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\vspace*{12pt}

\noindent
{\large {\bf    Notes on Discounting and Dynamic Programming }
}

\noindent
9 October  2009\\
Eric Rasmusen, erasmuse@indiana.edu

The present  value of a one-time payment of $X$, received at the end of one year, is, if the discount rate is $r$ per year, 
 \begin{equation} \label{e1}
V = \frac{X}{1+r}
\end{equation}
   As a result, the value of $X$ received  at the end of  $N$ years is $\frac{X}{(1+r)^N}$.  The value of $X$ received immediately is, of course, $X$. 

 The value of a flow of $X$ per year, received at the end of each year,  is 
 \begin{equation} \label{e1}
V(consol) =  \frac{X}{r}. 
\end{equation}
 The value if the payment is made at the beginning of each year is just the consol value plus the $X$ received immediately, $X + \frac{X}{r}$. 

The value $r$ is the discount rate. The discount factor is 
 \begin{equation} \label{e1}
     \delta = \frac{1}{1+r}
\end{equation}

 I remember being irritated by the redundancy of this terminology when I was a student. 
 The reason for having to remember two terms is that sometimes $r$ makes the equations neater and easier to write, sometimes $\delta$. 

 Here are two ways to think about the consol formula. 

(1) It says that if the interest rate is 5 percent, the value of a payment of 1,000 dollars per year is 20,000 dollars. That makes sense, because if you started with a lump sum of 20,000 dollars, you could put it in the bank and earn interest of  (.05)(20,000) = 1,000 dollars per year. The flow of 1,000 and the stock of 20,000 have equal value. 

(2) Use dynamic programming.  Let $V$ be the present value of the stream of $X$ per year.  It is equal to the amount $X$, discounted because it is received at the end of the year, plus at the end of the year you'll still have the value $V$. Thus, 
 \begin{equation} \label{e1}
 V =    \frac{X }{1+r} + \frac{V }{1+r} 
\end{equation}
 Thus, $(1+r)V = X +V$, and $rV = X$ and $V = \frac{X}{r}$. 

  This last formulation is useful in thinking of the value   of a growing flow. Suppose the flow starts at $X$ and grows at $\gamma$ per year.  That means the value of the asset rises at $\gamma$ per year too. Thus, 
 \begin{equation} \label{e1}
 V =    \frac{X }{1+r} +  (1+\gamma) \left(\frac{V }{1+r} \right)
\end{equation}
 Then $\frac{1+r}{1+\gamma} V = X +V$, and    $\frac{1+r - (1+\gamma)}{1+\gamma} V = X $ and    $(r- \gamma)V = X$ and  
 \begin{equation} \label{e1}
 V =   \frac{X}{r-\gamma}
\end{equation}

  If $ r< \gamma$, however, the equation does not apply. The value of the asset is infinite, so it is a false step to say that it rises at $\gamma$ per year. 

   It's interesting to apply this  growth idea to the Klein-Leffler reputation model. If the market is not growing, and marginal cost of high quality is $c$ and  $r = .10$, the price is $p = (1+r)c$  ten percent above marginal cost. If the market is growing at 8 percent per year, however, the price is only $p = (1+r-\gamma)c$, just 2 percent above marginal cost. 

   
  

 


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