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Robinson's technology for producing coconuts (x) is represented by x = 1/2 , where is labor,in hours per day.
1. Robinson’s technology for producing coconuts (x) is represented by x = ㅣ1/2, where ㅣ is labor,in hours per day. His preferences for coconuts and labor are given by the utility function u(ㅣ, x) = x− ㅣ /2. Assume Robinson is the only consumer of coconuts, and the owner of theonly firm which produces coconuts. Suppose the price of coconuts is set at 1.
(a) Calculate the Pareto efficient allocation of this simple production economy.
(b) Derive the competitive equilibrium of this economy. Find Robinson’s consumption of coconuts, his labor supply, the market wage rate, and the firm’s profits.
2. Consider Robinson from Question 1. Suppose his technology for producing coconuts (x)changes to x = ㅣ2/3. His utility function remains the same: u(ㅣ, x) = x− l /2.(a) Calculate the new Pareto efficient allocation.(b) Derive the competitive equilibrium of this economy. Find Robinson’s consumption ofcoconuts, his labor supply, the market wage rate, and the firm’s profits.
