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\centerline{MATH 134 -- Calculus with Fundamentals 2}
\centerline{Discussion Day 1 -- Consumer and Producer Surplus}
\centerline{February 6, 2018}
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\noindent
{\it Background}
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In a free market economy, the \emph{demand curve} is the graph of the function $D$
that represents the relationship between the demand for a quantity and its 
price.  It is traditional in economics to think of $D$ as a function of $q$ and to draw 
the graph with the \emph{quantity} $q$ on
the \emph{horizontal} axis and the \emph{price} on the vertical axis
as in Figure 1 on the back of this page.  So then the graph is $p = D(q)$  
Similarly the \emph{supply curve} is the graph $p = S(q)$ that represents
the relationship between the amount of a quantity producers are willing
to supply to the market as a function of the price.  

The point $(p^*,q^*)$ where the demand and supply curves cross is 
called the \emph{equilibrium point}, where supply and demand balance.  In theory, 
that price $p^*$ and that quantity $q^*$ will be ``found'' by 
forces of the free market (possibly after some initial ``settling down.'')\footnote{
Producers who set their price too high won't sell their product; producers who
set their price too low will realize they could have set their price higher when
they see competitors making more money! Consumers won't buy products if the 
prices are too high and they will buy at lower prices when they can.}

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\noindent
{\it Questions}

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\begin{enumerate}
\item[(1)] The area with $p^* \le p \le D(q)$ (see Figure 1) represents the 
\emph{savings} that consumers realize if they buy the quantity at
the market price $p^*$ rather than the higher price they would have 
been willing to pay at $q < q^*$.  This amount is called the \emph{consumer
surplus}.  Thinking of the area in the picture, write this as a formula using 
definite integrals and other area formulas.
\item[(2)] The area between the supply curve and the horizontal line $p = p^*$ is called
the \emph{producer surplus} -- it represents the extra money producers
make by selling their product at price $p^*$ rather than the lower prices
some would be willing to accept.  Thinking of the area in the picture, write this as a formula using 
definite integrals and other area formulas.
\item[(3)] A variety of local stores sell inexpensive thermal underwear (``long johns'') 
for those cold winter days.
Suppose the supply curve for one package of the underwear 
is $p = S(q) = 2q + 3$ ($p$ in dollars per package and $q$ in thousands of pieces)
and the demand curve
is $\displaystyle p = D(q) = \frac{21}{q + 1}$.  Determine the point $(p^*,q^*)$.  Then 
determine the consumer surplus and the producer surplus when the underwear is
sold at the price $p^*$.  
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\noindent
{\it Assignment: }  Group writeups due by the end of class.
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\begin{figure}
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\includegraphics[width=.8\linewidth]{SD.pdf}
\caption{Supply and Demand Curves, Equilibrium Point, Consumer and Producer Surplus}
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\end{figure}
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