The Magical Mathematics of Mr Piketty Part 2

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 2

The philosopher Friedrich Nietzsche wrote: “There is no more dangerous error than confounding consequence with cause: I call it the intrinsic depravity of reason.” In economics the problem of confusing cause and effect is rife and frequently leads to disastrous policy mistakes.

The more I think about the logical framework of Thomas Piketty’s Capital In The Twenty-First Century, the more I become concerned that Mr Piketty has fallen into Nietzsche’s trap. I fear that Piketty has got his causes and effects the wrong way round.

To explain where I think the problem lies, I am going to use a simple thought experiment:

***

Consider an isolated island kingdom. Half of the island is covered in productive farmland and half is covered in unproductive land which is too rocky to farm. The island is a perfect feudal society. The King owns all of the land and therefore all of the capital of the island. All of the island’s inhabitants work for the King.

Fortunately the island’s farmland is fertile enough to produce more food than is needed to pay the wages of the farmworkers who tend the crops. As a result, in each year, the King enjoys a surplus value of food which he uses to pay the wages of additional workers. The King employs these additional workers to clear the rocky-land thereby turning it into new productive farmland.

It so happens that each acre of farmland requires exactly 9 people to farm it, yet in each year it produces enough food to pay the wages of exactly 10 workers. It also happens that it takes exactly 20 years of labour to turn each acre of rocky land into productive new farmland.

The upshot of these convenient numbers is, for each twenty acres of land owned by the King he is able to employ enough workers to convert exactly one additional acre of land into new productive farmland each year. As a result the King’s farmland, and therefore the island’s economy, grows at a steady rate of 1/20, or 5%, per year.

It also so happens that the King’s hobby is accountancy – stay with me! The King likes to idle away his time tallying his earnings and wealth. For convenience he chooses to do this by accounting for everything in units of ‘years of labour’.

The King notes that, each year, each of his acres of farmland produces enough food to purchase 10 years of labour. He also notes that, in order to receive this 10 years of labour, he must spend 9 years of labour to pay the wages of the farm workers. He therefore considers that each acre of land generates a profit to himself of 1 year of labour.

He also notes that it takes 20 years of labour to turn an acre of rocky-land into productive farmland and therefore considers that each new acre of land costs 20 years of labour. Once cleared of rocks the new farmland is no more nor less productive than the old farmland. He therefore considers old and new farmland to be of equal value and accounts for each of his acres as being worth 20 years of labour.

As each acre of land produces a surplus value of 1 year of labour and is valued as being worth 20 years of labour the King considers that he receive a return on his farmland of 1/20 or 5% per year.

The King notes that the yield he receives on his farmland, r, and the rate of growth of his acreage, g, are both equal, r = g, at 5%.

The King is so intrigued by the coincidence between the 5% return on his farmland and the 5% rate of growth of his acreage that he commissions two of his most venerated priests to investigate the phenomenon. The two priests, Karl and Adam, set about their task of investigating the r = g conundrum.

After years of research the two priests are summoned to present their findings to the King’s court.

Karl presents his report first. His is an enormous work running to hundreds of pages and is packed with charts and tables.
Karl begins speaking: “My lord, I have researched the r = g conundrum and I can conclude that it is just an incredible coincidence, there’s no natural force behind this incredible coincidence of pushing the growth rate of the economy toward the rate of return on capital.”

The King is clearly disappointed by this finding but Karl continued: “Nevertheless, my studies have lead me to discover new and important laws and relationships governing the workings of our island’s economy. I have studied all the records of this land, through all of its known history, and I can tell you that the return on farmland has always been 5% per year. So reliable has been this rate of return I am forced to conclude that a 5% return on capital must now be considered to be something close to a universal constant of economics.
Furthermore, I have discovered the ratio of the value of the capital of the economy relative to the value of its annual production is governed by a new fundamental law of capitalism. This law states that the ratio of capital to income is the same as the ratio of the savings rate to the growth rate of the economy.”

The King looked rather more pleased with these exciting new findings but his enthusiasm was soon dashed by the terrible news that Karl delivered next: “Unfortunately, my Lord, these new findings lead me to conclude that the island is about to suffer a terrible famine.”

At this point Karl pauses for dramatic effect and to give time for the assembled audience of high priests to nod and mutter their approval. One of the high priests becomes too excited to contain himself. He leaps to his feet and declares: “If you think you have found an obvious hole, logical or empirical, in Karl then you’re very probably wrong.”

Karl continues: “I have been up to the north of the Island to survey the rocky-lands that are yet to be cleared. The news is very dire. I can report that this land is much rockier than any we have cleared so far. I estimate that the cost of clearing this land will be not 20 years of labour per acre but 40 years of labour.”

Again he pauses for effect as the priests gasp at this terrible news.

Karl continues:” Once we are forced to start clearing the very-rocky land I am afraid to say the whole island will be plunged into a dreadful famine.”

At this point the King, who is becoming increasingly bemused, interjects with a question. The King asks: “You are saying that today we are able to feed ourselves quite amply and also able to clear the less rocky land but that in the future, when we start clearing the rockier land, there will suddenly be famine. How can this be so? Will the island not still have the same farmland that it had before?”

Karl responds: “Let me explain. It is my new laws of capitalism which show the inevitability of this famine. Clearing the rockier land will take not 20 but 40 years of labour. As a result the price of purchasing each new acre of land will double. Once this happens we will be forced to revalue the rest of Your Majesty’s capital. The value of your capital will double, to reflect the new higher cost of purchasing new land.
It is this higher valuation of capital that will cause the dreadful famine. Due to my newly discovered law, showing that the return on capital is always 5%, your income will double so that you will now receive not 1 but 2 years of surplus value per acre. This doubling of income being necessary to keep the yield of your land at 2/40 or 5%.

As a consequence there will be just 8 years of labour available to pay for the 9 farmworkers required to tend each acre of land. Incomes will fall sharply and sadly people will starve.”

At this point the King, loses his temper and turns to the second priest demanding: “Adam, is this true? Are the farmers facing famine?”

Adam, who was becoming increasingly agitated during Karl’s presentation, hands the King his own report – a single sheet of paper marked with just one expression “r ≡ g”.

Adam begins speaking: “My Lord, I fear that my most esteemed colleague, Karl, is talking poppycock. There will be no famine. The problem is, he has got his cause and effect the wrong way round.”

The priests in the gallery, who sense their beautiful crisis slipping away, start hissing and catcalling.

Adam continues: “The relationship between growth and the return on land is not a coincidence. The two are tied together by nothing more nor less than your own accounts, My Lord.
Each acre of land produces 1 year of surplus value and each acre of land is valued at the price, in years of labour, of clearing a new acre of land. It follows that yield of your land is simply 1 divided by years of labour needed to clear new land. However that same calculation also gives you the fraction of each new acre that can be cleared with the surplus generated from each existing acre. You are choosing to calculate the yield on your existing land from the cost of purchasing new land and so the two numbers, r and g, are the same.
Let me explain with some examples.

If it takes 10 years of labour to clear an acre of land, each acre will be worth 10 and will have a yield of 1/10. And each year of labour will pay for 1/10th of a new acre of land. The yield and the rate of growth will be 10% in both cases.

If it takes 20 years of labour to clear an acre of land, each acre will be worth 20 and will have a yield of 1/20. And each year of labour will pay for 1/20th of a new acre of land. The yield and the rate of growth will be 5% in both cases.

If it takes 40 years of labour to clear an acre of land, each acre will be worth 40 and will have a yield of 1/40. And each year of labour will pay for 1/40th of a new acre of land. The yield and the rate of growth will be 2.5% in both cases.

In all cases r will be equal to g simply because, My Lord, you are choosing to value your existing capital at the same price as the cost of acquiring new capital.

I have heard rumours that, in distant lands, this practice is referred to as the no-arbitrage condition or the law of one price and is considered by some to be a fundamental law of capitalism.”

The King pauses for a moment before asking: “And what of the famine and my surging income and asset prices?”

Adam replies: “The price of your land will double, when accounted for in years of labour, but each acre will still produce the same 1 year of surplus value, so the yield on your land will be halved. Your incomings and outgoings will remain unchanged and your workers will be paid just as before. However the rate at which you accumulate new land will be halved.

There will be no famine.”

***

I may be missing something obvious in how I am thinking about Piketty’s thesis. He is arguing we face a low growth future. To my mind, this means that we are faced with a situation where it becomes more expensive to purchase additional economic activity. That is to say the return on new investment must be lower. Yet at the same time he is also saying that the return on the existing stock of investment will remain high even in this new low growth world. I am struggling to understand how the markets will not arbitrage away the different returns available on new and existing capital.

The question I am asking myself is: Does Piketty’s thesis require the impossible situation of having two different prices for interchangeable, fungible, assets?

Clearly the story above is not a model of a modern developed economy. Nevertheless it is a useful aid to assist thinking about the relationships between: return on capital, economic growth, savings rates and capital valuations. Hopefully it helps to demonstrate that it is unreasonable to consider these variables as being independent of one another.

Piketty is of the view that r, the return on capital, and, g, the rate of economic growth are quite independent of one another – to quote: “There’s no pilot in the plane, there’s no natural force that will make this incredible coincidence of pushing the growth rate toward the rate of return happen, and so we need to find another plan in case this does not happen.”

Reasonable people can take a different view on this. Piketty warns of a future low growth world with much higher levels of capital and as a consequence capital’s share of income rising significantly: “experience suggests that the predictable rise in the capital/income ratio will not necessarily lead to a significant drop in the return on capital… With a capital/income ratio of seven to eight years and a rate of return on capital of 4-5 percent, capital’s share of global income could amount to 30 or 40 percent, a level close to that observed in the eighteenth and nineteenth centuries, and it might rise even higher.”

My questioning of Piketty’s thesis is not an attempt to defend the status quo. I have myself written on the flaws in economic theory and offered suggestions on how we may go about reforming the science of economics. Both our economic policies and our economic theories have failed us terribly in recent years. The science of economics needs a radical overhaul, but leaping from one flawed theory to another, without thoroughly testing its logic, is not the way to proceed.