<p>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.</p>

<p>In response to your edit: <br>
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 $G_2$.</p>

<p>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.</p>

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

CC BY-SA 3.0
2015-09-15T18:05:59.053Z
In reality there are many countries. Is it possible that there is no theory for this case?
Sergei Akbarov
CC BY-SA 3.0
2015-09-15T18:43:20.107Z
@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.
Giskard
CC BY-SA 3.0
2015-09-15T22:37:39.863Z
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.
Sergei Akbarov
CC BY-SA 3.0
2015-09-16T04:11:33.027Z
@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.
Giskard
CC BY-SA 3.0
2015-09-16T08:28:46.780Z
I edited again.
Sergei Akbarov
CC BY-SA 3.0
2015-09-15T13:14:36.193Z
2015-09-15T21:50:45.567Z
2015-09-15T21:50:45.567Z
Giskard
<p>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.</p>

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

CC BY-SA 3.0
2015-09-18T12:15:26.073Z
2015-09-18T12:15:26.073Z
Revoltic
<p>In textbooks on economics the <a href="https://en.wikipedia.org/wiki/Comparative_advantage" rel="nofollow noreferrer">Ricardo theory of comparative advantages</a> 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?</p>

<p>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.</p>

<ol>
<li>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:
$$
\begin{matrix}
 &amp; G_1 &amp; G_2 \\
A: &amp; 1 &amp; 1 \\
B: &amp; 2 &amp; 4 \\
C: &amp; 4 &amp; 2 \\
\end{matrix}
$$
(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). </li>
</ol>

<p>According to Ricardo, </p>

<ul>
<li><p>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 $B$,</p></li>
<li><p>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 $C$.</p></li>
</ul>

<p>And the problem appears, </p>

<blockquote>
  <p>what the country $A$ should produce?  </p>
</blockquote>

<ol start="2">
<li>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:
$$
\begin{matrix}
 &amp; G_1 &amp; G_2 \\
A: &amp; 1 &amp; 1 \\
B: &amp; 2 &amp; 4 \\
\end{matrix}
$$
(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:
$$
\begin{matrix}
 &amp; \text{man-hour} &amp; G_2 \\
A: &amp; 1 &amp; 1 \\
B: &amp; 1/2 &amp; 2 \\
\end{matrix}
$$
(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 $G_1$). </li>
</ol>

<p>And the Ricardo trick gives the conclusion that </p>

<ul>
<li><p>the contry $B$ must abandon the production of the good $G_2$ (in favour of its import from the country $A$), and</p></li>
<li><p>the country $A$ must "abandon the production of its own workforce (in favour of its import from the country $B$)".</p></li>
</ul>

<p>Of course this is impossible. So the question arises, </p>

<blockquote>
  <p>how is this logical paradox resolved in economic theory? </p>
</blockquote>

<p>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). </p>

<p>So my question is, </p>

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

<p>NEW EDIT. I asked this question in a <a href="https://mathoverflow.net/questions/218456/logical-paradoxes-in-economic-theory">mathematical forum</a>.</p>

CC BY-SA 3.0
2015-09-16T09:31:24.890Z
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.
Giskard
CC BY-SA 3.0
2015-09-16T09:47:32.410Z
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?
Sergei Akbarov
CC BY-SA 3.0
2015-09-16T09:55:46.913Z
I suspect that somethiong is not clear because of its abstractness. I can widen the explanation in the examples, or add some other ones.
Sergei Akbarov
CC BY-SA 3.0
2015-09-16T11:05:16.720Z
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.
Giskard
CC BY-SA 3.0
2015-09-18T07:50:01.267Z
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".
Giskard
CC BY-SA 3.0
2015-09-21T08:27:21.087Z
Pardon, a wrong verb. Edit: @denesp, my question here was not whether I understand the theory correctly or misinterpret it. I asked if there is a "theory for several countries with several goods". In the book you suggested there are no mentionings of several countries, only of several goods. So I received the answer about this not here, but in MO. However, if you want to add something to what I ask here or there, you are welcome.
Sergei Akbarov
CC BY-SA 3.0
2015-09-21T08:30:35.653Z
I decided to choose a different way of explaining my concerns.
http://economics.stackexchange.com/questions/8354/ricardos-theory-of-comparative-advantage
Giskard
CC BY-SA 3.0
2015-09-21T08:36:31.010Z
denesp, Revoltic and ejQhZ: after the explanation in MathOverflow, it is not important, how those paradoxes are resolved in that initial version of the theory. It is much more interesting, if they have traces in the modern theories. I would be grateful if somebody could answer the last questions in my post in MO.
Sergei Akbarov
CC BY-SA 3.0
2015-09-21T08:41:31.203Z
Unfortunately I still do not see any paradoxes.
Giskard
CC BY-SA 3.0
2015-09-21T08:42:39.270Z
It does not matter, how you call them. You can just answer the questions.
Sergei Akbarov
<international-trade><free-trade><comparative-advantage>
Ricardo's theory of comparative advantage for many countries
Sergei Akbarov
# Ricardo's theory of comparative advantage for many countries

(Asked by: username_0 on 2015-09-15)

In textbooks on economics the [Ricardo theory of comparative advantages](https://en.wikipedia.org/wiki/Comparative_advantage) 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?

2. 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](https://mathoverflow.net/questions/218456/logical-paradoxes-in-economic-theory).

**Tags:** international-trade, free-trade, comparative-advantage
> **Comments:**
> **username_1**: 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.  
> **username_0**: 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?  
> **username_0**: I suspect that somethiong is not clear because of its abstractness. I can widen the explanation in the examples, or add some other ones.  
> **username_1**: 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.  
> **username_1**: 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".  
> **username_0**: Pardon, a wrong verb. Edit: @denesp, my question here was not whether I understand the theory correctly or misinterpret it. I asked if there is a "theory for several countries with several goods". In the book you suggested there are no mentionings of several countries, only of several goods. So I received the answer about this not here, but in MO. However, if you want to add something to what I ask here or there, you are welcome.  
> **username_1**: I decided to choose a different way of explaining my concerns. http://economics.stackexchange.com/questions/8354/ricardos-theory-of-comparative-advantage  
> **username_0**: denesp, Revoltic and ejQhZ: after the explanation in MathOverflow, it is not important, how those paradoxes are resolved in that initial version of the theory. It is much more interesting, if they have traces in the modern theories. I would be grateful if somebody could answer the last questions in my post in MO.  
> **username_1**: Unfortunately I still do not see any paradoxes.  
> **username_0**: It does not matter, how you call them. You can just answer the questions.  

---

## 2 answers

### Answer by username_1 on 2015-09-15. Score: 1

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.

> **Comments:**
> **username_0**: In reality there are many countries. Is it possible that there is no theory for this case?  
> **username_1**: @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.  
> **username_0**: 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.  
> **username_1**: @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.  
> **username_0**: I edited again.  

---

### Answer by username_2 on 2015-09-18. Score: 0

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


---

