A Short Note on Marshall's Third Law of Derived Demand : Why Does Labor's Share Interact with the Elasticity of Substitution to Decrease the Elasticity of Labor Demand?

The third Marshall-Hicks-Allen rule of elasticity of derived demand purports to show that labor demand is less elastic when labor is a smaller share of total costs. As Hicks, Allen, and then Bronfenbrenner showed, this rule is not quite correct, and actually is complicated by an unexpected negative relationship involving labor's share of total costs and the elasticity of substitution. The standard intuitive explanation for the exception to the rule, due to Stigler, describes a situation rather different than the one described in the rule. In this paper, I present an example that illustrates the peculiar negative impact of labor's share, operating via the elasticity of substitution. I then explain why the unexpected relationship between labor's share of total cost, the elasticity of substitution, and the elasticity of labor demand holds