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How to Maximize Profit with Derivatives

You can use calculus to maximize the total profit equation. Because total revenue and total cost are both expressed as a function of quantity, you determine the profit-maximizing quantity of output by taking the derivative of the total profit equation with respect to quantity, setting the derivative equal to zero, and solving for the quantity.

The market demand curve for the good your monopoly produces is

image0.jpg

where q is the market and firm’s quantity demanded, and P is the market price in dollars.

Using the demand equation to derive total revenue as a function of q requires the following steps:

  1. Add 200P to both sides of the demand equation.

    image1.jpg
  2. Subtract q from both sides of the equation.

    image2.jpg
  3. Divide both sides of the equation by 200.

    image3.jpg
  4. To determine total revenue, multiply both sides of the demand equation by q.

    image4.jpg

This equation tells you how much total revenue equals given any value for quantity, q. Thus, total revenue is a function of q.

If your total cost equation is

image5.jpg

total profit, ð, is determined by subtracting total cost from total revenue, or

image6.jpg

After you have the total profit equation, the following steps enable you to determine the profit-maximizing quantity and price:

  1. Take the derivative of the total profit equation with respect to quantity.

    image7.jpg
  2. Set the derivative equal to zero and solve for q.

    This is your profit-maximizing quantity of output.

    image8.jpg
  3. Substitute the profit-maximizing quantity of 2,000 into the demand equation and solve for P.

    image9.jpg

    Or you should set a price of $40 for the good.

  4. Finally, total profit is determined by substituting 2,000 for q in the total-profit equation.

    image10.jpg

    Your total profit equals $18,000.

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