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The Ramsey Suboptimality of Perfect Purveyance
In Section V we showed that full Ramsey optimality requires a positive usage fee. This is not too surprising since we are accustomed to Ramsey optimal prices being above the corresponding marginal costs, and here the marginal cost of an additional library reader is zero. The usual reason for this result is that with increasing returns to scale in production, prices must generally be above marginal costs for revenues to cover total costs.
Here, we show that perfect purveyance of excludable public goods is Ramsey suboptimal for a new and different reason. We consider a rule that each library must finance the proportion a of the subscription price PL by means of a use fee whose size then varies over libraries:
Under this regime, with a > 0, the library does not perfectly purvey the noncongested journal copy, and the use payments do not contribute directly to cover the publisher's total costs. Yet, we shall see that the Ramsey optimal value of a is positive. The sole effect of the introduction of a positive a is to raise profit by the mark-up on the personal subscription sales to the former marginal potential subscribers. Because the profit constraint is binding, this profit increment enables Ps and p, to be lowered, bringing profit back down to its previous level, and increasing consumer welfare.
To establish this result, we note first that given (A1), m* is implicitly defined by
As before, ǝWP/ǝPu
LR, and, from (A1), at a = 0, ǝpu/α = PL/LR. Substituting these facts into (A3) gives, at a = 0,
The derivatives of consumer welfare, V, with respect to a,
at a = 0, can be calculated from (39), recognizing that now Pu is the function of both a and m given by (A1).
given by (42) and (5), again remembering that (A1) and (A2) give new interpretations to p,, and m*. Here, in view of the critical (A4), calculation shows that at a =
+ (1+x) ≥0. This inequality is strict whenever there
are any marginal prospective subscribers in any of the subscribing libraries. In this case, Ramsey optimized net or consumer welfare is strictly increased by the imposition of a positive a.
It is interesting to note that each library population would prefer to circumvent the positive a rule and to pay p1 solely out of the lump sum taxes characteristic of perfect purveyance. However, each library population benefits from the decreases in PS and PL which results from collective adherence to the rule.
Multidimensional Characterization of Library Populations
Throughout the paper, we have characterized library populations by the scalar m, and have assumed that the willingness to pay is increasing in m. This is an overly restrictive formulation which, however, finds frequent use in the literature. Here we show how the model can be considerably extended to allow for a multidimensional characterization of library populations, without at all affecting the power of the onedimensional approach.
Let each library population be characterized by the vector m = (my, m2, ..., mn), where m, represents the number of agents in the population m of type i. Type i agents are themselves characterized by the density function g; (B,T). Thus, the histogram function of the population m is
All population specific structural functions have their analogues defined for each agent type. Thus, here, for example, we have an analogy to (3) the willingness to pay of a unit population of type i, wpi. Then,
Because WP (m) is increasing in each component, by (A8), a population purchases a library subscription if
Here, we have arbitrarily chosen to normalize on m1. Now, m* plays the same central role as that played throughout the paper by m*. Thus, for example,
f(m,, . . .,m) dm, dm2...dmn.
0 m† (m2 • • • •,mn )
A specific analytic gain from this more general specification is the replacement throughout of the number of potential subscribers in the marginal library," PNS (m*), by "the average number of potential subscribers in the marginal libraries." To see this, calculate from (A10):