1.7.3.md

Solution to Review Question

by Qiang Gao, updated at May 20, 2017

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Chapter 1 Finite-Sample Properties of OLS

Section 7 Application: Returns to Scale in Electricity Supply

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Review Question 1.7.3 (Recovering technology parameters from regression coefficients)

Show that the technology parameters ($$\mu$$, $$\alpha_1$$, $$\alpha_2$$, $$\alpha_3$$) can be determined uniquely from the first four equations in (1.7.5),

$$ \begin{align} \beta_1 & = \mu, \tag{1.7.5a} \\ \beta_2 & = \frac{1}{r}, \tag{1.7.5b} \\ \beta_3 & = \frac{ \alpha_1 }{ r }, \tag{1.7.5c} \\ \beta_4 & = \frac{ \alpha_2 }{ r }, \tag{1.7.5d} \end{align} $$

and the definition $$r \equiv \alpha_1 + \alpha_2 + \alpha_3$$. (Do not use the fifth equation $$ \beta_5 = \alpha_3 / r $$.)

Solution

From (1.7.5a) we have

$$ \mu = \beta_1. \tag{1} $$

Substituting (1.7.5b) into (1.7.5c) we have

$$ \alpha_1 = \frac{ \beta_3 }{ \beta_2 }. \tag{2} $$

Similarly, substituting (1.7.5b) into (1.7.5d) we have

$$ \alpha_2 = \frac{ \beta_4 }{ \beta_2 }. \tag{3} $$

Finally, using the definition of $$r$$ and (1.7.5b),

$$ \alpha_1 + \alpha_2 + \alpha_3 = \frac{1}{\beta_2}, \tag{4} $$

substituting (2) and (3) into (4) and rearrange terms,

$$ \alpha_3 = \frac{1}{\beta_2} - \frac{\beta_3}{\beta_2} - \frac{\beta_4}{\beta_2} = \frac{ 1 - \beta_3 - \beta_4 }{ \beta_2 }. \tag{5} $$

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Copyright ©2017 by Qiang Gao