As is well known by now, Thomas Piketty argues in Capital in the Twenty-First Century that the historical record shows the rate of return on capital in market economies consistently exceeding the growth rate of national income in those economies, usually by a sizeable margin. Piketty finds typical values for the pre-tax rate of return on capital during the 19^th century, for example, of around 5%, with typical income growth rates of around 1.5%. He encapsulates this historical regularity with the formula r > g.

Yves Smith, commenting on a post at her blog Naked Capitalism, is convinced that that Piketty’s claim is absurd:

There’s no sound basis for Piketty’s contention that capital makes a steady 5%, or perhaps more important, a 4% premium to GDP growth. Pull out a calculator. It does not take that long before capital that consistently showed that much of a return premium would eat the entire economy. Then by definition it would be the entire economy and unable to earn a premium. His r>g as some sort of constant is absurd, tantamount to “trees grow to the sky.”

And she adds later:

Folks, this is basic compounding, and that’s where the fallacy in his reasoning lies. It’s bloomin’ obvious. That’s why it being a materially higher rates than GDP growth has to break down, and not in a very long period of time either.

I find it somewhat depressing that an informed economics blogger like Yves Smith could be in this case so deeply uninformed about Piketty’s fundamental conceptual framework, to the extent that she thinks Piketty must be some kind of bungling duffer who has flunked basic accounting algebra. But her claims here can easily be shown to be false. Piketty’s framework is simple, elegant and coherent, and working within that framework one can easily see how the typical values Piketty finds for r and g can be sustained indefinitely without capital overwhelming the economy and without capital income devouring all other income.

Piketty’s general framework for thinking about income and wealth at the national level makes use of the following basic notions:

W_i – the nation’s accumulated capital stock in year i – its national wealth. Piketty defines national wealth as “the total market value of everything owned by the residents and government of a given country at a given point in time, provided in can be traded on some market.”

Y_i – the national income in year i.

s_i – the national savings rate for year i; i.e the percentage of national income whose value is equal to the net addition to the capital stock in year i.

YC_i – income generated from capital in year i; this includes profits, rents, dividends, interest, royalties, capital gains and any other form of income generated from the ownership of wealth rather than from labor.

r_i – the rate of return on capital: i.e. the ratio YC_i/W_i of income generated from capital in year i to total wealth in year i.

g_i is the rate of growth of national income in year i

Here’s the rule that states how wealth accumulates from one year to the next:

W_i+1 = W_i + sY_i

In fact, this formula can be taken as an implicit definition of the national savings rate. For a given year i, the national saving in that year is equal to (W_i+1 – W_i)/Y_i, the change in wealth as a proportion of national income.

We also need the rule for income growth:

Y_i+1 = (1 + g)Y_i

And finally, Piketty focuses on two other important ratios:

The capital-to-income ratio β_i for any year i is equal to W_i/Y_i.

The capital share of income α_i in year i is YC_i/Y_i, which we can see is also equal to r_i times β_i as a matter of algebraic necessity.

Now, assume initial values of $600 for W₁ and $100 for Y₁. Assume a constant savings rate of 10%, a constant growth rate of 2% and a constant rate of return to capital of 5%, which means that the income from capital YC₁ in year one is $30. The initial capital-to-income ratio β₁ can then be seen to be 600%, and the initial capital share of income α₁ is clearly 30%. Now do an extended time series computation for α_i and β_i, using the wealth accumulation rule and the income growth rule. You will see that that neither figure grows without bound, but that α_i converges to 25% and β_i converges to 500%. If we assume a 1% annual growth rate instead of 2%, then α_i converges to 50% and β_i converges to 1000%. In neither case does capital “eat the entire economy.”

Smith appears to have in mind a scenario in which the entire return to capital is rolled over into the capital stock each year, where all of it then earns a return at the rate of return r along with all of the previously accumulated capital. If we assume that that income from capital is the only income added to the capital stock then we get a simple compounding problem: If we begin with a capital stock of $600 and an annual income of $100 in year zero, and a rate of return on capital of 5% and income growth rate of 1%, then the capital stock grows at the same rate as the rate of return on capital. In Year N, national income will be $100 x (1.01)^N, the capital stock will be 600 x (1.05)^N and capital income will be 600 x (1.05)^N x (.05). Thus by Year 31, capital income will exceed total income.

But this way of looking at things completely ignores the national savings rate and the relationship of capital income to total income. Capital income is only a portion of total income, and only a relatively small fraction of national income is typically added each year to the national wealth. When we apply the appropriate analytic framework, we get the convergence phenomenon described above, rather than an indefinitely increasing share of capital income in national income.

I have created an Excel workbook that allows the user to perform these calculations quickly and observe the evolution of wealth, income and the related variables over a 50-year time span. The sheet contains a section labeled “adjustable parameters” within which the user can experiment with different values. The user can also see the formulas applied in each of the other cells by clicking on the cell and looking in the function field. I have found that using calculation tools of this kind helps to build up the mathematical intuitions needed to follow Piketty’s arguments with ease.

But One Warning: you can understand all of the concepts and computations in the basic framework presented on the spreadsheet without understanding almost any of the most important arguments Piketty presents related to the structure of inequality. All of the matters dealt with on the spreadsheet are explained in the first six chapters of Piketty’s book. But Piketty’s discussion of inequality is contained in chapters 7 through 12 , covering about 240 pages. The impression exists among many reviewers and commentators that Piketty’s basic story about inequality is that increasing inequality during a given period follows directly from r being greater than g during that period, or from the growth of capital during that period, or from the growth of the capital share of income during that period. But while while each of these phenomena is a contributing factor in the growth of inequality, none of them is suffient in itself to produce greater income inequality and wealth concentration, and the story Piketty tells is significantly more complicated.

Piketty argues that several forces driving increased inequality have predominated in recent decades, and that these forces will likely continue to predominate in the 21st century (without policy changes to counteract them). The argument does not depend on a continually rising rate of return to capital, since inequality can continue to surge even if the rate of return falls somewhat. The growth of inequality and increased concentration of wealth are determined by several factors having to do with the initial unequal distribution of capital; with the fact that some capital owners earn much higher returns than others; with the fact that some capital owners save at significantly higher rates than others; with the way in which globalization is producing a closer relationship between the rate of return earned by a portfolio and the size of that portfolio; with the fact that there are sometimes significant differences in labor income driven by hierarchical power relationships and bargaining power – a factor especially pronounced in the United States; and with the fact that policies favorable to capital accumulation by the rich have been enacted in recent years. But perhaps the most important factor is related to demographic trends. Almost all models are predicting very low population growth in the 21st century. If population growth is very low than income growth will likely be very low, even with increases in per capita productivity. Thus, even if r falls somewhat, r – g can remain very large. And if r – g remains large, the main precondition exists for all of those other factors to come into play.