By way of illustrating the preceding discussion consider the data given in Table 8.8. Attempting to fit the Cobb-Douglas production function to these data, yielded the following results:

lrfGoP, = -1.6524 + 0.3397 ln Labor, + 0.8460 ln Capital, (8.7.13)

where RSSUR is the unrestricted RSS, as we have put no restrictions on estimating (8.7.13).

We have already seen in Chapter 7 how to interpret the coefficients of the Cobb-Douglas production function. As you can see, the output/labor elasticity is about 0.34 and the output/capital elasticity is about 0.85. If we add these coefficients, we obtain 1.19, suggesting that perhaps the Mexican economy during the stated time period was experiencing increasing returns to scale. Of course, we do not know if 1.19 is statistically different from 1.

To see if that is the case, let us impose the restriction of constant returns to scale, which gives the following regression:

ln(GDP/Labor), = —0.4947 + 1.0153 ln (Capital/Labor), (8.7.14)

t = (—4.0612) (28.1056) p value = (0.0007) (0.0000)

where RSSR is the restricted RSS, for we have imposed the restriction that there are constant returns to scale.



EXAMPLE 9.3 (Continued)

Since the dependent variable in the preceding two regressions is different, we have to use the Ftest given in (8.7.9). We have the necessary data to obtain the Fvalue.

Note in the present case m = 1, as we have imposed only one restriction and (n — k) is 17, since we have 20 observations and three parameters in the unrestricted regression.

This Fvalue follows the Fdistribution with 1 df in the numerator and 17 df in the denominator. The reader can easily check that this F value is not significant at the 5% level. (See Appendix D, Table D.3.)

The conclusion then is that the Mexican economy was probably characterized by constant returns to scale over the sample period and therefore there may be no harm in using the restricted regression given in (8.7.14). As this regression shows, if capital/labor ratio increased by 1 percent, on average, labor productivity went up by about 1 percent.

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