A great deal of unnecessary confusion continues to surround Thomas Piketty’s Second Fundamental Law of Capitalism. I have already dealt with the most common source of confusion here, and refer the reader to that post. But the same misreading has reappeared frequently in the ongoing debate over Piketty’s Capital in the Twenty-First Century, so it is worth returning to the topic briefly.

James Hamilton writes:

On page 168 of Piketty’s book the reader is introduced to “the second fundamental law of capitalism” according to which β = s/g, where β denotes the capital/income ratio, s the economy’s saving rate, and g the overall economic growth rate. Note that a curious corollary of this “law” is the claim that if the economy is not growing (g = 0), the capital/income ratio β has to be infinite.

But that is a misreading. Piketty’s Second Fundamental Law is not an identity or an approximate identity. It is, as Piketty makes clear at some length on pages 166-170 of the book, a long-term asymptotic law. For the benefit of the general reader, Piketty abbreviates the statement of the law in the form  “β  = s/g”. But it might be stated more carefully this way: “For a fixed savings rate s and growth rate g, the capital-to-income ratio β converges over time to s/g.” He might have abbreviated it in more conventional fashion in the form β → s/g. Piketty sketches an elementary proof of this convergence theorem in his online technical appendix. Like any limit theorem proved over the real numbers, it requires implicit restrictions on the range of the variables to avoid singularities.

However, if you augment the real numbers with a point-at-infinity ∞, you could certainly prove a version of the theorem for the extended reals, in which case the theorem also says that β → ∞, for cases where g = 0; or in other words, the function β(t) fails to converge, where t ranges over years starting with some arbitrarily chosen year 0. That reading makes perfect sense in this case. In a society which maintains a constant annual savings rate while at the same time having a zero rate of national income growth, then the capital-to-income ratio would rise without bound rather than converge to a limit. So the law doesn’t say that if g = 0, then β is infinite; rather it says that if g is zero, then β “goes to infinity”: that is, β increases without bound over time.

I have prepared an interactive workbook illustrating Piketty’s basic wealth and income framework to accompany an earlier post. The reader is invited to download the workbook, and use it to experiment with different values for s and g in modeling changes in wealth, income and the capital-to-income ratio over time. A short time with that workbook should be enough to persuade anybody that these convergence phenomena exist, and are described by the Second Fundamental Law.

Hamilton also gets into the weeds over Piketty’s concept of national savings and the national rate of savings. I think the best and most straightforward way to understand Piketty’s concept of the national savings rate is just to use his wealth growth rule

W_i+1 = W_i + s_i*Y_i

as an implicit definition of s. So s_i is by definition equal to (W_i+1 – W_i)/Y_i. It’s nothing but the change in wealth as a percentage of national income. Note also that Piketty’s concept of national income already includes the subtraction of depreciation. He defines national income as national output plus income from abroad minus depreciation. And he provides a definition of national wealth as well. So the concept of national savings is well-defined according to the definition I have just given.

If a society was holding steady with a stable wealth-to-income ratio β, and with stable savings rate s and growth rate g, and then growth suddenly plunged to some very small rate g’ just above zero, the Second Fundamental Law does not imply that the wealth-to-income ratio β would suddenly soar to an astronomical value. What it says is that β would begin to increase, at a rate depending on the size of s, toward the new convergence value s/g’. But as Piketty clearly notes in the passage cited above, and as I explained further in the earlier post I referenced, it might take many, many years to reach values near that convergence value, even if s remained constant.

And in the real world, of course, there is no reason to expect s to remain constant over that long a period of time. The Second Fundamental Law isn’t intended as some way of calculating the current wealth-to-income ratio β as a function of the current rate of savings and the current rate of growth. Rather it is intended to describe the direction in which β is moving at any given time, with implications about the rate at which it is moving, given the current savings and growth rates. This is relevant to Piketty’s later arguments, because an increasing β corresponds to an increase in the strength of the forces for income and wealth divergence connected with income form capital.