Since is not going to be easy to measure the rate of K/H to distinguish between poor and rich, then including the human and physical capital how is going to help estimating the convergence among the countries. if we show the (Y=F(K/H,AL) and (K/H)dot=s(K/H)-depreciation rate*(K/H))
Parente and Prescott (1994 and 2000) analyzed a similar issue but they were focused on RBC and TFP,. In particular, they assumed that technology is increasingly more easily disseminated throughout the world and were focused on quantifying the level of barriers that each country was facing relatively to the tech leader. In this realm, given that you are interested in labor efficiency as well, not only capital per worker, their papers might give you additional insights because their analysis is based on paths of long-run convergence envisioned in terms of capital per worker. These are the refs:
Parente SL, Prescott EC (1994) Barriers to technology adoption and development. J Polit Econ 102(2):298–321
Parente SL, Prescott EC (2000) Barriers to riches. MIT Press, Cambridge - it is a small book but it shows detailed derivations of their model.
I humbly also recommend you to check my co-authored paper. There you will find detailed derivations of the model adopted and insights about the calibration procedure as well as data sources:
https://www.researchgate.net/publication/225409892_Barriers_to_technological_adoption_in_Spain_and_Portugal
Article Barriers to Technological Adoption in Spain and Portugal
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