Lawrence Summers, in what is generally a very favorable review of Thomas Piketty’s Capital in the Twenty-First Century, pushes back on some of Piketty’s arguments in favor of the conclusion that capitalism shows an inherent tendency toward increased inequality. The review raises a number of important questions, many of which deserve careful future attention. But in this post, I just want to touch on two areas where I think Summers has misconstrued Piketty’s argument in some important ways, misconstruals that lead Summers to underestimate the argument’s overall strength. I will deal first with what I think is the less important of the two misconstruals, and then turn to the more important one.

At one point in the review, Summers says this:

His [Piketty’s] argument is that capital or wealth grows at the rate of return to capital, a rate that normally exceeds the economic growth rate. Thus, economies will tend to have ever-increasing ratios of wealth to income, barring huge disturbances like wars and depressions.

But this isn’t quite right. Piketty does, indeed, believe that the average rate of return to capital in an economy normally exceeds the economy’s growth rate. But he does not argue that the rate of capital growth is always equal to the rate of return to capital, nor does he argue that capitalistic economies will tend to have ever-increasing ratios of wealth to income. The rate of capital growth will always be equal to s/β, where s is the savings rate and β is the ratio of capital to income. If β is less than s/g, where g is the growth rate, then capital will grow more rapidly than the economy, and β will increase. But if β is greater than s/g, then capital will grow more slowly than the economy and β will decrease.

Piketty also says that, for given rates of growth g and savings s, the capital-to-income ratio β will converge toward s/g so long as those rates are stable. He calls this assertion The Second Fundamental Law of Capitalism. But this is a long-term asymptotic law, and the process of convergence can continue for  many, many years before the capital-to-income ratio reaches values that are meaningfully “nearby” the value of s/g.

Later in the review Summers makes a claim similar to those he made in the preceding quote:

Piketty argues that as long as the return to wealth exceeds an economy’s growth rate, wealth-to-income ratios will tend to rise, leading to increased inequality.

But again that is not true. The return to capital can exceed the economy’s growth rate even while the wealth-to-income ratio is falling. For example, suppose that an economy that has been growing slowly for many years, with a savings rate of 9% and growth rate of 1%, and has achieved a capital-to-income ratio near the long-term convergence value of 900%. Now suppose that the economy increases its rate of growth to 1.5%. If it maintains this growth rate indefinitely with the same savings rate as before, the capital-to-income ratio will gradually fall toward 600%. But the process would take a very long time. Indeed, it would require 170 years for the capital-to-income ratio to fall below 625%.

But the more important way in which I believe Summers has misinterpreted Piketty’s argument lies in the phrase “leading to increased inequality” with which the previous quote ends. Summers suggests in that passage that it is the growth in the wealth-to-income level that drives the growth in inequality. But Piketty’s analytic framework shows that even a stable wealth-to-income level can lead to growing inequality under fairly ordinary conditions. Further developing his reading of Piketty’s analysis of capitalism’s tendency to produce economic inequality, Summers writes:

This rather fatalistic and certainly dismal view of capitalism can be challenged on two levels. It presumes, first, that the return to capital diminishes slowly, if at all, as wealth is accumulated and, second, that the returns to wealth are all reinvested. Whatever may have been the case historically, neither of these premises is likely correct as a guide to thinking about the American economy today.

We will return to the Summers’ first challenge later. But as for the second, I believe Summers is simply in error. Piketty does not assume that in order for the forces favoring inequality to prevail in an economy, the returns on wealth need be entirely, or almost entirely, reinvested. For example, Piketty writes the following in discussing the conditions favoring an “inheritance society” characterized by a very high concentration of wealth and the persistence of large fortunes from generation to generation:

Now, it so happens that these conditions existed in any number of societies throughout history, and in particular in the European societies of the nineteenth century. As Figure 10.7 shows, the rate of return on capital was significantly higher than the growth rate in France from 1820 to 1913, around 5 percent on average compared with a growth rate of around 1 percent. Income from capital accounted for nearly 40 percent of national income, and it was enough to save one quarter of this [emphasis mine] to generate a savings rate on the order of 10% (see Figure 10.8). This was sufficient to allow wealth to grow slightly more rapidly than income, so that the concentration of wealth tended upward. (p. 351)

So even if capital consumes 75% of its income from capital, and reinvests only 25%, that is sufficient to increase the concentration of wealth. In fact, it seems to me that the important point here is not whether wealth is growing more rapidly than national income, but whether there is some wealthy class of capital owners whose wealth increases at a rate greater than the overall wealth increase. This can happen, it seems to me, under a wide variety of fairly normal conditions.

To see how this can be the case, let’s suppose a hypothetical society in which there exists some class of rentiers, the entirety of whose income comes from the return on their capital. These are people who live entirely off their rent. Let W_i be the total national wealth in year i, and let R_i the total wealth of the rentiers in year i. Also let Y_i be the national income in year i, r the average return on capital, s the national savings rate, and ρ the rentiers’ own savings rate. Begin with the simplifying assumption that the rate of return on capital is exactly the same value r for all capital owners. We can then describe the way national wealth and rentiers’ wealth in year i + 1 depends on the values of these various quantities in year i:

1. R_i+1 = R_i + ρrR_i
2. W_i+1 = W_i + sY_i

Under what conditions is the rentiers’ wealth growing at a faster pace than total national wealth? That is, under what conditions is the following inequality true?

3. R_i+1/R_i > W_i+1/W_i

Using equations 1 and 2, we see that equation 3 is equivalent to:

4. (R_i + ρrR_i)/R_i > (W_i + sY_i)/W_i

And, recalling that the capital-to-income ratio β_i is W_i/Y_i , by further reduction and inference we get:

5. 1 + ρr > 1 + s(Y_i/W_i)
6. ρr > s/β_i
7. ρ > s/rβ_i

We can also reformulate the inequality 8 as

8.  ρ > s/α_i

where α = rβ_i, the total return on capital in year i.

So, if the inequalities 7 and 8 obtain, the rentiers’ wealth will be growing faster than total national wealth, and rentiers’ share of national wealth will be increasing. Suppose r = 5%, s = 10% and β_i = 6, then ρ would have to be greater than only 1/3 for the rentier share of wealth to be growing faster than overall wealth. It is not at all necessary that the rentiers be saving nearly all of their rental income. And if β_i = 8, then ρ would only have to be greater than 1/4 for the rentier share of wealth to be growing faster than overall wealth. This is in fact the scenario that Piketty describes in the passage quoted above, since the total capital share of income is 40% when β_i = 8 and r = 5%.

Now, as we recall, in the long run β approaches s/g according to the Second Fundamental Law of Capitalism. Let’s suppose that our hypothetical nation has only a 1% annual growth rate, and that it has also reached a capital-to-income ratio close to its long-run convergence value s/g, which is in this case 10.  So our inequality ρ > s/rβ_i is approximately equivalent under these conditions to ρ > g/r. Now if r = 5% and g = 1%, then g/r = 1/5. So, if ρ > 1/5, we again find a rising rentier proportion of wealth.

This brings us, though, to the first of Summers’s two challenges to Piketty’s argument. It is true, as Summers says, and as Piketty acknowledges, that the rate of return on capital r is likely to fall in the long run as the capital-to-income ratio rises. So consider the possibility that it falls from 5% to 3%. Even in these circumstances, rentiers who are able to save more 33.3% of their capital income will see their share of national wealth increasing. And note that the fall in r will occur gradually. Before it reaches a level at which the rentier share of wealth is no longer increasing, rentiers may be able to accumulate massive shares of national wealth, shares that will subsequently decline only slowly once the capital return rate falls below its critical level.

And note that the forces tending toward inequality might actually be even stronger than these mathematically simplified scenarios suggest. Throughout all of these calculations we have made the simplifying assumption that the rate of return on capital is identically equal to the average return on capital for all classes of capital owners. But in the real world some owners of capital are able to earn much higher returns on capital than others. For example, the New York Times reports that hedge fund clients are “disappointed” over these funds’ 9.1% average return rate, part of a five-year “slump” as far as hedge funds go.  In such scenarios, where privileged but significantly large classes of capital owners have access to rates of return far higher than the average rate, the rentier share of wealth could rocket upward quite dramatically.

And finally, we should also consider cases in which the national savings rate might fall to levels significantly lower than 10%. This might happen in a very low growth, low intensity, steady-state kind of economy with relatively low depreciation. Some might feel that the developed world is already moving toward such an economy. In such a world, the rentier savings rate ρ could be much lower than the rates described above, but still high enough that rentiers can maintain a continually increasing share of national wealth. This kind of neo-feudalist outcome is a nightmare that Piketty’s analysis, I believe, shows we must at least take seriously.

 

NOTE: Thanks to Matt Breunig for spotting an error in my first statement of the Second Fundamental Law of Capitalism. The error has been corrected.