Profit Maximization

The monopolist maximizes profit when its marginal cost equals its marginal revenue.

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 = 1352 - 8Q`.

Her production costs are `72 Q+ 67`.

What quantity of lemonade should Zoe produce to maximize profit?

Zoe maximizes profit when producing `Q=80`L of lemonade.

With a cost function `72 Q+ 67`, Zoe's marginal cost is

$$ MC \left( Q \right) = \frac{d C ( Q )}{d Q} = 72 $$

With an inverse demand `P = 1352 - 8Q`, Zoe's revenue is

$$ R \left( Q \right) = (1352 - 8Q) Q = 1352 Q - 8Q^2 $$

Her marginal revenue is

$$ MR \left( Q \right) = \frac{d R ( Q )}{d Q} = 1352 - 8 \times 2 Q = 1352 - 16Q $$

The profit-maximizing quantity satisfies

$$ \begin{align*} MC \left( Q \right) &= MR \left( Q \right) \\ 72 &= 1352 - 16Q \\ 16 Q &= 1352 - 72 \\ 16 Q &= 1280 \\ Q &= 80 \\ \end{align*} $$