# Summary of Piketty, Part 1: Capital/Income Ratio and the Capital Share of Income

[This is the second of five posts on Capital in the 21st Century. The first is here.]

The first part of Piketty’s Capital in the 21st Century is about the Capital/Income ratio and the split of Income between Capital Income and Labour Income. Piketty finds that Capital/Income ratios are increasing, and predicts they will continue to do say. Alongside this increase in Capital/Income ratios, and as a result of it, the share of income going to Capital appears to be increasing, and is also predicted to continue to do so.

Increasing Capital/Income ratios are the result of two forces. The first is a recovery to higher historical levels after the destruction of large amounts of capital in the first-half of the 20th century, mainly as a result of the World Wars. The second is decreasing income growth rates — due to both slowing technological progress and slower population growth — which mechanically means that the same rates of savings will result in higher Capital/Income ratios.

Increasing Capital/Income ratios automatically result in an increasing share of income going to Capital, unless they are fully offset by falling interest rates. The evidence suggests that interest rates do fall as Capital/Income ratios increase, but that they do not fall enough to fully offset the increasing importance of capital.

### Capital/Income Ratios:

Main point: Capital/Income ratios are rising, and will continue to do so. There are two main reasons. The first is a return to historical norms after a massive fall in Capital/Incomes due to the destruction of capital in the World Wars. The second is a slowing growth rate of income (partly due to falling population growth), so that the same rate of savings leads to substantially higher Capital/Income ratios.

Return to historical norms after a massive fall in Capital/Incomes due to the destruction of capital in the World Wars: The focus here will be on France & Britain, simply because the data for these two countries goes back much further. The observations follow directly from

Capital in Britain , 1700-2010

Capital in France, 1700-2010

Here we see that the Capital/Income Ratio was high, until the two World Wars (and high inflation between them) substantially reduced the ratios. Since World War II they have been on an upward tendency. The first of the main points here is that Capital is currently becoming more important; and that this represents a return to historical norms after the destruction of the early 20th Century.1

The growth rate of national income is an important determinant of the levels at which the Capital/Income Ratio will settle: If a country saves a fraction $$s$$ of income each year, and has per-capita income growth of $$g$$, population growth of $$n$$, and capital deprecates (wears-out) at rate $$\delta$$ then the Capital/Income Ratio ($$K/Y$$) of that country will eventually stabilize at

$$\frac{K}{Y}=\frac{s}{g+n+\delta}$$

(think of the Solow growth model; Piketty calls this relationship $$\beta=s/g$$). Population growth rates have fallen over recent decades, it is possible that per-capital income growth may be slowing as well. Thus we expect that the Capital/Income ratio will rise. As a rough guide, we might expect it to rise from levels historically experienced in countries such as the US that have had higher population growth (2.5 million people in 1776 to 300 million in 2006), to the levels historically experienced in countries such as France that have not seen so much population growth (25 million people in 1789 to 61 million in 2006). The US Capital/Income ratio has historically been about 4, while that for France has historically been around 7.

Capital in France, 1700-2010 (again)

Capital in the United States, 1770-2010

Some further points on the Capital/Income ratio:

• Most capital is privately held: public (Government) holdings of wealth are small compared to private holdings, especially net public capital (assets-debt), and in many countries has been further decreasing in importance over recent decades (think privatization). (See F3.3, F3.4, F3.5, F3.6, F4.4, F5.1)
• Some of the fall in the Capital/Income Ratio during the mid-20th reflects physical destruction of capital, some reflects falling prices. Piketty addresses this decomposition and finds that both play important roles. Thus, some of the rise since World War II simply reflects ‘price recovery’. (See Chapter 5)
• Most capital is domestic: the difference between domestic wealth and national wealth, that is the importance of how much one country owns in foreign countries, is small. The only notable exceptions are from when countries, such as Britain and France, had large overseas Empires. While net foreign wealth remains small, gross holdings have increased over recent decades. (See F5.2, F5.7)
• The only issue not covered in-depth is what determines the savings rate, $$s$$. Since we expect that in the long-run the Capital/Income ratio will converge to $$K/Y=s/(g+n+\delta)$$ the savings rate is important in determining where the Capital/Income ratio is going to. Piketty does discuss it briefly and provides some Tables (T5.1, T5.2, T5.3, T5.4) showing that it varies across countries. The question is why it varies across countries, and over time, and will it change in the future? This is a weakness in Piketty’s predictions for increasing Capital/Incomes, but given the history and current trends of Capital/Income ratios it is unlikely to be a major one (F3.1, F3.2, F4.6, F5.3).
• Capital/Income ratios for some other rich countries, F5.3.
• Interesting aside: Slaves were an important part of the ‘wealth’ of the US in the late eighteenth century, F4.10.

### The Share of Income going to Capital

Main point: Increasing Capital/Income ratios automatically result in an increasing share of income going to Capital, unless they are fully offset by falling interest rates. The evidence suggests that interest rates do fall as Capital/Income ratios increase, but that they do not fall enough to fully offset the increasing importance of capital.

The income of capital is just the amount of capital times the income per unit of capital; the latter is called the return on capital or the interest rate. Thus the share of income going to capital, $$\alpha$$ — the Capital Share of Income — is simply the Capital/Income ratio times the interest rate $$r$$, that is

$$\alpha=r \frac{K}{Y}$$

There is little to say here other than simply to give the graphs of the Capital Share of Income for Britain and France and simply observe that the capital share of income is not constant, and that it is higher when the Capital/Income ratio is higher.

The capital-labor split in Britain, 1770-2010

The capital-labor split in France, 1820-2010

Over the last decades the Capital/Income ratio has been rising, and the Capital Share of Income has risen along with it. The conclusion that the Capital/Income ratio will continue to rise leads us to conclude that the Capital Share of Income will continue to rise.

Some further points on the Capital Share of Income:

• The return on capital (interest rate) does vary, but is comparatively stable. This fits with our earlier observation that, by accounting identity, the only way an increase in the Capital/Income ratio will not lead directly to an increase in the Capital Share of Income is if the return on capital falls enough to offset it. (F6.3, F6.4)
• An increasing share of capital is seen in many countries, F6.5.2
• Technical note I: The point about whether $$r$$ falls enough when $$K/Y$$ rises to keep $$\alpha$$ constant is, in the language of Economics, a question of whether the elasticity of substitution between capital and labour is equal to one. The empirical evidence presented suggests that $$r$$ falls a little, but not enough to keep $$\alpha$$ constant; that the elasticity of substitution between capital and labour is greater than one.
• Technical note II: Economists often use a ‘Cobb-Douglas production function’. This function assumes that the capital share of income is constant; that the elasticity of substitution between capital and labour is equal to one. One can easily fix this by using a ‘Constant Elasticity of Substitution (CES) production function’ which allows for an elasticity of substitution between capital and labour greater than one.

Part 2: Income Inequality and Wealth Inequality

1. The observation about falling Capital/Income Ratios, and a return to ‘normal’ ratios since, is much more relevant to countries such as Britain and France that were devastated by the wars, than to countries such as US that were largely (physically) unaffected. This is evident comparing the graphs of Capital/Income Ratios for these countries. []
2. For more, see Karabarbounis and Nieman (2014) – The Global Decline of the Labor Share. []