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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")?