Fully-informed producer and consumers

As a reference point, we analyze the case where the consumers are fully informed about their individual values of , the producer knows the distribution , and the producer and the consumers act so as to maximize their expected gain.

Integrating g, we obtain the cumulative distribution . From Eq. 6 we can compute the conditional distribution

 

The average valuation for this conditional distribution is . Using Eqs. 5 and 9, we obtain the expected number of purchased articles and the expected surplus (assuming the consumer subscribes):

 

To compute the producer's expected profit as a function of and F, we can substitute Eqs. 12 into Eq.10, which yields:

 

where is defined as and is defined as the unique solution to .

Eq. 13 can be visualized as a profit landscape in which the expected profit is plotted as a function of F and . It is convenient to define a normalized expected profit and a normalized fee f = F/N. Fig. 1 illustrates the landscape for two different production costs: and .

  
Figure 1: Profit landscape . h is a uniform distribution with , . a) Production cost . b) Production cost .

In each such landscape, there are two ridges. The lower ridge, which demarcates the boundary beyond which the producer attracts no consumers and thus makes no profit, is defined by . The upper ridge is described by a piecewise joining of two nonlinear curves, the simpler one being described by the relation . This portion of the ridge has the following characteristics:

• it is formed by a discontinuous derivative with respect to f and (it changes abruptly from positive to negative)
• for price settings along this ridge, all consumers subscribe
• if the production cost , the optimal setting of occurs along this part of the ridge. For the examples of Fig. 1, the critical production cost is found  to be approximately 0.28, so that Fig. 1a ( ) is in this regime, while Fig. 1b ( ) is not.

The other portion of the ridge is defined by a more complex nonlinear relation between f and . It is less sharp, resulting from derivatives with respect to f and being zero rather than jumping discontinuously from positive to negative .

Figure 2 shows the dependence of the optimal f and upon the production cost . There is a discontinuous derivative at , due to the switchover between the two nonlinear curves that define the upper ridge in the landscape. As one might guess, the profit decreases monotonically with the production cost. The proportion m of consumers that subscribe is 1 for all ; for exceeding this threshold the proportion of subscribers is strictly less than one, and is given by .

  
Figure 2: Optimal price-per-item , normalized subscription fee f, normalized profit , and fraction of subscribers m vs. item production cost , where h is a uniform distribution with , .