Some of the leading economists of our days, like Paul Krugman and Brad DeLong, have praised Piketty’s hypotheses, which are underpinned by a vast amount of empirical data.
There are other commentators, some economists among them, who have been much more critical of Piketty’s theories and his approach. A lot of this criticism, however, has been misguided. It seems that some of the critics have not even bothered to read more than the introduction of the 600-page bestseller. As a consequence, they criticize the author for claims he has never made. Brad DeLong summarizes a vast amount of criticism to Piketty’s book and shows that most of it is based on a misunderstanding of his work. Especially, the right-wing critics never seem to have made an effort understanding the economic mechanisms, which Piketty describes in his bestseller.
See here: http://www.project-syndicate.org/commentary/j--bradford-delong-is-surprised-by-the-poverty-of-conservative-criticism-of-capital-in-the-twenty-first-century).
In any case, it is clear that Piketty’s work is one of the most important contributions to the theory and empirics of economic growth and inequality. In what follows, I will summarize and comment on some of Piketty’s most important findings and claims.
The easiest way to start is to note that total GDP (Y) can be divided into total labor compensation (w*L), where w is the average wage rate and L is the total labor stock, and total capital income (r*K), where r is the average rental rate and K is the entire capital stock of the economy.
Y = w*L + r*K
The labor share is then simply total labor compensation divided by total output while the capital share is total capital income divided by output. Obviously, both shares must sum up to one:
1 = (w*L)/Y + (r*K)/Y
Historically, economists have believed that the two factor shares have been relatively constant over time. Consequently, many growth models have used the Cobb-Douglas production function, which exhibits the aforementioned property of constant factor shares.
However, Piketty’s data forcefully illustrates that constant factor shares are only an idealization in the mind of economists, but have little to no empirical foundation in reality. Indeed, the labor share and capital share of income have fluctuated wildly in many countries over the last centuries, thus implying that the Cobb-Douglas specification is probably an overly simplistic representation of the economy.
In our master thesis, my coauthor and I calculated the labor share for the U.S. since 1947. We have used data from the National Product and Income Accounts (NIPA) to obtain the total wage compensation of all employees for all industries. Obviously, one also has to take into account the income of the self-employed whose total income is also reported in the NIPA. The problem that arises for the self-employed is that it is not entirely clear which part of their income accrues solely to labor and what share of their income is the return on a risky entrepreneurial and/or financial investment.
To calculate a proxy for the labor income of the self-employed, we have thus simply assumed that they earn, on average, the same wage as regular employees in the same industry. This assumption might obviously be untrue. It is possible that the self-employed earn considerably more (or less) than their employee counterparts in the same industry. Any bias resulting from our assumption, however, is likely to be small since the number of self-employed is relatively low: In 2012, less than 8.5% of all workers in the private sector were self-employed. The variation in between different industries is relatively large, ranging from 0% in utilities and 2.5% in mining to 20% in construction and even 40% in agriculture.
The figure below shows the adjusted labor share (adjusted for the self-employed) we calculated for the entire U.S. economy, but also for the private sector only. Indeed, total labor income in the U.S. economy has actually been relatively stable from the late 1950s until the late 1980s at around 61%. The minor fluctuations observed during that period are probably a result of the business cycle. The figure somewhat confirms the ‘constancy’ of factor shares during the first decades. However, ever since the 1990s labor income has dropped significantly. Indeed, in the private sector total labor compensation fell by about 6 percentage points within a short period of time, from its 1981 to 1998 average of around 59% to a record-low of 53% in 2012. Our calculations thus seem to support at first glance Piketty’s hypothesis that capital has become more important in recent years. Indeed, the fall in the labor share implies an equivalent increase in the capital share. Since the distribution of capital ownership is highly unequal within society, a rise in the capital share also implies a more inegalitarian society. Indeed, capital ownership is much more unequally distributed than labor income, an issue I will discuss later on.
What Piketty calls the first fundamental law of capitalism is that the capital share, denoted alpha, is equal to the average rental rate of capital r times the capital/income ratio beta. This equation is a simple accounting identity and it holds at all times.
The first thing one should note is that equation 1 includes the term r, the average return on capital. This concept is obviously to some extent an abstract from reality. In very economy, there are many different financial and real assets that yield very different returns. Piketty notes, for example, that normal savings accounts usually yield a real return of only 1 to 2% in normal times. Meanwhile, the rental return on housing averages typically at around 3 to 4%. The return on stocks, which includes dividends and capital gains, can be as much as 6 to 7% in the long-run. These differences in returns between different asset classes obviously reflect the fact that the underlying assets are fundamentally very different in their structure. Above all, investors are compensated for additional risk, which obviously explains why stock returns are higher than the return on saving accounts, at least in the medium to long-run.
Despite the fundamental differences that exist across the many asset classes, leading to highly variable yields depending on the underlying asset, it is mathematically possible to calculate an average return on capital. Piketty notes that this average return on capital has historically been at around 4 to 5% in all industrialized countries during the 20th century.
Note that a capital/income ratio of 600% combined with an average return on capital of 5% leads to a capital share of income of 30% (600*0.05 = 0.3).
Equation 1 also clearly illustrates that an increase in the capital/income ratio leads to a higher capital share of income, given that the return on capital remains unchanged.
The crucial assumption here that r remains unchanged, however, is clearly invalid. One of the most fundamental laws in economics is the one of diminishing returns. As more and more capital gets accumulated, that is as the capital/income ratio rises, the average return on capital must decrease. That is because every additional unit of capital is slightly less productive than the previous unit (capital becomes less productive at the margin). The fundamental question is by how much the return on capital will fall.
It turns out that the decrease in the return on capital will be determined by the elasticity of substitution between capital and labor, that is how easy it is to replace labor by capital, and vice-versa. Intuitively, if labor is easily replaceable by capital, that is if the elasticity of substitution is high, then capital remains quite productive even as more capital gets accumulated. Consequently, the return on capital will not decrease by that much as the capital/income ratio rises. If the elasticity of substitution on the other hand is low (i.e. labor is not that easily replaceable by capital), then additional units of capital are not very productive and, as a consequence, the return on capital will fall much faster. It turns out that there are three possible scenarios in the case of further capital accumulation, i.e. an increase in the capital/income ratio.
1) An elasticity of substitution smaller than 1
A very low elasticity of substitution implies that the returns to capital will fall quite substantially as more capital is accumulated. In fact, even as the capital/income ratio beta rises, the return on capital r will fall by so much that the capital share of income alpha actually decreases.
Let’s assume an initial scenario in which beta is equal to 6 and r is equal to 5%, implying a capital share of income exactly equal to 30%. If the capital/income ratio to rose from 6 to 8 (an increase of 33%) and if the return on capital fell to 3.5% (a decrease of 30%), then the capital share would actually fall to 28%.
This example just given above actually corresponds to an elasticity of substitution between capital and labor equal to about 0.8, somewhat smaller than the unity case (see below).
2) An elasticity of substitution equal to 1 (the unity case)
The elasticity of substitution equal to unity (=1) corresponds to the special case in which the economy is characterized by a Cobb-Douglas production function. In that case, the capital share of income always remains constant, no matter what the capital/income ratio is. Let’s again assume beta rises from 6 to 8. In the Cobb-Douglas case, the return on capital would then fall from 5 to 3.75%, so that the capital share remains at exactly 30% of annual income.
3) An elasticity of substitution larger than 1
The last case is the one in which the elasticity of substitution is larger than one, implying that labor is easily replaceable by capital. Further accumulation of capital will then lead to a much smaller decrease in the return on capital since additional units of capital are still quite productive at the margin. Again increase in beta from 6 to 8, combined with a decrease in the return on capital from 5 to 4.2%, for example, implies an increase in the capital share. In this case, the capital share rises from 30 to 33.6% of annual income. This example corresponds to an elasticity of substitution of about 1.65, which is considerably larger than the unity case.
Note that only in the special case in which the elasticity is equal to unity the economy can be described by the rather simplistic Cobb-Douglas production function. If the elasticity is larger or smaller than one, economists usually use the more complicated constant elasticity of substitution (CES) production function to describe the economy.
The ultimate question now is, of course, which of the three scenarios from above is the one that is the most likely to occur. Or rather which of the three scenarios is the one industrialized economies are subject to right now. The level of the elasticity of substitution between capital and labor is ultimately a matter of empirics. Piketty argues that this elasticity is larger than one, implying that capital and labor are highly substitutable. His claim is backed up by other studies, such as Karabarbounis and Neiman (2013), who also argue that the elasticity is larger than one, and more specifically, that it is roughly at about 1.25. This, of course, implies that the capital share of income rises as the capital/income ratio increases. That is because the return on capital r only falls somewhat as more capital is accumulated. This is exactly the process Piketty has documented for many industrialized countries in recent decades: An increase in the capital/income ratio combined with an increase in the capital share of income.
As Piketty explains in his book, there are good reasons to believe that the capital/income ratio and consequently the capital share of income will increase even further in the future. One should note that this process would inevitably lead to a more unequal society since wealth is much more unevenly distributed than income.
Since this post is already far too long, I will explain in my next writing what the fundamental economic forces are that determine the capital/income ratio in the long-run. I will then also come back to the distribution of income and wealth within society, a topic that is also discussed at great length in Pikettys’s book.
· Grenestam, E. & Probst, J. (2014). Rising markups and the fall of the labor share: Predicting the effects of industry market power in the U.S. Master thesis, Lund University.
· Karabarbounis, L. & Neiman, B. (2013). The global decline of the labor share. NBER Working Paper, 19136, 1-46.
· Piketty, T. (2014). Capital in the twenty-first century. (1 ed., Vol. 1, p.43). Cambridge, Massachusetts London, England: The Belknap Press of Harvard University Press.