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In under grad intermediate macro classes they often teach the Solow model. While intuitively it makes sense, it seems like it no longer is really taken seriously anymore (I could be wrong).

From my understanding (which could I be wrong of course), there appears to be a move toward more endogenous growth theories that try to explain technology growth rather than taking it as a constant like the Solow model.

Can anyone provide me some examples of growth models that, at the moment, are considered the most complete or correct?

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Oded Galor's interesting work on Unified Growth Theory subsumes the Solow model, the Malthusian model and endogenous growth theories in a very stylized way. This model is of course "long term" and ignores institutions.

It is a OLG model where the main problem from the household side is a choice between quantity versus quality of offspring (human capital intensity or number of children), increasing human capital is costly. From the factors side there is a fixed factor land, that plays the role of the Malthusian restriction when technology growth is non existent or low, human capital, and a labor augmenting technological or productivity factor that grows exogenously, this is understood as technological progress.

The model generates a first malthusian trap, that is only escaped after there is some population reduction shock (think black death) or some technological shock. Then the Solow stage, and then a endogenous growth phase. Population dynamics plays a vital role, in the malthusian stage population is stagnant and dependent on the fixed land factor, then in the second stage it grows exponentially and in the third stage households choose quality over quantity and the population decreases. Think of Middle Ages, 20th century US and Japan in this century respectively as examples of this stages. Anyway I am a little rusty and it is not my field but this work is amazing.
Slides: Unified Growth Theory and Comparative Economic Development

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There exists also a tendancy to treat growth models with heterogeneous agents. I think the question is not to use the right growth model but contribute to the theory by trying to change "representative agent" framework. That's why there exists so many growth models with heterogeneous agents in recent years.

I don't think at all that endogenous growth models are better than Solow-Swan exogenous growth models. Using one of these models depends on what you are trying to show as an economic mechanism. For example, if you are interested by some economic questions as "how it could be possible to maintain a growth path in a developping country ?", sure that it will be more appropriate to use an endogenous model.

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