1. Introduction

The Heckscher-Ohlin model is valuable property not only in international trade theory, but also in the microeconomics. There are a lot of theorems embedded in the model. It is sometimes confusing. Hence, I propose to use simple matrix algebra exercise to gain deeper understanding of it.

2. Model Construction

The HO model is essentially a mathmatic problem, with the assumptions of zero-profit (due to constant returns to scale), and full employment. The two conditions can be expressed in the language of matrix algebra. First, we denote \(A\) as the matrix of technology coefficients.

A <- matrix(c(0.7,0.3,0.5,0.5), nrow = 2, byrow = T)
A
##      [,1] [,2]
## [1,]  0.7  0.3
## [2,]  0.5  0.5

The matrix shows that it takes 0.6 unit of labor and 0.4 unit of capital to produce 1 unit of goods X. The labor-capital ratio is 1.5 for X, and 1 for Y. Hence, X is labor-intensive goods. In a closed economy, we assume the labor and capital endowment for a home country is:

endow_home <- matrix(c(10,8), nrow = 2)
endow_home
##      [,1]
## [1,]   10
## [2,]    8

Without specifying the price vector, we assume the home market is cleared, and the production matrix can be soly determined by the production technology and endowment.

production <- solve(A) %*% endow_home
production
##      [,1]
## [1,]   13
## [2,]    3

Hence, the economy will produce 10 units of X and 8 units of Y.

3. Rybczynski theorem

The theorem claims that an increase in an factor endowment will increase the output of goods which uses that factor intensively, with higher proportion. Here, we assume that the labor will increase by 10% to 11, and solve for the output again.

endow_home_2 <- matrix(c(11,8), nrow = 2)
production_2 <- solve(A) %*% endow_home_2
production_2
##      [,1]
## [1,] 15.5
## [2,]  0.5

The output of X will increase to 15 by 15%, and the output of Y will decrease to 0.5 by 83%. As we know, we have assumed that the production technology is CRS, which results in zero-profit, and we also know that the CRS function is h.o.d. 1. How does the allocation of factor inputs changes?

factor_input_1 <- matrix(c(0.7 * 13, 0.3 * 3, 0.5 * 13, 0.5 * 3), nrow = 2, byrow = T)
factor_input_1
##      [,1] [,2]
## [1,]  9.1  0.9
## [2,]  6.5  1.5
factor_input_2 <- matrix(c(0.7 * 15.5, 0.3 * 0.5, 0.5 * 15.5, 0.5 * 0.5), nrow = 2, byrow = T)
factor_input_2
##       [,1] [,2]
## [1,] 10.85 0.15
## [2,]  7.75 0.25

We clearly see that the labor and capital used to produce X will increase to 10.85 and 6.5 from 9.1 and 7.75, respectively. The percentage increase is:

(10.85-9.1)/9.1 
## [1] 0.1923077
(7.75-6.5)/6.5
## [1] 0.1923077

It conveys two important messages: first, the labor used to produce X will increase by more than 10%; second, the capital will increase by the same proportion. Based on the Euler’s Theorem, the output of X will increase by more than 10%.
WHY? Most people are told that it is becasue there is more labor while the wage is constant at equilibrium. To remain the wage, more capital shall be used to enhance the productivity of labor. Nonsense! We do not even need wage in the model. All we need is the full employment condition. Actually, since there is more labor available, we have to distribute them in the two sectors to fully employ them. A right-away strategy is to distribute them accordingly based on the original ratio, holding capital allocation constant. However, it is not possible, because the technology does not allow for it. If X has more labor, the output of X will increase. At the same time, more capital is required! And goods Y has to decrease the use of capital. A further question is why labor used in X increase by so much, more than 10%? That is because X is labor intensive, if you do not use more than 10%, Y would not do it, because Y is capital intensive. You cannot rely on Y to absorb the increase in labor, because it does not really “like” labor. The same proportion increase is due to the technology as well. Hence, we conclude that the essence of Rybczynski theorem is the full employment assumption and factor intensity. A by-product of the exercise is that we understand that the “curse of dimensionality” is actually a mathematic concern. The problem is solvable when the matrix A has the inverse. And, it will have the inverse when there are more factors than the number of goods.

To be continued…