＊） Gakushuin University (Professor Emeritus), Tokyo.

＊＊） Gakushuin University, Tokyo (Email: junichi-noro@nifty.com)
Mr. Kawashima has been responsible for the theoretical analysis in this paper, and Mr. Noro for the computational and chart-making tasks through Mathematica. The authors express their gratitude to Professor Joshua M. Drucker, Professor Hisayoshi Morisugi and anonymous referees of this journal for their precious comments. The usual disclaimer applies. They also owe to Mathematica by Wolfram Research Inc. for the computer software which has enabled the authors to calculate rather sophisticated mathematical functions as well as to draw charts some of which graphically express the functions obtained through the computation by Mathematica.
This paper is a revision of a part of its earlier draft papers entitled “Bell-shaped Demand Curve and Marginal Social Benefit Curve and Croissant-shaped Demand Curve: N-external Economies and P-external Economies” which was presented at the 57^th Annual North American Meetings of the Regional Science Association International (2010 NARSC Meetings) in Denver, CO, USA, on November 10-13, and “Utility Surface, Demand Surface, Derived Demand Curve and Marginal Social Benefit Curve: N-external Economies and P-external Economies” which was presented at the 24^th Annual Conference of the Applied Regional Science Conference (ARSC) at Nagoya University on December 4-5, 2010.

1） This assumption originates from the paradigm of “Theory of Clubs” by Buchanan (1965).

2） The existence of positive external economies and that of external diseconomies are defined as following where M stands for the equilibrium demand level for the service and N for the demand level for the service:

(i) Existence of positive external economies stemming from M: The magnitude of positive external economies increases as the value of M increases for given N.

(ii) Existence of external diseconomies generation from M: The magnitude of external diseconomies increases as the value of M increases for given N.

3） It does not necessarily mean that the derived demand curve always exhibits a bell-shaped form even when the T-external economies exist. Depending on the shape of the demand surface, the derived demand curve can have the shape of decreasing curve as the conventional demand curve.

4） This condition ensures that u₀(x) ≧ 0.

5） This condition together with the condition x ≧ 0 ensure that u₁(x, M) ≧ 0. On the other hand, theoretically speaking, the value of M is supposed to be positive. We, however, set M ≧ 0 (or＋0) to avoid the practical intricacy in mathematical handling of equation (2).

6） More precisely speaking, this demand level is the “expected resulting equilibrium demand level.”

7） Precisely speaking, positive external economies.

8） The external economies and diseconomies mentioned here respectively correspond to the bandwagon effects and snob effects discussed in Leibenstein (1950, pp.190ff and 199ff).

9） This numéraire assumption enables the income level Y to become equal to the amount of the composite goods x when the consumer spends all her/his income in purchasing the composite goods exclusively.

10） This relationship means that the utility levels of both points remain equivalent.

11） This inequality is derived from the relationship u₀(Y) ＜ u₁(Y−P, M).

12） The initial value of M means the beginning value of M which moves from 0 towards its larger values.

13） The demand surface shows the relationship among the demand level N for the service, the equilibrium demand level M for the service and the price level P of the service.

14） This number 25 is the maximum value of Y in equation (9).

15） This explicit function has been obtained through the approximate-value method.

16） The concept of the DD is similar to that of “actual market demand” in Leibenstein (1950, P.192).

17） For the discussion in this sub-section, the price function does not have to be specified. The price function is often referred to as the average social cost function or private cost function.

18） This case would take place when the price and DD curves both have upward convexity in a bell-shaped from and when the convexity of the price curve is sharper than that of the DD curve.

19） An example of the demand surface function P=h(N,M) which is neutral to the T-external economies is P=a+bN+0×M (for a >0, b<0, N ≦−a/b). The form of the cross-section curves on such a demand surface cut by M remains the same for any M.