Consider an economy with 2 firms and a mass 1 of labor input that can be...

 

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Consider an economy with 2 firms and a mass 1 of labor input that can be allocated to either firm (that is, labor supply is 1 unit for every possible level of the wage rate. This assumption - perfectly inelastic labor supply is a popular one when you want to focus on labor demand. All the qualitative conclusions from the problem will still hold if you assume an upward sloping labor supply curve.). Each firm i=1,2 produces using the production function Y = An (note the use of lowercase letters for firm-specific choices of factor inputs). Assume A₁ > A₂ (you'll need this assumption in (c)). 1. A Planning Problem. Suppose a hypothetical benevolent social planner wants to maximize total output for this economy by choosing how much labor each firm gets. This planner solves the following problem: max An+Ana 11.2 subject to n₁ + n₂ = 1. Solve the planner's problem and find the optimal choices of n₁, 7₂ (hint: it's more direct if you substitute for n₂ using the constraint and solve the unconstrained maximization problem for 7₁, and then solve for n₂=1-₁.). Call these n. n. Calculate the ratio n/n as a function of A₁, A2 only. We'll use this ratio in part (c). 2. A Decentralization of the Planning Problem. Now let's consider how the market economy would allocate resources if it were allowed to function freely. Suppose labor must be hired at a wage w. Each firm i = 1,2 solves the problem max An n₂ 1-0 ww (a) Solve for firm is optimal labor input as a function of w. A, only. (b) Calculate the total demand for labor at the wage rate w. (e) Use Market Clearing in the labor market to obtain the equilibrium wage u as a function of A₁, A₂, a only. Recall that the total mass of labor input is 1. (d) Substitute the market clearing wage back into each firm's labor demand curve to get the equilibrium inputs of labor for each firm ni, ng. How do these compare with ni, n? Provide some intuition for why the market and the planning solution are related in the specific way you find. Consider an economy with 2 firms and a mass 1 of labor input that can be allocated to either firm (that is, labor supply is 1 unit for every possible level of the wage rate. This assumption - perfectly inelastic labor supply is a popular one when you want to focus on labor demand. All the qualitative conclusions from the problem will still hold if you assume an upward sloping labor supply curve.). Each firm i=1,2 produces using the production function Y = An (note the use of lowercase letters for firm-specific choices of factor inputs). Assume A₁ > A₂ (you'll need this assumption in (c)). 1. A Planning Problem. Suppose a hypothetical benevolent social planner wants to maximize total output for this economy by choosing how much labor each firm gets. This planner solves the following problem: max An+Ana 11.2 subject to n₁ + n₂ = 1. Solve the planner's problem and find the optimal choices of n₁, 7₂ (hint: it's more direct if you substitute for n₂ using the constraint and solve the unconstrained maximization problem for 7₁, and then solve for n₂=1-₁.). Call these n. n. Calculate the ratio n/n as a function of A₁, A2 only. We'll use this ratio in part (c). 2. A Decentralization of the Planning Problem. Now let's consider how the market economy would allocate resources if it were allowed to function freely. Suppose labor must be hired at a wage w. Each firm i = 1,2 solves the problem max An n₂ 1-0 ww (a) Solve for firm is optimal labor input as a function of w. A, only. (b) Calculate the total demand for labor at the wage rate w. (e) Use Market Clearing in the labor market to obtain the equilibrium wage u as a function of A₁, A₂, a only. Recall that the total mass of labor input is 1. (d) Substitute the market clearing wage back into each firm's labor demand curve to get the equilibrium inputs of labor for each firm ni, ng. How do these compare with ni, n? Provide some intuition for why the market and the planning solution are related in the specific way you find.

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