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\begin{center}
{\bf Some Common Confusions about Hyperbolic Discounting } \\
\bigskip
\bigskip
8 February 2008 \\
\bigskip
Eric Rasmusen \\
\bigskip
{\it Abstract}
\end{center}
\vspace*{-12pt}
There is a lot of confusion over what ``hyperbolic discounting'' means.
I try to clear up that confusion.
\bigskip
\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/papers/hyperbolic-rasmusen.pdf}.
\newpage
\noindent
{\bf What Hyperbolic Discounting Is}
\begin{LARGE}
\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}
\end{LARGE}
$t$= absolute years ( $\delta_t <1$)
$\tau$ = relativistic ``years in the future''
(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 2001 instead of
20001, his utility function
will be:
\begin{equation} \label{e3}
U_{2001} = C_{2001}
+ \delta_{2002}C_{2002} + \delta_{2002}\delta_{2003}C_{2003}.
\end{equation}
\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}
At time 1 (
year 2001) the person would maximize $U_{1} = C_{1} +
\delta_{1} C_{2} + \delta_{1} \delta_{2} C_{3}$, not $U_{1}' = C_{1}
+ \delta_{2} C_{2} + \delta_{2} \delta_{3} C_{3}$.
\newpage
\begin{equation} \label{e3}
U_{2009} = C_{2009}
+ \delta_{2010}C_{2010} + \delta_{2010}\delta_{2011}C_{2011}.
\end{equation}
where $\delta_\tau <1$, using $\tau$ now instead of $t$ because time is
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 $U_{1}' = C_{1}
+ \delta_{2} C_{2} + \delta_{2} \delta_{3} C_{3}$.
For example, it may be that the
person is expecting a big income bonus in 2002. In year 2000, he might
want to spread that income's consumption between 2002 and 2003
because though he highly values year 0 consumption, he is relatively
indifferent between years 2 and 3. By the time 2002 arrives, however,
year 2002 {\it is} year 0, and he would want to consume the entire bonus
immediately.
\newpage
Hyperbolic discounting is a useful idea.
First, it can
explain revealed preferences that are inconsistent with exponential
discounting.
Second, 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.
\newpage
\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.
\bigskip
\noindent
(b) ``Hyperbolic discounting'' does not, as commonly used, mean
discounting using a hyperbolic function.
\bigskip
\noindent
(c) Hyperbolic discounting really isn't about the shape of the discount
function anyway.
\bigskip
\noindent
(d) Hyperbolic discounting is not about someone being very impatient.
\bigskip
\noindent
(e) Hyperbolic discounting is not necessarily about lack of self-
control, or irrationality.
\bigskip
\noindent
(f) Hyperbolic discounting does not depend delicately on the length of
the time period.
\newpage
\noindent
{\bf (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.
}
(1) it makes the per-period discount rate change over
time.
(2) it bases discounting on relativistic time rather than
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. That's still absolute-time discounting.
It's 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.
\newpage
\noindent
{\bf
(b) ``Hyperbolic discounting'' does not, as commonly used, mean
discounting using a hyperbolic function.
}
\begin{center}
\includegraphics[width=6in]{hyperbolic1.jpg}
{\bf Figure 1 \\
The Shapes of Exponential (solid), Hyperbolic (dotted), and Quasi-
Hyperbolic (dashed)
Discounting }\\
($\delta_{exp} = .92$, $ \frac{1}{1+.1\tau}$,
$\beta= .8$ or $H=1.25$ and $\delta_{qh} = .96$)
\end{center}
\newpage
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_1+ \delta^3 C_2 +...,
\end{equation}
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_1+\beta \delta^3 C_2
+...
\end{equation}
True hyperbolic utility has the
form $f(\tau) = \frac{1}{ 1+\alpha \tau}$:
\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}
The supply side, the budget constraint must use absolute time-- $t$,
not $\tau$.
\begin{Large}
\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) \cdot \cdot \cdot \left(
\frac{1}{1+r_t} \right)C_t \leq W_0,
\end{equation}
\end{Large}
where $W_0$ is the present value of wealth at time 0.
\newpage
Quasi-hyperbolic utility (also called ``Beta-Delta Utility'') has the
form:
\begin{equation} \label{e5}
U_0= C_0 + \beta \delta C_1 + \beta \delta^2 C_1+\beta \delta^3 C_2
+...
\end{equation}
Quasi-hyperbolic discounting's functional form can
also be written as:
\begin{equation} \label{e8}
U_0= H*C_0 + \delta C_1 + \delta^2 C_1+ \delta^3 C_2
+...,
\end{equation}
where $H >0$, and where $H>1$ for a person who distinguishes
sharply between current consumption and future consumption. The
marginal rate of substitution between consumption at time $0$ and time
$\tau'$ is $H/\delta^\tau$, as opposed to the
$(1/\beta)/\delta^\tau$ from
equation (\ref{e5}), but they represent exactly the same consumer
preferences.
\newpage
\noindent
{\bf (c) Hyperbolic discounting really isn't about the shape of the
discount
function anyway.}
\noindent
{\bf Example.} We will use the hyperbolic discounting function from
Figure 1:
\begin{equation} \label{e6b1}
\delta_\tau= \frac{1}{(1+ .1 \tau) }.
\end{equation}
\begin{Large}
\begin{center}
\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 $t$: $f(t)$ or
$ f(\tau)$ & 1 & .91
&.83 & .77 &.71 & .67& .63 \\
& & & & & & & \\
\hline
& & & & & & & \\
The Discount Rate for $t-1$ to $t$: $\rho $ &
--- & 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}
{\bf Table 2: Exponential Discounting with a Hyperbolic Shape
(listed to 2 decimal places) }
\end{center}
\end{Large}
\newpage
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$, ...
$ 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}
\newpage
\begin{Large}
\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 $t$: $f(t)$ or
$ f(\tau)$ & 1 & .91
&.83 & .77 &.71 & .67& .63 \\
& & & & & & & \\
\hline
& & & & & & & \\
The Discount Rate for $t-1$ to $t$: $\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) }
\end{center}
\end{Large}
Note in Table 2 how the exponential discount 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.
\newpage
\noindent
{\bf (d) Hyperbolic discounting is not about someone being very
impatient.}
A
person can have high time preference even under standard
exponential discounting.
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.
In addition, see Figure 1.
\newpage
\noindent
{\bf
(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.
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.
\newpage
\noindent
{\bf (f) Hyperbolic discounting does not depend delicately on the length
of the time period. }
If we mean ``discounting using a hyperbolic
function'' then it is actually true that the model depends heavily on
the length of the time period. Double the units in which you measure
time, and you change the shape of the discounting function.
But if we
mean ``quasi-hyperbolic discounting'' that criticism
fails to apply. Recall:
$$
U_0= H*C_0 + \delta C_1 + \delta^2 C_1+ \delta^3 C_2
+...,
$$
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).
\newpage
\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).
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