My coauthor sent me some Swedish capital stock estimates for 1850-2000 which allows me to estimate the production function equation that explains economic output in terms of labor, capital and energy in addition to the energy cost ratio equation I estimated on its own before. Having two equations which share many of the same parameters makes estimating those parameters accurately a lot easier. I got some moderately sensible results pretty much straight away. But this is a pretty tricky econometric problem for the following reasons:
1. The production function we are using (CES function) is non-linear and non-linear econometrics is always harder than linear econometrics.
2. Variables like capital, GDP, and energy are "stochastically trending" i.e. they are random walks (now you know why this blog has the title it does :)). Econometrics with trending or random walk variables is tougher than with more classically behaved variables that just fluctuate around an average value.
3. The variables on the right hand side of our equations are almost certainly endogenous in the economic system. The capital stock is affected by the level of GDP and not just vice versa. Econometrics with endogenous variables is trickier too.
4. The rate of technological change has not necessarily been constant over the last 200 years. In fact it almost certainly hasn't. We need to deal with that too.
From the preliminary econometric results and simulations I've done in Excel it seems that the elasticity of substitution between capital and energy is between 0.3 and 0.7. This is roughly the range that Koetse et al. found in their meta-analysis of the elasticity of substitution between capital and energy. The traditional Cobb-Douglas model used very extensively in empirical and theoretical applications assumes that it is 1.0 instead, which would mean that energy availability isn't much of a constraint on economic output and growth. But, instead, it seems that it is.
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