# What if growth isn’t exponential, but linear?

I have said before that I think we are on the cusp of a macro revolution where more and more of the beliefs that we have held for decades are not only being scrutinised, but replaced with new, more accurate paradigms. The last time I wrote about how inflation expectations may not be all that relevant for inflation after all. That of course means that monetary policy has focused on the wrong metric to manage inflation. Now let’s look at technological progress and the fundamental assumption that productivity grows exponentially.

The fundamental assumption in all growth models is that productivity growth is constant over time, which means that in the long run productivity grows by some average rate (say 1% per year). Of course, there are bursts of growth, for example during the industrial revolution and after the invention of the internal combustion engine when productivity growth is above that long-term average for a decade or two. But whether you look at labour productivity (i.e. units of output generated per unit of work) or total factor productivity (the famously unexplained residual term in classic growth models that are often interpreted as the rate of technological progress) the mystery has been that empirically, growth rates decline over time. No matter whether you look at industrialised countries or developing countries, productivity growth slows down over time and is seemingly going to zero.

That’s a major problem because with population growth declining as well, it means that real GDP growth is constantly declining, and we face the prospect of growth slowing down to Japanese levels and eventually zero. And if real GDP growth rates are declining towards zero, earnings growth for equities is declining as well while real interest rates remain low. These two factors together imply that equity market returns should decline over time as well.

In a fascinating paper, Thomas Philippon from NYU has empirically tested whether productivity growth is constant over time. He found that productivity itself should not be modelled by an exponential curve (i.e. constant growth rates), but by a linear curve. A linear productivity function means that the same amount of productivity is gained each year, not the same growth. If productivity is linear, then growth rates for productivity naturally decline over time. The chart below shows how much better the forecasts for productivity growth are based on a linear model rather than an exponential growth model.

Forecasts for productivity growth based on a linear and exponential growth model

Source: Philippon (2022). Note, this chart uses total factor productivity (TFP) but results for labour productivity are shown in the note as ell with the same result.

For the UK, we have an extremely long historic time series of productivity growth and once again, a linear model works much better than the assumption of exponential growth. Annual gains in productivity were low before the first industrial revolution. From 1650 onwards, the age of enlightenment led to the widespread adoption of the scientific method and thus to more inventions that increased productivity. It culminated with the industrial revolution in the late 18th and early 19th centuries.

The second acceleration in annual productivity gains appeared with the second industrial revolution in the mid-19th century when mass production became more widespread and improved hygiene led to a decline in lost workdays due to sickness and death. Essentially, this age continues to this day.

Productivity growth in the UK

Source: Philippon (2022).

It seems then, that we have to maybe reconsider how growth works. As I said above, if productivity growth is additive, growing by the same increment each year as opposed to the same growth rate, then productivity growth will continue to decline and eventually approach zero. This in turn means that as long as we don’t get a sudden burst in population growth, real GDP growth will slow down more and more over time. As I have said above, this implies steadily declining equity returns.

But it would also have some consequences that may be more important than lower equity returns. Remember that some time ago I showed the empirical evidence that real interest rates are on a long-term decline? If Philippon’s theory is correct and productivity follows a linear curve, not an exponential curve, then real rates MUST decline over time and eventually become zero or negative. Below is the long-term trend in UK and US real rates. Both show a similar decline over the (very) long run.

Real rates in the UK and the US

Source: Schmelzing (2019)

It all fits together nicely. If productivity is linear, growth rates drop over time and as a result real rates drop as well, as long as population growth is not compensating for the decline in productivity growth. But with real rates closing in on zero and eventually becoming negative, the problem arises that saving is becoming less and less attractive. And that means that in the long run, investments decline as well since we can only invest, what has been saved by consumers. But if investment growth is declining that feeds back into lower economic growth and the vicious cycle continues.

How can we break this vicious cycle of ever-declining growth and low real rates? I have hinted at one “solution” above: have more children. Of course, that creates all kinds of other problems like how to feed more people and how to build a sustainable economy that is able to support so many people.

The second option, as far as I can tell, is to hope for a third industrial revolution where the annual productivity gains increase. That would postpone the inevitable slowdown by a generation or two, but not forever.

Finally, the third solution is not one that seems particularly desirable either. That is to provide so much free capital that people cannot help but invest it. If people get free money and face low or negative real rates, they obviously will consume most of it. And that boosts growth in the short term. But human nature being what it is, people will still not consume all of their money but instead want to put something aside, particularly in times of deteriorating social safety nets. And that means that once enough capital has been “force fed” to the system, investment activity will increase again simply because people don’t know what else to do with the money. If that sounds like the last decade or so, I don’t think it is a coincidence. It’s not a nice future, but probably the most sustainable of the three I have described.