# A Micro Founded Model in Which Trade Causes Higher Productivity Growth

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.

Knowing nothing more than the small introduction above, I hazard a speculation.

In the closed system of value added chains the equilibrium tends to equalize all paths in length so each good has the same path length and thus the distribution net is balanced. Alternatively, each value chain is at least integer factor of the typical chain. These conditions are the conditions that minimize transaction counts which is what we need.

I should mention the issues revealed by chain analysis. Government, in the USA does not keep a balanced distribution. A town in California is four steps removed from DC, but in Montana it is two steps. his is an artifact of us not being a proportional democracy, but a republic. So in a Republic, DC can only engage in activities that are essentially multiples of two in chain length, and that means mainly things like social security, a short path.

Programs like No Child left behind can never find a balance until it is reduced to direct payment on a student attendance basis. Adding intermediate institutions makes the distribution unbalanced as the large states adapt, but the small cannot.

Tangentially related – Virtually every technological society from chipped flint blades on has conducted trade of some kind. Better stone for stone tools, better alloys for bronze, luxury goods of all kinds are found far from their source by 6000BCE, if not before. Without trade, we would not have the bronze age. Europe and the middle east would have been a hodge-podge of bronze, copper and stone societies operating in isolation. The wheel is only an important invention if you want to make it easier to transport something.

I hate to stink up your blog, but this is a different kind of analysis, logistics as an aggregate problem. It is a good idea to understand the mathematics a bit.

Value chains, in theory, meet a container algebra with boundary conditions on a finite chain. Consider iScrew, a train load of iron ore enters the chain, a box of screws exit the chain at the hardware store. At each step you have integer factorization, one container in yields n items out, which has its own container ratio, m. This is a finite, combinatorial boundary problem.

ut you have conditions on inventory,. At any given node, if inventory goes to zero the chain breaks. Or if inventory is full and you leave something on the dock you break conservation of goods. It becomes a queuing problem on finite directed graphs.

Matt:

If capacity of one node is greater than the output of the entire line, zero inventory at that node is ok. Most supply chains and production lines are not balanced. Typically, you will have one or more bottle necks which will constrain the rest of the production or vice versa.

Could you have used K’s own Big Push model to argue against him? The logic seems quite similar: http://web.mit.edu/krugman/www/dishpan.html