# How to pin down the bias and its/their sources in derived technical change indices?

Using the following CES function I backed out factor-augmenting indices $A_1$ and $A_2$ using OECD country level macro data. This process is akin to Solow's (1956) method for deriving technology residuals. I want to examine the bias in the direction of technical change by examining the temporal evolution of the factor-augmenting indices.

$$Y=[((A_1L^\alpha) K^{(1-\alpha)})^\sigma+ A_2E^\sigma]^{1/\sigma}$$

The elasticity of substitution parameter $\sigma$ is 0.1 (Estimated using MLE). $Y$ is GDP. $E$ is energy flow.

There is no evidence of trend in the two technical change indices until the last few decades of the sample. From there on, the labour-augmenting index grows rapidly. The other index augmenting energy declines gradually. This divergence in the indices coincides with the sudden energy price shocks.

It might look pretty obvious that there was a labour saving and energy using bias in technical change. Though, I am stuck at this point, how would I link the derived factor-augmenting indices with factor prices and factor ratios to impart the technical change biases in a systematic way?

I tried a slightly different formulation because of the nesting of the Cobb-Douglas function in CES. The evidence pointed out by @luchonacho in previous posts shows that Cobb-Douglas is not an apropriate fit to the data given that there is little potential for $K$ and $L$ to be perfect substitutes.

An alternative formulation could be $$Y=[(A_lL^\nu+ A_kK^\nu)^\frac{\sigma}{\nu}+ A_eE^\sigma]^{1/\sigma}$$ However, estimating $\nu$ and $\sigma$ would not be easy due to the nonlinearities in the relationship. Thus, the first formulation was adopted. Any suggestions welcome! Alternative formulation, methods etc.

• Interesting research there. Some comments/questions: (i) what is your estimate of $\sigma$? The interpretation of labour-saving/augmenting depends on its value. (ii) have you tested for changes in $\sigma$? (iii) ideally you want your production function to be normalised. (iv) why assuming elasticity of substitution between capital and labour equal to one? Most of the evidence I have seen point toward $\sigma_{K,L}<1$. Maybe use a translog production function. (v) can you clarify your question? – luchonacho Oct 13 '17 at 12:08
• @luchonacho Many thanks, I've edited the post. Is it good enough? – london Oct 13 '17 at 13:12
• 0.01? Really? So basically, no substitution between K-L bundle and energy? is this in line with other research? As far as I know the translog allows for quite a lot of freedom in terms of substitution. You do not face the nonlinearity problem. – luchonacho Oct 13 '17 at 13:18
• Sorry, I hastily typed the original edit and ended up making a number of typos due to having to step out of my office. I've edited the post again. The elasticiy value is the average for several countries. Please have a look. – london Oct 13 '17 at 14:49
• What is Y in your data? Notice value added might not be a good proxy for Y, because this measures contribution from non-contemporaneously produced factors of production and not from intermediate inputs like energy, which are also produced by K and L. This is, $E=f(K_E,L_E)$, both of which are included in $K$ and $L$. – luchonacho Oct 13 '17 at 15:03