3. Consider a group of homeowners around a lake who want to improve the quality of the shared access road to their community. Imagine that each owner gets the same benefit, B(Q), from an access road, where B is a benefit function with diminishing marginal returns (B' > 0, B" < 0), and Q is the total amount of money spent on the road. Thus, Q = E1 + E2 + . . . + EN, where Ei is the dollar amount contributed by household i to the road budget.

Mathematically describe the socially optimal contribution of each homeowner, and each homeowner’s predicted contribution, when each is simply asked to contribute as much as the homeowner wants. Provide exact solutions for the case where the utility function is given by B(Q) = ln(Q), showing that in this case, each individual will contribute exactly 1/Nth of what the homeowner should.