Monopoly price decisions, a question and answer

Here is a quick economics question and answer regarding monopolies:

A monopolist has two types of customers. There are 100 Type A, who will each pay up to $10 for a single unit of the good, and 50 of Type B, who will each pay up to $8. Neither is willing to purchase additional units at any price. If it must charge a uniform price, find that price.

A) Assume that spending $80 on advertising will attract 100 more Type B customers. Should the monopolist advertise? If so, what happens to price?

The trick here is to remember that firms want to profit maximize (look here for a discussion on monopoly and profit maximization).  Since we have no idea what the firm's costs are, we can assume that they want to revenue maximize.

To answer the first part of this question, we will consider two different prices.  If the monopolist charges $10, then they maximize their revenue with Type A customers, but lose out on business with type B customers.  If they charge $8, then they sell to both Type A and Type B customers, but they miss out on charging Type A customers $10.  So what is their revenue for each decision?

If they charge $10, they can sell 100 units (to Type A people) and therefore make $1,000.

If they charge $8, they can sell 150 units (to Type A and Type B people) and therefore make $1,200.

So it is in their best interest to lower the price, and sell more product.

Another way to approach this problem is to look at the decision on the margin:  by lowering the price $2, they give up $200 in revenue from Type A people when their price is lowered.  However, they gain $400 revenue by selling 50 more units to the Type B people.  The net gain in this case is $200, identical to the difference shown above.

If the monopolist can spend $80 on advertising to attract 100 more Type B customers, they will absolutely do it.  Remember that firms try to maximize profit, so if spending $80 results in revenue of $800 (100*$8), they gain a net profit of $720.

Because they already has their price set to $8, the price will not change if they decide to advertise because they still want to include Type B people in their market.
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