After the previous two Decision Reports you now have a full understanding of the demand for your product --- that is, you've modeled the relationship between the price point you charge \(p\) and the total number of sales quantity in the national market at that price \(q\text{,}\) and that relationship is captured in the demand function \(p = D(q)\text{.}\)

The next step is to go beyond demand and construct function models for **sales revenue**, defined as the total dollar amount that would be collected from selling \(q\) units at a price of \(p\) dollars each:

\begin{equation*}
R(q) = q\cdot p = q\cdot D(q)
\end{equation*}
We'll also ultimately want a function model for the **total cost** of production \(C(q)\text{,}\) so that at any given production quantity we can compare revenue and cost to determine the net **profit**, which is defined to be their difference:

\begin{equation*}
P(q) = R(q) - C(q)
\end{equation*}
(In other words, profit is the amount of money your firm can "keep" after paying a total cost of \(C(q)\) dollars to produce \(q\) units and then collecting \(R(q)\) dollars of revenue from selling those units.)

The key tool in the next decision is "marginal analysis:" the study not only of the functions' values themselves, but of how those values *change* in response to changes in quantity. At a certain level of production \(q\text{,}\) would increasing production have a positive effect on sales revenue (if so, we would say that "marginal revenue is positive")? Or a negative effect ("marginal revenue is negative")?

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Proposition3.0.1Decision Report 3

At which of your target markets' price points is the marginal sales revenue for your product positive? Negative? In which market(s) are the prices and quantities closest to those that would lead to a *maximum* in sales revenue?