Both Plague & War Cut Capital Share?

I just finished reading Walter Scheidel’s The Great Leveler: Violence and the History of Inequality from the Stone Age to the Twenty-First Century, and found myself agreeing with Scheidel against his critics. Scheidel is a historian who says that inequality has mainly risen in history when income increased, making more inequality physically possible, and when scale and complexity increased, creating more and bigger chokepoints (e.g., CEO, king) whose controllers can demand more rents.

Big falls in inequality have mainly come from big collapses, such as big wars, revolutions, plagues, and state collapses, which are usually associated with violence. This suggests that a big inequality fall is unlikely anytime soon, and we shouldn’t wish for it, as it would likely come from vast destruction and violence. All of which I find very plausible.

While usually big wars via mass mobilization didn’t change inequality much, in the mid 1900s such wars seemed to have gone along with a big taste for redistribution and revolution. This happened to a lesser extent in Ancient Greece and Rome, and fits a story wherein more forager-like cultures care more about redistribution, especially when primed by visible mass sacrifice.

I noticed one puzzling pattern, however. Income in the world goes to owners of capital, to owners of labor, and to those who can take without contributing to production. As the rich usually get more of their income from capital, compared to labor, one thing that can cause less inequality is a change that makes capital earn a smaller share of total income. The puzzling pattern I noticed is that even though big plagues and big wars should have opposite affects on the capital share, both of them seem to have cut inequality, and both apparently in part via cutting the capital share of income! Let me explain.

Big plagues cut the number of workers without doing much to capital, while big wars like WWI & WWII destroy a much larger fraction of capital than they do of labor. Which event, big plague or big war, reduces the share that capital earns? The answer depends on whether capital and labor are complements or substitutes. If they are substitutes, then destroying capital should cut the capital share of income. But when they are complements, it is destroying labor that should cut the capital share.

The simple middle position between complements and substitutes is the power law (a.k.a. “Cobb-Douglas”) production function, where output Y = La*K1-a, for Labor L, capital K, and constant a in (0,1). (Partial derivatives set wages w = dY/dL and capital rent r = dY/dK.) In this situation, the capital share of income r*K/(r*K+w*L) = 1-a, and so never changes.

If, for example, labor L falls by a factor of 2, while capital K stays the same, then wages rise by the factor 21-a while rents fall by the factor 2a, with the product of these factors being 2. Compared to this simple middle position, if labor and capital are instead complements, then in this example wages would rise and rents would fall by larger factors. If labor and capital are instead substitutes, the factors would be smaller.

Economic papers based on data over the last century usually find labor and capital to be complements, though there are notable exceptions such as Thomas Pietty’s blockbuster book. That fits with data on the Black Death. In the century from 1330 to 1430, Europe’s population fell roughly in half, wages doubled, and rents fell a lot. In England, wages tripled. Similar behavior is seen in other large ancient plagues – wages rose by a factor of four in Mexico! This looks more like what you’d see with complementarity than with a simple power law.

World War I (WWI) killed about 1% of the world population, while the concurrent 1918 flu killed about 4%. World War II (WWII) killed about 3%. But capital was cut much more. The ratio of private wealth to national income fell by a factor of two world wide, and by even larger factors in the main warring nations (source):
Now for the puzzle. If capital and labor were still complements during WWI & WWII, then destroying a lot more capital than labor should have resulted in rents on capital rising by a factor so big that product of the two factors increases the capital share of income. Is that what happened? Consider Japan, where 5% of the population died:

Real [Japanese] farm rents fell by four-fifths between 1941 and 1945, and from 4.4% of national income in the mid 1930s to 0.3% in 1946. .. By September 1945, a quarter of the country’s physical capital stock had been wiped out. Japan lost 80% of its merchant ships, 25% of all buildings, 21% of household furnishings and personal effects, 34% of factory equipment, and 24% of finished products. The number of factories in operations and the size of the workforce they employed nearly halved during the final year of the war. p.121

Gains from capital almost disappeared during the war years: the share of rent and interest income in total national income fell from a sixth in the mid-1930s to only 3% in 1946. In 1938, dividends, interest, and rental income together had accounted for about a third of the income of the top 1%, with the remainder divided between business and employment income. By 1945, the share of capital income had dropped to less than an eighth and that of wages to a tenth; business income was the only significant revenue source left to the (formerly) wealthy. p.122

In 1946, real GNP was 45% lower than it had been in 1937. p.124

The sharp drop in top income shares .. were caused above all by a decline in the return on capital. .. Most of these changes occurred during the war itself. p.128

Consider also France and Germany (which lost 2% & 11% of people in WWII, respectively):

During WWI, .. a third of the French capital stock was destroyed, the share of capital income in national household income fell by a third, and GDP contracted by the same proportion. ..In WWII, .. two-thirds of the capital stock was wiped out. .. real rents fell by 90% between 1913 and 1950. p.147

[German] rentiers lost the most: their share of national income plummeted from 15% to 3% even as entrepreneurs were able to maintain their share .. real national income was a quarter to a third lower in 1923 than it had been in 1913. p.152

Maybe I’m missing something, but I don’t see how this is remotely consistent with labor and capital being complements. Yet complementarity seems a good fit to big ancient plagues and more recent empirical studies. What gives?