I don't know if I'm re-inventing the wheel. I don't normally do this sort of micro, or any micro.
Talking about "Public Club Investment goods" will sound oxymoronic to an economist. But the investment goods I'm talking about are a bit of both. They look like club goods from one angle, and look like public goods from a different angle. It's the club itself that is a public good, so let's call it a "Public Club". (I probably need a better name.)
[For non-economists: I am using the words "Public" and "Club" the way economists use them, not in the way you use them. A good is "non-rival" if the marginal cost of an additional person consuming the good is zero; a good is non-excludable if it is impossible to prevent an additional person consuming the good without paying. Both Club goods and Public goods are non-rival; but Club goods are excludable and Public goods are non-excludable. Cable TV is a Club good; broadcast TV is a Public Good (it costs nothing to let an additional person watch an existing show, but the cable company can charge and the broadcast company can't charge). Don't argue and split hairs; that was just an example.]
Take a simple two-period example. In the first period the club decides how much to invest. In the second period the club members enjoy the benefits of that investment. Non-members are excluded from enjoying the benefits. But can non-members be excluded from joining the club at the beginning of the second period?
You can see the problem. If club members suffer net costs in the first period, and enjoy net benefits in the second period, each individual has an incentive to postpone joining the club until the beginning of the second period. The investment never gets made. Even if there are positive net benefits for some members in the first period (because the club has some other benefits), the members will under-invest in the first period if they cannot charge new members an additional premium in the second period.
If capital markets are perfect, and in a world of certainty, this problem can be resolved. In the first period the members borrow the funds to finance the investment, so it costs them nothing. In the second period all members (new and old) have to pay dues sufficient to cover the costs of that investment. But this means the public club has zero net equity (a 100% debt/equity ratio), so it won't work if there is uncertainty.
If new members can be charged a one-time initiation fee, that existing members don't have to pay, this problem can be resolved. That gives existing members an incentive to build equity in the club, so they can issue new shares and sell them to new members at a positive price. But that would mean the club is a Private (excludable) club, and not a Public club.
What happens in a multi-period model, where each period the members enjoy the benefits of previous periods' investment and invest for future periods' enjoyment? The hypothetical social planner would make everyone a club member, and require the club to invest up to the point where the Present Value of the sum (across all future members) of the marginal benefits equals the current marginal cost. A net flow of new members is like investing in an appreciating asset from the Planner's perspective. But if the existing club members themselves vote on how much to invest, they will ignore the benefits that will accrue to new members who join after the investment has been made, and will discount the future returns if they themselves might leave the club. And they will want to postpone investment until the new members actually arrive and share the costs. The existing members will only solve the Planner's Problem if there are no new members.
Take an extreme case, in continuous time, where a discrete investment has to be paid for by existing members at a point in time, but yields a continuous flow of future benefits. That investment will never be made. The proof is simple. If the investment were made at time t, each individual member would resign at t minus epsilon, and rejoin the club at t plus epsilon. They get nearly all the benefits without paying the costs.
The equity in a Public Club is a Public Good. Everyone wants to wait for someone else to put their equity into the lighthouse.
I'm trying to think of a simple but reasonably general way to set up a model with an interior solution that lets us see what the amount of under-investment depends on. No luck yet.
[This is related to my previous post on Open Borders, but I thought it best to try to think more generally and abstractly about the underlying problem.]