First, the studies of Bhagwati, LS, and SS I call “consistency checks.” That is, these studies analyze whether observed product-price changes were consistent with rising wage inequality in the sense that the relative price of skilled-labor-intensive products rose relative to those of unskilled-labor-intensive products. Bhagwati simply assumes that exports (imports) employ more-skilled (less-skilled) labor relatively intensively. LS, and SSh following LS, refine this assumption by using disaggregated data to identify the pattern of relative factor usage.
These first studies have some important limitations in terms of the distance between SS theory and the empirical analysis and in terms of appropriately accounting for data complexities. One important limitation is that a consistency check on product prices cannot make the important link to factor prices. It cannot answer the important question of how much product-price changes might have contributed to actual factor-price changes. Another major limitation is that these regressions of product-price changes on factor-employment levels is not a tight implication of the SS theorem’s zero-profit logic as summarized in equations (1) and (2). Equation (1) relates price levels to factor-employment levels (i.e., to the A matrix) while equation (2) relates price changes to factor cost shares (i.e., to the 0 matrix). Regressing price changes on employment levels seems to capture the broad intuition of the SS theorem, but it uncomfortably mixes the levels and changes versions of the zero-profit conditions central to the SS theorem.
There are some data limitations as well. One is that these studies assume that the United States does not affect world prices: changes in domestic (or export or import) prices reflect changes in world prices triggered abroad that are communicated to the price-taking U.S. economy. Another limitation is that these studies ignore capital and any other primary factors of production. Given this, their results are best interpreted as a test of the General, Restrictive, or Essential Versions of the SS theorem under the maintained assumption that other primary factors don’t matter. A third limitation is they measure only direct factor intensities: they do not account for factor usage embodied in intermediate inputs.