The fourth key point is that all six SS versions do not consider intermediate inputs. In reality intermediate inputs matter a lot: for nearly 40 years, in U.S. manufacturing input purchases have accounted for well over 50% of the value of final shipments. Accordingly, any empirical work must account for the fact that firms hire both primary factors and intermediate inputs. Rewriting equation (1) to account for intermediate inputs suggests alternative ways of doing this. Define matrix B as the (N x N) matrix of intermediate input requirements whose bij element tells the number of units of intermediate input i required to produce one unit of product j. Then the set of zero-profit conditions in (1) can be rewritten as follows.
where I is an (N x N) identity matrix.
This suggests two alternative ways to account for intermediate inputs when linking product prices to factor prices. Following equation (3′), one can measure factor usage not in direct terms but rather in total terms accounting both for direct factor usage and indirect factor usage through intermediate inputs. That is, one can relate factor prices to [(I – B)-1 x A] rather than just to A. Alternatively, following equation (3”) one can continue using just direct factor usage as regressors but construct the regressand to be gross-output prices less input prices weighted by the B matrix. Link
To summarize this theoretical preview of the empirical work, these alternative versions of the SS theorem provide some helpful guideposts. Assume, as most researchers have, that the U.S. economy has less-than-free trade with abroad, more than two products and two factors, and intermediate inputs. Given these assumptions, the most appropriate version of the SS theorem for guiding empirical work is probably the Correlation Version. That is, for any given change in product prices and factor prices consistent with equation (2), in the data one should try to demonstrate the following.
All this requires a substantial amount of data. In terms of the “explanatory variable,” one needs some systematic way to identify the effect of one or more exogenous characteristics of “international trade” on domestic product prices while controlling for non-trade influences on these prices. To identify how this exogenous force of international trade affects relative wages, one also needs industry-level data on domestic product prices and on the prices and quantities of inputs and factors employed.