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.
