR to an SER in the distant future. We
will demonstrate this phenomenon below. Assume the following preferences: In period 0,
the individual's marginal utility from money is 10. For each dollar she gains, her marginal
utility diminishes by 1. As explained in the introduction, the individual perceives future
wealth as fluctuating between periods, but not falling below current wealth.
The agent's perception about future wealth is specified by the following
probabilities: In the first period, the individual perceives that she might gain a dollar with
probability m. If she does not gain the dollar, she might gain the dollar with the same
probability in the next period. However, if she gains a dollar in the first period, in the
next period she might gain an additional dollar with probability p, lose a dollar with
probability q, and neither win or lose with probability r, where p + r + q = 1 . Table 1
below describes the individual's probabilities for having various utilities from an
additional dollar in the two following time periods, as perceived from period 0. For
example, the probability that additional utility will be 9 in period 1 is m, and the
probability that additional utility will be 8 in period 2 is mp. The bottom row summarizes
the individual's VNM expected utility from an additional dollar. For example, the
6
expected utility in period 1 is 10 − m . Since 10 > 10 − m > 10 + m2 − 11m + m(8 p + 9r + 10q) ,
expected additional utility from a dollar diminishes over time.
Additional
utility from $1
10
Period 0
Period 1
Period 2
1
1− m
(1 − m ) 2 + mq
9
0
m
(1 − m )m + mr
8
0
0
mp
Expected
additional
utility from $1
10
10 − m
10 + m 2 − 11m
+ m(8 p + 9 r + 10q )
Table 1: The columns indicate the probabilities of having various utilities from an additional
dollar in each period. The bottom row indicates expected utility from the additional dollar in each
period.
Proposition 1: The marginal utility from money is hyperbolic when p < m + q ≤ 1 + p .
Proof: Note that the marginal utility is hyperbolically discounted when the individual
perceives future additional utility as decreasing at a decreasing rate as a function of time.
Expected utility from an additional dollar in period i is defined as Eui . The difference
between expected additional utility in period i-1 and period i is defined as ∆i , thus
∆i ≡ Eui −1 − Eui . In order to show that the discounting is hyperbolic, we must show that
the difference between expected additional utility in period i-1 and i decreases as i
increases, that is, ∆1 − ∆ 2 > 0 . Substituting the probabilities from Table 1, we get
∆1 = m and ∆ 2 = m(1 + p − m − q) . The marginal utility from money is hyperbolic when
∆1 − ∆ 2
and
∆1 are positive and ∆ 2 is not negative. ∆1 is always positive. ∆ 2 is
not negative when m + q ≤ 1 + p , while ∆1 − ∆ 2 is positive when p < m + q .
7
The intuition behind the condition m + q ≤ 1 + p is that p must be large enough so
that expected additional utility will not increase in the second period. The intuition
behind p < m + q is that p must be small enough so that expected additional utility will
not decrease much, causing the decrease in expected additional utility between periods 1
and 2 to be smaller than between periods 0 and 1.
In order to demonstrate inconsistency of preferences provided by the experimental
results on hyperbolic discounting, we need to show that an individual prefers an SER to
an LLR in the near future, but that the converse is true in the distant future. Recall that
Table 1 depicts the probabilities for having various utilities from an additional dollar
because of various levels of wealth. Table 2 shows the utility from an additional little
more than a dollar (a dollar and ten cents)3 under the same conditions as in Table 1. The
only difference between the tables is that, in Table 2, we increased utility by 0.5 for each
possible level of wealth.
3
The values of a dollar for the SER and a dollar and ten cents for the LLR are arbitrary. The
results depend only on the utilities attached to these values. These utilities are subjective and can,
thus, differ across individuals.
8
Additional utility
from $1.1
Period 0
Period 1
Period 2
10.5
1
1− m
(1 − m) 2 + mq
9.5
0
m
(1 − m )m + mr
8.5
0
0
mp
Expected
additional utility
from $1.1
10.5
10.5 − m
10.5 + m 2 − 11m
+ m(8.5 p + 9.5r + 10.5q )
Table 2: The columns indicate the probabilities of having various utilities from an additional
dollar and ten cents in each period. The bottom row indicates expected utility from the additional
dollar and ten cents in each period.
Proposition 2: Inconsistency of preferences will prevail (the individual will prefer the
SER in the near future, but the LLR in the distant future) when the probability of winning
the first dollar is large enough, m > 0.5 , and the probability of losing a dollar is large
enough, q > p + 0.5 − 2 .
Proof: We need to show that Eu ($1) 0 > Eu($1.1)1 and Eu ($1.1) 2 > Eu($1)1 . Substituting
expected utilities from Tables 1 and 2, the first condition holds when m > 0.5 , and the
second condition holds when q > p + 0.5 − 2 .
The intuition behind this result is that the probability of gaining a dollar in the
first period (m) must be large enough to diminish the expected additional utility for the
LLR in period 1. In that case, the individual will prefer the SER in period 0 to the LLR in
period 1, which is the near-future tradeoff. The probability of losing a dollar in the second
period (q) must be large enough so that the expected additional utility in the second
9
period will not be too small. Thus, the individual will prefer the LLR in the second period
to the SER in the first period, which is the distant-future tradeoff.
3. Infinite horizon case
We now extend our discussion to a multi-period environment. We assume that the
marginal utility from money in period t, ut , decreases linearly as a function of wealth in
period t, xt . Specifically,
(1)
ut = A − x t .
When A is a constant parameter.
Let pr0 ( xt = x ) be the probability in period 0 that xt = x . We assume that pr0 ( xt = x ) is
derived by the following procedure: The individual begins with wealth b in period 0. She
can win, lose, or neither win nor lose a dollar in each period. But when her wealth is b,
she can not lose a dollar4; thus, she has a lower boundary for the amount of wealth she
might hold. The expected marginal utility in period t as seen from period 0's perspective
is:
∞
(2)
E 0 ut = ∑ ( A − x t ) pr0 ( x = x t )
x =0
In order to demonstrate that the expected marginal utility is hyperbolic, we now turn to
numerical analysis and compute equation (2) for each period using a computer program
in Mathematica 35.
4
We are not considering a situation where wealth is lower than b because, in this model, such a
situation can not arise.
5
The computer program is in the appendix.
10
Figure 1 depicts the individual's marginal utility from money 400 periods into the
future in a situation where A=15, the probability to win a first dollar is 0.5. When the
individual's wealth is more than her initial wealth of b=0, she can win or lose a dollar
with probability 0.25. Figure 1 shows that, in such a situation, the marginal utility from
money will hyperbolically decrease in time.
Utility
Figure 1. Expected utility from an additional dollar as a function of time. The probability of
winning a dollar when the individual has no wealth is m=0.5, the probability of winning or losing
a dollar when the individual has some initial wealth is p=q=0.25 , and the probability of neither
winning nor losing a dollar when the individual has some initial wealth is r=0.5 .
Figure 2 shows that individual's choices are inconsistent under the dynamic
described in this section6. For each time period, t, the line marked LLR indicates expected
utility from a larger sum of money to be received in period t+1, while the line marked
SER indicates expected utility from a smaller sum of money to be received in period t.
The expected utility from a large reward is greater than the expected utility from a small
6
The computer program is in the appendix.
Time
11
reward in any specific time period7. Figure 2 shows that, in the near future, the agent
prefers the SER, while the agent prefers the LLR in the distant future. This result is
congruent with the experimental evidence [Thaler (1981), Benzion, Rapoport and Yagil
(1989), Kirby and Herrnstein (1995), Kirby (1997)]. The reason for the inconsistency in
choices is that the expected additional utility from a monetary reward decreases faster in
the near future than in the distant future. The fast decrease in the near future causes the
individual to prefer an SER to an LLR because the LLR decreased by a large amount. In
the distant future, the decrease between periods is small and the individual prefers the
LLR to the SER.
Utility
SER
LLR
Figure 2. Expected utility from an additional dollar in period t (SER) and expected utility from an
additional dollar and ten cents in period t+1 (LLR).
7
For demonstrating the arguments in this paper, the level of additional utilities in the program are
arbitrary.
Time
12
4. Concluding remarks
We show that a hyperbolic discount function for money can be derived when
individuals assume that, in each period, their wealth might alter under the constraint that
the probability that their wealth will deteriorate below the current level is small. People
contemplate how much money they are likely to have in the future and evaluate their
marginal utility from money accordingly. When the marginal utility as a function of
wealth itself is hyperbolic, the utility will be hyperbolic in time when there is a positive
probability of earning money in each period without the need to have a positive
probability of losing money in each period. In such a situation, a positive probability of
losing money and returning to previous levels of marginal utility enhances the hyperbolic
discounting phenomenon. The framework provided here shows how hyperbolic
discounting can exist even when the marginal utility as a function of wealth is not
hyperbolic. Seemingly impulsive behavior is shown to be rational. Dynamic
inconsistency is an outcome of the hyperbolic discounting phenomenon. But, again,
individuals are only seemingly inconsistent. They alter their choices for the future
because they know that their preferences might change.
References
Azfar, O. (1999) Rationalizing hyperbolic discounting. Journal of Economic Behavior
and Organization. 38, 245-252
Benzion, Uri, Amnon Rapoport and Joseph Yagil (1989) Discount Rates Inferred from
Decisions: An Experimental Study. Management science, 35,270-84.
Kahneman, Knetsch and Thaler (1992) The Endowment Effect, Loss Aversion, and
Status Que Bias in The winner's Curse Richard H. Thaler, Ed. Princeton: Princeton
University Press, 63-78.
13
Kirby, K. N. (1997) Bidding on the Future: Evidence Against Normative Discounting of
Delayed Rewards. Journal of Experimental Psychology: General, 126, 54-70
Kirby, K. N. and R.J. Herrnstein (1995) Preference Reversals due to Myopic Discounting
of Delayed Reward. Psychological Science, 6, 83-89.
Laibson, G. (1997) Golden Eggs and Hyperbolic Discounting. Quarterly Journal of
Economics, 112(2), pp. 443-77
Rabin (1998) Psychology and Economics. Journal of Economic Literature, 36, 11-46.
Rubinstein, A. (2000) Is It "Economics and Psychology"? The case of Hyperbolic
Discounting. Unpublished
Thaler, Richard H. (1981) Some Empirical Evidence on Dynamic Inconsistency.
Economic Letters, 8, 201-7.
14
Appendix
The following Mathematica 3 program was written for the situation described in section
3, in which the individual contemplates her marginal utility for money, but the
individual's wealth does not fall below her current wealth. The program that generates
Figure 1 is:
W U [ IL P S + U/ s + IL/ T IL S
D] 7DEOH# W ' D] 3UHSHQG# D] ' D]] D]
P 7DEOH# D 7DEOH# D]# # L' ' U D]# # L ' ' T D]# # L ' ' S L W ' D 3UHSHQG# D D]# # ' ' U D]# # ' ' T D]# # ' ' P ' D 3UHSHQG# D D]# # ' ' + P / D]# # ' ' T' D $SSHQG# D D]# # W' ' S' D] D W ' P 3UHSHQG# P D]]' RX 5DQJH# [ [ W ' X P RX
/LVW3ORW# X'
15
The program that generates Figure 2 is:
W U [ [O IL P S + U / s + IL / T IL S
D] 7DEOH # W ' D] 3UHSHQG # D] ' D]] D]
P 7DEOH # D 7DEOH # D] # # L ' ' U D] # # L ' ' T D] # # L ' ' S L W ' D 3UHSHQG # D D] # # ' ' U D] # # ' ' T D] # # ' ' P ' D 3UHSHQG # D D] # # ' ' + P / D] # # ' ' T ' D $SSHQG # D D] # # W ' ' S ' D] D W ' P 3UHSHQG # P D]] ' RX 5DQJH # [ [ W ' RXO 5DQJH # [O [O W ' X P RX
XO P RXO
V /LVW3ORW # X 3ORW6W\OH ! 5*%&RORU # ' 3ORW-RLQHG ! 7UXH ' VO /LVW3ORW # XO 3ORW-RLQHG ! 7UXH ' X 'HOHWH # X ' GX X XO
V /LVW3ORW # X 3ORW6W\OH ! 5*%&RORU # ' 3ORW-RLQHG ! 7UXH ' VO /LVW3ORW # XO 3ORW-RLQHG ! 7UXH ' 6KRZ # V VO ' /LVW3ORW # GX 3ORW5DQJH ! $OO 3ORW-RLQHG ! 7UXH '