The golden ratio and Fibonacci sequence have long been associated with financial markets, particularly in the analysis of price movements and market timing. A new paper by Bert de Groot and Rene Segers takes the concept in a much broader and rather fascinating direction.
In The Golden Ratio as Organizing Principle of the Economy, the authors examine economic cycles ranging from just a few years to well over a century. Their research finds relationships between the lengths of these cycles and the Fibonacci sequence and golden ratio. More importantly, they offer a mathematical explanation for why these relationships might occur.
Their conclusion is intriguing: The unique mathematical properties of the golden ratio may help multiple economic cycles coexist with greater stability.
Economic cycles and the Fibonacci sequence
We generally think of the business cycle as the familiar pattern of economic expansion, slowdown, recession and recovery. In reality, researchers have identified many overlapping cycles operating over different periods of time.
De Groot and Segers review 165 years of research on these cycles. When viewed together, their approximate lengths reveal a familiar pattern:
| Approx. years | Economic cycle | In simpler terms |
|---|---|---|
| 3 | Kitchin | Inventory adjustments |
| 5 | Business | Economic expansions and contractions |
| 8 | Juglar | Credit and investment |
| 13 | Bootstrap | Production and capacity |
| 21 | Kuznets | Construction and infrastructure |
| 34 | Societal | Social and behavioral patterns |
| 55 | Kondratieff | Long economic and innovation waves |
| ~90 | Four-generation | Generational change |
| ~145 | War and hegemony | Global power and conflict |
The longer cycles have been associated with phenomena ranging from construction and innovation to generational change and shifts in geopolitical power.
Look again at the approximate cycle lengths:
3 → 5 → 8 → 13 → 21 → 34 → 55 → 89 → 144
These are consecutive numbers in the Fibonacci sequence.
As readers of this site will know, divide consecutive Fibonacci numbers and their ratios approach the golden ratio:
8 / 5 = 1.600
13 / 8 = 1.625
21 / 13 = 1.615
34 / 21 = 1.619
55 / 34 = 1.618
The limiting value is φ = 1.6180339887…
That makes for an interesting pattern, but the authors have evidence that goes beyond simply matching historical economic cycles to Fibonacci numbers.
The golden ratio in GDP cycles
In earlier research, De Groot and colleagues analyzed GDP growth in 25 OECD countries and Europe. They identified between two and five economic subcycles in individual countries, generally ranging from about 3 to 15 years.
When they compared the lengths of these cycles, the average ratio was 1.62.
Compare:
Economic cycle ratio ≈ 1.62
Golden ratio φ ≈ 1.618
Finding φ in economic data is interesting, but it raises a more important question:
Why should the golden ratio appear there at all?
This is where I found the new paper particularly insightful.
The golden ratio and resonance
Consider two cycles with periods of 10 and 20 years. Their ratio is exactly 2:1, so they regularly synchronize.
Or consider cycles of 10 and 15 years. Their 3:2 relationship also causes them to repeatedly return to the same relative positions.
When cycles have simple numerical relationships like these, they can resonate, with their effects reinforcing one another. We see resonance in everything from musical instruments and playground swings to mechanical systems and planetary orbits.
Resonance can also create instability.
Cycles whose lengths do not have simple numerical relationships are less likely to synchronize and reinforce one another. This is where one of the golden ratio’s more unusual mathematical properties becomes important.
The golden ratio is an irrational number, meaning that it cannot be expressed exactly as a fraction of two whole numbers. More significantly, φ is unusually difficult to approximate closely with simple fractions. For this reason it is sometimes called the “most irrational” number.
This makes cycles whose periods are related by φ particularly resistant to resonance.
In very simple terms, the authors’ reasoning is:
φ relationship → less resonance → greater stability
The mathematical foundation for this comes from Kolmogorov-Arnold-Moser (KAM) theory, which deals with the stability of systems containing interacting cycles. De Groot and Segers show that their mathematical model of economic investment meets the conditions necessary to apply this theory. Non-resonating cycles can remain stable despite disturbances, while resonating cycles are more susceptible to instability.
The mathematics gets considerably more involved from there, and interested readers can explore those details in the authors’ paper. The important insight for our purposes is that the golden ratio isn’t simply a number that happens to appear in their analysis. Its particular mathematical properties provide a reason that φ-related cycles could be favored for their stability.
From an interesting pattern to a possible explanation
This distinction is what makes the research particularly interesting to me.
It’s one thing to observe that Fibonacci numbers or the golden ratio appear in market data. Such observations have been made for decades.
It’s quite another to propose a mathematical mechanism that might explain why the pattern appears.
De Groot and Segers bring together several pieces of evidence. Economic research has identified multiple cycles operating at different timescales. Many reported cycle lengths cluster around Fibonacci numbers. Their earlier analysis of GDP subcycles across 25 OECD countries and Europe found an average cycle-length ratio of about 1.62. And their mathematical model provides a reason that cycles related by highly irrational ratios should be more stable.
The authors therefore propose that the golden ratio serves as an “organizing principle” of the economy.
This doesn’t mean that φ causes recessions, construction booms, innovation or wars, or that every economic event occurs according to a Fibonacci timetable. Rather, the authors propose that among the many cycles that can develop within a complex economic system, those with relationships that minimize resonance may be more likely to remain stable and persist.
And the golden ratio is uniquely suited to do just that.
For me, that’s the most interesting idea in the paper. The appearance of 1.618 is not itself the surprise. The intriguing part is that a mathematical property that makes the golden ratio unique may help explain why it appears in the first place.
For readers interested in the economic research, mathematical model and KAM theory behind these findings, I encourage you to read the complete paper by Bert de Groot and Rene Segers:
Download: The Golden Ratio as Organizing Principle of the Economy
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