Profit Maximization

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