Suppose that there are only two inputs. They are complementary if D2fx1x2 > 0. Show that Dw1x2 < 0 if the factors are complementary and Dw1x2 > 0 otherwise. This is special case of example 6.10.
With explicit constraints, it can be more straightforward to linearize the first-order conditions before deducing the comparative statics, as we did in example 6.14. Again, we emphasize that the theoretical foundation for this analysis remains the implicit function theorem. The difference is merely a matter of computation.