# 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.

## 7 thoughts on “Piketty and Growth Economics”

1. Hi, thanks for this review, which among other things helps answer a question I had about the alleged Cobb-Douglasness of the production function. However, could you clarify how your sentence in bullet point 1 (saying that economic profit has risen but accounting profit has not) sits with the content of para 5 (which as far as I can see says that what has risen is the return to capital)? My textbook (Mankiw, ‘Macroeconomics’) says that accounting profit includes the return to capital but economic profit does not.
I’m fairly new to this so I must be looking at something the wrong way. I hope it’s not something obvious but I would be very grateful if you could drop a note to put my mind at rest. Thanks!

• Sarah – thanks for reading. In bullet point 1 my statement should have been something like “…with economic profits (meaning true profits not just accounting profits) rising in the late 20th century.” The return to capital is measured in national accounts as the total income that goes to owners of capital, but does not tell us whether those are economic profits or accounting profits. My point was that perhaps the rising return to capital perhaps reflects a rise in economic profits.

In paragraph 5, I was talking about the share of total income that goes to owners of capital (whether economic or accounting profits). Kaldor suggests it’s fixed at about 1/3, but Piketty questions this number, and says it has fluctuated over time. Taking Piketty’s data as correct, then this share is getting higher. This implies that the Cobb-Douglas may not be the right production function to use.

Hope that helps.

2. I haven’t yet read Prof. Piketty’s book – that will have to wait until the end of the academic year. So I don’t know how he is measuring the capital share or labor share over time. But for the record, it is not at all straightforward to read these income shares from national accounts data. The “true” labor and capital shares cannot be easily inferred from macro data – and they are also difficult to pin down in firm-level micro data.

Even then, making any link from income shares (often called the “functional distribution of income”) to income distribution (the “size distribution of income”) is problematic. Not all capital income accrues to rich people, and not all labor income goes to the poor or the working classes.

The national income and product accounts for most countries report something called employee compensation. This sounds like labor income, but it leaves out some important forms of labor income, such as the labor income of the self-employed. Employee compensation also includes labor income earned by some very high-income individuals (e.g., corporate CEOs), so the employee compensation share is an inadequate measure for thinking about the size distribution of income.

Movements over time in the employee compensation share of national income may thus reflect changes in the structure of employment rather than in the underlying degree of inequality. For instance, high income individuals may prefer under some tax codes to take their income as wages; under other regimes, they may choose partnership arrangements or dividend income that would be treated differently in the national accounts.

This suggests that caution is warranted in interpreting movements in the functional distribution of income. These may reflect changes in the underlying labor and capital shares. But they may also (or instead) reflect changes in tax policy, employment regimes, or the structure of corporate incentives.

The bottom line is that it is not easy to demonstrate that there has been a change in the capital share of income – as opposed to the “measured share,” which is what finds its way into the national accounts. And even if the capital share could be measured clearly, it would still be challenging (or indeed impossible) to arrive at a clear inference about the implications for the size distribution of income.

The unfortunate truth is that the national income and product accounts do not provide any unambiguous information about the size distribution of income. They were not devised for this purpose, and they are not an appropriate data source for thinking about income inequality.

I’ll leave it to others to discuss the feasibility of using the national income accounts to calculate rates of return to capital. My impression is that this is equally problematic, but I have not looked at this as carefully. Perhaps others can comment on this?

• Doug – thanks for the comments. You have probably just bought yourself a full post as a reply, as I need to digest all your thoughts.

3. Hi, I really enjoy reading your posts. I came across this particular post only today. I was wondering about your reaction to the argument put forward recently by Kanbur and Stiglitz in their VoxEU post (URL below). They seem to be arguing that since all of the ‘capital’, as defined in Piketty’s book, need not be going into physical production, we need not resort to a production function with a elasticity of substitution (between capital and labor) that is greater than 1, in order to reconcile Piketty’s finding of increasing share of ‘capital’ in total income with rising K/L ratios predicted by standard growth-theory models.
URL: http://www.voxeu.org/article/wealth-and-income-distribution-new-theories-needed-new-era.

• I need to read their post, but yes, Piketty’s numbers are about “capital” meaning the sum of financial assets (which includes claims on physical capital, like shares in companies), while growth models are about “capital” meaning the stock of useful, productive assets. And those need not be similar, and the whole idea of an elasticity of substitution between Piketty’s “capital” and labor doesn’t actually mean much then.