The world of the weird: r is greater than g and inequality is not increasing? Can it happen?

I was recently discussing Piketty’s book (yet again) with my friend Mario Nuti. I mentioned to him that I think it is possible to have the rate of return on capital (r) be greater than the rate of growth of income (g) and still have a non-increasing inequality—under the simplest possible conditions and within the very narrow confines of Piketty’s model. After I explained it, Mario encouraged me to publish it in a blog. So this is what I am doing now, essentially transcribing the notes I took while reading Capital in the 21^st century.

No, you don’t need to run away thinking that I will try to prove this by using a second-order  differential equation and a very fancy growth model (I would not know how to do it anyway). Not even by assuming (as we can) that while r>g holds, the distribution of capital becomes more equal and thus offsets, and possibly overturns, the pro-inequality effect of the rising capital share in total income. No, none of that. Much more simply; in effect, disappointingly simply… 

Just as a reminder: as we all know by now, r>g implies that the stock of capital is increasing faster than net income. Then, If the rate of return on capital does not fall proportionately (and Piketty thinks that it may not fall at all), the share of income from capital in total income will rise, and since capitalists are generally the rich guys, inter-personal inequality will increase too. So, that’s the basic story we all know. 

But now let’s go behind the mirror and assume that rate of return falls to 0 while the rate of growth of the economy is negative (a situation not too dissimilar from the one experienced by the Eurozone countries today). What happens then? Obviously, capital stock will not increase since net saving is zero. But capital/income ratio (Piketty’s β) will rise because income—the denominator—is going down. The share of capital income in total income (α)  will remain unchanged at zero.  Thus we can have (1) r>g, and (2) a rising β while—and this is strange—(3) α is constant. 

This is interesting because the general interpretation to which we are used, perhaps because we are used to living in or dealing with growing economies, is that r>g implies a rising β and a rising α. Not so in a declining economy. The last part does not hold. 

It is indeed a degenerate case, due the fact that α is bounded from below. But, as a general proposition, it is nevertheless true that we can have a rising β, a constant r, no change in the distribution of capital assets, and –surprisingly—a non-increasing share of capital, and presumably, non-increasing inter-personal inequality.

(To some extent, this issue goes back to Keynes’s great chapter in “The General Theory…” dealing with the special nature of money. It alone among all “commodities” has no “carriage cost”—decrease in value due to simple passage od time—so its lowest “own rate of return” is zero. If it could go below zero, there would be a decreasing capital/income ratio, negative α and a decreasing inter-personal inequality. The “paradox” to which we pointed out above would have disappeared.)