¹ Olson, Mancur Jr., The Logic of Collective Action: Public Goods and the Theory of Groups (Cambridge, Mass.: Harvard University Press, 1965)Google Scholar. A somewhat different and more general formulation of Olson's analysis subsequently appeared in an article co-authored by Olson and Zeckhauser, Richard, “An Economic Theory of Alliances,” Review of Economics and Statistics, 48 (11, 1966), 266–279 Google Scholar.

² See Samuelson, Paul, “The Pure Theory of Public Expenditure,” Review of Economics and Statistics, 36 (11, 1954), 387–389 CrossRefGoogle Scholar; “Diagrammatic Exposition of a Theory of Public Expenditure,” Review of Economics and Statistics, 37 (11, 1955), 350–356 CrossRefGoogle Scholar; and Head, John, “Public Goods and Public Policy,” Public Finance, 17 (11 3, 1962), 197–219 Google Scholar.

³ Olson, p. 2.

⁴ Ibid., p. 36.

⁵ Ibid., p. 44.

⁶ See Head, “Public Goods and Public Policy,” for a full discussion of the characteristics of collective goods.

⁷ Samuelson makes the following distinction between the polar cases of private and pure collective goods. For a private good,

while for a pure collective good x ¹ = x ² = … = xⁱ = … = xⁿ = X, where xⁱ is the consumption of individual i, and X is the amount of the good produced. See Samuelson, , “The Pure Theory of Public Expenditure,” p. 387 Google Scholar.

⁸ Olson, pp. 36-42.

⁹ It is important that the nature of this process be understood. In particular, it is important to note how the “imperceptibility” of an individual's actions is treated. The process is one of dynamic adjustment, in which each individual reacts to the behavior of others on the assumption that they will not perceive his action (more correctly, will not change their behavior as a result of his action), even though in reality this is not the case. The model is equivalent to a Cournot duopoly model. See Cohen, Kalman and Cyert, Richard, Theory of the Firm: Resource Allocation in a Market Economy (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1965), chapter 12Google Scholar, for one discussion of this model. It is important to distinguish this treatment of “imperceptible” actions from the cases in which the effects of an individual's actions are either imperceptible to himself because of uncertainty, lack of information, etc., or are imperceptible to others for the same reasons. The analysis in Olson's book and in the article by Olson and Zeckhauser assumes that each individual has perfect knowledge of the (past) actions of all individuals and of the production function for the good in question. For an examination of the problem of an actor's behavior under uncertainty, see Frolich, Norman and Oppenheimer, Joe, “I Get By With a Little Help From My Friends,” World Politics, 23 (10, 1970), 104–120 CrossRefGoogle Scholar. For a discussion of Olson's “exploitation” hypothesis (that the small exploit the great), see Olson, p. 35.

¹⁰ For a discussion of this classification, see Cohen and Cyert, pp. 75-76 as well as Olson and Zeckhauser, p. 270, footnote 15.

¹¹ In Figure 4, the curves are drawn in such a way that it appears that a good is always inferior (or normal, or superior). This is generally not the case. The good may fall in different classes for different individuals, and as the level of consumption of the good rises for an individual, the good may move from one class to another.

¹² In The Logic of Collective Action, Olson ignores income effects, considering only the case in which the slope of the reaction curve is — 1 (the case of no income effects). Olson and Zeckhauser consider income effects more fully, particularly on p. 270, footnote 15. If collective good is a superior good in the framework under consideration, groups will be led either to invest all of their resources in the provision of the good (if the slope of the reaction curve is positive) or to provide an amount of the good in direct proportion to group size (if the slope of the reaction curve is infinite). Since such behavior is highly unlikely, superior goods will be ignored in the analysis to follow.

¹³ Under the assumptions of the model, an individual will change his behavior if the point (x_A, x_B ) corresponding to the amounts of the collective good provided by the individuals does not lie on the individual's reaction curve. Since the intersection of the two reaction curves is the only point at which neither individual will change his behavior, this is the only possible equilibrium. If both reaction curves are continuous, an equilibrium will exist. Sufficient assumptions for a continuous reaction curve are that the individual have a continuous utility function and that the indifference curves derived from this function obey the condition of strong convexity (i.e., if two possible allocations are indifferent, then their weighted average with positive weights is preferred to them). The particular configuration shown in Figure 5 is not guaranteed, but is the most likely case. Other possibilities are shown and discussed in the Appendix. It should be noted that this equilibrium also results if the problem is viewed as a noncooperative game rather than as a Cournot duopoly problem.

¹⁴ This assumption greatly simplifies the analysis to follow without affecting the substance of the results.

¹⁵ Olson, p. 48, footnote 68.

¹⁶ Olson, Mancur, “Increasing the Incentives for International Cooperation,” International Organization, 25 (Autumn, 1971), 866–874, at p. 866, footnote 1CrossRefGoogle Scholar.

¹⁷ Olson, , (The Logic of Collective Action [p. 22]Google Scholar) points out the importance of fixed costs as a barrier to collective action. He suggests that cooperative action may be necessary to overcome this barrier. In making use of a budget line, the present analysis ignores fixed costs. It should be emphasized that the inability of a group to overcome these costs via non-cooperative action will not be a function of group size.

¹⁸ Ibid., p. 23.

¹⁹ It may well be true, although no empirical evidence is available, that there are differences between the types of collective goods desired by small and large groups, and that these differences are such that an amount of one good may be an “adequate” amount for the small group but an amount of another good may be “inadequate” for its group, a large one. Such a possibility seems of sufficient interest to merit further thought.