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# Chapter3Total & Marginal Revenue

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")?