(Note that if the buyers can trade between themselves then this strategy may come unstuck: buyers with low reservation values may be able to buy many units of the good, and re-sell some of them to buyers with high reservation values. In some cases, a monopolist can prevent such trades. For example, in the case of admission to a movie, admission may be granted to an adult only if she is in possession of an adult ticket.)
How much should the monopolist produce in this case? So long as there is a buyer whose reservation price exceeds the monopolist's marginal cost, it is in the interest of the monopolist to sell to that buyer. Thus the monopolist will produce the output y for which MC(y) is equal to P(y). That is,
the optimal output of a perfectly discriminating monopolist is Pareto efficient!In this outcome the monopolist gets all the surplus, so unless the monopolist is needy the outcome is not likely to be equitable---but it is Pareto efficient.
Suppose that the monopolist can separate the market into two parts, one in which the inverse demand function is P1 and one in which it is P2. Then its total revenue functions in the two markets are TR1(y) = yP1(y) and TR2(y) = yP2(y). The monopolist's profit-maximization problem in this case is to choose outputs y1 and y2 for the markets to solve
maxy1,y2 (y1,y2) = maxy1,y2 [TR1(y1) + TR2(y2) TC(y1 + y2)].
At a solution to this problem the value of y1 must maximize profit, given the value of y2, and the value of y2 must maximize profit, given the value of y1. Thus the following conditions must be satisfied:
MR1(y1*) MC(y1* + y2*) = 0We can write these conditions alternatively asMR2(y2*) MC(y1* + y2*) = 0.
MR1(y1*) = MR2(y2*) = MC(y1* + y2*).(As in the case of a monopolist setting a single price, values of y1* and y2* that satisfy these equations may correspond to minimum profit rather than maximum profit, and if the monopolist's profit is negative when it chooses y1* and y2* then it should instead produce nothing.)
p1[1 1/|1(y1*)|] = p2[1 1/|2(y2*)|].Suppose that the elasticities are constant, independent of demand. Then the condition is
p1[1 1/|1|] = p2[1 1/|2|],or
p1 1 1/|2| = p2 1 1/|1|
Thus if |1| > |2| then p1 < p2. [Try 1 = 4 and 2 = 2: then p1/p2 = (11/2)/(1 1/4) = (1/2)(4/3) = 2/3.]
That is:
a monopolist charges a lower price in a market in which demand is more elastic.Examples and exercises on a price-discriminating monopolist