Answer to Question #113598 in Statistics and Probability for nadzirah

Question

Economists at the Wilson Company are interested in developing a production function for fertilizer plants. They collected data in 15 different plants that produce fertilizer ( see in the following table).

QUESTION
1. Estimate the Cobb-Douglas production function Q= αLβ₁Kβ₂, where Q = output; L = labor input; K =capital input and α, β₁, and β₂ are the parameters to be estimated
2. Test whether the coefficients of capital and labor are statistically significant.
3. Determine the percentage of the variation in output that is “explained” by the regression equation
4. Determine the labor and capital estimated parameters and give an economic interpretation of each value.
5. Determine whether this production function exhibits increasing, decreasing, or constant returns to scale. (Ignore the issue of statistical significance.)

Answer 1

1. We estimate the model

Q= αLβ1Kβ2

We take log to make the model loglinear and make it suitable for OLS regression. The regression output is given by:

SUMMARY OUTPUT

 

 Regression Statistics

 

Multiple R 0.973748

 

R Square 0.948186

 

Adjusted R Square 0.93955

 

Standard Error 0.038937

 

Observations 15

ANOVA

significance

df SS MS F F

Regression 2 0.332922 0.166461 109.7978 1.94E-08

Residual 12 0.018193 0.001516

Total 14 0.351115

 

Standard

Coefficients Error t Stat  P-value Lower 95%

Intercept -2.06496 0.349946 -5.90079 7.24E-05 -2.82742

logK 0.415207 0.134517 3.086656 0.009421 0.12212

Log L 1.078004 0.249335 4.323522 0.00099 0.53475

The estimated model is given by:

Log Q==2.06+.41log K+1.07logL

Here, p value of log K&L are<.01. thus, the coefficient of capital and labor are statistically significant at 1% level.

2. Here, p value of log K&L are<.01. thus, the coefficient of capital and labor are statistically significant at 1% level.

3 . Here, R^2=.94, thus 94 percentage of the variation in output is explained by the regression equation

4 .

EK = β1 = 0.515 ( 1% increase in K yield a 0.415% increase in Q)

EL = β2 =1.078 ( 1% increase in K yield a 1.078% increase in Q)

5 .

   Since the sum of the exponents of the capital and labor inputs exceeds 1.0 (1.493), the production function exhibits increasing returns to scale.