Matt Rognlie April 10, 2014 at 2:45 am, wrote :
« Krugman correctly highlights the importance of the elasticity of substitution between capital and labor, but like everyone else (including, apparently, Piketty himself) he misses a subtle but absolutely crucial point.
When economists discuss this elasticity, they generally do so in the context of a gross production function (*not* net of depreciation). In this setting, the elasticity of substitution gives the relationship between the capital-output ratio K/Y and the user cost of capital, which is r+delta, the sum of the relevant real rate of return and the depreciation rate. For instance, if this elasticity is 1.5 and r+delta decreases by a factor of 2, then (moving along the demand curve) K/Y will increase by a factor of 2^(1.5) = 2.8.
Piketty, on the other hand, uses only net concepts, as they are relevant for understanding net income. When he talks about the critical importance of an elasticity of substitution greater than one, he means an elasticity of substitution in the *net* production function. This is a very different concept. In particular, this elasticity gives us the relationship between the capital-output ratio K/Y and the real rate of return r, rather than the full user cost r+delta. This elasticity is lower, by a fraction of r/(r+delta), than the relevant elasticity in the gross production function.
This is no mere quibble. For the US capital stock, the average depreciation rate is a little above delta=5%. Suppose that we take Piketty’s starting point of r=5%. Then r/(r+delta) = 1/2, and the net production function elasticities that matter to Piketty’s argument are only 1/2 of the corresponding elasticities for the gross production function!
Piketty notes in his book that Cobb-Douglas, with an elasticity of one, is the usual benchmark – and then he tries to argue that the actual elasticity is somewhat higher than this benchmark. But the benchmark elasticity of one, as generally understood, is a benchmark for the elasticity in the gross production function – translating into Piketty’s units instead, that’s only 0.5, making Piketty’s proposed >1 elasticity a much more dramatic departure from the benchmark. (Keep in mind that a Cobb-Douglas *net* production function would be a very strange choice of functional form – implying, for instance, that no matter how much capital is used, its gross marginal product is always higher than the depreciation rate. I’ve never seen anyone use it, for good reason.)
Indeed, with this point in mind, the sources cited in support of high elasticities do not necessarily support Piketty’s argument. For instance, in their closely related forthcoming QJE paper, Piketty and Zucman cite Karabarbounis and Neiman (2014) as an example of a paper with an elasticity above 1. But K&N estimate an elasticity in standard units, and their baseline estimate is 1.25! In Piketty’s units, this is just 0.625. »
Source : http://marginalrevolution.com/