Conversely, equation 10 implies that region r uses relatively fewer non-production workers than region s. These ratios might incorrectly lead one to assume that RFPEQ fails across these two regions and that region r has higher true relative wages.
Without extra information on relative factor quality, neither observed wages nor observed factor quantities can help us disentangle the problem of RFPEQ versus differing input quality. To tackle this problem, we use data on both wages and employed factors and multiply the terms in equations 10 and 11. This gives us
The numerator in left hand side of equation 12 represents total wages paid to non-production workers in region r while the denominator is the wages paid to production workers in region r. On the right, we have the product of two terms, the non-RFPEQ adjustment factor and the ratio of the nonproduction to production worker wage bill in region s,
With this formulation, we do not have to separately identify the wage for the quality adjusted input and the quality adjustment factor.
For intuition, consider the following. Suppose RFPEQ holds and production workers and capital are of identical quality in the two regions. Then firms in the two regions will want to hire the same number of production workers as well as the same amount of capital per unit of output. This can be seen in equation 3. Now the firms decide how many non-production workers to hire. The first order conditions tell us that the firms will hire the same amount of quality adjusted non-production labor and more importantly they will pay the same total amount to the non-production workers for that quality adjusted input. Neither the number of non-production workers per unit of output nor the wage per non-production workers will be the same in the two regions unless the qualities are also identical.