The economic complexity index or ECI is an estimate of the factors present in an economy built on a simple recursive observation. An economy is as complex as the activities it can do, and an activity is as complex as the places that can do it.

Complexity here doesn’t mean complicated or anything esoteric. It follows the standard dictionary definition of honoring the idea that economic systems are composite (production involves multiple inputs, capabilities, or factors).

If we use averages to formalize this idea, we get the economic complexity index (ECI), which is the standard formula used in this literature. That means we want to find the eigenvector, ECI, of the following matrix (Mcc’)

Empirically, ECI has been shown to be a robust predictor of future economic growth. By why does it work in theory? Does it really estimate the availability of factors in an economy? If so, what factors? Any factors? How can that be?

Consider each economy is endowed with factors that activities require. In that world, an economy’s potential output in an activity is the probability it is endowed with the factors required by that activity.

We can write that output as one minus the probability that an activity requires a factor that the economy doesn’t have.

For an arbitrary number of factors, this formula gets a bit more complicated, but the intuition will be the same.

So, in math terms, our question is: can we recover the factor endowment matrix r if we only observe the output matrix Y? The answer is yes. Economic complexity is a monotonic estimator of those factor endowments that is robust to variations in size and noise.

For one factor, it is not hard to derive the Mcc’ matrix associated with this production or output function. That is a matrix separating economies with a higher than average probability of having a factor from those with a lower than average probability.

The shape of the matrix is simple when the number of economies and activities is even.

And becomes more complicated when they are odd. But the result is the same.

The second eigenvector of this matrix tells you which cluster an economy belongs too. That is, we have used the known spectral clustering property of the second eigenvector of a matrix to estimate economic factors without having to define them.

This result can be generalized numerically to an arbitrary number of idiosyncratic factors even when these are assigned with substantial levels of noise (50%).

Basically, ECI is a monotonic estimator of the first eigenvector of the matrix of factor endowments r.

Now, you might be wondering if there are simpler ways to recover r from Y. And for the simplest version of this model you can. But once you add heterogeneous countries and products (eg China is way bigger than Paraguay :), plus noise, you need something as far removed from Y as

The bonus is that this same theory explains differences in networks of related activities.

For instance, if you connect products that are co-exported you get a network where the high complexity products are in the center. Why?

We can derive that by assuming factors are correlated across economies (eg Sweden is well endowed with most factors & Mozambique is not). So we can interpret the fact that products at the center of the product space are high complexity as evidence of correlated factors.

When we look at networks of research activities we observe a ring structure. Why?

Because the factors required by a research field are only reusable among a few neighboring fields, so factor endowments and requirements follow a Toeplitz circulant matrix.

Economics is a field that has been proud to embrace math, but eigenvectors are by no means a canonical or core theoretical concept (as they are in physics). They are used mostly as a statistical tool.

In physics Eigenvector or Eigenfunctions are not just “PCA” components used to summarize data. They are the solutions to key equations, such as Schrödinger’s equation. The energy levels of an atom are eigenvalues, and the orbitals or states are the eigenvectors.

Using eigenvectors to summarize data is cool. But theories where you estimate the eigenvectors of mechanistic models are profound. To describe systems theoretically, without abusing aggregation, sooner or later your theory must embrace eigenvectors.

Should economics embrace the math of modern physics? You probably know where I stand on this issue. If you want to get a taste for these ideas, you will find a good summary on our latest working paper on the theory of economic complexity. https://arxiv.org/abs/2506.188...

@cesifoti Eigenvectors is very standard Math in Economics

@FranNunesEcon Not as much as in other fields or as I would like. Unfortunately, I have spent the better part of the last 20 years presenting this work across the world, and the standard reaction I get to using an eigenvector as a way to explain economic growth potential is like I pulled a

@cesifoti Why you are locked in a lapalcian world (past dictates the future) and trying to shoehorn in uncertainty, I (Physicist) treated the problem like a Quantum problem set. With tiny data sets, the VERY crude tests put trad econ to shame. PCE - https://x.com/jrfuisz/status/1...

@cesifoti And loss of complexity causes aging and death — I think there’s some parallels for economics: https://www.nature.com/article...

@cesifoti Eigenvectors is standard math in economics, @cesifoti! C/c @ben_golub

@cesifoti Eigenvectors are "exotic" math??? This is high school linear algebra..

@cesifoti physics math solving economics? neat

@cesifoti Agreed

@cesifoti This is why quants dominate Wall Street. Complex math gives you an edge where others see noise. The real money is in understanding patterns most people cant even see. Math becomes power