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Reading left to right, the integrals measure the net benefits of the personal subscribers, the potential subscribers, and the perusers, respectively.
Here, without a library subscription, the only readers are those with personal subscriptions. Finally,
Thus, the personal subscribers contribute nothing to WP, and the perusers are willing to pay their full benefit, net of inconvenience, B - T. However, the potential subscribers add only the difference Ps T between their evaluations of the library inconvenience and the money cost of a personal subscription,
Using the willingness to pay concept, we can identify the libraries which are just indifferent to acquiring the journal. Such marginal libraries will be denoted by the index m*, with
For convenience, we take m to be a scalar index defined so that the
at our results with a mathematically more satisfying representation of multidimensionally differentiated libraries.) Letting f(m) be the number of population groups with characteristic m,
is the total number of subscribing libraries.
Denoting by NS the total number of personal subscribers, publisher profits are
Total social welfare generated by the journal in question, given by the sum of producer's surplus and consumers' surplus, 5 is denoted by W = V + T, where
Now, we can turn to the choice of Ps and PL which maximizes W subject to the constraint that a > 0. Forming the Lagrangian, L = W + 10 = V + (1+1), we investigate the necessary first order conditions for positive optimal prices:
However, because of the definitions of m* and WP, (3) and (4), the second term is zero and we are left with this familiar version of Roy's Law
Routine differentiation of the profit function (6) gives this solution to the simultaneous equations of (8), where c denotes the marginal cost C'(NS+NL):
Here, subscripts S and L denote partial derivatives with respect to Ps and PL Of course (11) is the standard Ramsey rule for optimal deviations of prices from marginal costs under the nonnegative profit constraint.? If the cross demand partials are zero, then (11) reduces to the familiar inverse elasticity rule. 8 In the present form, (11) is not very illuminating. A more useful formulation can be derived by substituting into it detailed relationships among the partial derivatives of demand extracted from the underlying model.
aWP Note that > 0, by construction, so that (12) and (13) imply that
am <0. Turning back to the definition of WP in (3), we calculate
This is just the number of potential subscribers who frequent each
marginal library. Together, (12), (13), and (14) yield
The number of personal subscribers in a population, m, with a
with respect to PL, which affects only the set of subscribing libraries,
Now, together with (15) and (12), (19) reveals that
nę PNS (m*)N20.
The relationships in (20) are both surprising and useful. Personal and library subscriptions are gross substitutes, provided there are potential subscribers in the marginal libraries. 9' Despite the fact that the demand for library subscriptions is determined by the şimultaneous collective decisions of many population groups, while N results from the individual decisions of the agents, the Slutsky symmetry of the demand cross-partials (with no income effects) is preserved.
It remains only to investigate the behavior of NS with respect to changes in Ps. Working from (18),
We denote by is the negative terms in the brackets which represent the derivative of N° with respect to ps, holding constant the set of subscribing libraries. Using (13), (14), and (19), we have