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The arithmetic of Piketty's second fundamental law of capitalism

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A surprising thing about Thomas Piketty’s economic treatise “Capital in the Twenty-First Century” is that it is a good read. It wasn’t to be expected that a book of over 600 pages on economics translated from a foreign language would be so readable. 200,000 copies of the English translation sold is evidence of the book’s readability. But even many highly professional readers will not be intimately familiar with asymptotic, algebraic expressions as used in Piketty's second fundamental law of capitalism. And the law is central to Piketty's thesis about the likely rise in accumulated wealth holdings relative to annual incomes into the twenty-first century. So we expand on the example Piketty uses to expound on the law to make the math in that example more explicit.

Piketty uses the Greek letter β to represent the ratio of a nation's capital to its level of annual national income. Let's use the letter 'K' to represent the stock of capital and the letter 'Y' to represent a nation's total annual income. Piketty states his law algebraically as

β = s/g, where

's' is the proportion of income that is saved, or the rate of savings, 'g' is the rate of growth of national income.

Since β is the ratio of K to Y, we can also write the law as

K/Y = s/g.

Let's go through these expressions term by term.

Capital (K)

By capital, Piketty means assets that can be owned and exchanged at market value. It includes real property (land and its resources, buildings, including housing, etc.), financial capital (bonds, bank accounts, stocks, mutual funds, insurance policies, pension funds etc.) and productive capital (plants, equipment, machinery, electronics, infrastructure, etc.); and it includes precious metals such as gold and jewelry, and works of art.

In the way Piketty is using the term capital, it is equivalent to a nation's national wealth, whether owned by the private sector or government. So we could just as well use the letter 'W' rather than 'K' for Piketty's capital, but let's stick with 'K'.

For Piketty's purposes, which is an analysis of wealth and its distribution among nations and social classes over long historical periods, this is legitimate, since in the long run, all forms of wealth can be interchanged at market prices, e.g., precious art can be sold to buy plant and equipment to start a business or to expand an existing company. Or a business can be sold and the proceeds used to buy fine art.


Piketty's measure of national income is closely related to GDP or gross domestic product, but is adjusted for annual depreciation of a nation's capital from wear and tear and obsolescence; and for a nation's net income earned in foreign countries.

The proportion of income that is saved

Piketty uses the small case letter 's' to represent the proportion of a nation's annual income that is saved rather than consumed. So 's' is the national rate of savings. For example, if a nation in the aggregate spends 90 percent of its annual national income on consumption goods and services then it has saved 10 percent of that income. It is important to realize that 's' is not the absolute amount of savings, but the proportion of income that is saved. To know the actual amount of savings we need to think of the product of income and the proportion of that income that is saved. It is useful to have a symbol for the actual amount that is saved. Let's use the capital letter 'S' for the amount of savings. Then we can write

S = s times Y, or S = sY.

The annual rate of growth of national income

The letter 'g' represents the annual rate of growth of a nation's income. The rate of growth of national income in any given year is simply the change in this year's income compared to last year's divided by what total income was at the beginning of the year. If we use the symbol 'Δ' to represent 'change in' then we can say that

g = ΔY divided by Y, or ΔY/Y.

Piketty's example for his second fundamental law, β = s/g

Let's go through Piketty's concrete example of his law on page 166 of his book.

In the example, Piketty uses a savings rate, s = 0.12, or 12 percent in percentage terms. And he uses an annual rate of growth in Y of two percent per year or g=0.02. Thus, for his example, Piketty announces that β = K/Y = 0.12/.02 = 6. By this he means that a nation that has been saving 0.12 of its income and whose national income has been growing at 0.02 times its level of income per year, year after year, will, after a long period of time, end up with total wealth holdings equal to six times its national income. We say 'after a long time' because the relationship is a long term one, not necessarily true in any given year. So it is best to think of 's' as the long term annual average savings rate and 'g' as the long term annual average rate of growth of national income.

How can we make this more transparent?

Remember the symbol Δ represents "change in," so that

ΔY represents the change in Y, and

ΔK represents the change in K (capital or wealth).

We know that, in this example, s/Y = 0.12, i.e., the proportion of income that is saved.

But we want to use the expression for the absolute amount of savings rather than the proportion of income saved. Remember that we are using capital letter 'S' for the amount of savings. We know that it is simply the proportion of income saved times the amount of income or

S = 0.12 times Y.

But what is saved is added to capital or wealth, so

S = 0.12Y = ΔK, the change in wealth.

Let's start with the case where β = K/Y is not yet equal to 6, let's say it is only equal to 5.75 in year 1, where K1 = 115 and Y1=20. We could start at a much earlier point, say when K/Y is only equal to 2.75, but while it would take longer to get there, we would get the same result, that eventually if 'S' remains at 0.12Y and 'g' remains at 0.02, then K/Y will be 6.

Let's do the arithmetic starting in year one with Y1 = 20, K1 = 115 and K1/Y1 = 5.75, then

In year 2:

Y2 = Y1 + 0.02Y1

Y2 = 20 + 0.02 times 20 = 20.4

K2 = K1 + ΔK = 115 + 0.12Y1

K2 = 115 + 0.12Y1 = 117.4 and

K2/Y2 = 117.4/20.4 = 5.7549

In year 3:

Y3 = 20.4 + 0.02 x 20.4 = 20.808

K3 = 117.4 + 0.12 x 20.4 = 119.848


K3/Y3 = 119.848/20.808 = 5.7597

You see where this is going. β1 = 5.75, β2 = 5.7549 and β3 = 5.7597. you can repeat the calculations for years 4,5,6 ... if you wish.

You will see that if the process keeps going year after year, β = K/Y will get closer and closer to 6. The reason, of course, is that year after year, capital or wealth grows by 0.12Y, which is faster than the growth in income of 0.02Y, so the ratio β = K/Y closes in on that ratio of s/g = 6. It never really gets there entirely, but after enough steps, it is close enough to round off to 6.

As long as 's', the proportion of national income saved remains at 0.12, and the rate of growth of national income remains at 0.02, the capital-income ratio of a country will move towards the number 6. This is so even if the capital-income ratio were to start from a number higher than 6. For those readers patient enough they can try it out for themselves. For example, start with the numbers in year one where β1 = 7, K1 = 105 and Y1 = 15. You will see that as long as you keep 's' equal to 0.12 and 'g' equal to 0.02, the capital - income ratio will gravitate towards the number 6. It may take many steps to get there because in those first steps β will be even higher than 7. But it will then gravitate downward towards 6.

Piketty's data shows that 's' and 'g' can vary at different times and in different countries. But once a country breaks out of its development stage and becomes one of the highly developed nations of the world its savings rate and its growth rate averages over long periods of time can be remarkably stable, and hence have long run predictive value for the ratio of wealth to income.

This is what Piketty means when he says his second fundamental law of capitalism is valid in the long run. But of course that is what Piketty's book is all about, the long run trends in capitalist economies.