1.7.7.md

Solution to Review Question

by Qiang Gao, updated at May 20, 2017

---

Chapter 1 Finite-Sample Properties of OLS

Section 7 Application: Returns to Scale in Electricity Supply

...

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

$$ \log \left( \frac{ TC_i }{ p_{i2} } \right) = \beta_1 + \beta_2 \log ( Q_i ) + \beta_3 \log \left( \frac{ p_{i1} }{ p_{i2} } \right) + \beta_5 \log \left( \frac{ p_{i3} }{ p_{i2} } \right) + \varepsilon_i. \tag{2} $$

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.

---

Copyright ©2017 by Qiang Gao