Revisiting Marshall's Third Law: Why Does Labor's Share Interact with the Elasticity of Substitution to Decrease the Elasticity of Labor Demand?

Abstract

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 presented by Stigler and referenced in many textbooks describes a situation rather different than the one described in the rule. The author presents an example that illustrates the peculiar negative impact of labor's share operating via the elasticity of substitution and then explains why the unexpected relationship between labor's share of total cost, the elasticity of substitution, and the elasticity of labor demand holds.

Keywords:

• labor demand
• Hicks–Marshall rules

JEL codes:

• J010
• J200

Saul D. Hoffman is professor of economics and chair of the Department of Economics at the University of Delaware (e-mail: [email protected]).

Notes

^aProduction function for Case 1 is Q = L ^.8 K ^.2.

^bProduction function for Case 2 is Q = L ^.2 K ^.8.

1. This article was inspired when my graduate students in labor economics asked me for an explanation. Despite having taught labor economics for nearly three decades, I had no explanation whatsoever to offer, and I could not find one in any standard labor economics sources or on a Web search. This provides additional evidence that teaching does give rise to research, as asserted by Becker and Kennedy (2006).

2. Bronfenbrenner (1961) showed that the derivations and explanations of the various authors are not identical, although the resulting rules are. Marshall's rule relating to ease of substitution predated the development of the elasticity of substitution, and thus was not originally stated in those terms.

3. Bronfenbrenner (1961) noted that Allen's equation is a special case of the far more complicated equation from Hicks, corresponding to a situation in which the supply of the other factor of production is perfectly elastic. Analytically, this case might correspond to the demand response of a firm that takes the price of the other input as given or of a competitive industry for which the other factor is not a specialized input. In either case, the supply of the other input is perfectly elastic, so that changes in input choices will not have effects on the price of the other factor that must be considered.

4. A Google search of the Hicks–Marshall rules uncovers some very nice lecture notes, but no convincing explanations. The account by Hicks (1961) is not very helpful. This particular relationship is possibly the only element of labor demand not explained in Hamermesh (1993, 24–25, n. 2, which refers back to Stigler).

5. Bronfenbrenner (1961, 258) showed that neither Hicks nor Robertson, both of whom offered explanations, got it correct.