I see the tags "economics" and "mathematical-economics" in MathOverflow, so I believe it will be reasonable to address this question to mathematicians (instead of [economists][1], as I tried before, the more so that, I think, this sounds more like a mathematical question).
I have just read the book by a norwegian economist, [Erik Reinert][2], ["How Rich Countries Got Rich… and Why Poor Countries Stay Poor"][3]. I think it is very interesting for people who want to understand how modern economic works, and, in particular, where poverty comes from. And, what is unexpected, the problem, as Reinert it presents, seems to be quite simple from the point of view of a mathematician.
Reinert blames in everything the [Ricardo theory of comparative advantages][4]. In short, this is a scheme of reasonings that allows to understand, which commodity is more profitable in this or that country. Reinert writes that it is exactly this theory that justifies poverty in the modern world. He gives numerous examples, when arguments based on Ricardo's theory made some countries extremely poor, without hope for correcting the situation in future. He paints horrible pictures of the situation in Mongolia, Peru, Equador, Haiti, Russia, etc. Simultaneously, from what he writes it becomes clear why there are so many migrants in the modern world, so many jobless people, so many refugees, etc. -- so I recommend his book to everyone.
To explain the mathematical component of what he writes, I have to explain briefly the Ricardo theory. It is simple for any mathematician, I wouldn't even call it a theory, but just a
Ricardo example
------------------
Suppose we consider two countries, $A$ and $B$, and each of them produces two goods, $G_1$ and $G_2$, and the table of expenses is as follows:
$$
\begin{matrix}
& G_1 & G_2 \\
A: & 1 & 1 \\
B: & 2 & 4 \\
\end{matrix}
$$
(this means that 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).
Both goods, $G_1$ and $G_2$, are cheaper in the coutry $A$, so it seems evident, that it is more profitable to produce both $G_1$ and $G_2$ in the country $A$. But Ricardo notices that the difference in the comparative expences changes the situation: if the country $A$ conveys the production of $G_1$ to the country $B$, while $B$ conveys $G_2$ to $A$, and they begin to trade, this becomes more profitable for both countries, since
- in $A$ each man-hour still gives one unit of $G_2$, but when selling it to $B$, $A$ takes two units of $G_1$ (instead of one unit of $G_1$, as it was when they did not trade),
- in $B$ each man-hour still gives $1/2$ unit of $G_1$, but when selling it to $A$, $B$ takes $1/2$ unit of $G_2$ (instead of $1/4$ of $G_2$, as it was when they did not trade).
This justifies the following
> **Ricardo law:** if in a given country $A$ the expences in the production of a good $G_2$ in comparison with another good, $G_1$, is less that in a country $B$, then it is more profitable (for both countries) to produce $G_2$ in the country $A$ (and to trade).
What about many countries with many goods?
--------------------------
This way of thinking seems to be reasonable, when we have two countries with two goods. But in reality there are many countries with many goods, what does the theory tell about them? It is easy to see (this is a simple excercise), that the situation dramatically changes.
1. First, we can come to a situation when it is impossible to understand, what a given country must produce. For example, consider three countries with two goods and with the following table of expences:
$$
\begin{matrix}
& G_1 & G_2 \\
A: & 1 & 1 \\
B: & 2 & 4 \\
C: & 4 & 2 \\
\end{matrix}
$$
Here, 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 un unsolvable question (a logical paradox in this theory) appears,
> what the country $A$ should produce?
2. Second, sometimes it becomes impossible to understand, where a given commodity must be produced. For example, let us consider two countries with three goods like this:
$$
\begin{matrix}
& G_1 & G_2 & G_3\\
A: & 1 & 1 & 1 \\
B: & 2 & 4 & 1 \\
\end{matrix}
$$
Here
- in comparison with $G_2$ the expenses for producing $G_1$ are less in the country $B$, so the good $G_1$ must be produced in the country $B$,
- but in comparison with $G_3$ the expenses for producing $G_1$ are less in the country $A$, so the good $G_1$ must be produced in the country $A$.
So we obtain another unsolvable question (another paradox):
> in which country the commodity $G_1$ must be produced?
3. Moreover, if we look more carefully at Ricardo's example above, we can notice that nothing prevents us to consider workforce as another, third commodity in this situation, and to look at the comparative expences in its production. We can just change the unit of measure, and use the good $G_1$ instead of "man-hours", then the table of expenses becomes the following:
$$
\begin{matrix}
& \text{workforce} & G_2 \\
A: & 1 & 1 \\
B: & 1/2 & 2 \\
\end{matrix}
$$
(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 an amazing conclusion:
- the country $A$ must "abandon the production of its own workforce" (i.e. make all its citizens jobless, and import the workforce from $B$).
I am not sure that this can be called a logical paradox (since formally $A$ can abandon both $G_1$ and workforce), but I am sure everyone will agree that this shows one more problem in the theory (and this is a problem in real life, an illustration to what Reinert writes).
Where is the science?
---------------------
Of course, those examples mean that the process of making decisions in this field of human activity can't be that simple. So one can expect that there is a general theory where those difficulties are overcame. Reinert writes that the decisions in economics are still made on the base of the Ricardo theory (if you have no time to read his book, you can look at the chapter "Criticism" in the [Wikipedia article on Ricardo theory][4]). On the other hand, my [attempts to find the generalizations without those paradoxes][1] did not yield to success. So my question is
> Did anybody study these logical paradoxes in economic theory, which corrections for overcoming them were found, and where is this written?
P.S.
----
I ask my colleagues-mathematicians not to send me to economists again. This is a social issue, if [economists can't give answer][1], then mathematicians must do this (or at least emphasize the problem), I believe, this is our professional duty.
[1]: http://economics.stackexchange.com/questions/8290/ricardos-theory-of-comparative-advantage-for-many-countries?noredirect=1#comment10419_8290
[2]: https://en.wikipedia.org/wiki/Erik_S._Reinert
[3]: http://www.amazon.com/How-Rich-Countries-Poor-Stay/dp/1586486683
[4]: https://en.wikipedia.org/wiki/Comparative_advantage