The Piketty Curve and the Elasticity of Substitution

Thomas Piketty’s (2014) recent book has been the occasion of a good deal of discussion, much of it centered around his references to the elasticity of substitution between labor and capital. He presents evidence that suggests that capital’s share rises as the capital-labor ratio rises. In a very simple growth model, this would mean that the elasticity of substitution between capital and labor is greater than one. However, many economists probably concur with a criticism by Lawrence Summers: (2014) “I know of no study suggesting that … the elasticity of substitution is less than one, and I know of quite a few suggesting the contrary.” A number of such studies are reviewed in Leon-Ledesma et. al (2010, see esp. Table 1, p. 1335) and most do indeed estimate elasticities of substitution less than one; however, they also assume that technical progress is either Hicks-Neutral or factor-augmenting and, if factor-augmenting, that the rates of factor augmentation are exogenously given1. Piketty does not split definitional hairs, referring sometimes to capital claims and sometimes to the imputed competitive income of the capital input. As Summers notes, he abstracts from depreciation and, barring a few mostly negative comments, does not treat technical progress nor human capital as important determinants of the income shares. Thus, clearly, the studies referenced by Leon-Ledesma et. al do not apply to Piketty’s argument. On the other hand, there is some room for conjecture as to how Piketty’s thinking might be represented in mathematical economic models of economic growth. This paper will briefly sketch a model influenced by Stokey (1991), Romer (1986), and Duffy et al (2004). in which trends like those reported by Piketty can arise depending on the differences between the elasticities of substitution among physical capital, human-technological capital, and raw labor. For this model, “capital” will be understood in a strictly neoclassical way, as an index of an aggregate of heterogeneous durable produced means of production, treated “as if” a divisible and homogenous input. The paper will then reconsider the relation of Piketty’s writing to a model of this kind.