%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Applications to Economics">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sat Mar 29 21:21:21 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=Reference.cst}
%TCIDATA{PageSetup=72,72,72,72,0}
%TCIDATA{<LINK REL="contents" HREF="..\Reference\ReferenceIndex.tex#Reference Index">}
%TCIDATA{<LINK REL="next" HREF="cal04_09.tex">}
%TCIDATA{<LINK REL="parent" HREF="cal_contents.tex">}
%TCIDATA{<LINK REL="previous" HREF="cal04_07.tex">}
%TCIDATA{AllPages=
%F=36,\PARA{038<p type="texpara" tag="Body Text" >\hfill \thepage}
%}


\input{tcilatex}
\begin{document}


\section{Applications to Economics\label{Applications to economics}\label%
{Marginal cost}\label{Marginal revenue}}

Several concepts in economics that have to do with rates of change can be
effectively described with the methods of calculus. Among these are marginal
cost and marginal revenue.

\subsubsection{Marginal Cost}

If the \textbf{cost function} $C(x)$ is the cost of producing $x$ units of a
certain product, then the \textbf{marginal cost} is the rate of change of $C$
with respect to $x$. In other words, the marginal cost function is the
derivative, $C^{\prime }(x)$, of the cost function.

The \textbf{average cost function} 
\[
c(x)=\frac{C(x)}{x} 
\]
represents the cost per unit when $x$ units are produced. Now $c^{\prime
}(x)=0$ when $xC^{\prime }(x)-C(x)=0$ and this gives 
\[
C^{\prime }(x)=\frac{C(x)}{x}=c(x) 
\]
Therefore, if the average cost is a minimum, then the marginal cost equals
the average cost.

\subsubsection{Marginal Revenue}

Let $p(x)$ be the price per unit that the company can charge if it sells $x$
units. Then $p$ is called the demand function (or \textbf{price function})
and we would expect it to be a decreasing function of $x$. If $x$ units are
sold and the price per unit is $p(x)$, then the total revenue is 
\[
R(x)=xp(x)
\]
and $R$ is called the revenue function (or \textbf{sales function}). The
derivative $R^{\prime }$ of the revenue function is called the \textbf{%
marginal revenue function} and is the rate of change of revenue with respect
to the number of units sold.

If $x$ units are sold, then the total profit is 
\[
P(x)=R(x)-C(x) 
\]
and $P$ is called the profit function\textbf{.} The marginal profit function
is $P^{\prime }$, the derivative of the profit function. In order to
maximize profit we look for the critical numbers of $P$, that is, the
numbers where the marginal profit is 0. But if $P^{\prime \prime
}(x)=R^{\prime }(x)-C^{\prime }(x)=0$ then $R^{\prime }(x)=C^{\prime }(x).$
Therefore, if the profit is a maximum, then the marginal revenue equals the
marginal cost.

To ensure that this condition gives a maximum you could use the Second
Derivative Test. Note that $P^{\prime \prime }(x)=R^{\prime \prime
}(x)-C^{\prime \prime }(x)<0$ when $R^{\prime \prime }(x)<C^{\prime \prime
}(x)$, and this condition says that the rate of increase of marginal revenue
is less than the rate of increase of marginal cost. Thus the profit will be
a maximum when $R^{\prime }(x)=C^{\prime }(x)$ and $R^{\prime \prime
}(x)<C^{\prime \prime }(x)$.

\noindent \hrulefill

\begin{center}
\begin{tabular}{ccc}
{\small \hyperref{\FRAME{itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Computing Techniques}{}{}{%
../../Help/DMindex.tex}} & {\small \hyperref{\FRAME{itbpF}{13.9375pt}{%
14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "PICT";valid_file "F";width
13.9375pt;height 14.4375pt;depth 3pt;original-width
24.0625pt;original-height 24.8125pt;cropleft "0";croptop "0.9959";cropright
"0.9937";cropbottom "0";filename 'Nblogo.wmf';file-properties "XNPEU";}}%
General Information}{}{}{../../Help/Index.tex}} & {\small \hyperref{\FRAME{%
itbpF}{13.9375pt}{14.4375pt}{3pt}{}{}{nblogo.wmf}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"PICT";valid_file "F";width 13.9375pt;height 14.4375pt;depth
3pt;original-width 24.0625pt;original-height 24.8125pt;cropleft "0";croptop
"0.9959";cropright "0.9937";cropbottom "0";filename
'Nblogo.wmf';file-properties "XNPEU";}}Reference Library}{}{}{%
../ReferenceIndex.tex}}
\end{tabular}
\end{center}

\noindent \hrulefill

\end{document}
