Teaching links: a sketch of Piketty from Robert Solow's book review

From the brilliant review by Solow.

Piketty's logic.

Step 1. what's the long run captial-income ratio of an economy?

"Imagine an economy with a national income of 100, growing at 2 percent a year (perhaps with occasional hiccups, to be ignored). Suppose it regularly saves and invests (that is, adds to its capital) 10 percent of national income. So, in the year in which its income reaches 100 it adds 10 to its stock of capital.

We want to know if the capital-income ratio can stay unchanged for next year, that is to say, can stabilize for the long run. For that to happen, the numerator of the capital-income ratio must grow at the same 2 percent rate as the denominator. We have already said that it grows by 10; for that to be 2 percent of capital, capital must have been 500, no more, no less. We have found a consistent story: this year national income is 100, capital is 500, and the ratio is 5. Next year national income is 102, capital is 510, the ratio is still 5, and this process can repeat itself automatically as long as the growth rate stays at 2 percent a year and the saving / investment rate is 10 percent of national income.

Careful attention to this example will show that it amounts to a general statement: if the economy is growing at g percent per year, and if it saves s percent of its national income each year, the self-reproducing capital-income ratio is s / g (10 / 2 in the example).

Step 2.  g might slowdown, and with s constant, the capital/income ratio must therefore rise.

Piketty suggests that global growth of output will slow in the coming century from 3 percent to 1.5 percent annually. (This is the sum of the growth rates of population and productivity, both of which he expects to diminish.) He puts the world saving / investment rate at about 10 percent. So he expects the capital-income ratio to climb eventually to something near 7 (or 10 / 1.5).

Step 3. Inequality in the sense of capital income to total income

The key thing about wealth in a capitalist economy is that it reproduces itself and usually earns a positive net return. That is the next thing to be investigated. Piketty develops estimates of the “pure” rate of return (after minor adjustments) in Britain going back to 1770 and in France going back to 1820, but not for the United States. He concludes: “[T]he pure return on capital has oscillated around a central value of 4–5 percent a year,

Now if you multiply the rate of return on capital by the capital-income ratio, you get the share of capital in the national income. For example, if the rate of return is 5 percent a year and the stock of capital is six years worth of national income, income from capital will be 30 percent of national income, and so income from work will be the remaining 70 percent.

Step 4.  Since the captial/income ratio is going to rise, then if the return on capital stays the same, the share of capital in the economy (an "inequality" measure called the "functional" distribution of income) will rise.

Solow's critique:

Suppose we accept Piketty’s educated guess that the capital-income ratio will increase over the next century before stabilizing at a high value somewhere around 7. Does it follow that the capital share of income will also get bigger? Not necessarily: remember that we have to multiply the capital-income ratio by the rate of return, and that same law of diminishing returns suggests that the rate of return on capital will fall. As production becomes more and more capital-intensive, it gets harder and harder to find profitable uses for additional capital, or easy ways to substitute capital for labor. Whether the capital share falls or rises depends on whether the rate of return has to fall proportionally more or less than the capital-income ratio rises.

There has been a lot of research around this question within economics, but no definitely conclusive answer has emerged. This suggests that the ultimate effect on the capital share, whichever way it goes, will be small. Piketty opts for an increase in the capital share, and I am inclined to agree with him. Productivity growth has been running ahead of real wage growth in the American economy for the last few decades, with no sign of a reversal, so the capital share has risen and the labor share fallen. Perhaps the capital share will go from about 30 percent to about 35 percent, with whatever challenge to democratic culture and politics that entails.

Step 5.   What if growth slows down so that the rate of return is greater than the growth rate?  Then we get inequality in the sense that owners of capital just rack up returns just by sitting on their capital and waiting for the effects of compound interest

Suppose it has reached a “steady state” when the capital-income ratio has stabilized. Those whose income comes entirely from work can expect their wages and incomes to be rising about as fast as productivity is increasing through technological progress. That is a little less than the overall growth rate, which also includes the rate of population increase.

Now imagine someone whose income comes entirely from accumulated wealth. He or she earns r percent a year. (I am ignoring taxes, but not for long.) If she is very wealthy, she is likely to consume only a small fraction of her income. The rest is saved and accumulated, and her wealth will increase by almost r percent each year, and so will her income. If you leave $100 in a bank account paying 3 percent interest, your balance will increase by 3 percent each year.

This is Piketty’s main point, and his new and powerful contribution to an old topic: as long as the rate of return exceeds the rate of growth, the income and wealth of the rich will grow faster than the typical income from work. This interpretation of the observed trend toward increasing inequality, and especially the phenomenon of the 1 percent, is not rooted in any failure of economic institutions; it rests primarily on the ability of the economy to absorb increasing amounts of capital without a substantial fall in the rate of return. This may be good news for the economy as a whole, but it is not good news for equity within the economy.

Step 6. Policy: to tax wealth

So Piketty’s foreboding vision of the twenty-first century is slower growth of population and productivity, a rate of return on capital distinctly higher than the growth rate, the wealth-income ratio rising back to nineteenth-century heights, probably a somewhat higher capital share in national income, an increasing dominance of inherited wealth over earned wealth, and a still wider gap between the top incomes and all the others.

Piketty’s strong preference is for an annual progressive tax on wealth, worldwide if possible, to exclude flight to phony tax havens. He recognizes that a global tax is a hopeless goal, but he thinks that it is possible to enforce a regional wealth tax in an area the size of Europe or the United States. An example of the sort of rate schedule that he has in mind is 0 percent on fortunes below one million euros, 1 percent on fortunes between one and five million euros, and 2 percent above five million euros. (A euro is currently worth about $1.37.) Remember that this is an annual tax, not a onetime levy.

He estimates that such a tax applied in the European Union would generate revenue equal to about 2 percent of GDP, to be used or distributed according to some agreed formula. He seems to prefer, as would I, a slightly more progressive rate schedule. Of course the administration of such a tax would require a high degree of transparency and complete reporting on the part of financial institutions and other corporations.

A formal critique of Piketty is here by Rognlie.  My summary

1. More capital has in the past lowered the return to capital so that the share of capital will not rise.
2. Trends in wealth are driven mostly by housing, rather than the compounding effect of wealth. 
3. The key issue is not therefore inequality in the sense of capital to labour income, but inequality within labour income.  That is likely driven a combination of trade, technology and schooling being the factors that determine the competitive pressure and technical demand for workers of different skills.   That is set out in David Autor, "Polanyi's Paradox and the Shape of Employment Growth," MIT, September 3, 2014.