Piketty and Growth Economics

Reviews of Thomas Piketty's "Capital in the 21st Century" are second only to cat videos on the internet, it seems. Not having any cats, I am unable to make a video, so you're stuck with a review of Piketty's book.

I was particulary struck by the implications of this work for economic growth theory. The first section of the book studies capital/output ratios, one of the core elements of any model of growth that includes capital. Piketty provides a long time series of this ratio, showing that in Europe it tended to hover around 7 during the 1800's and early 1900's, then dropped dramatically following World War I, stayed at around 3 until the 1970's, and now is rising towards 6. In the U.S., it has been less variable, going from around 4.5 in the 1800's to about 3 in the 1960's, and now is back up to about 4.5.

The projection that Piketty makes is that the capital/output ratio will tend to be about 6-7 across the world as we go into the future. The main reason is that he expects population growth to decline, and the capital/output ratio is inversely related to population growth. In a standard Solow model with a fixed savings rate ${s}$, the capital/output ratio is ${K/Y = s/(n+\delta+g)}$, where ${n}$ is population growth, ${\delta}$ is depreciation, and ${g}$ is the growth of output per worker. You can see that as ${n}$ goes down, the ${K/Y}$ ratio rises.

By itself, this doesn't imply much for growth theory, in that the expected ${K/Y}$ ratio in the future is entirely consistent with Piketty's claim regarding population growth. He might be wrong about population growth, but if ${n}$ does in fact fall, then any growth model would have predicted ${K/Y}$ will rise.

The interesting implication of Piketty's work is on the returns to capital. In particular, the share of national income that goes to capital. His figures 6.1-6.3 document that this share has changed over time. From a share of about 35% in the 1800's in both Britain and France, the share dropped to about 20-25% in both countries by the mid-20th century. Most recently, the capital share is starting to rise across many countries, going up about 10 percentage points between 1970 and 2010.

One of the bedrock assumptions made in most growth models is a Cobb-Douglas production function, which implies (under conditions of perfect competition) that capital's share in output is fixed by a technological parameter, typically called ${\alpha}$ and typically assumed to be ${\alpha = 0.3}$. Over time, the share of output going to capital is constant at this value of ${\alpha}$. Growth economists lean on this assumption because of work done by Nicholas Kaldor, who established as a ``stylized fact'' that capital's share in output is constant at about 0.3--0.35. As Piketty points out, though, Kaldor established this fact using a very small time series of data from a particularly unusual time period (roughly the mid-20th century).

The fact that capital's share of output has changed distinctly over long time frames means that this baseline assumption is called into question. What does it mean? I have two immediate thoughts.

• Perfect competition is not a good assumption. This is probably trivially true; there is no such thing as a perfectly competitive economy. But what Piketty's data would then indicate is that the degree of imperfection has possibly changed over time, with economic profits (not accounting ones) rising in the late 20th century. We have lots of models of economic growth that allow for imperfect competition (basically, any model that involves deliberate research and development), but we do not talk much about changes in the degree of that competition over time.
• The production function is not Cobb-Douglas. Piketty talks about this in his book. The implication of rising capital shares that coincide with rising capital/output ratios is that the elasticity of substitution between capital and labor is greater than one. For Piketty, this contributes to increasing inequality because capital tends to be owned by only a small fraction of people. For growth economists, this raises interesting possibilities for what drives growth. With a sufficiently large elasticity of substitution between capital and labor, then growth can be driven by capital accumulation alone. To see this, imagine perfect substitutability between capital and labor in production, or ${Y = K + AL}$, where ${A}$ is labor-specific productivity. Output per worker is ${y = K/L + A}$. As the capital/labor ratio rises, so does output per worker. This continues without end, because there are no longer decreasing returns to capital per worker. Even if technology is stagnant (${A}$ does not change), then output per worker can go up. We tend to dismiss the role of capital per worker in driving growth, but perhaps that is because we are wedded to the Cobb-Douglas production function.

The remainder of Piketty's book is very interesting, and his own views on the implications of rising inequality have been subject to an intense debate. But from the perspective of growth economics, it is the initial section of the book that carries some really interesting implications.