Wanna see my quick draw?

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In the standard RCK growth model, studied by any Masters economics student at an Anglo-Saxon university, three equilibrium conditions are:

1) The growth rate of output = g + n

2) ρ > n + (1 – θ) x g

3) The rate of return on capital (which is risk-free in this model) = ρ + θ x g

where

ρ = the household discount rate (pure time preference)

g = the rate of technical progress

n = the rate of population growth

θ = the coefficient of relative risk aversion

θ → 1 ⇒ utility tends to a standard log form. That means equations 2 and 3 become

2) ρ > n + (1 – 1) x g = n

3) The rate of return on capital = ρ + g

Suppose that we abstract from population growth, so set n = 0. Then equations 1 to 3 become:

1) The growth rate of output = g

2) ρ > 0

3) The rate of return on capital = ρ + g > g = The growth rate of output

That means that in an RCK model (a representative agent model in which there is never any inequality) with log utility and no population growth, along the equilibrium growth path the risk-free rate of return exceeds the growth rate of output.

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Wanna see it again?

The past few weeks folk keep telling me that someone or other has written some book or other using 200 years of data and 700 pages to prove that the rate of return on capital exceeds the growth rate of output, and that this demonstrates that there will always be rising inequality and therefore there is some kind of pathology at the heart of capitalism. But I don’t need 200 years of data to prove that r > g. I need about 10 seconds, during which I recall that “r > g” is the almost-immediate consequence of the standard equilibrium conditions of the standard long-term macroeconomic model that just about every masters’ student educated in the UK or US learns. Furthermore, about another five seconds later I recall that this cannot possibly in itself imply anything about rising inequality (not that I would care if it did), since in the standard growth model every household is identical (economists say it is a “representative agent model”) so there is no inequality in this model whatever, and none ever arises.

In fact, the result is so obvious that I can even explain it to you intuitively without any maths. First, so as to avoid spending all our time converting everything to “per capita” equivalents, let’s assume there is no growth in population. And let’s focus on the very long term underlying situation and assume that everyone is identical, so we only need to consider how a typical person would behave. Next, economists normally assume that people “smooth their consumption across the life-cycle”. What that means is, if you knew you were going to earn £50,000 one year and £10,000 the next year, you would be unlikely to choose to consume £50,000 in the first year and £10,000 in the second. Instead, you would prefer to even out your consumption at least a bit. If you like to smooth out your consumption totally, then (if we forget risk and interest returns and impatience and stuff like that) you would prefer to consume £30,000 in each of those two years. Let’s assume everyone were like that – preferring (once we’d ignored impatience, interest rates, risk etc.) to smooth consumption perfectly. (For you technical nerds, this is equivalent to assuming that utility takes a log form or, which is almost the same thing, assuming that the coefficient of relative risk aversion tends to 1.)

Next let’s assume that society has not achieved, and never will achieve, perfect bliss (at least in any timescale relevant to anyone alive today), and that people’s happiness never becomes infinite. If we would all perfectly smooth consumption, abstracting from patience/impatience, that means that once we bring patience back in we have to be at least a little bit impatient about the future – preferring to have things today to waiting to have the same thing tomorrow. Suppose, for example, we preferred to have things in the future to having them today. Then every year we could save up, and then at some far distant date consume an enormous amount, and because we thought it better to wait than to have it today, consuming that enormous amount would seem absolutely fantastic. Doing this sort of thing enough, with an infinite horizon, would deliver infinite bliss to us. But we’re assuming we can’t achieve infinite bliss. So that means we must be at least a tad impatient about the future. (Again for you technical nerds, that’s what’s going on in equation (2) above.)

Next, let’s think about the return on capital. Suppose the future is certain (so there is no risk – returns on capital are “risk-free”). Then there will be two key factors that decide whether it is worth investing in a project today. The first factor is growth. The faster the economy grows, the greater will be total future consumption, and vice versa. That means that, for a given level of GDP today, the faster growth is the greater will be the temptation to borrow today (paying it back out of future growth) instead of saving. So if we are indeed to be persuaded to save and invest a little extra today, the interest rate we are paid for doing so will have to be higher if the growth rate is higher. If (absent impatience) we are the kind of folk that prefer to smooth consumption perfectly, this effect is one for one. In other words, if we are to give up a unit of consumption today so as to get back extra in the future, that extra has to grow at the same rate as the economy. If we were totally patient and our investments, even when risk-free, yielded more rapid growth than the economy as a whole, we ought to give up even more today so as to get even more back in the future. Any slower, and we ought to prefer to consume the extra today and not bother investing.

The second factor that counts is the degree of impatience. The more impatient we are, the more we prefer to have extra consumption now to waiting until later. If we had negative impatience, we would prefer to consume later to consuming today.

So the long-term return on capital here – which is risk-free – will be equal to the sum of the long-term growth rate and the degree of impatience. But we have already said that the degree of impatience must be positive, if we are not all to be able to achieve infinite bliss but simply deferring consumption long enough. So, since the long-term risk-free return on capital is equal to the sum of the degree of impatience and the long-term growth rate, that means it must be higher than the growth rate.

Recall that in coming to this result, I assumed that everyone is alike – there is no inequality whatever. Furthermore, no inequality ever arises in the model either. So we can have (we do have) the result that the risk-free return on capital exceeds the growth rate of GDP even absent any inequality or any rising inequality.

To get here, I haven’t even needed to assume anything about risk. Obviously, most investment is not in risk-free projects, but instead in risky ones. In the UK historically, very low-risk projects have yielded inflation-adjusted returns of about 2.5 per cent, whilst risky projects have an average return of about 7.5 per cent. Let’s be generous and assume that risk-free returns could constitute around 30 per cent investment (they’re probably much less than that – indeed, virtually 0 per cent in fact – but let’s set that aside for now). Then we would expect average returns on capital to be around 6 per cent. That compares with a historic average UK growth in per capita output of some 1.5-2 per cent.

None of this, in itself, delivers us any implications whatever for inequality. In order to generate any implications for inequality, those advocating the idea that “r > g” has some implications for the stability of capitalism require all sort of additional exotic, strong, and very probably wrong assumptions. For example, some commentators appear to believe that r > g implies that the stock of capital grows faster than the economy. For some of them, this (to be frank) arises from the embarrassing error of confusing the level of something (the return on capital) with the growth of something (the growth in output). That returns to capital are r does not in itself imply anything about how rapidly the stock of capital rises or falls. Marginally more sophisticated advocates of the idea claim that the wealthier are likely to consume only a tiny fraction of their total capital income (regardless of their labour income), reinvesting the rest, so that the total capital stock will rise approximately in line with the rate of return to capital. So if you are very wealthy but have almost no labour income you consume virtually nothing, whereas if you have high labour income but few assets, you save almost nothing (e.g. no investment banker that had started out with no assets would ever buy a house). This is somewhere between “plain wrong” and “silly”. For example, it would imply that we should expect that the more unequal a society is, the higher will be its investment as a proportion of total GDP — so, one should expect the proportion of investment in GDP to be higher in, say, the US or Brazil than in, say, Sweden or France. This is a straightforwardly testable hypothesis which is, as it happens, false. For example, in 2012, according to the CIA World Factbook, gross fixed investment was 12.8 per cent of GDP in the US and 18.1 per cent in Brazil, but 19.8 per cent in France and 18.9 per cent in Sweden.

None of this is to deny that inequality has been increasing in recent decades. It’s hardly surprising that this is so, given that most inequality arises when someone finds a way to add value – e.g. by creating a Windows or Facebook or Harry Potter series or cool football skill or a better bond-trading algorithm or whatever – and the fruits of that new idea or method is not shared equally amongst everyone (and why on earth should it be?). In an increasingly globalised economy, the gains from a new computer system or football skill or book or whatever is leveraged over vastly more people than in the past. That creates inequality precisely because the adder of the value is benefitting an enormous number of other people, so mutual gains from each trade result in a little gain for a large number of people and an enormous gain for one person – generating inequality.

Now some inequality may well be caused by other things – corrupt capture of the political process in some developing economies; market and regulatory failures in the financial sector; inadequate planning regulation boosting land and house prices. But none of this is remotely a result of “r > g”, which is a fundamental feature of capitalism, not some failure of it. So next time someone tells you “Aha, but Capitalism is flawed because r > g!” you can say: “That’s a very dull, old and obvious result that doesn’t need 200 year of data and 700 pages of text to demonstrate. Wanna see my quick draw?”

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