How does monopolists maximize profits?

Answer 1

A monopolist maximizes profits by choosing an output such that marginal revenue equals marginal cost. This is in contrast to a perfect competition where firms maximizes profits at price equal to marginal cost. Mathematically, the monopolist profit can be calculated as such:

π = pq-cq, where π is the profits, p is the price, c is the marginal cost and q is the quantity

The price reflects the inverse demand function; p=a-bq, where a is a constant and b is the slope (e.g. p=100-2q)

If we insert this expression into the profit function it can be written as follows:

π = (a-bq)q-cq

Taking first order conditions (The derivative of π with respect to q) and put the condition equal to zero (Finding the stationary point (maximum) for the function):

dπ/dq= 0 -->

-bq+a-bq-c=0

Rewriting the expression:

a-2bq=c, where a-2bq is the marginal revenue, and c is the marginal cost

Solving this expression with respect to q:

q=(a-c)/2b

This is the optimal output of the monopolist, to find the price we insert this expression inte to expression for p we had earlier:

p=a-b(a-c)/2b

Rewritten:

p=(a+c)/2

This is the optimal price of the monopolist. To find the profit we now insert the expression for optimal output and optimal price into the profit expression:

π = (a-bq)q-cq --> π = ((a+c)/2-c)(a-c)/2b

Simplifying and rewriting the expression we get:

π = (a-c)^2/4b

This might look a bit challenging if you are not used to working with algebra on general form, however, if you insert your specific numbers in the beginning it is quite straight forward.