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.
- 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?
- 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.
