Question

A monopolist demand function is given by; P=Q2-10Q+28 and cost function; C=Q2. If the firm is causing a pollution to the society valued at MC=3, solve for the social optimal level of output and price. What should be the tax level?

Answer 1

The social optimal quantity is estimated by equating the price to be equal to the marginal cost.

$\text{P=MC}$

Therefore:

$3=Q^2-10Q+28$

$Q^2-10Q+28-3$

$Q^2-10Q+25$

$\dfrac {-b \pm \sqrt{b^2-4ac}}{2a}$

$\dfrac {10 \pm \sqrt{ (-10)^2-4\times 1 \times 25}}{2\times 1} =5$

The social optimal level is therefore, 5 units.

The optimal price is:

$p=5^2-10\times 5+28$

$p=3$

Or

$\text{P=MC}$

$\text{P=3}$

At social optimum level, there will be no taxation instead the government should instead give subsidies to the monopolist.