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In textbooks on economics the Ricardo theory of comparative advantages is explained on the example of two countries with two goods. I suppose, there must be generalizations of Ricardo's theory for several countries with several goods. Can anybody advise me reading on this topic?

EDIT. I see that I have to explain, why I think that there must be a more complicated theory, than just the trick with the comparison of two countries with two goods. I have two reasonings.

  1. If we consider three or more countries, we can come to a situation of uncertainty on what a given country should produce. For example, suppose we have 3 countries, $A$, $B$, $C$, and each of them produces two goods, $G_1$ and $G_2$, and the table of expenses is as follows: $$ \begin{matrix} & G_1 & G_2 \\ A: & 1 & 1 \\ B: & 2 & 4 \\ C: & 4 & 2 \\ \end{matrix} $$ (i.e. in the country $A$ one unit of $G_1$ costs 1 man-hour, and the same for $G_2$, in the country $B$ one unit of $G_1$ costs 2 man-hours, while one unit of $G_2$ costs 4 man-hours, and in the country $C$ one unit of $G_1$ costs 4 man-hours, while one unit of $G_2$ costs 2 man-hours).

According to Ricardo,

  • the least comparative expenses for producing the good $G_1$ are in the country $B$, so the good $G_1$ must be produced in the country $B$,

  • at the same time the least comparative expenses for producing the good $G_2$ are in the country $C$, so the good $G_2$ must be produced in the country $C$.

And the problem appears,

what the country $A$ should produce?

  1. Even if we consider two countries which produce two goods, there must be an explanation of what must be done with the third good, the workforce (that exists everywhere). For example, suppose we have 2 countries, $A$ and $B$, and each of them produces two goods, $G_1$ and $G_2$, and the table of expenses is the following: $$ \begin{matrix} & G_1 & G_2 \\ A: & 1 & 1 \\ B: & 2 & 4 \\ \end{matrix} $$ (i.e. in the country $A$ one unit of $G_1$ costs 1 man-hour, and the same for $G_2$, and in the country $B$ one unit of $G_1$ costs 2 man-hours, while one unit of $G_2$ costs 4 man-hours). We can change the unit of measure, and use the good $G_1$ instead of the "man-hours", then the table becomes the following: $$ \begin{matrix} & \text{man-hour} & G_2 \\ A: & 1 & 1 \\ B: & 1/2 & 2 \\ \end{matrix} $$ (this means that in the country $A$ one man-hour costs one unit of $G_1$ and the same for one unit of $G_2$, and in the country $B$ one man-hour costs $1/2$ unit of $G_1$, while one unit of $G_2$ costs 2 units of $G_1$).

And the Ricardo trick gives the conclusion that

  • the contry $B$ must abandon the production of the good $G_2$ (in favour of its import from the country $A$), and

  • the country $A$ must "abandon the production of its own workforce (in favour of its import from the country $B$)".

Of course this is impossible. So the question arises,

how is this logical paradox resolved in economic theory?

NEW EDIT. My point is that if the scheme of reasoning is as simple as it is presented in textbooks on economics (and up to now I see nothing contradicting to what I say in the book by Krugman, Obstfeld and Melitz) -- "just look at the comparative expences and you'll see what is more profitable!" -- then nothing prevents us to consider workforce as another commodity, and to look at the comparative expences in its production. And logically, purely by the Ricardian scheme, we come to a conclusion that for some countries it is much cheaper to abandon the production of workforce (i.e. to make all or at least most of their citizens jobless, this is my example 2). Moreover, some contries must disappear at all (or isolate themselves from the international trade), since all the production there is unprofitable from the point of view of Ricardian theory (this is my example 1).

So my question is,

who studied these logical paradoxes in Ricardo's theory, which corrections for overcoming them were found, and where is this written?

NEW EDIT. I asked this question in a mathematical forum.

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  • $\begingroup$ I have now absolutely lost track of what you are asking. First you asked about many countries, now you are asking about the production of workforce. Please consider posting a new question that where you ask exactly what you mean to ask and provide the appropriate definitions. $\endgroup$ – Giskard Sep 16 '15 at 9:31
  • $\begingroup$ From the very beginning I am asking about general theory that considers many countries and many goods. The example with the workforce is an ilustration of this (the workforce is a third good in this example). There is no need to separate the question about many countries and the question about many goods, that would be an unnecessary complexification. There must be a general theory with both parameters: countries and goods. Does it exist? $\endgroup$ – Sergei Akbarov Sep 16 '15 at 9:47
  • $\begingroup$ I suspect that somethiong is not clear because of its abstractness. I can widen the explanation in the examples, or add some other ones. $\endgroup$ – Sergei Akbarov Sep 16 '15 at 9:55
  • $\begingroup$ I have answered that question. I don't think widening the explanation will do any good, it is already too long. Making the question succinct might be good. $\endgroup$ – Giskard Sep 16 '15 at 11:05
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    $\begingroup$ Well it appears that mathematicians gave you the same answer: you misinterpret the theories of Ricardo (even for two countries). It is a shame economists couldn't "give an answer". $\endgroup$ – Giskard Sep 18 '15 at 7:50
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International Economics: Theory and Policy by Krugman, Obstfeld, Melitz meets part of your requirement. The chapter on comparative advantages discusses two countries with several goods. I have to admit I am not sure what adding more than two countries would bring to the mix unless you want to change the equilibrium concept.

In response to your edit:
Actually the modell does not conclude what you claim in 2. The actual claim is that if country B abandons the production on any good, it will be $G_2$. But it could also be that it produces both goods. This occurs if at the price where country B is indifferent between producing $G_1$ and $G_2$, (all effective production choices maximize the value of country B's production) the entire labor force of country A is not be sufficient to meet the aggregate demand of $G_2$.

In your example this price ratio between $G_1$ and $G_2$ is $\frac{1}{2}$. It is not possible to say whether country A can meet the demand at this price level, this depends on the demand functions and the size of the labor pools.

With respect to claim 1.:
Again the demand and supply of the goods at price ratio 1 would determine the production of country A.

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  • $\begingroup$ In reality there are many countries. Is it possible that there is no theory for this case? $\endgroup$ – Sergei Akbarov Sep 15 '15 at 18:05
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    $\begingroup$ @SergeiAkbarov I am not sure what you want to show. It seems to me the same analysis would show the same result for multiple countries. I have never seen it written down and I think that is because it would yield no new results. $\endgroup$ – Giskard Sep 15 '15 at 18:43
  • $\begingroup$ I did not understand you. First, you introduce new parameters: demand functions, labour pools, supply... Are they elements of the Ricardo theory? Even if so, why can't we assume that these parameters allow to do what I say? Second, as far as I understand, I can't claim that "country A must abandon the production of its own workforce". Why? Logically it is possible in this scheme of resonings. $\endgroup$ – Sergei Akbarov Sep 15 '15 at 22:37
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    $\begingroup$ @SergeiAkbarov I would like to point out that I actually did not say anything about the part you quoted, I don't even understand what "production of its own workforce" means. But I think the main issue is that if you find these "new parameters" then your textbook has failed you. Again I recommend reading International Economics: Theory and Policy by Krugman, Obstfeld, Melitz. $\endgroup$ – Giskard Sep 16 '15 at 4:11
  • $\begingroup$ I edited again. $\endgroup$ – Sergei Akbarov Sep 16 '15 at 8:28
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i think you need at least as many goods as you have countries for the theory to work, in your case, country A ends up with no comparative advantage in any goods, its like the case where one country is excatly twice as productive for every good and so there is no comparative advantage to be gained.

If a third good existed, where A=1 B=2 C=2, A would be able to have a comparative advantage for that good

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