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 Law6

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.7 If the cross demand partials are zero, then (11) reduces to the familiar inverse elasticity rule. 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.

Note that > 0, by construction, so that (12) and (13) imply that

(14)

< 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

Personal

The relationships in (20) are both surprising and useful. 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 simultaneous collective decisions of many population groups, while NS 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 NS 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

am* f (m*) (NS (m*) - NS (m*)) = -NCPNS (m*).

aps

Thus, (21) can be rewritten

(22)

N§ = - N=PNS (m*) + ѧ ≤ ѧ < 0.

Together, (22) and (18) yield considerable insight into the structure of demand and the optimal prices. Note first that the Jacobian,

NS NS

PL

of the map giving NS and NL as functions of ps and p is an NP matrix (i.e., the principal minors alternate in sign from negative, for the 1x1 minors, to positive). This is so because both NS and N are negative and, using (20) and (22),

Thus, the interrelated demands for library and personal subscriptions are "normal" in the sense of Sandberg [11]. If ps and PL change, the demand for at least one of the goods moves normally, in the opposite direction to the movement in its price. For example, if both prices rise, both demands cannot simultaneously increase. We think the fact that NS and NL comprise a normal demand system is a confirmation of the plausibility and applicability of the model. Further, the NP property of the demand Jacobian may be a useful restriction on estimated demand equations.

Turning to the optimal price rule, we note first that at the profit constrained welfare optimum, both ps and PL are strictly above the marginal cost, c. This follows from (11) in that NSNL - NENS < 0 and -N5N2 + N1⁄2NS ‹ 0; N2NS - NN > 0, by (23); and finally ≥ 0 by

<

the Kuhn-Tucker conditions. Moreover, if λ were 0, PS = c = PL, which, by the assumed increasing returns, would leave costs uncovered and violate the constraint ≥0. Thus > 0, Pg > c and > PL

= d

> C.

Now we can delve into the determination of the optimal ratio,

PL-C

Ps-c'

and rewrite the basic equation (11) several ways to expose the roles of the underlying variables of the model. Rearrangement of

Now, substituting (20) into (25) and rearranging yields

Thus, PNS (m*) > 1 would immediately imply that p > 1, that the optimal

library price exceeds the optimal personal subscription price.

Now, to contrast the formula for p with the classic inverse elasticity rule, divide the numerator and denominator of (24) by N NL and use (20) to get

Of course, if PNS (m*) = 0, then the cross-elasticities vanish and