In this section, we first analyze the expected surplus
and number of articles purchased per subscription period
for a rational, fully-informed
consumer with a given valuation distribution g.
Then, we derive an expression for a monopolistic producer's
expected profit as a function of its price schedule and
the distribution h of valuation parameters 
across the consumer population, assuming that all consumers
are rational and fully-informed about their valuation
distribution. A rational, fully-informed producer would
choose its price schedule so as to maximize its expected
profit.
Suppose that a given consumer has its valuations w
drawn from a probability density function 
.
(The distribution parameters may vary from one consumer to another.)
For the sake of simplicity, we assume
that the producer constrains itself to a linear price schedule:
. Then a rational consumer will purchase
k articles, where k is
the number of articles with valuations exceeding
the threshold 
. The probability
for exactly k articles to have valuations 
is
where 
represents the
cumulative distribution function that corresponds
to , i.e. it is the probability for an article
to be valued at less than w. From this we can compute the
expected number of articles purchased:
where the last equality follows from simple manipulations of binomial coefficients.
The expected surplus 
can be obtained from
Eq. 2,
provided that we first compute 
, the
expected surplus given that exactly k articles prove
to have valuations .
The conditional probability distribution for a single draw
from given that is
where 
represents the step function, equal to
1 if x>0 and 0 otherwise. The expected valuation for this
distribution is
The expected sum of k draws from this conditional distribution
is 
. Subtracting the price
, we obtain:
Inserting this result into Eq. 2, we obtain:
Note that the consumer will subscribe if and only if 
is greater than the subscription fee F.
Now we take the producer's perspective.
If we assume (for simplicity) that the cost of producing and
delivering k items is 
,
then the expected profit is
where 
is the probability distribution for
the consumers' parameters, as defined previously.
In effect, the last approximation replaces the actual realized set
of consumer distribution parameters 
with an ensemble average
over all possible realizations of a set of M consumers generated
from the distribution h, and this approximation grows increasingly
accurate in the limit of large M.
Note that a producer that knows h can compute this profit landscape.
A fully knowledgeable and rational producer would set 
and F
so as to maximize 
.
