
{\Large Using functions for models} \pz

Functions are basic tools in using mathematics to model situations
in the physical sciences, the social sciences, engineering, finance, and 
other places where quantitative analysis is used. \pz
We will be developing functional models throughout the course,
using them to motivate and illustrate almost all of the topics we cover. \pz
As with graphing functions, we will not dwell on this topic now.  You should
read section 1.4, and we can refer back to this section if we encounter
problems with understanding functional models later in the course. \pb
%The one topic we will cover now is market equillibruim. \pb

{\large Economic modeling \pz}

\begin{definition}
\begin{itemize}
\item A cost function $C(x)$ measures the cost of producing $x$ units of a commodity. \pz
(Question: why is this not always linear?) \pz
\item A demand function $p = D(x)$ is a function which gives the price $p$
for a commodity so that a quantity $x$ of that commodity will ``exactly sell out''. \pz
In other words, it is the price consumers are willing to pay, per item, 
for a commodity when exactly $x$ units are available. \pz
(Thus, as $x$ gets large, $D(x)$ must go to zero.) \pb
\item A total revenue function $R(x)$ calculates the money earned by
producing and selling $x$ units of a commodity. \pz 
\begin{align*}
R(x) &= {\rm (number \; of\; items \;sold)} \; \times \; {\rm (price \;per\; item) } \\
	&= x p = x D(x).
\end{align*} \pz
\item A profit function is the difference between revenue and cost functions. \pz
$$ P(x) = R(x) - C(x) = xD(x) - C(x).$$ \pz
\item A supply function is the price $p = S(x)$ at which producers are willing
to supply $x$ units. \pz
\item The market equilibrium is the number of units $x_e$ for which $S(x_e) = D(x_e)$. \pz
\end{itemize}
\end{definition}

\pb

The first step in calculating how much of a good to produce is to 
find the market equilibrium, \pz since the law of supply and demand dictates that
suppliers and consumers will
alter their behavior until the market is in equilibrium. \pz (After one finds the 
market equilibrium, one must then try to calculate one's own market share
in order to know how much to produce.) \pz

\begin{example}
Graph the supply and demand functions and find 
the equilibrium production level
for $S(x) = 3x + 150$ and $D(x) = -2x + 275$; \pz   
also $S(x) = x^2 + 240$ and $D(x) = 40 - 10 x$. \pb
\end{example}
 