Piketty’s (2014) book Capital and associated articles contains a lot of fascinating stuff and has inspired a lot of interesting debate. Here are some summary notes. Please comment on errors. If you want to change the world, you need to know how it works.

The critique: Data and theory don’t support the r > g story, Piketty’s 2nd law, that low growth and high returns on capital will make K/Y increase in the future.

• Net savings rate does not seem to be stable in the long run and does not seem to correlate negatively with growth.
• Adjusted for valuation, K/Y doesn’t seem to have increased in the last decades, contrary to some of Piketty’s main findings.
• Piketty means that the elasticity of substitution between K and L ($\sigma$) is above 1, based on, among other things, data on K/Y. But if K/Y hasn’t increased, Piketty might also be wrong about $\sigma$. A lot of earlier empirical analysis indicates $\sigma$< 1. Also, for Piketty’s theory to hold, $\sigma$ might need to be much higher than Piketty suggests.

Piketty

Presents two “fundamental laws of capitalism”:

1. The basic definition of capital income share to national income: $\alpha = r \times \beta$, where r is real rate of return on capital, and $\beta = K/Y$, K=capital, Y=real national income (GDP).
2. In the long run $\beta = K / Y = s / g$, where s=net savings rate, g=growth rate of the economy. So long run $\alpha = rs/g$. Assuming r and s are long run stable.

“When the rate of return on capital exceeds the rate of growth of output and income, as it did in the nineteenth century and seems quite likely to do again in the twenty-first, capitalism automatically generates arbitrary and unsustainable inequalities”. I.e., when r > g. Piketty guesses that the difference will be huge in 100 years. If I get this right, the term “unsustainable” here seems to mean socially unsustainable – so it’s a problem because people don’t accept it. But it’s not a problem for employment, GDP etc. Theoretical details are described in the technical appendix (p. 37-39) to the book, and Piketty and Zucman 2015. Some parts are standard, others less standard. The appendix describes a production function with constant elasticity of substitution:

$Y=F\left(K,L\right)=\left[aK^{\frac{\sigma-1}{\sigma}}+(1-a)L^{\frac{\sigma-1}{\sigma}}\right]^{\frac{\sigma}{\sigma-1}}$

where Y=net GDP, K=capital, L=labor, $\sigma$=elasticity of substitution, a=K-share of national income. The marginal product of

$K=F'_{K}=\frac{\sigma}{\sigma-1}\left(\frac{\sigma-1}{\sigma}\right)aK^{-\frac{1}{\sigma}}\left[aK^{\frac{\sigma-1}{\sigma}}+(1-a)L^{\frac{\sigma-1}{\sigma}}\right]^{\frac{\sigma}{\sigma-1}-1}$ $=a\left(\frac{Y}{K}\right)^{1/\sigma}=a\left(\frac{K}{Y}\right)^{-\text{1/\ensuremath{\sigma}}}=a\beta^{-1/\sigma}$.

Assuming K recieves marginal product, then rate of profit and interest is $r=F'_{K}$. K share of income $\alpha=r\beta=a\beta^{\frac{\sigma-1}{\sigma}}$. So $\alpha$ is an increasing function of $\beta$ if $\sigma>1$ (increasing capital accumulation correlates with increasing capital share of national income). Piketty and Zucman 2015: In the long run, with steady net savings rate after deprecation: K/Y $\rightarrow$ s/g, where s=S/Y=net savings rate and g=growth rate of economy.

Based on wealth data, Piketty suggests that $\sigma$ is around 1.3 – 1.6. Piketty and Zucman 2014 argues that we cannot exclude even higher $\sigma$ in the future, but also “stress that our discussion of capital shares and production functions should be viewed as merely exploratory and illustrative.”

Piketty and Zucman use a non-standard definition of capital:

The main difference between our work and the growth accounting literature is how we measure capital. Because of the lack of balance sheet data, in the growth literature capital is typically measured indirectly by cumulating past investment flows (…). By contrast, we measure capital directly by using country balance sheets in which we observe the actual market value of most types of assets: real estate, equities (which capture the market value of corporations), bonds, and so on.

Elasticity of substitution, $\sigma$, is a measure of how the relative demand/use of two or more products changes when their relative prices change. For a constant returns to scale production function, using K and L inputs with positive diminishing returns:

$\sigma = - \frac{ d [log (K/L)] } { d[ log(F'_{K} / F'_{L} ] }$

Thus, how demand for K and L change in relation to their marginal products, given by $F'_{K}$ and $F'_{L}$. If $\sigma\rightarrow 0$, K and L becomes perfect complements (Leontief production function). If $\sigma\rightarrow \infty$ the $F(K,L)$ becomes linear, K and L are perfect substitutes. If $\sigma\rightarrow1$, the $F(K,L)$ becomes the Cobb-Douglas production function. Pedagogic examples here.

Constant gross or net saving? The r > g story

Krusell och Smith (2015): The r>g story doesn’t add up. The usual assumption nowadays is that gross savings rate is constant in the long run, and net savings rate is endogenous in the model. Piketty assume constant net savings rate, and endogenously given gross savings in the model. Standard textbook model:

$\frac{k_{t}}{y_{t}}=\frac{s}{g+\delta}$

where k=capital, y=GDP, s=gross savings rate, g=growth rate of the economy and $\delta$=deprecation rate. Net expression: $\frac{k_{t}}{y-\delta k}=\frac{1}{\left(g+\delta\right)s-\delta}=\frac{s}{g+\delta\left(1-s\right)}$. Net savings rate $\tilde{s}=\frac{sg}{g+\delta(1-s)}$. So $\lim_{g\rightarrow0}\tilde{s}=0$

Piketty and Zucman:

\begin{aligned} \underbrace{k_{t+1}-k_{t}}_{\text{net investment}} & =i-\delta k_{t} = \underbrace{\tilde{s}}_{\text{constant net savings rate}} ( \underbrace{y_{t}-\delta k_{t})}_{\text{net output}} \end{aligned}

and $k_{t+1}=k_{t}+\tilde{s}\tilde{y}_{t}$ which gives Piketty’s second law of capitalism: $\frac{k_{t}}{\tilde{y_{t}}}=\frac{\tilde{s}}{g}$. Gross savings rate in this case is given by the continuous function

$s(g)=\frac{\tilde{s}\left(g+\delta\right)}{g+\tilde{s}\delta}$, so if economic growth (g) goes to zero or close, all income in the economy is saved $s\left(g\right)=\frac{\tilde{s}\delta}{\tilde{s}\delta}=1$, with savings rate = 100%. Then $k/\tilde{y}=\infty$. This “seems implausible”. See also Piketty and Saez in Science 2014.

Krusell and Smith also points out that the data from Piketty and Zucman (2014) and elsewhere indicate that growth rates correlates somewhat positively with net and gross saving. Both net and gross s fluctuates, gross fluctuates less. “To sum up […] data speak quite strongly against Piketty’s model” (cf. fig. 4 below).

Graph taken from Krusell och Smith (2015).

Acemoglu and Robinson 2015 discusses general laws of capitalism in general: new data is great but why doesn’t Piketty present any regressions on r-g, inequality? Acemoglu and Robinson run regressions on 1-, 5-, 10- and 20-year averages for 18-28 countries going back to 1870 at most. But instead of the positive correlation Piketty suggests (low growth more inequality), their results indicate no, or a negative, relation: increasing growth sometimes correlates with increasing inequality.

The definition of capital

Piketty and others claim that K/Y have increased in recent decades, and uses a non-standard definition of K. Rowthorn (2014): Pikettys non-standard definition of capital is problematic, since housing is not always included in the production process, and there might be a issue with valuation. Let W be the market value of K, and the valuation ratio $v=\frac{W}{K}$, which for public companies is Tobin’s Q. In Piketty’s case

$\beta=W/Y=vK/Y$.

Growth rates satisfy $g_{\beta}=g_{K/Y}+g_{v}$, where Piketty then assumes $g_{v}=0.$ Rowthorn:

Given [Piketty’s] finding that $\beta$ has increased by a great deal in recent decades, Piketty concludes that K/Y must have increased by a similar amount. However, evidence (…) indicates that K/Y has been falling since around 1981-2 in the United States and has been roughly constant in most of Europe. Indeed, this is just what Piketty and Zucman (2013) find when they correct the wealth-income ratio for valuation changes (capital gains) [cf. figure A133 below].

Graph taken from Piketty Zucman 2013 chartbook to “Capital is back”.

Rognlie (2015): The net K income share increase since 1948 comes entirely from the housing sector. Contribution from other sectors zero or negative. The development is not easy to understand with the common theories on bargaining power and technology. Data might be tricky – countries with fewer homeowners might have lower housing capital income.

Piketty and others explain the rise in the K income share through a rise in the value of reproducible capital relative to aggregate income. This should also be true within sectors. By decomposing the K income share to observed value of K, net user cost, and firm’s markups over costs not attributable to the user cost, Rognlie compares how different factors contribute to the development of capital income: “Figure 6 shows how the net capital income for the U.S. corporate sector (…). In other words, contrary to [Piketty and other’s claims], time-series shifts in the capital share in the corporate sector cannot be explained by parallel shifts in the measured value of capital.”

Graph taken from Rognlie (2015).

The elasticity of substitution between capital and labor

Piketty and Zucman claims that the elasticity of substitution ($\sigma$) between K and L is 1.3-1.6. Rowthorn (2014) again: The share of capital income, $\alpha=a\left(\frac{K}{Y}\right)^{\frac{\sigma-1}{\sigma}}$. In growth rates this satisfy

$g_{\alpha}=\left(\frac{\sigma-1}{\sigma}\right)g_{K/Y}$

meaning that $g_{\alpha}$ and $g_{K/Y}$ have the same sign if $\sigma>1$ and opposite signs if $\sigma<1$, which is a standard result. But if, for instance, K/Y hasn’t increased over the last decades, $\sigma$ is probably lower than 1.3-1.6.

Rognlie (2015) again: Piketty’s claim that increasing capital accumulation has led to increase of K income share since the rising quantity of capital is not fully offset by a fall in the returns on K. This only holds if $\sigma$ is much higher than what Piketty and others claim. Theoretically: Standard neoclassical definition of $\sigma$ as above gives

$\frac{d\left[\log\left(F'_{K}K/F\right)\right]}{d\left[\log\left(K/F\right)\right]}=1-\frac{1}{\sigma}$

So if $\sigma > 1$, K income share will increase as K/F increase. And if $\sigma < 1$, K income share will decrease as K/F increase. This could describe both a gross and net production function, which is important difference. Net $\sigma$ equals gross $\sigma$ times the ratio of the net capital share $\left(F'_{K}K-\delta K\right) / \left(F-\delta K\right)$ and the gross capital share $F'_{K}K/F$. Since net K share is always less than gross K share, net $\sigma$ is always below gross $\sigma$, due to depreciation. If return on capital equal its marginal productivity, $r=F'_{K}$, net savings rate s is exogenous, then

$\frac{\partial\left(r-g\right)}{\partial g}=\frac{r}{g}\sigma_{net}^{-1}-1$

which is negative (r-g increase when g decrease) if r/g < $\sigma_{net}$ (net elasticity of substitution). Data indicate r/g=3. Net $\sigma$ < 3, means gross $\sigma$ < 4.05.

Rognile, Rowthorn and Acemoglu and Robinson discuss earlier empirical literature on $\sigma$. Most studies suggests it is < 1. Very few > 1. Surveys include Rowthorn 1999, Klump et al 2007, Chirinko 2008. Also, see Semieniuk 2014 on data and $\sigma$.

Summers, Krusell and Smith Swedish: Berge, Lindbeck