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.)

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