Inside of Coase Theorem

(ii)(if exclusion is possible)
utility of leaving club become zero. Thus, one can sustain the optimal club size.

(iii)(if exclusion is not be credible, but menbers can adjust number of menbers)
if optimal size is not reached, menbers can offer marginal benefit of additional person for outsider.


If one joins at time zero, one can gain
(marginal benefit of optimal size) - pay
If one joins at time one, one can gain
(offer for additional person)

Therefore, encompassing club (n menbers) is sustainable if and only if

(marginal benefit of optimal size)-pay> (offer for additional person)


Summary:
if exclusion is impossible, club is break because free-riding is inevitable.
if exclusion is possible, free-riding is prevented, but optimazation depends on barganing ability of club.


Discussion:

Coase Theorem: if there are no transaction cost and property right, efficiency is reached voluntary.

This is almost true, because barganing power affects negotiation.
(but ongoing work suggest Coase Theorem holds in the end)

(i)limits of punishment and (ii)asymmetric information
is inefficiency that Coase Theorem hides.