Section 3.4 More About Marginal Analysis
Marginal analysis was introduced in Section 2.3. Recall that the marginal cost, marginal revenue, and marginal profit functions are the (instantaneous) rates of change of the cost, revenue, and profit functions respectively. In other words, the marginal functions are the derivatives of their respective business functions, and approximate how these functions change as the input variable changes by one unit. If \(C(x)\text{,}\) \(R(x)\text{,}\) and \(P(x)\) are the cost, revenue, and profit functions respectively, then their marginal functions are:
- \(\displaystyle MC(x) = C'(x)\)
- \(\displaystyle MR(x) = R'(x)\)
- \(\displaystyle MP(x) = P'(x)\)
To tie this into the our previous work with these functions, note that the derivative of a linear function is its slope. Indeed, if \(C(x) = mx + b\) is a linear cost function, then
\begin{align*}
MC(x) = C'(x) \amp = \lim_{h \rightarrow 0} \dfrac{C(x+h) - C(x)}{h} = \lim_{h \rightarrow 0} \dfrac{m(x+h) +b - (mx+b)}{h} \\
\amp = \lim_{h \rightarrow 0} \dfrac{mh}{h} = \lim_{h \rightarrow 0} m = m
\end{align*}
In other words, if \(C(x) = mx + b\text{,}\) then \(MC(x) = C'(x) = m\text{.}\) See Example 2.3.5 as an example illustraing this idea. In general, the marginal functions \(MC(x)\text{,}\) \(MR(x)\text{,}\) and \(MP(x)\) approximate the change in cost, revenue, and profit as \(x\) increases by one unit.
- Find and interpret the average revenue generated from producing 100 pairs of speakers. Solution.\begin{equation*} \bar{R}(100) = \dfrac{R(100)}{100} = \dfrac{10(100) - .002(100^2)}{100} = 9.8 \end{equation*}When ECCO sells 100 pairs of speakers, they make an average of $9,800 in revenue per pair of speakers sold.
- Find and interpret the average rate of change of revenue when production increases from 100 to 110 pairs of speakers. Solution.\begin{equation*} \textrm{AROC} = \dfrac{R(110)-R(100)}{110-100} = \dfrac{1075.8 - 980}{10} = 9.58 \end{equation*}As production increases from 100 to 110 pairs of speakers, ECC0's revenue increases an average of $9,580 per pair.
- Find and interpret the marginal revenue of producing 100 pairs of speakers. Solution.We are asked to find and interpret \(MR(100) = R'(100)\text{.}\) We first need to find the marginal revenue function.\begin{align*} MR(s) \amp = \lim_{h \rightarrow 0} \dfrac{R(s+h) - R(s)}{h} \\ \\ \amp = \lim_{h \rightarrow 0}\dfrac{10(s+h) - .002(s+h)^2 - (10s -.002s^2)}{h} \\ \\ \amp =\lim_{h \rightarrow 0} \dfrac{10s + 10h -.002s^2 - .004sh - .002h^2 -10s +.002s^2}{h} \\ \\ \amp = \lim_{h \rightarrow 0} \dfrac{h(10 -.004s - .002h)}{h} = \lim_{h \rightarrow 0}(10-.004s -.002h) \\ \\ \amp = 10 - .004s -.002(0) = 10 - .004s. \end{align*}So, \(MR(s) = 10-.004s\) which means that \(MR(100) = 9.6\text{.}\) This means: When When ECCO produces \(s=100\) pairs of speakers, their revenue is increasing at an approximate rate of $9,600 per additional pair produced. In other words, the revenue generated from producing the \(101^{\textrm{st}}\) pair of speakers is approximately $9,600.
