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\noindent
 {\bf  Some  Common Confusions about  Hyperbolic Discounting   }



\noindent
\begin{tiny}

8  February 2008.
Eric Rasmusen.
  \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}.


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\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.\\
 (b) ``Hyperbolic discounting'' does not, as commonly used, mean
discounting using a hyperbolic function.\\
 (c) Hyperbolic discounting really isn't about the shape of the discount
function anyway.\\
 (d) Hyperbolic discounting is not about someone being very impatient.\\
 (e) Hyperbolic discounting is not necessarily  about  lack of self-
control, or irrationality.\\
(f) Hyperbolic discounting does not depend delicately on the length of
the time period.
 \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}
 Exponential utility with a constant discount factor $\delta$ has
the  form:
 \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 (with $0  \leq \beta \leq 1$):
 \begin{equation} \label{e5}
 U_0= C_0 + \beta \delta C_1 +  \beta \delta^2 C_1+\beta  \delta^3 C_2
+...
\end{equation}
 The  functional form in equation (\ref{e5}) can
also be written  as:
$
 U_0=  H*C_0 +   \delta C_1 +   \delta^2 C_1+   \delta^3 C_2
+...,
$
    where  $H >0$,  and where $H>1$ for a person  who distinguishes
sharply between   current consumption and future consumption.
True hyperbolic utility would have the form:
 \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}






An exponential utility function with the  hyperbolic 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/cdot /cdot /cdot
\delta_t$.
so  we can calculate
$
   \delta_t =\frac{f( t)}{ f(t-1)}$,
    and
since   $    \delta_t =\frac{1}{1+ \rho_t}, $  we can calculate
$    \rho_t =\frac{1 - \delta_t}{ \delta_t}. $

}


\begin{center}
 \includegraphics[width=4in]{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}


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