Suppose a monopolist is facing a single buyer 1 whose willingness to pay is given by P₁(Q) = 10 Q. For simplicity, we assume that the cost of production is 0. (So in this question, the monopolist's revenue is equal to its profit.)

(a) If this monopolist charges a single price, what is the optimal price P and maximum profit ?

(b) Now consider the following pricing strategy of the monopolist:

Charge a lump-sum fee F for a loyalty card, which allows the buyer to shop in his store. Then charge a per-unit price P for the product.

Find the optimal lump-sum fee Fand per-unit price P, and the maximum profit with this pricing strategy. (Hint: The monopolist wants to extract surplus from the buyer as much as possible. A graph will help you Note that F corresponds to an area in the graph. The answer to this question is surprisingly simple.)

(c)Suppose there is another buyer 2 whose willingness to pay is given by P₂(Q) = 10 2Q. The monopolist is not able to distinguish between these two buyers, and still uses a pricing strategy described in part (b). Find the optimal lump-sum fee Fand per-unit price P, and the maximum profit with this pricing strategy. (Hint: The lump-sum fee cannot be too high in this case, because otherwise buyer 2 will be deterred from buying.)