The division of labor is limited by the extent of the market.
The model is a modified version of the simplified Romer 90 model. The modification is that there is a minimum efficient scale for the production of intermediate goods.
Gross output in the growing sector is (sum i = 1 to N of x_i^alpha)L1^(1-alpha) where x_i is the amount of the ith intermediate good used. There is also another way to produce the final product 1 for 1 from labor output = L2. L1+L2 = L which is fixed.
intermediate goods can be made from the final good one for one, but one must make at least one unit.
There is a small closed economy with (alpha)(L^(1-alpha)) <1. So in this economy it is not efficient to use or produce any intermediate goods. So N is fixed at zero and there is no growth. With free trade and no transporation costs, the relevant L is the world labor force, so it makes sense to make intermediate goods. They have to be invented and intellectual property is protected. Except for the minimum efficient scale of 1 unit, this is Barro and Sala i Martin's simplified version of Romer's 1990 model. Well also the number of inventions is a whole number, because making it a continuum is silly. Value added is proportional to N. So is the real wage. Increased N is technological progress and is the engine of growth and increasing produtivity. N only grows if L is large enough. L is world labor supply if there is free trade. Under autarchy small countries have no growth (which costs more than 1% of potential GDP). The model is very simple and actually very old. Here I come to an embarrassing conclusion. I think the minimum efficient scale isn't even needed at all -- it just makes the result extreme. In fact, I think the model as presented in the textbook has the effect high L causes high growth. There is no minimum efficient scale and no backstop no intermediate goods technology. It was decided that the scale effect was unreasonable, so the model was changed to eliminate it. This eliminated the effect of trade on productivity growth. I don't think it was difficulty of finding a model. It was a consensus on what is a reasonable thing for a model to do. In particular, there was a habit in (not so good) empirical work of treating each country as independent. I mean the standard work horse model confronted with data assumed no trade. Then it implied big countries grow faster than small countries. Ooops. So the model was modified. Then removing a counterfactual implication of the counterfactual no trade assumption removed all effects of trade on productivity growth, and at least one very smart person decided that theory suggested that there was no effect of trade on growth. In fact very old simple theory suggested cases where there could be no growth without trade. And I have 12 minutes left. I will not waste your time suggesting you read more just so that I can present the model in 30 minutes.