# Oberfield and Raval on Capital/Labor Elasticity of Substitution

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

I was in Boston for the NBER summer institute on Friday, sitting in on what it typically called either “Growth day” or “Jones/Klenow” after the organizers. Regardless, here’s the program. It’s a chance to see what is some of the cutting/bleeding edge research in economic growth.

The first paper I saw was by Ezra Oberfield of Princeton and Devesh Raval of the Federal Trade Commission (I missed the Grossman/Helpman paper because I like to sleep, and didn’t get to Boston until 10:15am – sue me). They were doing two things. (1) providing an estimate of the aggregate elasticity of substitution (EOS) between capital and labor and (2) using that to try and account for the decline in labor’s share of income over the last 30-40 years.

On (1), they made the point that the aggregate EOS is not a technological constant, but rather is an artifact of the micro-level EOS. Specifically,

$\displaystyle \sigma^{agg} = (1-X)\sigma^{micro} + X \epsilon \ \ \ \ \ (1)$

where ${\sigma^{micro}}$ is the EOS at the plant level. The weighting term ${X}$ reflects the variation in capital shares across firms. ${\epsilon}$ is the elasticity of demand for plant output. The demand elasticity is in to account for the fact that some of the response to a change in factor prices is to move demand away from the plants that tend to use the more expensive factor.

Regardless, Ezra and Devesh provide evidence that ${X}$ is really close to zero, so essentially this demand adjustment is negligible, and the aggregate EOS is roughly equivalent to the micro EOS. They estimate this from plant-level data, and find something like 0.52, meaning that capital and labor are not easily substituted for each other. Over time, the aggregate EOS is roughly stable at around 0.70, based on their values for ${X}$ and ${\epsilon}$.

On (2), given their aggregate EOS, the implication is that the decline of labor’s factor share is biased technical change. Increased automation, IT investment, and offshoring, among other things, have driven down labor’s share of output down over time.

Changes in factor prices alone (wages and rental rates) would have raised labor’s share of output over this period, they find. The force of biased technical change was so strong it overcame that tendency.

It’s worth noting how important finding the EOS1, then firms can switch easily from labor to capital. Relatively cheap capital is substituted for labor, and labor’s share drops. If EOS>1, then the decline in labor share is driven in part by more expensive labor, and hence the implied degree of biased technical change is smaller.

# Piketty and Growth Economics

NOTE: The Growth Economics Blog has moved sites. Click here to find this post at the new site.

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.