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Section 7 Application: Returns to Scale in Electricity Supply
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Review Question 1.7.7
A more realistic assumption about the rental price of capital may be that there is an economy-wide capital market so $$p_{i2}$$ is the same across firms. In this case,
(a) Can we estimate the technology parameters?
(b) Can we test homogeneity of the cost function in factor prices?
Solution
(a) When $$p_{i2}$$ is the same across firms, there will have the perfect multicollinearity problem, such that Assumption 3 does not hold. So it will not be possible to estimate the 5 parameters $$ ( \beta_1, \ldots, \beta_5 ) $$ simultaneously from unrestricted regression (1.7.7).
But recall that $$ ( \beta_1, \ldots, \beta_5 ) $$ are not 5 independently free parameters,
$$
\beta_3 + \beta_4 + \beta_5 = 1,
\tag{1}
$$
It is safe to disregard $$ \beta_4 $$ and estimate $$ ( \beta_1, \beta_2, \beta_3, \beta_5 ) $$ from the restricted OLS regression
Then the estimate of $$\beta_4$$ can be calculated from (1).
So the answer is yes. Even though there will be prefect multicollinearity problem if $$p_{i2}$$ is the same across firms, $$ ( \beta_1, \ldots, \beta_5 ) $$ can be estimated.
(b) No, because when the price of capital is constant across firms we are forced to use the adding up restriction $$ \beta_3 + \beta_4 + \beta_5 = 1 $$ to calculate $$ \beta_4 $$ (capital's contribution) from the OLS estimate of $$\beta_3$$ and $$\beta_5$$. After all, we can't test the assumption which can't be relaxed.