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How to Calculate Maximum Profit in a Monopoly

Profit is maximized at the quantity of output where marginal revenue equals marginal cost. Marginal revenue represents the change in total revenue associated with an additional unit of output, and marginal cost is the change in total cost for an additional unit of output.

Therefore, both marginal revenue and marginal cost represent derivatives of the total revenue and total cost functions, respectively. You can use calculus to determine marginal revenue and marginal cost; setting them equal to one another maximizes total profit.

The monopolist’s demand curve

image0.jpg

generated the total revenue equation.

image1.jpg

Also assume your total cost equation is

image2.jpg

Given these equations, the profit-maximizing quantity of output is determined through the following steps:

  1. Determine marginal revenue by taking the derivative of total revenue with respect to quantity.

    image3.jpg
  2. Determine marginal cost by taking the derivative of total cost with respect to quantity.

    image4.jpg
  3. Set marginal revenue equal to marginal cost and solve for q.

    image5.jpg
  4. Substituting 2,000 for q in the demand equation enables you to determine price.

    image6.jpg

    Thus, the profit-maximizing quantity is 2,000 units and the price is $40 per unit.

The profit-maximizing quantity and price are the same whether you maximize the difference between total revenue and total cost or set marginal revenue equal to marginal cost.

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