In his new book, Piketty has a rather lucid interpretation of Marx’s hypothesized “tendency of the rate of profit to fall” that also helps illuminate his hypothesis that capital’s share of income will grow in the next century.

For Marx, the logic of capitalism would either cause the rate of profit to fall endlessly or capital to gobble up an increasing share of national income endlessly, either of which would generate political instability that would unwind the whole system.

Piketty plugs Marx’s somewhat vague prosaic presentation of this theory into the two formulas Piketty relies upon to describe the dynamics of capital.

- β = s/g
- α = r*β

The first formula states that, in the long run, the capital/income ratio (β) of a nation’s economy is equal to the savings rate (s) divided by the growth rate (g). By capital/income ratio, Piketty means the total level of national capital divided by the total flow of national income. A nation with $6 of total capital that has an annual national income of $1 therefore has a β of 6 (or 600%). By savings rate, Piketty means the percentage of national income added to the stock of capital each year (each period really). A nation with a national annual income of $100 that saves $12 has an s of 0.12 (or 12%). By growth rate, Piketty means growth in the national income caused both by population growth and per capita productivity growth. A nation with $100 national income one year that has $102 national income the next year, has a g of 0.02 (or 2%).

So the formula is β = s/g. If s = 12, and g = 2, then β is equal to 6. That would be the long-term capital/income ratio, which is to say this is the capital/income ratio a nation would tend towards and eventually stop at.

Piketty then uses a second formula to relate β to capital’s share of the national income. This formula states that capital’s share of the national income (α) is equal to the rate of return on capital (r) multiplied by the capital/income ratio (β). So the formula is α = r*β. If r = 0.05 (5%) and β = 6 (600%), then α is equal to 0.30 (or 30%). That is, capital’s share of the national income would be 30%. Out of every dollar of national income, 30 cents would go to capital.

To repeat:

- β = s/g
- α = r*β

Piketty claims that Marx’s view (along with others in the period) is that growth (g) is primarily a function of capital deepening: you get more output because you add more capital. Today, we have the concept of Total Factor Productivity that captures the fact that productivity can grow without or in excess of additional capital inputs.

But without TFP, you may believe that growth in national income would eventually run towards zero. In that zero-growth world, if savings continued to be non-zero (capitalists continued putting aside some of their income to build out more capital), then the capital/income ratio escalates towards infinity. In formula terms, since β = s/g, if g is at 0, and s is above 0, β asymptotically approaches infinity.

If β asymptotically approaches infinity, that then generates problems for the second formula α = r*β. If the capital/income ratio (β) is constantly increasing, then either the rate of return (r) will have to constantly decrease towards zero, or capital’s share of income (α) will have to constantly increase. That’s the only way to make the equation balance.

The case where r decreases to compensate is the “tendency of the rate or profit to fall” scenario. The rate of return on capital trends towards zero and that causes all sorts of political instability because of capitalists tearing each other apart to try to scratch out a return. The case where α increases to compensate is a world where capital slowly gobbles up more and more of the national income until it gobbles up all of it, a scenario which will surely generate worker revolution.

So, if we assume growth or near-zero growth will eventually result after capital deepening has essentially run its course (Piketty’s account of what Marx may have had in mind), capitalism does appear to self-destruct from internal contradictions. It is only the existence of TFP-driven growth elements that allows us to stave off this particular conclusion, something Marx would not have been aware of.

**Piketty’s View**

Using the concepts from above, Piketty’s own view about the future dynamics of capital becomes easy to explain.

Recall again the two formulas:

- β = s/g
- α = r*β

Because population growth, and catch-up growth, and recovering-from-20th-century-shocks growth will subside, the rate of growth (g) will decline, though it wont race towards zero: it will be stable, just lower. If savings (s) hold steady, that then means that the capital/income ratio (β) will increase.

Moving to the second formula (α = r*β), the operative question is how will the rate of return on capital (r) respond. As the amount of capital increases, the marginal productivity of capital will decline, meaning that r will decline. But will r decline enough to offset the increase in the capital/income ratio (β) in such a way as to keep capital’s share of income (α) from climbing? Piketty doesn’t believe so.

Therefore, a decline in growth will cause the capital/income ratio to increase, an increase which will not be totally offset by falling returns on capital. Consequently, capital’s share of total income will increase in the next century.