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Together, (22) and (18) yield considerable insight into the structure of demand and the optimal prices. Note first that the Jacobian,
of the map giving NS and N as functions of ps and PL is an NP matrix (i.e., the principal minors alternate in sign from negative, for the ix1 minors, to positive). This is so because both Ns and NZ are negative and, using (20) and (22), (23) NS
= NENE - NENS NG
Thus, the interrelated demands for library and personal subscriptions are "normal" in the sense of Sandberg . 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 (71) in that NNL - NINS < 0 and LNBNL + N_NS < 0; NINS - N?N5 > 0, by (23); and finally 130 by the Kuhn-Tucker conditions. Moreover, if i were 0, PS = C = PL, which, by the assumed increasing returns, would leave costs uncovered and violate the constraint a > 0. Thus 1 > 0, Ps > c and PL >
Now we can delve into the determination of the optimal ratio,
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 (mt) > 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 NSNE and use (20) to get
Of course, if PNS(m*) = 0, then the cross-elasticities vanish and
Otherwise, the needed modifications in (27) require only the number of potential subscribers and the ratio of the number of library to personal subscriptions.
Determining Best Price Adjustments from Current Data
There is considerable methodological difficulty in deriving from (26) and (27) insights that are relevant to current practices of journal pricing. The variables (elasticities, circulations, and number of potential subscribers) to which the formulae relate p are all to be evaluated at the to-be-determined prices. This endogeneity, endemic to necessary first order conditions, means that the optimal prices can only be determined as the solutions to simultaneous equations whose global behavior is almost impossible to deduce from available local data. Further, intuitions that we may have concerning current values of the variables governing e cannot be logically utilized via such first order conditions as (26) and (27) to illumine the optimal, prices. We cannot use a comparison, for example, of (NS/NS) / (N/NL) across journals to deduce from (28) a comparison of the corresponding optimal values of p. The relevant quantities to compare, holding other components of (27) equal, are the values of (NS/NS)/(N/NL) at the different optima. But these, themselves, are the objects of interest.
Fortunately, there is an analytic line of inquiry which circumvents these conceptual difficulties. We can ask for the direction of change from the current prices which is best for social welfare while preserving the current level of profit. It can be shown 10 that
P-C if the current o
is greater than the current value of x (defined
Ps-C in (24)), then the best, profit constrained, direction of change required that pi be lowered and ps be raised. Inversely, if, at current levels, o <y, Ehen PL should be raised and Ps lowered. It should be emphasized that these calculations do not necessarily indicate the relationships between the current and the optimal prices. Instead, they give the best local price adjustments that can be determined from strictly local information on the relevant functions.
From this point of view, W, calculated at current values of the variables, can indeed be meaningfully compared with the current ratio PL-C
Since both (26) and (27) give expressions equal to y, they can Ps-c serve as vehicles for the application of current data to the study of
present journal prices, yielding recommendations for the best direction of change. It now becomes meaningful to investigate the behavior of ♡ with respect to its component variables. This is not the standard comparative statics technique which requires consideration of the feedback between the underlying parameters and the consequent optimum at which the equations are evaluated. Instead, we study the level of We always evaluated at current prices, as a function of the values its parameters could take on as they pertain to different journals. Here, these parameters need not be viewed as functions of prices, as they must in comparative statics (with prices endogeneous), because the prices are themselves parametrically fixed at their currently realized values.
We shall first utilize this novel and powerful technique to establish conditions under which it can be unambiguously asserted that welfare would increase (without affecting profits) by introducing a positive margin between currently equal library and personal subscription prices. This assertion can be made if the current value of \ for a particular journal, with Ps. = PL, exceeds 1.
PL, exceed's 1. For this journal, the current p is equal tool, less than , indicating that PL should be raised and ps lowered.
For notational convenience, let
Using the representation of y given by the right hand side of (27),
Thus, y > 1 is equivalent to (k-1) + Z(1-n) > 0. This condition will be met whenever the circulation ratio, n, is less than 1 and the ratio of the elasticities, k, is greater than 1. The meager empirical evidence suggests that k is significantly larger than 2, for all journals studied. 11 Further, the best available data indicates that n<l for a majority of technical journals. 12 Thus a finding that y > 1 for a journal with ps = PL would not be surprising, and the policy recommendation to differentiate the subscription prices, PL > Ps, would be rigorously justified.
For journals already charging differentiated prices, the investigation of the best direction of price changes requires more current information. If k, n, and Z were known, then the test is just py. However, Z may be more difficult to estimate than are k or n. Nevertheless, we can use (28) to determine the minimum value of y, over all Z > 0, as a function of k and n. If for a particular journal it should be the case that p < Ymin, then surely psy and the recommendation to increase PL and decrease ps would follow.
Holding PL, Ps, k, and n constant, (28) shows that y is either monotone decreasing or increasing in Z as (PL!PS)nk is greater or less than one. In the latter case, vmin = (PL/PSJk. In the former case, we need an upper bound on Z to establish a Tower bound on W.
Substituting this uppen bound for Z into (28) gives