Task: Suppose that you have af(firm with a Cobb-Douglas production function in period t The onlytwist relative to our setup in the main text is that the firm does not use labor to produce output in period t + 1. The production function in that period is 1. Write down the optimization problem for the firm in this setup. It has o pay labor in period tuwr, and it discounts future dividends by The capital accumulation equation is standard. Assume, however that q+1 q 2. Using this specification of production, derive the first order optimality 3. Re-arrange the first order optimality conditions to derive explicit ex- 4. Re-arrange your answer from the previous part, making use of the 5. Create an Excel file. Assume the following values for exogenous conditions for the optimal choices of N and Kt+1 pression for the demand for N and the demand for Ki capital accumulation equation, to solve for an expression of . parameters: 1/3, = 0.1, At = 1, At+1 = 1, q 9-1 = 1, and K 2. Create a column of possible values of wi, ranging from a low of 1 to a high of 1.5, with a step of 0.01 between entries. For each possible value of wt, solve for a numeric value of N Plot w against the optimal value of N. Describe the resulting demand curve for labor. 6. Suppose that At increases to 1.1. Re-calculate the optimal value of Ne for each value of wi. Plot the resulting Nt values against W in the same plot as what you did on the previous part. What does the increase in At do to the position of the labor demand curve? Go back to assuming the parameter and exogenous values we star- ted with. Create a grid of values of ri ranging from a low 0.02 to a high of 0.1, with a space of 0.001 between. For each value of rt, solve for the optimal level of it. Create a graph with rt on the vertical axis and t on the horizontal axis. Plot this graph. Describe the graph. Task: Suppose that you have af(firm with a Cobb-Douglas production function in period t The onlytwist relative to our setup in the main text is that the firm does not use labor to produce output in period t + 1. The production function in that period is 1. Write down the optimization problem for the firm in this setup. It has o pay labor in period tuwr, and it discounts future dividends by The capital accumulation equation is standard. Assume, however that q+1 q 2. Using this specification of production, derive the first order optimality 3. Re-arrange the first order optimality conditions to derive explicit ex- 4. Re-arrange your answer from the previous part, making use of the 5. Create an Excel file. Assume the following values for exogenous conditions for the optimal choices of N and Kt+1 pression for the demand for N and the demand for Ki capital accumulation equation, to solve for an expression of . parameters: 1/3, = 0.1, At = 1, At+1 = 1, q 9-1 = 1, and K 2. Create a column of possible values of wi, ranging from a low of 1 to a high of 1.5, with a step of 0.01 between entries. For each possible value of wt, solve for a numeric value of N Plot w against the optimal value of N. Describe the resulting demand curve for labor. 6. Suppose that At increases to 1.1. Re-calculate the optimal value of Ne for each value of wi. Plot the resulting Nt values against W in the same plot as what you did on the previous part. What does the increase in At do to the position of the labor demand curve? Go back to assuming the parameter and exogenous values we star- ted with. Create a grid of values of ri ranging from a low 0.02 to a high of 0.1, with a space of 0.001 between. For each value of rt, solve for the optimal level of it. Create a graph with rt on the vertical axis and t on the horizontal axis. Plot this graph. Describe the graph
