Lags and holes in growth models

It is usual in empirical growth models to include a lag term for income. So the model looks like:

natural log of income this year = a + b*natural log of income last year + other terms like the amount of capital and knowledge in the country

a and b are constants. b is usually found to be less than one, so that writing the equation as
ln(income at t / income at t-1) = a + (b-1)*ln(income at t-1) + other terms
shows that as income increases, the rate of growth slows down, other things being equal.

A big problem with this model, which emerges even if pure economic analysis is ignored, is the amount of theoretical omission implied by the lagged term. There is no information why the lagged term occurs, it just occurs and is important. Why does growth slow down if people get richer? There is slowdown because of declining returns from capital and labour, but surely these are already included in the model? Why is a model like ln income = a + b*capital + d*education not sufficient?

One of the major empirical models (the MRW model, discussed in the "disproportionately useful" series in this blog) presents a derivation why the lag term occurs. It is essentially a statistical outcome of the instantaneous model ln income = a + b*capital + d*education, together with an equation for accumulating capital over time. However, the assumptions in the derivation have been tested and found to be unlikely. So even if the lagged model is correct for some reason, it probably isn't because of the MRW derivation and its assumptions.

Personally, I would say that some of the reason for slowing growth as countries become richer is because of technological convergence. When countries are poor, they lag behind other countries in technology and can receive significant economic benefit from technological transfers from abroad. As their income catches up, along with technology, they can receive fewer benefits from transfers.

Technological transfers do not explain the whole story, or the technological leaders would show no dependence on the lagged term. In fact, in the original MRW paper, OECD incomes show less dependence on the lagged term than poorer countries' incomes, but there is still significant dependence.

I would guess that network construction accounts for part of the explanation for the lagged term. That is, when countries are poor, people can easily find people who want to buy things and who can sell them things they want. As countries become richer, these opportunities become harder to find, as many of the sales links are already established. The probability of success in a random hunt for a new sales partner declines over time under a simple model with a fixed number of people in an economy and a fixed number of possible sales between them. This network construction could explain why income falls over time, and has a neat mathematical representation.

A difficulty with this explanation is that network formation is difficult to observe. While the model does develop a precise functional form of convergence, so that the model could be tested indirectly through implied model misspecification, the form would depend on the way that networks are built and the testing may be complex.