4. Consider an economy with 50 consumers and 5 firms and a single con- sumption good X. There is zero initial endowment of this good in the econ- omy. Each consumer has a Cobb-Douglas utility function for consumption , and leisure r : u(x, r) = x / /". This consumer's income consists of profits from his share of the firms and his wage income, with the latter coming from devoting a part of his total endowment of time T to working in the firm at an hourly wage rate of w. The consumer takes both the price of the good p as well as the wage-rate was given. [Note that the consumer has zero endowment of the consumption good X, and has endowment T of time which he can divide between working and leisure.] The ownership of the firms are equally divided among all the consumers, so that each owns s, of each firm. Each firm uses labor to produce the single consumption good X accord- ing to the following technology: x = f(1) = 2v. All firms are assumed to be price-taker in both the markets. (i) Derive each consumer's demand functions for good X and leisure r. (ii) Derive each firm's profit function. (iii) Calculate the equilibrium price to wage ratio . for this economy. The book works out a similar example in section 18.8, which you may wish to consult in doing this problem. But be sure you understand what's going on. 2