Question

Economists at the Wilson Company are interested in developing a production function for fertilizer plants. They have collected data on 15 different plants that produce fertilizer as shown in the table below.

Plant	Production	Capital	Labour
1	606	18891	700
2	566	19201	652
3	647	20655	823
4	524	15082	650
5	712	20300	859
6	488	16079	613
7	762	24194	851
8	443	11504	655
9	821	25970	901
10	398	10127	550
11	897	25622	842
12	359	12477	541
13	979	24002	949
14	332	8042	576
15	1065	23972	926

1. Estimate the Cobb-Douglas production function Q = αL^β1K^β2 where Q = output; L = labour input; K = capital input; and α, β₁, and β₂ are the parameters to be estimated. (Note: If the regression program on your computer does not have a logarithmic transformation, manually transform the preceding data into the logarithms before entering the data into the computer.)

2. Test whether the coefficients of capital and labour are statistically significant.

  

3. Determine the percentage of the variation in output that is "explained" by the regression equation.

4. Determine the labour and capital production elasticities 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

a.

We know that the logarithmic model of Cobb-Douglas production function is given by:

$lnQ = \alpha_0 + \beta_1lnL + \beta_2lnK + u$

where α₀ = ln(α)

So, calulating the log values of Q, L and K, we conduct the regression in Excel and obtain the results table as:

SUMMARY OUTPUT
Regression Statistics
Multiple R	0.97353
R Square	0.947761
Adjusted R Square	0.939054
Standard Error	0.089983
Observations	15
ANOVA
	df	SS	MS	F	Significance F
Regression	2	1.762813	0.881407	108.856	2.03E-08
Residual	12	0.097164	0.008097
Total	14	1.859977
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%	Lower 95.0%	Upper 95.0%
Intercept	-4.7488	0.809035	-5.86972	7.6E-05	-6.51154	-2.98607	-6.51154	-2.98607
lnK	0.414769	0.135089	3.070333	0.009711	0.120435	0.709103	0.120435	0.709103
lnL	1.077804	0.25046	4.3033	0.001026	0.532099	1.623509	0.532099	1.623509

So, the estimated double-log model is:

$\widehat{lnQ} = -4.75 +1.08*lnL + 0.41*lnK$

So, the estimated values for β1 and β2 are 1.08 and 0.41 respectively.

Since, α₀ = ln(α), we can calculate the value of αas α = antilog(α_0).

So, in this case,

$\widehat{\alpha } = antilog(-4.75) = 0.0086$

b.

By looking at the p-values in the regression table, we see that the p-value for the coefficient of lnK is 0.009. At 5% level of significance, since the p-value < 0.05, we reject the null hypothesis and infer that β2 is statistically significant.

The p-value for the coefficient of lnL is 0.001. At 5% level of significance, since the p-value < 0.05, we reject the null hypothesis and infer that β1 is statistically significant.

So, the coefficients of capital and labour are both statistically significant.

c.

The R-squared value is 0.947. This implies that 94.7% of the variation in output is explained by the regression equation.

d.

Since the above regression model is a double-log model, β1 and β2 represent elasticties of output with respect to labour and capital respectively.

$\widehat{\beta_1 } = 1.08$ represents the elasticity of output w.r.t labour.

This means that when the amount of labour increases by 1%, then on average, output increases by 1.08%, holding capital constant.

represents the elasticity of output w.r.t capital.

B2 = 0.41

This means that when the amount of capital increases by 1%, then on average, output increases by 0.41%, holding labour constant.

e.

In order to find if the production function exhibits increasing, decreasing or constant returns to scale, we need to calculate the sum of estimated coefficients of labour and capital.

So, $\widehat{\beta_1 } + \widehat{\beta_2} = 1.08 + 0.41 = 1.49$

Since, this value is greater than 1, we can infer that the production function exhibits increasing returns to scale.