Suppose a firm has a Cobb-Douglas production function Y=zKN- with capital and productivity fixed. The firm...

 

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Suppose a firm has a Cobb-Douglas production function Y=zK"N¹- with capital and productivity fixed. The firm pollutes the environment, so the government gives it a choice: pay a (proportional) taxi per unit of output or hire additional workers to clean up the pollution. In the second case, the firm would have to hire z additional workers per each unit of output in order to clean up the pollution. These additional workers are hired in the same labor market as the production workers. (a) Suppose the firm chooses to pay the proportional tax. Set up the firm's optimization problem and solve for the optimal labor demand N₁ and the maximal real profit T₁. (b) Suppose the firm chooses to hire workers instead. Set up the firm's optimization prob- lem and solve for the optimal labor demand N₂ and the maximal real profit #₂. (c) Under what conditions, if any, would the firm prefer to hire more workers instead of paying the tax? Show it mathematically and provide some intuition. Suppose a firm has a Cobb-Douglas production function Y=zK"N¹- with capital and productivity fixed. The firm pollutes the environment, so the government gives it a choice: pay a (proportional) taxi per unit of output or hire additional workers to clean up the pollution. In the second case, the firm would have to hire z additional workers per each unit of output in order to clean up the pollution. These additional workers are hired in the same labor market as the production workers. (a) Suppose the firm chooses to pay the proportional tax. Set up the firm's optimization problem and solve for the optimal labor demand N₁ and the maximal real profit T₁. (b) Suppose the firm chooses to hire workers instead. Set up the firm's optimization prob- lem and solve for the optimal labor demand N₂ and the maximal real profit #₂. (c) Under what conditions, if any, would the firm prefer to hire more workers instead of paying the tax? Show it mathematically and provide some intuition.