Reference no: EM133007756

Question 1: Consider an economy with two individuals (1 and 2), a dirty good (X), a clean good (T), and labor as the only input to the production. The utility of each individual is defined as U_i(X_i,T_i,E) for i = 1, 2 where X_i and T_i are the consumption levels of the two goods and E is an exogenous (to the individual) level of pollution emissions. Production of X releases the emissions, E. The production of X can be defined as X = f(L_x,E), where both labor input L_x and emissions E have a positive marginal products. This specification treats emissions as an input, as the reduction of pollution reduces the output of X by decreasing the productive factor. The clean good T is only produced by labor input following the production function T = g(L_T). Labor employed in the economy is constrained by the work time endowment L such that L_x + L_T = L.

Answer the following question in (A) (i)-(iv), (B) (i)-(vi), and (C).

(A) Given the following analytical presentation of the Pareto Efficiency problem of the economy, answer the questions in (i)-(iv).

Pareto Efficiency:
          max                           U₁(X₁,T₁,E) + λ_u [U₂ (X₂,T₂, E) - U₂‾] + λ_x.[f(L_x, E) - X₁ - X₂] + λ_T. [g (L_T) - T_l - T₂] - λ_L. [L - L_x - L_T]
x₁, X₂,T₁, T₂, L_x, L_T,E

where individual 1's utility is being maximized subject to the constraint that individual 2 obtains at least utility level of U₂‾, and where λ_IJ, λ_x, λ_T and λ_L are the Lagrangian multipliers of the constraints.
(i) Write down the first order conditions of Pareto Efficiency problem.
(ii) Write down the expression (equation) for "efficiency in consumption."
(iii) Write down the expression (equation) for "efficiency in production."
(iv) Write down the expression (equation) for "efficiency in product mix."

(B) Suppose Px and PT denote the prices of X and T, respectively. Furthermore, let the price of labor be W and the income of person i is Y_i.
(i) For individual i, write down the utility maximization problem and the first order conditions.
(ii) Write down the expression (equation) for "efficiency in consumption."
(iii) Write down the profit maximization problem of firm T and its first order condition.
(iv) Write down the profit maximization problem of firm X and its first order conditions.
(v) Write down the expression (equation) for "efficiency in production."
(vi) Write down the expression (equation) for "efficiency in product mix."

(C) Rewrite the first order conditions for emission (from part (A) and (B)) and then make a comparison of the efficient allocation of emissions in (A) and (B).