Economics Stack Exchange Archive

Why does the marginal revenue curve slope downward at twice the rate of demand in a monopoly?

The monopolist’s view of the market usually looks as follows:

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Why does the marginal revenue/benefit line always hit the Y-axis (Q) at the halfway point of the demand curve?

Answer 24

That particular result (MR = double slope of D) only holds when demand is linear. Here’s a mathematical argument for it:

Total revenue is the product of Price and Quantity:

TR = P*Q

Price is a function of Quantity, so the above can be more accurately written as:

TR = P(Q)*Q

In the case of linear demand, we can write P(Q) as:

P(Q) = a - b*Q

Plugging this into our expression for total revenue:

TR = (a - b*Q)*Q

Which can be simplified to:

TR = aQ - b*Q2

Taking the first derivative in Q of total revenue yields:

dTR/dQ = a - 2*b*Q

Using the definition of marginal revenue (which is the first derivative of total revenue in quantity), we can say:

MR = a - 2*b*Q

Which is what we wanted to show.

More intuitively, the reason is this: for a monopolist (or any firm facing a downward-sloping demand curve), a change in price leads to a change in total revenue in two components: first, lowering the price increases the quantity demanded. Since only one price may exist in a market at a given time (unless we allow for price discrimination) lowering the price means we must reevaluate the revenue from all of the previous units as well.

The monopolist’s marginal decision is therefore: is the increase in quantity demanded brought about by the decrease in price worth the reduction in revenue on all of the previous units? The monopolist requires a marginal cost curve to answer that question accurately.

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