How to Find Profit-maximizing Input

Theory

TheProfitMaximizingProductionLevel

The profit maximizing production level (PMPL) is the number of produced units $x$ where the profit $P\left(x\right)$ is maximized. This happens when

 ${I}^{\prime }\left(x\right)={C}^{\prime }\left(x\right)$

Note! PMPL is the production that yields the highest profit.

Theory

InterpretationofPMPL

• When ${C}^{\prime }\left(x\right)>{I}^{\prime }\left(x\right)$, the cost of producing one more unit, ${K}^{\prime }\left(x\right)$, is higher than the income from producing one more unit, ${I}^{\prime }\left(x\right)$. This is bad for business!

• When ${C}^{\prime }\left(x\right)<{I}^{\prime }\left(x\right)$, the cost of producing one more unit, ${C}^{\prime }\left(x\right)$, is lower than the income from producing one more unit, ${I}^{\prime }\left(x\right)$. This is good for business!

• At the point where ${C}^{\prime }\left(x\right)={I}^{\prime }\left(x\right)$, the cost of producing one more unit, ${C}^{\prime }\left(x\right)$, is equal to the income from producing one more unit ${I}^{\prime }\left(x\right)$. This is where you reach the profit maximizing production level.

Example 1

You are given the cost function

 $C\left(x\right)=15{x}^{2}-1180x+\text{}33\phantom{\rule{0.17em}{0ex}}200\text{}$

and the income function

 $I\left(x\right)=5{x}^{2}-140x+\text{}25\phantom{\rule{0.17em}{0ex}}000\text{}.$

Find the profit maximizing production level (PMPL).

Then, find the cost, the income and the profit at that production level.

You know that you find the PMPL when ${C}^{\prime }\left(x\right)={I}^{\prime }\left(x\right)$. That means you have to find ${C}^{\prime }\left(x\right)$ and ${I}^{\prime }\left(x\right)$ first:

$\begin{array}{llll}\hfill {C}^{\prime }\left(x\right)& =30x-1180\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {I}^{\prime }\left(x\right)& =10x-140.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Now you can set these two equal to each other, just like in the formula, and solve the equation you get for $x$:

$\begin{array}{llllll}\hfill 30x-1180& =10x-140\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 20x& =1040\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{1em}{0ex}}|:20\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =52\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

This means that the PMPL is at 52 units.

You find the cost and income at this production level by inserting the PMPL into the functions for cost and income you were given to begin with:

$\begin{array}{llll}\hfill C\left(52\right)& =15{\left(52\right)}^{2}-1180\left(52\right)+33\phantom{\rule{0.17em}{0ex}}200\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =12400,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill I\left(52\right)& =5{\left(52\right)}^{2}-140\left(52\right)+25\phantom{\rule{0.17em}{0ex}}000\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =31\phantom{\rule{0.17em}{0ex}}240.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llll}\hfill C\left(52\right)& =15{\left(52\right)}^{2}-1180\left(52\right)+33\phantom{\rule{0.17em}{0ex}}200=12400,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill I\left(52\right)& =5{\left(52\right)}^{2}-140\left(52\right)+25\phantom{\rule{0.17em}{0ex}}000=31\phantom{\rule{0.17em}{0ex}}240.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

The cost at the PMPL is $12\phantom{\rule{0.17em}{0ex}}400$, and the income at the PMPL is $31\phantom{\rule{0.17em}{0ex}}240$.

To find the profit, you subtract the cost from the income:

 $\text{Profit}=31\phantom{\rule{0.17em}{0ex}}240-12\phantom{\rule{0.17em}{0ex}}400=18\phantom{\rule{0.17em}{0ex}}840.$

That gives us a profit of $18\phantom{\rule{0.17em}{0ex}}840$ at the PMPL. Since this is the profit maximizing production level, no other production level achieves a higher profit than this one.