Jack Wong, a Tokyo investor, is considering plans to develop a primary steel plant in Japan. After reviewing the initial design proposal, he is concerned about the proposed mix of capital and labor. He has asked you to prepare several production functions using some historical data from the United States. The data file Metals contains 27 observations of the value-added output, labor input, and gross value of plant and equipment per factory.
a. Use multiple regression to estimate a linear production function with value-added output regressed on labor and capital.
b. Plot the residuals versus labor and equipment. Note any unusual patterns.
c. Use multiple regression with transformed variables to estimate a Cobb-Douglas production function of the form
Y = β0Lβ1Kβ2
where y is the value added, L is the labor input, and K is the capital input.
d. Use multiple regression transformed variables to estimate a Cobb-Douglas production function with constant returns to scale. Note that this production function has the same form as the function estimated in part c, but it has the additional restriction that β1 + β2 = 1. To develop the transformed regression model, substitute β2 as a function of β1 and convert to a regression format.
e. Compare the three production functions using residual plots and a standard error of the estimate that is expressed in the same scale. You will need to convert the predicted values from parts c and d, which are in logarithms, back to the original units. Then you can subtract the predicted values from the original values of Y to obtain the residuals. Use the residuals to compute comparable standard errors of the estimate.