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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.
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.