Estimate the normal equations, returns to scale, analyze the test statistic and degrees of freedom of the model.

Consider a Cobb-Douglas production function where output (Q) is made from three inputs:

K = capital
L = labour
M = materials
In (Qi) = γo + γ1 log(Ki) +γ2 log(Mi) + ui
Where E (ui) = 0 and
K, L, M are non-random

(i) prepare out exact expressions (in terms of Q, K, and L) for the OLS ‘normal equation' (4 equations in 4 unknowns) for a data set with 50 observations.

(ii) Returns to scale (RTS) in a 3-factor Cobb-Douglas production function are given by γ1 + γ2 + γ3 . If OLS is unbiased, will the sum of the OLS slopes in our model provide an unbiased estimator for returns to scale? describe.

(iii) Reprepare this model so which it will always exhibit constant RTS (i.e., γ1 + γ2 + γ3 =1 ). Simplify terms so which you end up with an equation of the form:
Yi = γo + γ1X1i + γ2X2i + ui;

Where Y, X1 and X2 are functions of the variables in the original variables

(iv) describe how you could use OLS regression results from the two versions of the model (with and without constant returns to scale) to formally test whether or not the true production function exhibits constant returns to scale. Specify null and alternative hypothesis, the exact form of the test statistic, and the exact degrees of freedom for the test.