1. Consider the one-person production economy we call the Robinson Crusoe model. Robinson Crusoe is a profit-maximising producer, who produces a consumption good from labour inputs. He is also a utility-maximising consumer who owns the profit-maximising firm and provides his labour to the firm. He has a decreasing returns to scale production function x = L^ 1/4 . Crusoe gains utility over consumption of x and leisure R through the Cobb-Douglas utility function u(x,R) = a ln x+ (1a) ln R. He is endowed with one unit of time, which can either be spent on labour L or leisure R.Friday is also endowed with one unit of time and works for Crueso.

There is no government in this country. Assume that the total output they generate is fixed, so that their incomes are fixed at w_c and w_f respectively. They decide to dedicate some of their incomes to a continuous public good. Their utility functions will now look like: u(x,R) = a ln x + (1 a) ln G . How much of the public good will Crusoe and Friday agree to purchase if they are operating independently of one another?