Example
Zoe, in her lemonade stand, faces the inverse demand `P = 10 - 2Q`. So her revenue is
$$ R ( Q ) = PQ = (10 - 2Q)Q = 10 Q - 2Q^2 $$Her marginal revenue is
$$ MR ( Q ) = \frac{d R ( Q )}{d Q} = 10 - 4 Q $$Zoe faces production costs equal to `C ( Q ) = 2Q`. So her marginal costs are
$$ MC ( Q ) = \frac{d C ( Q )}{d Q} = 2 $$The quantity Q that maximizes Zoe's profit solves
$$ \begin{align*} MC ( Q ) &= MR ( Q ) \\ 2 &= 10 - 4 Q \\ 4 Q &= 10 - 2 \\ 4 Q &= 8 \\ Q &= 2 \end{align*} $$Question
Zoe's satisfies an inverse demand `P = 139 - 1Q`.
Her production costs are `31 Q+ 18`.
What quantity of lemonade should Zoe produce to maximize profit?
Zoe maximizes profit when producing `Q=54`L of lemonade.
With a cost function `31 Q+ 18`, Zoe's marginal cost is
$$ MC \left( Q \right) = \frac{d C ( Q )}{d Q} = 31 $$With an inverse demand `P = 139 - 1Q`, Zoe's revenue is
$$ R \left( Q \right) = (139 - 1Q) Q = 139 Q - 1Q^2 $$Her marginal revenue is
$$ MR \left( Q \right) = \frac{d R ( Q )}{d Q} = 139 - 1 \times 2 Q = 139 - 2Q $$The profit-maximizing quantity satisfies
$$ \begin{align*} MC \left( Q \right) &= MR \left( Q \right) \\ 31 &= 139 - 2Q \\ 2 Q &= 139 - 31 \\ 2 Q &= 108 \\ Q &= 54 \\ \end{align*} $$