Question

Consider the following social welfare function:

SW=U_a(X^nc_a, X^c_a, X^c_~a, P_a)+U_b(X^nc_b, X^c_b, X^c_~b, P_b)+ U_d(X^nc_d, X^c_d, X^c_~d, P_d)+...

where: X=consumption bundle, P=pollution, nc=non-competitive consumption, c=competitive consumption, a, b, d = people, ~a, ~b, ~d = all other people besides a, b, d.

Question: Will an overall increase in the level of non-competitive consumption always increase social welfare?

Answer 1

Gievn the social welfare function

SW=U_a(X^nc_a, X^c_a, X^c_~a, P_a)+U_b(X^nc_b, X^c_b, X^c_~b, P_b)+ U_d(X^nc_d, X^c_d, X^c_~d, P_d)+...

where: X=consumption bundle, P=pollution, nc=non-competitive consumption, c=competitive consumption, a, b, d = people, ~a, ~b, ~d = all other people besides a, b, d.

Under such a utility function the increase in the stock of non-competitive goods will always unambiguously increase the social welfare.

To prove the above claim, if we consider the function only for the first consumer and differentiate the utility function with respect to nc, the marginal social welfare will always be a positive function depicting a direct relationship between the non-competitive consumption and the social welfare.