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It is interesting to note that each library population would prefer to circumvent the positive a rule and to pay p, solely out of the lump sum taxes characteristic of perfect purvefance. However, , each library population benefits from the decreases in Ps and PL which results from collective adherence to the rule.
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, m2: ...,
mn), where my 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 mis
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, WP;. 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 my. Now, m* plays the same central role as that played throughout the paper by m*.' Thus, for example,
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):
where we have used the fact that pns, the number of prospective subscribers in a unit population of type i. Substituting (A9) into the above gives
= PN° (m* (A12)
WP, WP 1
Thus, we have anyaps/an"/apL = Ph®(m*), the average number of prospective subscribers in the marginal libraries. When the basic model is specified this way, PNS(m*) replaces PNS (*) throughout. The same applies to all the concepts specific to marginal libraries.
This debate was stimulated by the celebrated case of Williams and
It was Ramsey  who first studied welfare optimal prices under such a constraint. See  for a cogent survey.
4. See Willig , for the development of this general approach. 5. Thus, throughout, we ignore distributional effects.
8. This rule was popularized by . 9. S. Berg's important study of journal demand , overlooked this
10. See Willig .
11. See Berg . Research in progress by Y. Braunstein et al. 
seems to indicate values of k significantly above 2. 12. See Fry and White .
These data for AER and El are annually released publicly. The editorial offices of JPE and JET specified NS + NL precisely, and offered estimates of N47NS. While Wiley, the new publisher of QJE, refused to give any information, the editorial office offered estimates of 1975 NS + NL and NL/NS.
14. This equation was estimated by Y. Braunstein , from a 1973
cross-section of 56 technical journals.
We assume here that the prices of all costly factors of journal production rose by 25 percent between 1973 and 1975. Both the Wholesale Price Index of book paper and the BLS index of printing trades wages did increase by approximately 25 percent between those dates.
16. See footnote 11.