Chapter 1 The theory of comparative advantage

1.1 Notations and hypotheses

Let us consider two person $P_1$ and $P_2$, able to produce the goods $G_1$ and $G_2$ by spending some time (the factor of production, noted $L$).

The amounts of time $L$ that $P_1$ has to use to produce one unit of the goods $G_1$ and $G_2$ are respectively $a^{P_1}_{G_1}$ and $a^{P_1}_{G_2}$. Similarly, the amounts of time $L$ that $P_2$ has to use to produce one unit of the goods $G_1$ and $G_2$ are respectively $a^{P_2}_{G_1}$ and $a^{P_2}_{G_2}$. We assume that this $a$ coefficients are strictly positive real numbers.

While considering a trade between $P_1$ and $P_2$, we can observe the amount of time $L$ that they actually chose to spend for the production of both good $G_1$ and $G_2$. The amounts of time $L$ the person $P_1$ spent on the production of goods $G_1$ and $G_2$ are noted respectively $L^{P_1}_{G_1}$ and $L^{P_1}_{G_2}$. Likewise, the amounts of time $L$ the person $P_2$ spent on the production of goods $G_1$ and $G_2$ are noted respectively $L^{P_2}_{G_1}$ and $L^{P_2}_{G_2}$. We assume that the $L$ coefficients are strictly positive real numbers.

From this we can deduce that the production of goods $G_1$ by person $P_1$ amounts to $\frac{L^{P_1}_{G_1}}{a^{P_1}_{G_1}}$ units, whereas the production of goods $G_2$ by person $P_1$ amounts to $\frac{L^{P_1}_{G_2}}{a^{P_1}_{G_2}}$ units. In the same way, the production of goods $G_1$ by person $P_2$ amounts to $\frac{L^{P_2}_{G_1}}{a^{P_2}_{G_1}}$ units, whereas the production of goods $G_2$ by person $P_2$ amounts to $\frac{L^{P_2}_{G_2}}{a^{P_2}_{G_2}}$ units.

After the trade has taken place, the person $P_1$ possesses a quantity $Q^{P_1}_{G_1}$ of goods $G_1$ and a quantity $Q^{P_1}_{G_2}$ of goods $G_2$, and the person $P_2$ possesses a quantity $Q^{P_2}_{G_1}$ of goods $G_1$ and a quantity $Q^{P_2}_{G_2}$ of goods $G_2$.

For each person, the quantity of goods given to or received from the other person can be deduced from the difference between the quantity they possess in the end and the quantity they produce before the trade took place.

For example, if the quantity of goods $G_1$ produced by the person $P_1$ exceeds the quantity of goods $G_1$ possessed in the end by the person $P_1$, it means that $P_1$ gave a quantity $\frac{L^{P_1}_{G_1}}{a^{P_1}_{G_1}} - Q^{P_1}_{G_1}$ to the person $P_2$. In that case, the quantity of goods $G_1$ produced by the person $P_2$ is inferior to the quantity of goods $G_1$ possessed in the end by the person $P_2$, meaning that $P_2$ received $Q^{P_2}_{G_1} - \frac{L^{P_2}_{G_1}}{a^{P_2}_{G_1}}$.

1.2 The theorem

What we want to show is that, if the trade is advantageous for both persons $P_1$ and $P_2$, then they should specialize in the production of the good for wich they have a comparative advantage. In mathematical terms:

Theorem 1.1 (Little theorem of comparative advantage)
Under the assumptions detailed in the previous section, and if the trade is benefitting both persons engaged in it, then:

$$\left(\frac{a^{P_1}_{G_2}}{a^{P_1}_{G_1}}-\frac{a^{P_2}_{G_2}}{a^{P_2}_{G_1}}\right)\left(Q^{P_2}_{G_2} - \frac{L^{P_2}_{G_2}}{a^{P_2}_{G_2}}\right) < 0$$

Such an equation means that the person ${P_2}$ is exporting goods ${G_2}$ if this person has a comparative advantage for the production of these goods, and is importing them otherwise.

Reciprocally, because goods ${G_2}$ can only be exchanged for goods ${G_1}$, the person ${P_2}$ is exporting goods ${G_1}$ if this person has a comparative advantage for the production of these goods, and is importing them otherwise.

The same applies to the person ${P_1}$.

Corollary 1.1
Trade allows to increase productivity, and the productivity increase is solely due to the comparative advantages.

Proof (Proof of the corollary). The fact that trade sometimes allows to increase productivity is shown by the classic example of England and Portugal exchanging wine and cloth. Furthermore, the above theorem shows that the comparative advantage is a necessary condition of a mutually beneficial trade.

1.3 Equations and inequalities

In order to be able to prove the theorem, we need to write the hypotheses in a mathematical way.

The first thing to notice is that there is, for each type of goods produced, an equality between the quantities produced by the two persons, and the quantities possessed by them in the end. This provides us with a system of equations:

$$\begin{cases} \frac{L^{P_1}_{G_1}}{a^{P_1}_{G_1}} + \frac{L^{P_2}_{G_1}}{a^{P_2}_{G_1}} = Q^{P_1}_{G_1} + Q^{P_2}_{G_1} \\ \frac{L^{P_1}_{G_2}}{a^{P_1}_{G_2}} + \frac{L^{P_2}_{G_2}}{a^{P_2}_{G_2}} = Q^{P_1}_{G_2} + Q^{P_2}_{G_2} \end{cases} \tag{1} \label{eqn:equations}$$

The second thing to notice is that we want to be sure that the trades we are analysing are beneficial to both persons, meaning that thanks to the trade each person must have spent strictly less time in producing than what would have been necessary to get this person the same amount of goods. this can be mathematically expressed as follows:

$$\begin{cases} a^{P_1}_{G_1}Q^{P_1}_{G_1} + a^{P_1}_{G_2}Q^{P_1}_{G_2} > L^{P_1}_{G_1} + L^{P_1}_{G_2} \\ a^{P_2}_{G_1}Q^{P_2}_{G_1} + a^{P_2}_{G_2}Q^{P_2}_{G_2} > L^{P_2}_{G_1} + L^{P_2}_{G_2} \end{cases} \tag{2} \label{eqn:inequations}$$

With these two systems, we shall be able to prove the theorem 1.1.

1.4 Proof of the theorem

We can rewrite the two inequalities as:

$$\begin{cases} \left(a^{P_1}_{G_1}Q^{P_1}_{G_1} - L^{P_1}_{G_1}\right) + \left(a^{P_1}_{G_2}Q^{P_1}_{G_2} - L^{P_1}_{G_2}\right) > 0 \\ \left(a^{P_2}_{G_1}Q^{P_2}_{G_1} - L^{P_2}_{G_1}\right) + \left(a^{P_2}_{G_2}Q^{P_2}_{G_2} - L^{P_2}_{G_2}\right) > 0 \end{cases}$$

We can now divide the first inequality by $a^{P_1}_{G_1}$, and the second by $a^{P_2}_{G_1}$ (the $a$ coefficients being strictly positive). The inequalities become:

$$\begin{cases} \left(Q^{P_1}_{G_1} - \frac{L^{P_1}_{G_1}}{a^{P_1}_{G_1}}\right) + \frac{a^{P_1}_{G_2}}{a^{P_1}_{G_1}} \left(Q^{P_1}_{G_2} - \frac{L^{P_1}_{G_2}}{a^{P_1}_{G_2}}\right) > 0 \\ \left(Q^{P_2}_{G_1} - \frac{L^{P_2}_{G_1}}{a^{P_2}_{G_1}}\right) + \frac{a^{P_2}_{G_2}}{a^{P_2}_{G_1}} \left(Q^{P_2}_{G_2} - \frac{L^{P_2}_{G_2}}{a^{P_2}_{G_2}}\right) > 0 \end{cases}$$

From system $\eqref{eqn:equations}$ (equality between production and consumption) we know that:

$$\left(Q^{P_1}_{G_1} - \frac{L^{P_1}_{G_1}}{a^{P_1}_{G_1}}\right) + \left(Q^{P_2}_{G_1} - \frac{L^{P_2}_{G_1}}{a^{P_2}_{G_1}}\right) = 0$$

So if we sum the two previous inequalities, we can easily get:

$$- \frac{a^{P_1}_{G_2}}{a^{P_1}_{G_1}} \left(Q^{P_1}_{G_2} - \frac{L^{P_1}_{G_2}}{a^{P_1}_{G_2}}\right) - \frac{a^{P_2}_{G_2}}{a^{P_2}_{G_1}} \left(Q^{P_2}_{G_2} - \frac{L^{P_2}_{G_2}}{a^{P_2}_{G_2}}\right) < 0$$

From system $\eqref{eqn:equations}$ we also know that the following quantity is equal to $0$:

$$\frac{a^{P_1}_{G_2}}{a^{P_1}_{G_1}} \left[\left(Q^{P_1}_{G_2} - \frac{L^{P_1}_{G_2}}{a^{P_1}_{G_2}}\right) + \left(Q^{P_2}_{G_2} - \frac{L^{P_2}_{G_2}}{a^{P_2}_{G_2}}\right)\right]$$

So we can add it to the previous inequality, and simply recognize that we proved the theorem 1.1:

$$\left(\frac{a^{P_1}_{G_2}}{a^{P_1}_{G_1}} - \frac{a^{P_2}_{G_2}}{a^{P_2}_{G_1}}\right) \left(Q^{P_2}_{G_2} - \frac{L^{P_2}_{G_2}}{a^{P_2}_{G_2}}\right) < 0 \tag{3} \label{eqn:trade}$$

1.5 Generality of the theorem

The proof of the theorem is more general than the ones usually given to buttress the theory of comparative advantage. The approach proposed is indeed the mathematical fundamental reason why it is not possible to consider a profitable trade without considering the comparative advantage.

Remark. Many of the hypotheses usually used to prove the theory of comparative advantage were not used:

• the international immobility of the factor of production
• the national mobility of the factor of production
• the negligible costs of trade
• the unicity of the price of each type of goods, nationally or internationally (depending on the case considered, autarky or free trade)
• the equality of hourly profitabilities for the production of goods $G_1$ and $G_2$ (labour market equilibrium).

1.6 Going further

It is possible to extend the approach to the case of:

• more than two people and two goods
• more than one factor of production
• not necessarily constant returns.

Theorem 1.2 (Theorem of comparative advantage)
A trade between several persons can be beneficial to all of them only if:

• comparative advantages are involved (at least two exportations take place according to a comparative advantage), or
• increased returns are involved.

Corollary 1.2
Trade:

• allows to increase productivity, and the productivity increase from trade is solely due to the comparative advantages
• implies some specialization, and the productivity increase from specialization is solely due to increased returns.

The proof has been published in French, but is not yet available in English. The proof contains the equivalent of equation $\eqref{eqn:trade}$ in the case of arbitrarily large numbers of persons and goods involved in a trade.