The Magical Mathematics of Mr Piketty Part 1

I have been asked to re-post four articles origionally written in April/May 2014 about the ideas of Thomas Piketty in his book Capital in the Twenty First Century: The Magical Mathematics of Mr Piketty Part 1 and Part 2, Credit in the Twenty First Century and The Horrible History of Mr Piketty

The Magical Mathematics of Mr Piketty Part 1

Capital in the Twenty-First Century is: “To put it bluntly, the discipline of economics has yet to get over its childish passion for mathematics…” p32

I could not agree more. But this does not mean we should dispense with mathematics entirely. Some problems in economics are easily formulated in mathematics, for those the equations can be a useful tool to test the validity of the underlying logic. This is true for the ideas in Piketty’s own book.

There are only three important mathematical relationships in Piketty’s book but I am having trouble reconciling them, especially in the low growth world that Piketty wants to analyse.

The three relationships are:

1. Piketty’s inequality page 25:

“This fundamental inequality, which I will write as r > g (where r stands for the average annual rate of return on capital, including profits, dividends, interest, rents, and other income from capital, expressed as a percentage of its total value, and g stands for the rate of growth of the economy, that is the annual increase in income or output), will play a crucial role in this book. In a sense, it sums up the overall logic of my conclusions.”

r > g

1. Piketty’s first fundamental law of capitalism page 52:

“I can now present the first fundamental law of capitalism, which links the capital stock to the flow of income from capital. The capital/income ratio β is related in a simple way to the share of income from capital in national income, denoted α. The formula is

α = r × β

Where r is the rate of return on capital.

For example, if β=600% and r = 5%, then α = r × β = 30%.

In other words, if national wealth represents the equivalent of six years of national income, and the rate of return on capital is 5 percent per year, then capital’s share in national income is 30 percent.”

1. Piketty’s second fundamental law of capitalism page 166:

“In the long run, the capital/income ratio β is related in a simple and transparent way to the savings rate s and the growth rate g according to the following formula:

β = s / g

For example, if s = 12% and g = 2%, then β = s/g = 600%.

In other words, if a country saves 12 percent of its national income every year, and the rate of growth of its national income is 2 percent per year, then in the long run the capital/income ratio will be equal to 600 percent: the country will have accumulated capital worth six years of national income.”

In summary the three key relationships in Piketty’s mathematical framework are:

The inequality r > g

The first fundamental law of capitalism: α = r × β

The second fundamental law of capitalism: β = s/g

Of these Piketty’s inequality has captured most attention. Piketty is at pains to emphasise that, r, the return on capital is always greater than, g, the growth rate of the economy. He also maintains that r is more or less a constant at around 4 to 5% and he expects growth to head lower toward around 1 to 1.5%.

We can explore what happens to these relationships as the rate of economic growth falls toward zero.

To keep the examples simple I will assume a constant return on capital of 5% and a constant savings ratio of 10%. This leaves the growth rate, g, as the only free variable in the system.

The following table shows the key variables under different growth scenarios.

As growth falls capital values rise pushing up the share of national income accruing to the owners of capital – one of Piketty’s key concerns. However as growth falls toward zero it becomes apparent that all is not well in this model. The capital/income ratio eventually rises to a point where more than 100% of the national income goes to the owners of capital - clearly an impossible scenario.

The problem arises because Piketty’s second ‘fundamental’ law of capitalism β=s/g contains a singularity , a divide by zero, which sends the value of capital toward infinity as the economy stagnates. When coupled with Piketty’s assertion that the return on capital remains above g, at around 4 to 5%, this sends the income from capital to infinity – another impossibility

Piketty’s equations simply cannot hold true in the low growth environment which he is trying to analyse.

The question is how to fix them. The most logical approach is to accept that the yields on assets fluctuate to reflect the growth rate of the economy. If growth is cut in half then asset prices will double but their yields will also be cut in half, a condition met when r = g.

If the scenarios are re-run with r = g we get the following results shown in the table below.

If we accept that the real return on assets floats with growth, r = g not, as Piketty claims, r > g, then there is no conflict with either of Piketty’s two fundamental laws of capitalism.

I expect the r = g assumption will make more intuitive sense to investors who have seen the real yields on, for example, inflation protected bonds collapse as growth has fallen. It also helps explain why pension funds are struggling to meet their funding targets and why the UK government has recently relaxed the requirement for pensioners to buy annuities – because annuity yields have fallen in line with economic growth.

However the r = g assumption causes a significant issue for Piketty’s case for a wealth tax. If r = g prevails in a low growth world then Piketty’s 2% wealth tax could push the return on capital into negative territory potentially crushing entrepreneurial activity.

In conclusion – Piketty’s own fundamental laws of capitalism appear at odds with the inequality on which much of his book is based. This is especially true in the low growth world he is concerned about.