A monopolist faces a demand curve D(p)=100-2p and has a cost function c(y)=2y. What is the...

In the below solution, p and y are the price and output, respectively; whereas, TR, MR, and MC represent the total revenue, marginal revenue, and marginal cost, respectively.

Calculation of optimal level of output and price:

{eq}\begin{align*} D\left( p \right) &= 100 - 2p\\ y &= 100 - 2p\\ p &= 50 - \frac{1}{2}y\\ TR &= 50y - \frac{1}{2}{y^2} \end{align*} {/eq}

{eq}\begin{align*} TR' &= MR\\ MR &= 50 - \frac{1}{2} \times 2 \times y\\ MR &= 50 - y \end{align*} {/eq}

{eq}\begin{align*} C\left( y \right) &= 2y\\ MC &= 2 \end{align*} {/eq}

Condition for profit maximization:

{eq}\begin{align*} MR &= MC\\ 50 - y &= 2\\ y &= 48 \end{align*} {/eq}

{eq}\begin{align*} p &= 50 - \frac{1}{2}y\\ p &= 50 - \frac{1}{2} \times 48\\ p &= 50 - 24\\ &= 26 \end{align*} {/eq}

Therefore, the optimal price and output are 26 and 48 units, respectively.