1.7.8.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 1.7.8

Taking logs of both sides of the production function (1.7.1), one can derive the log-linear relationship

$$ \log ( Q_i ) = \alpha_0 + \alpha_1 \log ( x_{i1} ) + \alpha_2 \log ( x_{i2} ) + \alpha_3 \log ( x_{i3} ) + \varepsilon_i, \tag{1} $$

where

$$ \alpha_0 \equiv \mathrm{E} [ \log ( A_i ) ], \qquad \varepsilon_i \equiv \log ( A_i ) - \mathrm{E} [ \log ( A_i ) ]. $$

Suppose, in addition to total costs, output, and factor prices, we had data on factor inputs. (a) Can we estimate $$ \alpha $$'s by applying OLS to this log-linear relationship? Why or why not? (b) Suggest a different way to estimate $$\alpha$$'s.

Solution

(a) The economic interpretation of the error term $$ \varepsilon_i $$ represents the firm's production efficiency relative to the industry's average efficiency. The input choice of the firm $$ ( x_{i1}, x_{i2}, x_{i3} ) $$ can depend on $$ \varepsilon_i $$, making regressors not be orthogonal to the error term. Applying OLS to log-linear relationship (1) will result in biased estimator, so we cannot estimate $$\alpha$$'s using OLS.

(b) Following microeconomic theory, under the Cobb-Douglas technology, input shares do not depend on factor prices.

This can be seen as follows. From equation (8) and (10) in solution to review question 1.7.1,

$$ \begin{align} \frac{ p_{i2} x_{i2} }{ p_{i1} x_{i1} } & = \frac{ \alpha_2 }{ \alpha_1 }, \tag{1} \\ \frac{ p_{i3} x_{i3} }{ p_{i1} x_{i1} } & = \frac{ \alpha_3 }{ \alpha_1 }. \tag{2} \end{align} $$

The input shares are calculated as

$$ \frac{ p_{i1} x_{i1} }{ p_{i1} x_{i1} + p_{i2} x_{i2} + p_{i3} x_{i3} } = \frac{1}{ 1 + \alpha_2 / \alpha_1 + \alpha_3 / \alpha_1 } = \frac{ \alpha_1 }{ \alpha_1 + \alpha_2 + \alpha_3 }, \tag{3} $$

similarly,

$$ \begin{align} \frac{ p_{i2} x_{i2} }{ p_{i1} x_{i1} + p_{i2} x_{i2} + p_{i3} x_{i3} } & = \frac{ \alpha_2 }{ \alpha_1 + \alpha_2 + \alpha_3 }, \tag{4} \\ \frac{ p_{i3} x_{i3} }{ p_{i1} x_{i1} + p_{i2} x_{i2} + p_{i3} x_{i3} } & = \frac{ \alpha_3 }{ \alpha_1 + \alpha_2 + \alpha_3 }. \tag{5} \end{align} $$

It is evident that these input shares are determined completely by parameters and do not depend on factor prices.

Under constant returns to scale, these shares equal to $$\alpha_1$$, $$\alpha_2$$ and $$\alpha_3$$ respectively. So we can estimate these parameters using sample averages of input shares.

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Copyright ©2017 by Qiang Gao