Suppose a monopolist's demand curve (D) is downward-sloping and linear

Suppose a monopolist's demand curve (D) is downward-sloping and linear. In this case, the marginal revenue (MR) curve will be a. twice as steep and below D, but with the same vertical intercept

Given the demand curve (D) is downward-sloping and linear, it has the following general form:

• P=−aQ+bP=−aQ+b

We derive the total revenue (TR) equation:

• TR=P×QTR=(−aQ+b)×QTR=−aQ2+bQTR=P×QTR=(−aQ+b)×QTR=−aQ2+bQ

The marginal revenue is the first derivative of the total revenue with respect to Q:

• MR=dTRdQMR=−2aQ+bMR=dTRdQMR=−2aQ+b

Comparing the equations of the demand curve and marginal revenue curve, we can see that:

• At Q=0Q=0, the vertical axis has the value of bb. This means that the marginal revenue curve is the same vertical intercept as D.

• The slope of the demand curve is −a−a while the slope of the marginal revenue curve is −2a−2a. This means that the marginal revenue is twice as steep and below D