Appendix A: Monopolistic Profit Curve

In Eq. 6, represents the profit per consumer vs. price for a monopolist broker. In the case where the consumer distribution is uniform, is given by Eq. 7. Differentiating this expression with respect to P, one can show that has a single peak which can be expressed algebraically as the only real solution to a cubic equation:

When and are both much smaller than V, the following approximation is reasonably good:

According to Eq. 7, consists of two pieces that join seamlessly at . Further analysis shows that for , brokers exercise the determining veto on marginal subscriptions by rejecting some consumers that would like to subscribe. For , the situation is reversed: consumers are more picky than brokers.

Figure 15 shows vs. P for as a heavy solid line. The two lighter solid lines show the continuation of the two pieces of Eq. 7 across the joint at . The peak occurs at . The approximation in Eq. 12 yields 0.594777, which is not far off even though and are sizeable fractions of V.

  
Figure 15: Monopolistic broker's expected utility per article per customer vs. price P (heavy solid line) for .

Many quantities can be computed from . For example, we can determine the conditions under which a consumer will subscribe to a monopolist broker that charges its optimal price The expected utility for a consumer with interest probability can be obtained by substitution of into Eq. 1: . In the example , and V=1, , and thus consumers with receive no articles; the utility of those with above this threshold increases linearly with .