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 = 818 - 4Q`.

Her production costs are `82 Q+ 51`.

What quantity of lemonade should Zoe produce to maximize profit?

Zoe maximizes profit when producing `Q=92`L of lemonade.

With a cost function `82 Q+ 51`, Zoe's marginal cost is

$$ MC \left( Q \right) = \frac{d C ( Q )}{d Q} = 82 $$

With an inverse demand `P = 818 - 4Q`, Zoe's revenue is

$$ R \left( Q \right) = (818 - 4Q) Q = 818 Q - 4Q^2 $$

Her marginal revenue is

$$ MR \left( Q \right) = \frac{d R ( Q )}{d Q} = 818 - 4 \times 2 Q = 818 - 8Q $$

The profit-maximizing quantity satisfies

$$ \begin{align*} MC \left( Q \right) &= MR \left( Q \right) \\ 82 &= 818 - 8Q \\ 8 Q &= 818 - 82 \\ 8 Q &= 736 \\ Q &= 92 \\ \end{align*} $$