I was in Boston for the NBER summer institute on Friday, sitting in on what it typically called either "Growth day" or "Jones/Klenow" after the organizers. Regardless, here's the program. It's a chance to see what is some of the cutting/bleeding edge research in economic growth.
The first paper I saw was by Ezra Oberfield of Princeton and Devesh Raval of the Federal Trade Commission (I missed the Grossman/Helpman paper because I like to sleep, and didn't get to Boston until 10:15am - sue me). They were doing two things. (1) providing an estimate of the aggregate elasticity of substitution (EOS) between capital and labor and (2) using that to try and account for the decline in labor's share of income over the last 30-40 years.
On (1), they made the point that the aggregate EOS is not a technological constant, but rather is an artifact of the micro-level EOS. Specifically,
$$ \sigma^{agg} = (1-X)\sigma^{micro} + X \epsilon $$
where ${\sigma^{micro}}$ is the EOS at the plant level. The weighting term ${X}$ reflects the variation in capital shares across firms. ${\epsilon}$ is the elasticity of demand for plant output. The demand elasticity is in to account for the fact that some of the response to a change in factor prices is to move demand away from the plants that tend to use the more expensive factor.
Regardless, Ezra and Devesh provide evidence that ${X}$ is really close to zero, so essentially this demand adjustment is negligible, and the aggregate EOS is roughly equivalent to the micro EOS. They estimate this from plant-level data, and find something like 0.52, meaning that capital and labor are not easily substituted for each other. Over time, the aggregate EOS is roughly stable at around 0.70, based on their values for ${X}$ and ${\epsilon}$.
On (2), given their aggregate EOS, the implication is that the decline of labor's factor share is biased technical change. Increased automation, IT investment, and offshoring, among other things, have driven down labor's share of output down over time.
Changes in factor prices alone (wages and rental rates) would have raised labor's share of output over this period, they find. The force of biased technical change was so strong it overcame that tendency.
It's worth noting how important finding the EOS1, then firms can switch easily from labor to capital. Relatively cheap capital is substituted for labor, and labor's share drops. If EOS>1, then the decline in labor share is driven in part by more expensive labor, and hence the implied degree of biased technical change is smaller.