Consider an economy with n consumers, one private good and one public good. We think of the private good as "all other goods," and measure it in terms of money. There are n individuals in the economy with utility functions Ui(xi , y) = i ln y + xi , where y is the public good, xi is the private good consumed by individual i = 1, . . . , n, and 0 < 1 < 2 < < n are parameters. Initial endowment of private good of each consumer is given by wi . We donote the aggregate endowment 1 of the private good by w = P i wi . The cost of producing the public good is c(y) = cy for some c > 0.

1.What is the calculation for optimal provision of the public good?

2.What is the calculation provided under voluntary contributions.