Consider an economy with population N. There is a club which is providing a level of provision G. When the cost of the club is shared equally and paid by members as a membership fee, the obtain utility is: Ui = (1/2)log(G^2) (n^2/2d) + M (G/n), where M is the individual income and d > 0 is a positive constant.

(a) If the utility of each member is maximized, what is the optimal level of membership, n, for the club?

(b) Now assume that the club chooses G optimally while its size is given, calculate the loss due to membership of a club with non-optimal size.

(c) What club size maximizes total utility produced by the club? If log(n) = while > 3/2 then contrast your result to the answer for part a. What happens to each member utility?

(d) Find the optimal club membership n which maximizes each member's utility when there are 2 identical clubs? What if there are 3 clubs? What you can say about increasing the number of clubs?