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.

What does this all mean for the Piketty’s central points – that total capital income rK/Y will increase, and that r-g will grow? His model imposes a constant, exogenous net savings rate ‘s’, which brings him to the “second fundamental law of capitalism”, which is that asymptotically K/Y = s/g. The worry is that as g decreases due to demographics and (possibly) slower per capita growth, this will lead to a very large increase in K/Y. But, of course, this only means an increase in net capital income rK/Y if Piketty’s elasticity of substitution is above 1, or if equivalently the usual elasticity of substitution is above 2. This is already a very high value, and frankly one to be treated with skepticism.

Meanwhile, it is even harder to get growth in r-g, which most readers take to be Piketty’s central point. Suppose that in recent decades, r has been roughly 5% while g has been 2.5%, and suppose that g will ultimately fall to around 1%. In Piketty’s framework, this implies an increase in steady-state K/Y of 2.5. If there is an elasticity of 1 (in Piketty’s units), this implies a decrease in r from 5% to 2%, and thus a *decrease* in the gap r-g from 2.5% to 1%. The point is that with this unit demand elasticity and the exogenous net savings assumption, it is the ratio r/g rather than the difference r-g that is constant, which means that a decline in g leads to a proportionate decline in r-g. (Note that Krugman’s review is ambiguous about this distinction.)

What would we need to obtain even a tiny increase in r-g in this setting – say, of half a percentage point? We would need r to fall from 5% to only 4% while g fell from 2.5% to 1%, increasing r-g from 2.5% to 3%. But given the 2.5-fold increase in K/Y, a decline in r by a factor of only 1/5th implies an elasticity of substitution (in Piketty’s sense) of nearly 4. This implies an elasticity of substitution in the *usual* gross production function sense of nearly 8, not plausible by any stretch of the imagination.

Unless I’m missing something, the formal apparatus in Piketty’s book simply is not capable of generating the results he touts. There are two very simple issues that break it quantitatively – first, the distinction between elasticities of substitution in the gross and net production functions; and second, the fact that as g falls, an extraordinarily high elasticity of substitution is necessary to prevent r from falling along with it and actually compressing the arithmetic gap between r and g. Perhaps there are modifications to the framework that can redeem it, but as it currently stands I’m baffled.

I believe Matt is correct. I would simply note that diminishing returns to capital — relative to other factors of production — are likely to hold in the long run. See also these earlier MR comments by Rognlie and Harless. And here are Piketty’s lecture notes.