To me this seemed a little too neat. Economists love the inverted U-shape relationship, or concave relationships in general, because it implies the existence of some kind of equilibrium - just set labour share correctly and you will maximise investment. Or something like that.
It turns out to produce the ‘inverted U-shape’ curve Latty used a time fixed effect and a non-linear estimator on data from AMECO. While this may be somewhat normal in the literature on this topic - for such reasons as removing the effects of international business cycles and to control for any secular trend in international growth rates - ultimately these model choices fundamentally determine the results. Like most observational data from complex systems, we are applying models as mere plausible stories, with no objective criteria for assessing their validity.
Often the details (including the outliers) in these types of international comparisons reveal much more of the story than the general trends. To satisfy my curiosity I constructed a similar chart from OECD and World Bank data.
Below is the scatterplot of my sample (you will notice the chart is animated to show country data connected through time) . Eyeballing the data there seems to be no general time-independent inverted U-shape. This is confirmed by regressions on the sample without controls, and indeed, with a quadratic specification we find a positive coefficient on the squared term.
Notice that each country seems to occupy a distinct section of the chart. Such divergence between countries should be of great interest to macroeconomists. In my smaller sample below Japan has consistently high labour share and investment share, while New Zealand is consistently low on both measures. Then we have the US and UK with consistently low investment accompanying a rather high labour share.
My expectation is that a high labour share of income would result in greater investment share, if only because owners of capital would have greater incentives to invest and innovate in order to gain a return, rather than simply hold existing assets. Who knows whether this is the case at all. But it’s my current speculation.
In broad terms the maturing nature of these economies may mean that such a downward trend in labour share is naturally expected. Or it could be that policy choices for the past four decades have directed national income towards owners of capital, which resulted in lower investment in some cases but not others. The answer still seems completely unclear. If you want more analysis, you can read the Productivity Commissions recent work on the topic.
Like many macroeconomic curiosities, the relationship between wage share and investment is a puzzle unlikely to find a satisfactory conclusion any time soon.