Here’s my latest Mises Canada post, in which I show how Brad DeLong actually (though it’s not his intent) concedes that Piketty’s entire case is built on quicksand.
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…
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!
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.
OK it’s past midnight as I type this out, so I reserve the right to change this later. But–especially if you follow the link to my Mises Canada post and see how DeLong paraphrased Rognlie’s concerns–I think this is what’s going on:
With standard estimates of the parameter values you plug into an aggregate production function, you get the result that if the market value of the capital stock keeps increasing relative to annual output (i.e. K/Y or “total wealth” expressed as a multiple of GDP), then the share of annual income going to the capitalists (i.e. rK/Y) goes down, meaning that the share of annual income going to the workers (i.e. wL/Y) goes up. That’s not the outcome
Rognlie Piketty wants; he needs inequality (in terms of the percentage of income earned by the “top x%”) to be increasing in order to scare everyone into supporting a global wealth tax.
Now to get that result, Piketty argues why we should opt for the higher end of the range of values on those parameters. But Rognlie is pointing out that Piketty is mixed up: he is making a basic mistake in going from one type of application to another. Once you adjust for depreciation, you realize that Piketty’s needed parameter values are *way* outside the range of plausibility. Hence Rognlie concludes: “Unless I’m missing something, the formal apparatus in Piketty’s book simply is not capable of generating the results he touts…Perhaps there are modifications to the framework that can redeem it, but as it currently stands I’m baffled.”