The most general pricing strategy is for the producer to choose a
price for each possible quantity, without restriction (except that
T(0)=0, since the consumer always has the option of exiting the
market). Maskin and Riley provide a method for calculating the optimal
continuous nonlinear pricing strategy [MR84]. The intuition
is this: a profit-maximizing producer wants each different customer
type ( 
) to purchase a ``constrained'' quantity 
at
price 
. Let some customer type choose to buy 
and
pay 
. Call the value this consumer would get from an
incremental unit 
. Now suppose there is another customer
with 
, who also chooses to buy . By definition, this
customer gets value from the next unit 
.
Therefore, the producer would make a greater profit by offering 
at 
where 
: the
second customer would prefer to purchase but the first
customer would not. Thus the problem can be stated as one in which
the producer sets prices to induce each customer to purchase its
profit-maximizing quantity, subject to the self-selection constraints
that ensure a consumer of type 
does not want to purchase the
intended quality for any other 
. The function is found
from solving a system of differential equations. To conserve space,
only the results are provided here.
We obtain the following optimal unrestricted, nonlinear price schedule, shown in Figure 1.
Figure 1: Optimal profit for nonlinear pricing
This yields profit, welfare and consumer surplus of
For example, when 
profit with nonlinear pricing is 32%
higher than either of the one-parameter models.