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Question: The following economy has two commodities, n consumers, and F firms. For i = 1, ..., n, consumer ...

Show transcribed image text The following economy has two commodities, n consumers, and F firms. For i = 1, …, n, consumer i has the endowment vector omega_i = (1, 0) and the utility function u_i(x^1_i, x^2_i) = log(x^1_i) + log(x^2_i). Each consumer owns equal share, 1/n of each firm. For f = 1, …, F firm f has a production function given by y^2_f = squareroot -y^1_f where y^1_i lessthanorequalto 0 reflecting the convention that inputs are negative outputs. (a) Define a competitive equilibrium for this economy. (b) Calculate the competitive equilibrium price vector and allocation, as a function of the parameters, n and F. (c) Is the utility received by consumers at the competitive equilibrium in- creasing or decreasing in F? Give the economic intuition for this result.

The following economy has two commodities, n consumers, and F firms. For i = 1, …, n, consumer i has the endowment vector omega_i = (1, 0) and the utility function u_i(x^1_i, x^2_i) = log(x^1_i) + log(x^2_i). Each consumer owns equal share, 1/n of each firm. For f = 1, …, F firm f has a production function given by y^2_f = squareroot -y^1_f where y^1_i lessthanorequalto 0 reflecting the convention that inputs are negative outputs. (a) Define a competitive equilibrium for this economy. (b) Calculate the competitive equilibrium price vector and allocation, as a function of the parameters, n and F. (c) Is the utility received by consumers at the competitive equilibrium in- creasing or decreasing in F? Give the economic intuition for this result.