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I have the model in my electronic spreadsheet (created by me) and its results quite suprised me. In short, complete specialization (I just call it specialization) in good where a country has comparative advantage CAN lead to inefficiency, to decrease of production of one of the goods (cheese in this case). This result suprised me and I would like it checked in case a mistake (in my reasoning or calculations) was made. (I can also share the spreadsheet if you want)

Suppose there are two countries, USA and France. Each country has industries: cheese production industry and wine production industry. Each industry in each country has exactly 1000 workers(the top is the American workforce, the bottom is the French workforce)

enter image description here

But technologies are different. enter image description here

From this we conclude that opportunity costs are following: enter image description here

France has comparative advantage in cheese, while USA hold comparative advantage in wine. So France should concentrate all its laborforce in its chesee industry while USA should concentrate all its labor force in wine industry.

... except it will be bad news for world cheese production.

Production of cheese and wine in both countries before trade and (complete) specialization (the top is the American production, the bottom is the French production):

enter image description here

Production of cheese and wine in both countries after trade and (complete) specialization (the top is the American production, the bottom is the French production):

enter image description here

Now France produces all cheese in the world. But its cheese production technologies are too labor-intensive to produce enough cheese to compensate for fall of cheese production in USA. In order to avoid this USA will have to employ at the least 500 of its workers in cheese production, despite it beign American comparative disadvantage.

Thus we can conclude that in this case full specialization in cheese is a bad idea.

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Its not correct to say that production will be lower in the solution to the Ricardian model (in your case full specialization of US is not equilibrium outcome). In general in Ricardian model the production after specialization will always be either the same (if the relative opportunity costs are exactly same) or production will be higher. The reason why you got it lower in your case is that your model omits demands for the two goods and by your assumption on production technology US is economically big country while France is economically small (meaning the production possibility frontier (PPF) of US severely outstrips the PPF of France).

In situations like these you will find that only small country fully specializes while the big country specializes only partially. The reason for this is that even if the small country produces on full capacity it simply wont be able to satisfy the world demand, so once the US produces enough wine to fill up the word demand it will be also allocating the rest of the resources to cheese.

Fully deriving the optimum solution with adding utilities would take too much space so instead of full proof consider the following numerical simulation with your own example.

As you have shown in autarky we will have the following production:

  • US Cheese: 500 US Wine: 1000
  • France Ch: 250 France W: 333
  • World Ch: 750 World W: 1333

Now consider possible scenario with full specialization France (all in cheese) and partial US (1500 labor in wine and 500 in cheese):

  • US Cheese: 250 US wine: 1500
  • France Ch: 500 France W: 0
  • world Ch: 750 world W: 1500

Now you can see the production of cheese is exactly the same as before plus world has more wine. With just one extra unit of labor in cheese in US you would still get more cheese and also more wine compared to autarky.

Now to get an exact optimal solution you need to explicitly include consumer utility and demand, and you would see that in this case there would be excess demand that France could not satisfy which would be allocated to US. For the full treatment see Feenstra, R. C. (2015). Advanced international trade: theory and evidence. Princeton university press.

The undergraduate treatment of Ricardian model does not include demand functions explicitly because as long as none of the two countries have production frontiers that do not strictly outstrips the PPF of the other at all points you will always get full and never partial specialization so the exact nature of demand function does not matter (notice that in undergraduate Ricardian model you wont even find the final distribution of the goods unless the final demands are assumed in the question - this is to simplify the model to make it easier to solve but it does not represent the full model in its modern specification)

PS: of course on off equilibrium path any combination of outputs that satisfies the resource constraint is possible

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    $\begingroup$ "The problem here is that you start with the position where both countries produce only cheese. " No, they both produce both goods! I just didn't show it! $\endgroup$ – user161005 Dec 12 '19 at 16:39
  • $\begingroup$ @user161005 how’s that possible if before trade they produce 500 cheese in US and 250 in France and if you say they only have 1000 labor and in US cheese costs 2 units of labor and in France 4. Then if they produce both cheese and wine before trade you made a mistake here $\endgroup$ – 1muflon1 Dec 12 '19 at 16:41
  • $\begingroup$ "if you say they only have 1000 labor" They have 1000 labor in each industry, i.e. 2000 labor total per country $\endgroup$ – user161005 Dec 12 '19 at 16:42
  • $\begingroup$ Can I just give you my spreadsheet? $\endgroup$ – user161005 Dec 12 '19 at 16:42
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    $\begingroup$ I have no issues with the current wording. $\endgroup$ – Giskard Dec 13 '19 at 4:55

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