Substitution and size effect for factor demand revisited

We reconsider the decomposition of the comparative statics effect of a factor price increase on (unconditional) factor demand into a substitution and a size (or level) effect. While for the own price effect the substitution effect and the size effect go into the same negative direction, the cross price effect cannot be signed unambiguously, in general. But for two cases of regularity, homotheticity and complementarity, the size effect is negative. We furthermore show by example that in both special cases the cross substitution effects are still ambiguous, but that under complementarity the unconditional effect is always negative. Hence, if the cross substitution effect happens to be positive, it is dominated by the negative size effect in this case. We further show by example that without such regularity assumptions the cross price effect is ambiguous. Finally, we study the impact of a factor price increase on the cost, which can also increase or decrease.