Price discriminating monopoly, solving for profit maximization

This post goes over the math required to solve for the profit maximizing price and quantity of a price discriminating monopoly operating in two markets. Consider the following problem:


A cable company sells subscriptions in San Francisco and Boston. The demand function for each of the two groups, which are separate and do not have the ability to re-sell to members of the other community, are Psf = 480 - 4Qsf and Pb = 400 -2 Qb. The cost of providing the cable service for the firm is TC = 500 + 4Q, where Q = Qsf + Qb. If the company can price discriminate between the two markets, what are the profit maximizing prices and quantities for the San Francisco and Boston markets?

To solve this problem, we need to review the steps for finding the profit maximizing price and quantity for a monopoly. We find that we need to find the price and quantity where marginal revenue (MR) is equal to marginal cost (MC).

We know that the slope of the MR curve is going to be twice the slope of the demand curve, so that means that MR for the SF market will be equal to MRsf = 480 - 8Qsf and that the MR for the Boston market will be MRb = 400 - 4 Qb. This simply means that MR is twice as sensitive to changes in price due to the fact that lower prices increase the quantity sold (raising revenue) but the lower price also plays a role in decreasing revenue.

Now that we have the equations for marginal revenue, we need to find the equation for marginal cost (MC). We can do this by taking the derivative of the total cost equation, or by noting that costs go up by 4 with each service that is provided. We now set MR = MC by setting our two previous MR equations equal to 4. This leaves us with:

4 = 480 - 8Qsf

and

4 = 400 - 4Qb

By subtracting for from both sides, and adding either 8Qsf or 4Qb (depending on the equation) and solving for the resulting quantity, we are left with:

Qsf = 59.5

and

Qb = 99

The intuition behind this result is that monopolies will continue to lower their price to sell more product as long as the change in revenue (MR) is greater than the change in cost (MC). Since it costs 4 for each new service, they will not lower their price to get more customers if the price reduction raises revenues by less than it costs to provide the service.

We can now plug these quantities back into the original equations to find the resulting price for each market:

Psf = 480 -4(59.5) = 242
Pb = 400 -2(99) = 202

We can see that the price discriminating monopoly will charge a higher price to the San Francisco market. This is due to two reasons, the first is that people in San Francisco have a higher willingness to pay (480 vs 400) and that their responsiveness to price changes is larger. As the monopoly provides more service to San Francisco, they will demand lower and lower prices making the profit maximizing point higher.