Sandra and Shehat defined the club in their articles on club theory as: "A group voluntarily * * * enjoys or * * * bears one or more of the following factors to obtain the same benefits: production cost, membership characteristics or products with exclusive benefits." James buchanan used this model for the first time to study the efficiency nature of voluntary clubs. In his model, there is such an assumption: a club excludes non-members without success; Members of the club will not be discriminated against by other members; Members share the same costs and benefits. Its analysis is carried out by investigating the behavior of representatives of club members (denoted by I). Suppose the personal utility function is: maxUi(Yi, x, s). Among them, Y I is the consumption of private products by the I-th individual, X is public products, and S is the group size. This leads to the following analysis problems: (1) determine the output of public goods that should be supplied; (2) Determine the optimal size of club members.

The first is the determination of the optimal supply of public goods. The condition of the optimal supply of public product X is called "Samuelson condition", which shows that the marginal cost (MRT) of the last unit producing X is exactly equal to the marginal income calculated by Y that all users get at the same time.

Secondly, the determination of the best number of club members. If the product scale and cost of the club are fixed, for a member P, with the increase of the number of members, the marginal cost is negative, because the increase of the number of members reduces the sharing cost. On the other hand, with the increase of the number of members, the marginal utility brought to a member is positive or zero at first, and then gradually negative. Therefore, in order to get the maximum benefit, each member must ensure that the marginal benefit brought by the total number of members is equal to the marginal cost. Because each member is homogeneous, the maximum utility of that member means that all members get the maximum utility, so the number of members who can meet the above conditions is the best number of club members under a given output situation. Buchanan's club theory explains the distribution of impure public goods. If the technology and preference of providing exclusive public goods are clustered, and many clubs with the best composition are formed in a given scale society, then the clubs formed by individual voluntary association are an optimal allocation of these exclusive public goods. But we should also consider the dynamic situation of many clubs or clubs with multiple products. Suppose the population size is n and a typical club has n members. So there are N/n clubs. If N/n is an integer, then everyone can join the club. But if N/n is not an integer, then some people don't belong to any club. They may set up their own clubs, so the existing club structure will be unstable. Because the peripheral staff of the club will always actively encourage the original club members to quit and join the new club, thus ensuring the appropriate scale of the new club. This process will last forever, so this equilibrium is unstable. This is called integer problem in club theory. In reality, there are few clubs with single products and many clubs with many products. For example, a sports club can offer tennis, swimming and other sports, not just one of them. In terms of pure economic efficiency, intuitively speaking, a club composed of members with the same preference is more efficient. For example, all members are charged the same membership fee. Once the difference of utilization level is not easy to determine, it is much more complicated to design the membership fee as a function of utilization level. At this point, mixed clubs can achieve efficiency, but a single club can't. For example, when personal differences are not reflected in the degree of utilization, but in use, in order to achieve efficiency, off-peak pricing and peak pricing are needed. In addition, only mixed clubs can always make more effective use of collective goods.