Consider the model of profit-making religions analyzed in Section 11.2.3. Suppose now that in a religion with n affiliated believers, a fraction of λ(n)=1 − 1 n2 , of the n believers are administrators, and that administrators do not pay fees. Assume that each affiliated believer who is not an administrator pays a fixed fee denoted by φ. Answer the following questions.

(a) Calculate the number of administrators in this religion.

(b) Calculate the number of believers who are not administrators, and conclude how much revenue is collected from the fees.

(c) Using Definition 11.5 infer whether this religion is efficient or inefficient.

(d) Suppose that all the revenue collected from the fees is spent on paying wages to administrators. Calculate the wage paid to each administrator. Conclude whether this religion is open or closed to new converts.