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\begin{center}
\begin{Large}
{\bf Some Common Confusions about Hyperbolic Discounting } \\
\end{Large}
\bigskip
\bigskip
25 March 2008 \\
\bigskip
Eric Rasmusen \\
\bigskip
{\it Abstract}
\end{center}
\vspace*{-12pt}
There is much confusion over what ``hyperbolic discounting'' means.
I argue that what matters is the use of relativistic instead of objective time, not the shape of the discount function.
\bigskip
\begin{small}
\noindent
\hspace*{20pt} Dan R. and Catherine M. Dalton Professor, Department of
Business
Economics and Public Policy, Kelley School of
Business, Indiana University. visitor (07/08), Nuffield College, Oxford
University. Office: 011-44-1865 554-163 or (01865) 554-163. Nuffield
College,
Room C3, New Road, Oxford, England, OX1 1NF.
Erasmuse@indiana.edu. \url{http://www.rasmusen.org}. This paper:
\url{http://www.rasmusen.org/special/hyperbolic-rasmusen.pdf}.
\noindent
Keywords: Time
inconsistency, hyperbolic discounting.
\noindent
I thank Kevin Du, Jeremy Tobacman, and participants in a workshop at Bath
University for helpful comments, and Nuffield College,
Oxford University for its hospitality.
\end{small}
\newpage
The term ``hyperbolic discounting'' comes up frequently nowadays in
economists's lunchtime discussions.\footnote{Two overviews are the
Frederick, George Loewenstein and Ted
O'Donoghue 2002 article in the {\it Journal of Economic Literature}
and the 2001 George-Marios Angeletos, David Laibson,
Andrea Repetto, Jeremy Tobacman and
Stephen Weinberg article in the {\it
Journal of Economic Perspectives.} Those papers go into detail about
motivations and applications for hyperbolic discounting, something I
will not do here. } Like ``adverse selection'' or
``method of moments'' it is one of those ideas that we all think we
should know about. The idea of hyperbolic discounting, however, is easy
to misunderstand. An example that I will explain below is that the term
is usually applied to models that don't use hyperbolic functions for
their discounting. There are other sources of confusion too, which I
will try to clear up. This note will, I am sure, contain no single idea
that is unknown to a great many scholars who have thought hard about
hyperbolic discounting. I nonetheless hope that even those scholars
could benefit from reading this, and could make use of this in
explaining hyperbolic discounting to non-experts. For non-experts, of
course, I hope this note clears away the most common misperceptions.
First, I will explain what hyperbolic discounting is, and how it
creates time inconsistency. Then, I will try to clear up various
confusions about it.
\bigskip
\noindent
{\bf What Hyperbolic Discounting Is}
People prefer to
consume more now rather than more later. The standard way to include
this in economic analysis is using a positive personal discount
rate in the utility function so consumption earlier adds more to
utility than consumption later. An example of such a utility function
is:
\begin{equation} \label{e1a}
U_{2008} = C_{2008} +
f(2009)C_{2009} + f(2010)C_{2010} +
f(2011)C_{2011},
\end{equation}
where the ``discount function'' is $f(t) <1$ and $f$ is declining in
$t$. Or, we could write the discounting in terms of per-period
``discount factors'' $\delta_t <1$, as in this example:
\begin{equation} \label{e1}
U_{2008} = C_{2008} +
\delta_{2009}C_{2009} + \delta_{2009}\delta_{2010}C_{2010} +
\delta_{2009}\delta_{2010}\delta_{2011}C_{2011}.
\end{equation}
We could but need not assume a constant discount factor $\delta$ and
``discount rate'' $\rho$, so that:
\begin{equation} \label{e2}
\delta_t=
\delta^t = \left(\frac{1}{1+\rho}\right)^t.
\end{equation}
The functional form for discounting in equation (\ref{e1a}) is
perfectly general (though of course the linear utility separable across
time is special). The functional form in equation (\ref{e1}),
however, is an example of
exponential discounting. Exponential discounting's key feature is that
the time subscripts for the discount factors are absolute years, which
I will denote by $t$, rather
than relativistic ``years in the future'', which I will denote by
$\tau$
(relativistic because they depend
on the year in which the person starts). As some would put it, the
difference is between the {\it date} $t$ and the {\it delay} $\tau$.
Because of using relativistic discounting, if we view our
person's decisions starting one year in the future, at 2009 instead of
2008, his utility function
will be:
\begin{equation} \label{e3}
U_{2009} = C_{2009}
+ \delta_{2010}C_{2010} + \delta_{2010}\delta_{2011}C_{2011}.
\end{equation}
\vspace{.5in}
\begin{center}
\begin{tabular}{|l|lll llll| }
\hline
& & & & & & & \\
The Time, $t$ & 0 & 1 &2 & 3 &4 & 5& 6 \\
& & & & & & & \\
\hline
& & & & & & & \\
The Discount Rate for $t-1$ to $t$, $\rho $ & --- & .10 &
.06
&.05 & .07 & .07 & .05 \\
& & & & & & & \\
The Discount Factor for $t-1$ to $t$, $\delta_t $ & --- &.909 &
.943
&.952 &
.935& .935 & .952 \\
& & & & & & & \\
\hline
& & & & & & & \\
The Discount Function for $0$ to $t$, $f(t) $
& 1 & .909
&.857 &
.816 &.763 & .713& .680 \\
& & & & & & & \\
\hline
\end{tabular}
\medskip
{\bf Table 1: Exponential Discounting (to 3 decimal places) }
\end{center}
\vspace{.5in}
A nice property of exponential discounting is that a person's
consumption path will be time consistent, meaning that if the person
maximizes his utility at time 2000 by the choices $(C_{2008}^*, C_{2009}
^*,C_{2010}^*,C_{2011}^*)$ then he will maximize them at time 2000 by
the same values $(C_{2009}^*,C_{2010}^*,C_{2011}^*)$ given his reduced
wealth as the result of the consumption in 2008. The person may regret
consuming so much in the year 2008, so his decisions across time are not
consistent in the sense of being the choices he would make ex post, but
the 2008 consumption is a sunk decision by 2009 anyway.
An alternative way a person might have time preference is to have
discount factors whose levels depend not on the year itself-- 2008,
2009, 2010-- but on how many years in the future the consumption will
occur--now, one year from now, two years from now. Looking at the
decision made in year 0, the functional form could be exactly the same,
e.g.
\begin{equation} \label{e4}
U_{0} = C_{0} + \delta_{1}C_{1} + \delta_{1}\delta_{2}C_{2} +
\delta_{1}\delta_{2}\delta_{3}C_{3},
\end{equation}
where $\delta_\tau <1$, using $\tau$ now instead of $t$ because time is
now relativistic rather than absolute. At time 1 (
year 2009) however, the person would maximize $U_{1} = C_{1} +
\delta_{1} C_{2} + \delta_{1} \delta_{2} C_{3}$, not $U_{1}' = C_{1}
+ \delta_{2} C_{2} + \delta_{2} \delta_{3} C_{3}$.
The function in equation (\ref{e4}) is just one of
many ways time preference could be non-exponential, but whatever
non-exponential form it takes, the person's decisions become time
inconsistent. The optimal choices $(C_{2008}^*, C_{20091}^*,C_{2010}
^*,C_{2011}^*)$ from the 2000 utility function will not match the
optimal choices using the 2001 utility function, $(C_{2009}
^{**},C_{2010}^{**},C_{2011}^{**})$.
For example, it may be that the
person is expecting a big income bonus in 2010. In year 2008, he might
want to spread that income's consumption between 2010 and 2011
because though he highly values year 0 consumption, he is relatively
indifferent between years 2 and 3. By the time 2010 arrives, however,
year 2010 {\it is} year 0, and he would want to consume the entire bonus
immediately.
Hyperbolic discounting is a useful idea for two reasons. First, it can
explain revealed preferences that are inconsistent with exponential
discounting. Second, and in my opinion more important, it can explain
certain observed behaviors such as people's commitments to future
actions when other explanations such as strategic positioning fail to
apply, e. g., a person's joining a bank's saving plan which penalizes
him for failing to persist in his saving.
\bigskip
\noindent
{\bf What Hyperbolic Discounting Is Not}
So much for what hyperbolic discounting is. What is it not?
I will make the following points:
\noindent
(a) Hyperbolic discounting is not about the discount rate changing over
time. A constant discount rate is not essential for time consistency,
nor does a varying discount rate create time inconsistency.
\noindent
(b) ``Hyperbolic discounting'' does not, as commonly used, mean
discounting using a hyperbolic function.
\noindent
(c) Hyperbolic discounting really isn't about the shape of the discount
function anyway.
\noindent
(d) Hyperbolic discounting is not about someone being very impatient.
\noindent
(e) Hyperbolic discounting is not necessarily about lack of self-
control, or irrationality.
\noindent
(f) Hyperbolic discounting does not depend delicately on the length of
the time period.
\bigskip
\noindent
{\it (a) Hyperbolic discounting is not about the discount rate
changing over
time. A constant discount rate is not essential for time consistency,
nor does a varying discount rate create time inconsistency.
}
Hyperbolic discounting makes not one but two changes from the
standard model. First, it makes the per-period discount rate change over
time. Second, it bases discounting on relativistic time rather than
absolute time. It is this second assumption which is the key one.
Both changes are necessary conditions to generate time inconsistency.
I view the second change as the important one, though, because the
assumption that the per-period discount rate is constant is merely a
simplifying assumption, not meant to change the essential behavior of an
economic model.
Comparison with interest rates on bonds is useful. We all know that
long-term and
short-term interest rates are different, and there is a large literature
studying the term structure of interest rates, yet in most theoretical
models we
assume a constant rate of interest, $r$. We do that not because we think
it realistic, but as a simplifying assumption, one that we often
discard when it comes to actual estimation.
In the same way, there is no reason for us to think of a person's rate
of time preference, $\rho$, as being constant, even in the standard
exponential-discounting model with absolute time. If I am planning for
the consumption of my 8-year-old daughter in 2008 I might use
$\rho_t= 10\%$ for each year in the interval [2008, 2015] and then use
$\rho_t= 5\%$ for the interval [2015, 2025] because I expect her degree
of impatience to change. Doing so is still absolute-time discounting.
It would become relativistic-time discounting only if in each of those
17
years I followed a policy of using 10\% for whatever years were the 7
next years from the present and 5\% for whatever years were the 8th to
17th year from the present.\footnote{
We would not even expect impatience to always
decline, in the standard model. Planning for myself, I might use
$\rho_t=3\%$ now, but $\rho_t=15\%$ for the year 2058, when I expect to
be in my second childhood and barely aware of future consumption. }
People change as they get
older. It is not implausible that I would use a different rate of time
preference at age 70 than at age 49. But that is fully compatible with
exponential discounting. The key is that exponential discounting treats
the parameter as ``Rasmusen's rate of time preference for when he is 70
in the year 2058'' whereas non-exponential discounting treats it as
``Rasmusen's rate of time preference for 21 years from the present.''
So long as the discounting uses absolute time, it will not give
rise to time inconsistency, even if the discount rate changes over time.
In 2008 I will be happy to have consumption from 2015 to 2016 discounted
at a rate of 5\%, and I will be equally happy with that plan in 2014.
It is also true that if my per-period discount rate is constant then it
does not matter whether I discount using absolute time or relativistic
time. If I use relativistic time, however, the assumption of a constant
per-period discount rate is no longer just a simplifying assumption. It
now becomes a substantive assumption, one necessary to avoid the changes
in behavior that time inconsistency would bring.
\bigskip
\noindent
{\it
(b) ``Hyperbolic discounting'' does not, as commonly used, mean
discounting using a hyperbolic function.
} The standard
way
to model hyperbolic discounting is with the ``quasi-hyperbolic
discounting'' of David Laibson
(1997), which is simpler to use than a hyperbolic function and has
similar properties,
though it is not a special case of the hyperbolic function.
Edward Phelps \& Pollack (
1968) had used the quasi-hyperbolic form to discount generations of
people over time. Laibson (1997) applied it to within the self and
named it ``quasi-hyperbolic''.\footnote{The term ``subholic'' has also
been suggested, since the function lies below exponential and hyperbolic
discounting for small values of $\tau$-- see Figure 1.} True
``hyperbolic'' discounting, proposed
by Shin-Ho Chung \& Robert Herrnstein
(1961) in connection with a particular theory of behavior, has similar
qualitative features but is more complicated to work with. Figure 1
shows how the shapes of the discounting over time differ.
\vspace{.5in}
\begin{center}
\includegraphics[width=7in]{hyperbolic1.jpg}
{\bf Figure 1: \\
The Shapes of Exponential (solid), Hyperbolic (dotted), and
Quasi-Hyperbolic (dashed)
Discounting }\\
($\delta_{exp} = .92$, $ f_h(\tau)= \frac{1}{1+.1\tau}$,
$\beta= .8$ or $H=1.25$ and $\delta_{qh} = .96$)
\end{center}
\vspace{.5in}
Exponential utility with a constant discount factor $\delta$ has
the form $f(t) = \delta^t$:
\begin{equation} \label{e7}
U_0= C_0 + \delta C_1 + \delta^2 C_2+ \delta^3 C_3 +...,
\end{equation}
which has the per-period discount rate
$\rho= \frac{1- \delta }{ \delta}$.
Quasi-hyperbolic utility (also called ``Beta-Delta Utility'') has the
form $f(\tau) = \beta \delta^\tau$:
\begin{equation} \label{e5}
U_0= C_0 + \beta \delta C_1 + \beta \delta^2 C_2+\beta \delta^3 C_3
+...
\end{equation}
with $0 \leq \beta \leq 1$.
This means that there is a lot of discounting
from the present to time 1 (discount factor $\beta \delta< \delta$,
and per-period discount rate $\rho=\frac{1-\beta \delta }{\beta
\delta}$),
but after that the discount factor between any two adjacent periods
falls to a constant $\delta$ (which is the per-period discount rate
$\rho= \frac{1- \delta }{ \delta}$). And, of course, this is
relativistic-time discounting, so the date of ``the present'' is
always changing.
True hyperbolic utility has the
form\footnote{Charles Harvey (
1986)
and George Loewenstein \& Drazen Prelec (1992) proposed a more
general form:
\begin{equation} \label{e6b}
f(\tau)= \frac{1}{(1+ \alpha \tau)^{\gamma/\alpha}},
\end{equation}
which has $f(\tau) = e^{-\gamma \tau}$ as its limiting case as
$\alpha$ goes to zero.} $f(\tau) = \frac{1}{ 1+\alpha \tau}$:
\begin{equation} \label{e6}
U_0= C_0 + \left(\frac{1}{1+\alpha} \right)C_1 + \left(\frac{1}{1+2
\alpha}
\right) C_2 +...
\end{equation}
Quasi-hyperbolic utility has some properties similar to hyperbolic
utility, but it is not a special case of it.
As with quasi-hyperbolic
discounting, what is most important is that hyperbolic discounting uses
relativistic time in the utility function.
All of these discounting
functions are for use on the demand side, for the utility function.
The supply side, the budget constraint, would be the same for all of
them, and must use absolute time-- $t$, not $\tau$, in the notation
here. Thus, even if the utility function is written in terms of $\tau$,
the budget constraint would be:
\begin{equation} \label{e6c}
C_0+ \left( \frac{1}{1+r_t} \right) C_1 + \left( \frac{1}{1+r_1}
\right) \left( \frac{1}{1+r_2} \right) C_2 + ... + \left( \frac{1}{1
+r_1} \right) \left( \frac{1}{1+r_2} \right) \cdots \left(
\frac{1}{1+r_t} \right)C_t \leq W_0,
\end{equation}
where $W_0$ is the present value of wealth at time 0.
Any functional form for discounting except exponential
discounting would give rise to time inconsistency. The
particular functional form does not matter much, except for
simplicity. This is
important because one might think that hyperbolic discounting was a
special case, which is false. Exponential discounting is the special
case. It would
be better to use the term ``nonexponential discounting'', but
``hyperbolic'' is what has become standard.
Hyperbolic discounting's functional form arises from the ``Matching
Law'' theory of decisionmaking. Quasi-hyperbolic discounting amounts to
adding to standard exponential discounting a parameter, $\beta$, that
makes the person distinguish sharply between ``Right now, time zero''
and ``All future times.'' Its functional form in equation (\ref{e5}) can
also be written as:
\begin{equation} \label{e8}
U_0= \gamma C_0 + \delta C_1 + \delta^2 C_2+ \delta^3 C_3
+...,
\end{equation}
with $\gamma >0$, where $\gamma =1$ for exponential utility,
and where $\gamma >1$ for quasi-hyperbolic utility. The
marginal rate of substitution between consumption at time $0$ and time
$\tau'$ is $\gamma/\delta^\tau$, as opposed to the
$(1/\beta)/\delta^\tau$ from
equation (\ref{e5}), but they represent exactly the same consumer
preferences.
I prefer form (\ref{e8}) to form (\ref{e5}), even though form
(\ref{e5}) is standard. In form (\ref{e8}), the parameter $\gamma$
represents
what
we might call ``hyperbolicity'' and allows $\gamma$ to increase to
infinity;
whereas in form (\ref{e5}) the parameter $\beta$ is declining in
hyperbolicity, with a value restricted to be no greater than 1. Form
(\ref{e5}) has the parameter $\beta$ in every term, whereas in form
(\ref{e8}) the parameter $\gamma$ only has to show up once. For that
reason, form
(\ref{e8}) better represents the intuition that the parameter concerns
the specialness of the present.
\bigskip
\noindent
{\it (c) Hyperbolic discounting really isn't about the shape of the
discount
function anyway.} Rather it is about relativistic versus absolute time:
how a person's discount rate for a given
time period changes as that time period gets closer. At any one time,
knowing the shape of the person's discount function for each period's
future consumption doesn't tell you whether he is using ``hyperbolic
discounting'' or not-- except for the special case in which his per-
period exponential discount rate is constant. This is really a
reprise of point (b), but I will expand on it here using examples.
It is easy to come up with examples where exponential and non-
exponential discount functions are identical.
\bigskip
\noindent
{\bf Example.} We will use the hyperbolic discounting function from
Figure 1:
\begin{equation} \label{e6b}
f(\tau)= \frac{1}{(1+ .1 \tau) }
\end{equation}
which generates the numbers
in Table 2.
\vspace{.5in}
\begin{center}
\hspace*{-1in} \begin{tabular}{|l|lll llll | }
\hline
& & & & & & & \\
The Time: $t$ or $\tau$ & 0 & 1 &2 & 3 &4 & 5& 6 \\
& & & & & & & \\
\hline
\hline
& & & & & & & \\
The Discount Function for $0$ to $t$ or $\tau$: $f(t)$ or
$ f(\tau)$ & 1 & .91
&.83 & .77 &.71 & .67& .63 \\
& & & & & & & \\
\hline
& & & & & & & \\
The Discount Rate for $t-1$ to $t$ (or $\tau-1$ to $\tau$): $\rho_t $ or $\rho_\tau $ &
--- & 10\% &
9\%
&8\% & 8\% & 7\% & 7\% \\
& & & & & & & \\
The Discount Factor for $t-1$ to $t$, $\delta_t $ &
--- &.91 & .92 &.92 & .93& .93 & .94 \\
& & & & & & & \\
The Quasi-Hyperbolic Delta Parameter for $t-1$ to $t$:
$\delta_\tau $
&
--- & .96 & .92 &.92 & .93& .93 & .94 \\
& & & & & & & \\
\hline
\end{tabular}
\bigskip
{\bf Table 2: Exponential Discounting with a Hyperbolic Shape
(to 2 decimal places, $f(\tau)= \frac{1}{(1+ .1 \tau) }$) }
\end{center}
\vspace{.5in}
An exponential utility function with the same shape can be derived from
$f(1) = \delta_1 $,
$ f( 2) = \delta_1 \delta_2$,
$ f( 3) = \delta_1 \delta_2\delta_3$, ...
$ f(t) = \delta_1 \delta_2\delta_3 \cdots
\delta_t$.
so we can calculate
\begin{equation} \label{e9}
\delta_t =\frac{f( t)}{ f(t-1)},
\end{equation}
and
since $ \delta_t =\frac{1}{1+ \rho_t}, $ we can calculate
$ \rho_t =\frac{1 - \delta_t}{ \delta_t}. $
Similarly, we can find a
quasi-hyperbolic utility function with the same shape if we are
allowed to vary the $\delta$ parameter. Let's set $\beta =.95$ We have
$ f(1) = \beta \delta_1 $,
$ f( 2) =\beta \delta_1 \delta_2$,
$ f( 3) =\beta \delta_1 \delta_2\delta_3$, \ldots
$ f(\tau) = \beta \delta_1 \delta_2\delta_3 \cdots
\delta_\tau$.
so we can calculate
\begin{equation} \label{e10}
\delta_1 = \frac{f(1)}{ \beta}
\end{equation}
and
\begin{equation} \label{e11}
\delta_\tau = \frac{f(\tau)}{ f(
\tau - 1) }.
\end{equation}
In equation (\ref{e11}) and Table 2 all but the first of the
$\delta$
parameters are the same for exponential and for quasi-hyperbolic
discounting.
Note in Table 2 how the exponential discount {\it rates} are declining
as
time passes. This is a general feature of the hyperbolic discounting
function with constant $\alpha$.
It is not a characteristic of the quasi-hyperbolic discounting function
with constant $\delta$, for which, of course, the discount factor is
constant at $\delta$ after the first period so the discount rate is also
constant.
\bigskip
\noindent
{\it (d) Hyperbolic discounting is not about someone being very
impatient.} A
person can have high time preference even under standard
exponential discounting. The key to hyperbolic discounting is that the
person's high rate of discounting for a given future year's utility
(say, utility in 2020) changes as that year approaches.
In theory, hyperbolic discounting could result in negative time
preference,
preferring future to present consumption. Someone might
always care little about the present year, but a lot about future years.
This would be one way to model a person who derives much of his utility
from anticipation of future consumption.
Patience of this kind would introduce time inconsistency too. In 2010
the person would want to consume a lot in 2015, but in 2015 he would
prefer to defer consumption. Thus, the essence of hyperbolic
discounting is not excessive impatience.
Figure 1 illustrates something else about the difference: although
quasi-hyperbolic discounting discounts the close future more than
exponential does, it discounts the distant future less. In Figure 1 the
amount of discounting is about the same at year 6, but the exponential
utility function is more patient at year 3 and less patient at year 30.
Thus, the question of which kind of discounting is more patient is ill-
formed.
\bigskip
\noindent
{\it
(e) Hyperbolic discounting is not necessarily about lack of
self-control, or irrationality.} It is one way to model lack of self-
control,
to be sure,
by having $0<\beta<1$ in the quasi-hyperbolic model. Lack of
self-control is just an interpretation of the equations, though, not a
necessary implication, and one might model lack of self-control in other
ways. Also, non-exponential utility can be used to model other kinds of
behavior not involving self-control. For example, we can imagine a
person who is afraid of poverty more than 10 years in the future, but
becomes braver as the poverty comes closer.
Or, we could simply take the preferences at face value.
The question of whether in a particular setting hyperbolic
discounting is being used to model (a) preferences that we usually don't
assume in economics, or (b) mistakes such as lack of self-control, is
important, especially for normative analysis. See Bernheim \& Rangel (
2008) or my own Rasmusen (2008) for two attempts to grapple with welfare
analysis when discounting is hyperbolic. But although hyperbolic
discounting does raise hard questions for welfare analysis, the hardest
questions do not arise because it implies irrationality. Indeed, if we
think of hyperbolic discounting as modelling irrationality, the welfare
analysis gets easier-- we simply try to undo the effect of the mistakes.
The question of whether hyperbolic discounting is
rational is separate from the deeper question of whether discounting in
general is rational or moral. This is an open question in moral
philosophy. Most modern economists seem to agree that positive time
preference is legitimate at the level of
the individual; most moral
philosophers seem to agree that it is illegitimate (and they even have
doubts about discounting in general). See the references in Rasmusen
(2008) for entry into the philosophy literature. The issue of what the
social discount rate should be is a separate one, which is even more
contentious, in both disciplines.
\bigskip
\noindent
{\it (f) Hyperbolic discounting does not depend delicately on the length
of the time period. } Recall from equation (\ref{e8}) that a quasi-hyperbolic utility function can be written
in the more intuitive form:
$$
U_0= H*C_0 + \delta C_1 + \delta^2 C_2+ \delta^3 C_3
+...,
$$
There is a big difference between the present and consumption at any
future time, but the units in which time is measured do not affect
tradeoffs between future time periods (though of course $\delta$ has to
be written in the new time units too, so its value will change). For example, take a quasi-hyperbolic discount function with $\beta=.8$ and $\delta=.96$ and units of one year. The discount function will look like this:
\begin{center}
\hspace*{-1in}\begin{tabular}{|l|lll llll |}
\hline
& & & & & & & \\
Time: & 0 & 1 &2 & 3 &4 & 5& 6 \\
& & & & & & & \\
Discount Function & 1 & .77 &.74 & .71&.68& .65& .63 \\
& & & & & & & \\
\hline
\end{tabular}
\end{center}
Now change the time period to 2 years:
\begin{center}
\hspace*{-1in} \begin{tabular}{|l|lll llll | }
\hline
& & & & & & & \\
Time: & 0 & 1 &2 & 3 &4 & 5& 6 \\
& & & & & & & \\
Discount Function & 1 & --&.74 & --&.68& --& .63 \\
& & & & & & & \\
\hline
\end{tabular}
\end{center}
The discount function values stay the same. We'd just change to $ \delta_{new} = \delta_{old}^2= .9216$ to match the new time period.
\bigskip
\noindent
{\bf Concluding Remarks}
Hyperbolic discounting is a useful idea for modelling people whose
rate of time preference is based on relativistic time-- distance in time
from the present-- rather than absolute time-- the date. As with
discounting using interest rates, we often model time preference with
just one or two parameters but more realistically we should expect the
rate of time preference to be constantly changing, whether we use
exponential discounting or non-exponential. Such time preference is not
necessarily irrational in the sense of being mistaken, but hyperbolic
discounting can be used to model either someone whose preferences are
based on relativistic time or someone who wishes he had more self-
control.
Either way, it gives rise to time inconsistency and can explain a
variety of observed behaviors in which people try to commit,
constraining their future selves.
\bigskip
\noindent
{\bf References}
Angeletos, George-Mariosm David Laibson,
Andrea Repetto, Jeremy Tobacman \&
Stephen Weinberg (2001) ``
The Hyperbolic Consumption Model:
Calibration, Simulation, and
Empirical Evaluation,'' {\it
Journal of Economic Perspectives,}
15(3): 47-68 (Summer 2001).
Bernheim, B. Douglas \& Antonio Rangel (2007) ``Beyond Revealed
Preference:
Choice Theoretic Foundations for Behavioral Welfare
Economics,'' NBER working paper 13737, \url{
http://www.nber.org/papers/w13737} (December 2007).
Chung, Shin-Ho \& Richard J. Herrnstein
(1961) ``Relative and Absolute Strengths of Response
as a Function of Frequency of Reinforcement,'' {\it
Journal of the Experimental Analysis of Animal
Behavior} 4: 267-272.
Frederick, Shane, George Loewenstein \& Ted O'Donoghue (2002)
``Time Discounting and Time Preference: A Critical Review,'' {\it
Journal of Economic Literature}, 40(2): 351-401 (June 2002).
Harvey, Charles M. (1986)
``Value Functions for Infinite-Period Planning,''
{\it
Management Science}, 32(9): 1123-1139 (September 1986).
Laibson, David (1997) ``Golden Eggs and Hyperbolic Discounting,''{\it
Quarterly Journal of Economics}, 112(2):
443-477 (May 1997).
Loewenstein, George \& Drazen Prelec (1992) ``Anomalies in
Intertemporal Choice: Evidence and an Interpretation,'' {\it
Quarterly Journal of Economics}, 107(2): 573-597 (May 1992).
Phelps, Edward S. \& R. A. Pollak (1968) ``On
Second-Best National Saving and Game-Equilibrium
Growth,'' {\it Review of Economic Studies,}. 35:
185-199.
Rasmusen, Eric (2008) ``Internalities and Paternalism: Applying Surplus
Maximization to the Various Selves across Time," working paper, \url{
http://www.rasmusen.org/papers/internality-rasmusen.pdf} (18 January
2008).
Strotz, R. H. (1956)
``Myopia and Inconsistency in Dynamic Utility Maximization,''
{\it
Review of Economic Studies}, 23(3): 165-180 (1955-56).
\end{document}