If you’re not an economist – or at least someone who took a macroeconomics course at the intermediate level or above – it’s unlikely you know what that means. First, however, the picture.

How national economies grow is a bit of a holy grail for Macroeconomists – the people who study the whole economy of a nation. Robert Solow came up with a beautifully simple model of economic growth in 1956 for which he got a Nobel prize in 1987 (technically, the The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel). Edmund Phelps, Nobel prize recipient in 2006, published a delightful fairy tale based on Solow’s growth model in 1961, from which we got the now-famous Golden Rule of economic growth. Many useful economic models came from these, and the basic Solow growth model is still used in contemporary macroeconomic analysis.

The Solow growth model and its many direct descendants assume that “growth happens”. That is, something makes the economy grow and we just put that into the model. We call these exogenous growth models (exo = outside, genous = created).

Some macroeconomists, however, were eager to find a model of the national economy that incorporated the thing that made it grow. We call these endogenous growth models (endo = inside). For example, a close descendant of the Solow growth model is the Ramsey–Cass–Koopmans model, which is partially endogenous.

An early proponent of endogenous growth theory is 2018 Nobel prize recipient Paul Romer. Like many great ideas, it started with an extremely simple idea. In 1986 Romer proposed an incredibly (as in, not expected to happen in real life) simple model where national output is a linear function of capital alone. Looking like this $$Y=AK$$ where $Y$ is national product (GDP), $K$ is total national capital stock, and $A$ is a constant that converts one into the other.

From a practical point of view, there are a few problems with this, not least which is the constant returns to capital. This, if nothing else, is a violation of the laws of thermodynamics, and we all know that physics always wins!

But what if this model – called the AK model for obvious reasons – really applied to the U.S. economy? That’s what the plot at the top of this post is about. The top solid line is real capital stock in billions of 2011 dollars – about 56 trillion dollars in 2017. The lower solid line is real GDP in billions of 2012 dollars – about 18 trillion dollars in 2017. The fact that they’re measured in dollars one year apart doesn’t really matter – there’s a lot of lag in the national economy. These are both plotted against the left axis.

The dashed line in the plot is the ratio of the two and represents A in the AK equation. This is plotted against the right axis. It’s coincidental that the capital stock plot appears to trace the average of the dashed line plot, but it helps to see the trend. That is, were the AK model to apply to the U.S. economy, it has certainly not been constant over the past 67 years, not even on average.

So here’s the endogenous growth part of the AK model. Clearly $$dY=AdK$$

where $dY$ is the change of production (growth in the economy) and $dK$ is the change of capital stock. What we know about the change of capital stock is that it’s a net change, with investment coming in and depreciation going out $$dK=sY-\delta K$$ Here *s* is the national saving rate (the fraction of income that households save) and $\delta$ is the depreciation rate (the fraction of capital stock that wears out or is used up each year).

Now this is another gross simplification, assuming, among other things, a closed economy (no imports or exports) so that total investment equals total household savings, *sY*. It also assumes no taxes or government spending (hard to picture, but taxes and government spending can be subtracted without a fundamental change to the model), and a constant rate of depreciation.

To get the economic growth rate, divide change of production by total production $$\frac{dY}{Y}=A\frac{sAK-\delta K}{AK}$$ where on the right side, *Y* has been replaced by *AK*, since they’re equal. This simplifies to $$g_Y=sA-\delta$$ where $g_Y$ is the economic growth rate, $\frac{dY}{Y}$.

To do something with this, we have to estimate household saving rate $s$ and deprecation rate $\delta$. For the past 40 years the saving rate has been below 10%, with a weird spike to 33% in April 2020 (https://fred.stlouisfed.org/series/PSAVERT). The national depreciation rate on fixed assets was about 5% of total capital in 2017 (https://fred.stlouisfed.org/series/M1TTOTL1ES000). Assuming $s=0.10$ and using $A=0.32$, the 2017 number from the plot above, we get $g_Y=0.1*0.32-0.05=-0.018$ or a negative 1.8%. All of these estimates except *A* are averages over the past few decades when growth has averaged about 3%. Using the 2017 estimate for *A* should have given us the highest possible growth rate.

Clearly, the basic AK model does **not** approximate the US economy very well at all. Are there national economies it better approximates? That’s a topic for another time. Additional topics for another time: how to incorporate human capital and intellectual capital. While human capital may have the diminishing returns we expect (physics expects!) from physical capital, there’s no reason for intellectual capital to exhibit diminishing returns. SPOILER ALERT: this may be a way to get a real economy to look a lot more like the AK model.