Recalculating Capital in the Twenty-First Century

1. Recalculating Capital in the Twenty-First Century

2. The missing link: depreciation It’s large enough to matter: over 15% of annual output and income (Y) goes to replace capital (K) that wears out. If capital has an average life of 25 years then 1/25th of capital, or 4% wears out every year (d = 4%) Total depreciation = 4% x K Depreciation as a percentage of annual output = 4% x K / Y Note that K / Y is the capital-output ratio that plays a major role in Piketty’s book. He is worried about its continuing growth which is also driving up the share of income accruing to capital owners (Piketty’s “rentiers”).

3. The Second Fundamental Law of Capitalism The Solow model: The capital-output ratio stops rising when total savings is exactly matched by the amount of investment necessary to cover depreciation (d) and keep the capital stock growing just as fast as the economy (g):sY = (d+g)K => K/Y = s / (d+g) In the Solow model, all of the right-hand terms are fixed constants. So if the savings rate = 24%, d = 4% and g = 2%, then the economy’s steady-state capital-output ratio = 24 / (4+2) = 4.0 But Piketty’s Second Fundamental Law looks different:K/Y = s / g What happened to the missing d? Piketty’s savings rate turns out to be the economy’s net savings rate (snet), which is the gross savings rate minus the percentage of output needed for depreciation:snet = [s – d(K/Y)] For Piketty, the net savings rate is fixed. He puts it at 10-12 percent. So, if snet = 10% and g = 2%, then K/Y = 5.0, so depreciation as a share of output = 4% x 5 = 20% and the gross savings rate (s) = 10% + 20% = 30%. Only one of these two alternatives can be correct in the real world.

4. Key result 1: When growth slows from, say 3% to, as Piketty predicts for the future, 1.5%, the impact on capital accumulation is much larger for Piketty than for Solow. K/Y and dK/Y double for Piketty, but rise by only 27% for Solow. s rises by 58% for Piketty, but is unchanged for Solow. snet is unchanged for Piketty, but falls by 36% for Solow.

5. The First Fundamental Law of Capitalism The income of capital as a share of total income (a) rises and falls with the capital-output ratio:a = rK/Y where r = the rate of return on capital wealth. Piketty assumes that the rate of return remains constant at its historical value of about 5 percent as capital accumulation rises. So, if K/Y rises from 6 to 7, as he predicts for Europe, the income share of capital rises by five percentage points, from 30% to 35%. This was pointed out in the review by Solow. Key result 2: In the Solow growth model, increasing the supply of capital lowers the rate of return capital receives. So, the income share of capital rises by less in the Solow model than Piketty predicts. How much less? Using a Solow growth model seeded with parameters from Piketty’s book yields the following results for a drop in the growth rate from 3% to 1.5%:(1) K/Y: rises from 3.8 to 4.9 (about the same increase as Piketty predicts for Europe)(2) r: falls from 8.6% to 7.4% (net r after 4% depreciation falls from 4.6% to 3.4%)(3) a: rises from 33% to 36%, 3 percentage points versus 5 for Piketty

6. The threat from increased capital accumulation Piketty does not elaborate, but it could be based on a concern that a higher relative income share would give capital owners disproportionately more resources to influence democratic policies (?). That line of reasoning requires that depreciation be removed from the capital share calculation since depreciation spending is not available for other purposes:anet= (rK – dK) /(Y - dK) = (r – d)K/Y) / (1 – dK/Y) What happens with the Solow model? (1) g=3%: K/Y=3.8; r=8.6% => anet= 21.0%(2) g=1.5%: K/Y=4.9; r=7.4% => anet= 20.8% Conceptually, all of the increase in the gross income share of capital gets offset by the extra depreciation expense related to the higher capital-output ratio. None of it is left over for political lobbying or other discretionary spending. Key result 3: In the Solow growth model, the higher income share for capital that comes with capital accumulation is accompanied by higher required spending on depreciation, so the capital income share net of depreciation barely changes.

7. The threat from Piketty’s capital taxation proposal Piketty states that the primary goal of his proposed tax on wealth is “to stop the indefinite increase of inequality of wealth” (518). To put a halt on further capital accumulation means at the same time to put a cap on the savings rate at exactly enough to cover current depreciation and future growth. To see how that works, recall the Second Fundamental Law of Capitalism:K/Y = s / (d+g)If s=27%, d=4% and g=3%, K/L = 3.8, the starting point we’ve used before. If g falls to 1/5%, but K/L is capped at 3.8, then we solve instead for s:s = K/Y x (d+g) = 3.8 x (4% + 1.5%) = 21%.Essentially, 6% of income that would have otherwise have been saved for more investment in capital gets diverted to consumption. The problem? With a lower capital stock, worker productivity and wages end up lower than they would otherwise be, by nearly 8% in the example here. Summers noted this consequence in his review. Key result 4: Discouraging further capital accumulation lowers the future productivity and wages of labor compared to what would happen otherwise.

8. Summary of key results: Piketty is no fan of economists’ “undue enthusiasm for simplistic mathematical models” (16). Nonetheless, models can help shed light on difficult controversies. In this case, the Solow growth model provides a lens for evaluating Piketty’s analysis, and here are some of the key results that emerge: If the savings rate for an economy is a fixed share of total output, then lower growth has a much smaller impact on the capital-output ratio than if the savings rate is a fixed share of output net of depreciation. If a larger supply of capital lowers the rate of return capital receives, the income share of capital rises by less than if the return remains constant. Even if the income share of capital rises with capital accumulation, little or none of the increase may be available for any spending except to cover the depreciation on the extra capital. Discouraging capital accumulation lowers the future productivity and wages of labor compared to what would happen otherwise. Summers lists many salient threats to the future prosperity of workers, but capital accumulation per se is not one of them.