# GLOBAL FACTOR TRADE: Theory 4

Think now about how previous tests have been implemented. Call the capital abundant country the US. Prior tests have used the US technology matrix to measure the factor content of trade. Consider how the input coefficients are constructed for the empirical US industry 1. Let BX be the column of average input coefficients for goods in the X sector and BN be the column of average input coefficients for goods in the the N sector. Then the measured input coefficients for sector 1 will be:

The weight R is determined by the X-sector’s weight in US output in sector 1 and we include the zero-weighted term BY to emphasize that it does not figure at all into calculation of the US technical coefficients. Note that the coefficients so estimated are a weighted average of the goods that the US actually exports (X) and goods with much more labor-intensive coefficients (N). That is, the estimated technology matrix will tend to understate the capital content and overstate the labor content of US exports. The consequence is to bias our measures of net factor trade toward zero. A parallel calculation for industry 2 would reveal the same downward bias in the US net factor content read only.

Now consider what happens if we apply the coefficients B1 taken from the US to exports by the labor abundant country. Again, Bx is a weighted average of US input coefficients in N and X. But the labor abundant country exports only 7 goods — which are more labor intensive than either X or N. That is, use of the measured US technology matrix will strongly overstate the capital content of the labor abundant country’s exports, while underestimating the labor content.

Use of the US technology matrix biases measures of the factor content of trade in both countries toward zero.

While the theoretical model is special in some respects, it does highlight two insights that we believe are more general than this example. The first is a pointed reminder that goods produced in different countries that are classified in the same industrial categories need not be the same goods at all. When we ignore this fact, we may well miss an important component of net factor trade. Second, insofar as trade in factor services is one motive for trade, when there are many goods that could embody this factor service trade, there will be an incentive to focus exports among those goods most intensively using the abundant factors. Hence average input coefficients for any country are likely to understate the true factor content of trade.

How would one know whether these theoretical problems are a real feature of the data? One consequence would be that industry factor usage will vary systematically with country capital abundance. We will explore this more fully below when we estimate the extent to which this affects factor ratios by industry across countries. The consequence here is twofold. First, we have to recognize that the technology matrices will differ systematically by country capital abundance, and so construct technology matrices that reflect this. Second, we will likewise need to recognize that the factor content of absorption must be measured bilaterally with the producing country’s technology matrix. With these two points in mind, it is relatively simple to derive the key expressions:

where the superscript in B reflects the fact that in the continuum of goods, Dornbusch-Fischer-Samuelson model, the unit input requirements in the tradable goods sectors will vary in accordance with the country’s capital to labor ratio.