Gross (with depreciation) versus net. Elasticity of substitution.
Piketty’s Fair-Weather Friends by Seth Ackerman
The problem for the book is that its gloomy forecast of a return to “patrimonial capitalism” is based on the prediction that over the next decades, the gap between r and g will widen due to a fall in the growth rate (g). No one has a problem with the prediction of falling growth — all else equal, this will happen simply if population growth slows, as it almost certainly will. The problem is that if growth slows while the rate of saving (i.e. investment) stays constant, capital will start accumulating faster than output is rising, meaning the measured capital-output ratio will increase.
But marginal productivity theory sees a rise in the capital-output ratio as an increase in the “supply of capital,” which, in classic supply-and-demand logic, ought to bring about a reduction in its “price” — that is, a fall in r. According to the theory, this should neutralize the effect on the r-g gap.
In his faint-praise review of Piketty’s book, Larry Summers was unyielding on this point: “Economists universally believe in the law of diminishing returns,” he insisted, in a line resounding with the thud of a fist pounding a lectern. Piketty’s forecast of rising r - g must be rejected, Summers concluded, because “as capital accumulates, the incremental return on an additional unit of capital declines.”
Piketty had of course been aware of this issue when he wrote the book, and he made an attempt to reconcile his argument with conventional theory. He contended that as growth slows and the capital-output ratio rises, r might decline (as theory predicts) but the magnitude of the decline might still be small enough to permit a net widening in the r - g gap.
The technical term for the quantitative relationship involved (that is, between the size of a change in the capital-output ratio and the size of the change in r that supposedly results, or vice versa) is the elasticity of substitution: the higher the elasticity, the smaller the “response” of r to a given change in the volume of capital. When the elasticity is higher, it’s taken to signify that the force of diminishing returns to capital is weaker, due to richer technological opportunities for labor-saving investment.
As Summers pointed out in his review, “economists have tried forever to estimate elasticities of substitution with many types of data,” so there’s a large literature on the subject. This is the so-called production function literature, which tries to apply marginal productivity theory empirically by estimating the supposed causal relationships between quantities of labor and capital inputs, on the one hand, and the quantity of output on the other. Piketty’s argument was that the elasticity needed for his forecast to come true isn’t all that much higher than at least a few of the estimates found in the existing production function literature.
But in saying so, he made a crucial error. He confused two different ways of measuring the elasticity: the usual (gross) measure, which counts depreciation as part of the return to capital, and his own (net) measure, which doesn’t.
When this discrepancy is accounted for, the elasticity Piketty requires turns out to be far, far higher than any known estimate. This was first pointed out in mid-April by Matt Rognlie, an MIT graduate student and blogger. The problem was ruefully acknowledged by Brad DeLong, the former Clinton administration economist, who is sympathetic to Piketty’s project.
Summers’s review a month later in the journal Democracy sealed the judgment: Piketty “misreads the literature by conflating gross and net returns to capital,” Summers wrote. “I know of no study suggesting that measuring output in net terms, the elasticity of substitution is greater than 1, and I know of quite a few suggesting the contrary.”
A reader at this point could be forgiven for feeling confused. Didn’t Piketty gather his own data? He did, of course. That Herculean effort by him and his team of international colleagues, compiling statistics on historical rates of return and capital volumes in many countries going back to the eighteenth century, is the one point on which all reviews are unanimous in their praise.
As Piketty makes clear, those data — which he’s made freely available on the internet for anyone to check — are indeed “explained” by a net elasticity of 1.3-1.6, which would indicate an extremely weak force of diminishing returns to capital. Yet it’s also true that this figure is far higher than any found in the existing literature — probably more than twice as high as the highest typical estimates.
What should we make of this?
First, Piketty’s estimate of the elasticity of substitution can’t really be compared with those in the literature. His is based on economy-wide data covering decades and centuries while estimates in the literature typically cover only a few years, and often just a few industries. Moreover, his pertain to all private wealth, while the literature focuses narrowly on production capital. These are very different concepts.
But most importantly, given the flawed marginalist theory behind it, and its even more flawed basis of measurement — a subject there’s no space to go into here, but which marks a fundamental critique of the production function literature advanced in an important book published just two months before Piketty’s — the elasticity of substitution simply cannot be regarded as a meaningful measure of an economy’s technology (or anything else), or as providing any clue to its future.
What’s essential, rather, is Piketty’s empirical demonstration that the rate of return on wealth has been remarkably stable over centuries — and, contraSummers, with no visible tendency to vary in any consistent way against the “supply of capital.”
Therefore, if we expect growth to slow, the most reasonable expectation is that the r - g gap will in fact increase, and inherited wealth will expand.
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