A Teaching Note on finding efficient allocations in a simple Public Good Economy

[Amit]

In this example, I have solved a simple problem of finding Pareto efficient allocations with two consumers - u_1, u_2, two goods - one public Y, one private X. 

u_1(x_1, y) = x_1 + y, 

u_2(x_2, y) = x_2y 

and set of feasible allocations is 

{(x_1, x_2, y)| x_1+x_2+y^2 = 30}. 

The approach discussed in the answer is different from the usual approach to solve such problems in two ways: - 

(1) It provides a graphical way of observing the set of feasible allocations and Pareto efficient allocations (just like the Edgeworth box in a two private goods economy) ; 

(2) The condition for efficiency is also presented differently.

https://economics.stackexchange.com/a/55050/11824

I'll be happy to answer your questions if you have any.

Best,

Amit