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La Bourse  /  Volume II  /  Nº II.02

Markets as Living Systems

Volume II — Foundations: The Paradigm and the Science

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THE PLATE

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THE LETTER

You have been told that the economy tends toward equilibrium, and you may have noticed that nothing you have ever worked inside behaves that way.

That mismatch is not a failure of your observation. It is a live question in economics, it has been live since at least 1972, and the mathematics on it is settled in a direction most people are never told about. This chapter is about that mathematics, and about what replaces equilibrium when you set it down.

It is the chapter a hostile reader will open first, so let us be exact about what is being claimed and what is not.

Claimed: that an economy is a system of heterogeneous agents adapting to an environment they jointly create, that its aggregate behaviour is not derivable from its components' rationality, that the statistical signatures of its outcomes — fat tails, power-law sizes, clustered volatility, path dependence — are what such systems generically produce, and that these facts are established in the literature rather than asserted by us.

Not claimed: that the economy is an organism. It is not, and the chapter's central arithmetic shows precisely where that metaphor breaks and by how much. Nor is it claimed that equilibrium models should be abandoned. They are the better instrument for a specific and namable set of questions, and those questions are listed here with the same care as everything else.

The living-systems frame earns its place by being more accurate about more of the economy, not by being warmer about it. If you came for a metaphor, this chapter will disappoint you in a way you may end up grateful for. A metaphor that cannot be falsified cannot be load-bearing, and the whole purpose of the second edition is to put weight on these ideas.

— The Editors

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DISCOVERY

What is already working

The appreciative question first: where is this already being done well, by named people, with results on the record?

Santa Fe, 1987. Kenneth Arrow and Philip Anderson — a Nobel laureate in economics and one in physics — convened ten economists and ten physical scientists for ten days to ask whether the tools of complex-systems science had anything to say about economies. The resulting volume, The Economy as an Evolving Complex System (Anderson, Arrow and Pines, 1988), is the founding document of the field, and it is not a manifesto. It is a proceedings, with disagreement in it.

What came out of the programme is the point. The Santa Fe Artificial Stock Market (Arthur, Holland, LeBaron, Palmer and Tayler, 1997) put heterogeneous traders, each carrying a bundle of forecasting rules, into a market and let them learn. Nobody wrote in fat tails. Nobody wrote in volatility clustering, or technical trading, or bubbles. All of them appeared — as a consequence of agents learning about a market their own learning was changing. That is the model's real contribution: it produced the stylised facts of real markets from adaptation alone.

Long House Valley, Arizona. Axtell, Epstein, Dean, Gumerman, Swedlund and colleagues (2002) built an agent-based model of the Kayenta Anasazi and ran it against eight hundred years of archaeological record, from 800 to 1350 CE. The model reproduced the population trajectory — the growth, the aggregation into settlements, the timing of the collapse — from household-level decisions about where to plant. It is the first agent-based social model validated against a real, long, independent time series, and it remains the standard anybody proposing this method should be asked to meet.

Threadneedle Street, 2016. The Bank of England built an agent-based model of the UK housing market — buyers, sellers, landlords, a banking sector, credit conditions (Baptista, Farmer, Hinterschweiger, Low, Tang and Uluc, Staff Working Paper 619). It was used to examine how loan-to-income limits would propagate, and it informed the Financial Policy Committee's thinking on exactly that instrument. A central bank is the least romantic institution in the world. It adopted this method because the method answered a question its existing models could not: what happens at the distribution, not the average.

Ecology into banking. Robert May spent his career on the stability of complex ecological networks. In 2011 he and Andrew Haldane, then the Bank of England's Executive Director for Financial Stability, published Systemic risk in banking ecosystems in Nature — importing the mathematics of food-web stability into the analysis of interbank exposure. The paper's influence runs through the post-crisis architecture of network-based stress testing. This is the ecological analogy at its most disciplined: not "banks are like species, isn't that interesting", but a specific stability criterion, carried across with its assumptions attached and tested on its new material.

Cambridge and the Atlas. Hidalgo and Hausmann (2009) treated an economy as a bipartite network of countries and the products they make, and derived a measure — economic complexity — from nothing but which country exports what. The measure predicts subsequent decade-scale growth better than standard governance and human-capital indicators. It is now used by finance ministries to choose industrial strategy. It came from asking what an economy's capability structure looks like as a network, which is a living-systems question asked with customs data.

Five programmes, forty years, three continents, and in every one the move is the same: stop assuming the aggregate, and model the interaction that produces it. That is the whole method, and everything below is the arithmetic of why it is necessary.

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THE ARITHMETIC

What works, what does not, and where the line sits

First, honour the opposition, because it is owed. The Walrasian general equilibrium programme is one of the great intellectual achievements of the twentieth century. Arrow and Debreu (1954) proved that under stated conditions a competitive equilibrium exists — a set of prices at which every market clears simultaneously. Hayek (1945) gave the deepest argument for why a price system is worth having at all: it transmits dispersed local knowledge that no central authority could assemble. Neither of these is being overturned here. Both are being read carefully.

Read carefully, they say something narrower than they are used to say.

Arrow and Debreu proved existence. They did not prove uniqueness, and they did not prove that any process reaches the equilibrium. Those are separate questions, and both were answered — in the negative — within twenty years.

Scarf (1960): the market that never arrives. Three goods, three consumers, each endowed with one good and wanting it paired with one other. Equilibrium exists at equal prices. Now run the textbook adjustment story: raise the price of whatever is in excess demand. The product of the three prices is conserved by that dynamic, so the price path lies on a closed orbit. Computed here from prices 25 percent off equilibrium and run two hundred thousand steps: the distance to equilibrium starts at 0.490, ends at 0.470, and its closest approach along the entire path is 0.457. Every axiom holds and the market never converges.

Sonnenschein–Mantel–Debreu (1972–74): what survives aggregation. Take an economy of impeccably rational utility maximisers with well-behaved preferences. Ask what their aggregate excess demand function can look like. The answer: almost anything. Continuity, Walras's law and homogeneity of degree zero survive aggregation — three properties. Uniqueness does not. Stability does not. Comparative statics does not. Any function satisfying those three is the aggregate demand of some perfectly orthodox economy.

Read the consequence slowly, because it is the load-bearing sentence of the volume: individual rationality places almost no restriction on aggregate behaviour. That is a theorem in the field's own journals, not a critique from outside. Kirman (1992) drew the corollary — the representative agent is not a simplification of a population; it is a different object, and it can be made to prefer what no member of the population prefers.

And the data agrees. Axtell (2001) fitted US firm sizes and found a Zipf distribution with Pareto exponent 1.059 across roughly 5.5 million firms. At that exponent the mean is finite and the variance is not. A representative firm is not a lossy approximation. It is a statistic the data does not possess.

The tail, priced two ways. On 19 October 1987 the S&P 500 fell 20.47 percent in a day. Against the daily standard deviation then prevailing — call it 1.0 percent, and the figure is the input, stated — that is a twenty-sigma move.

  P(that move or worse), Gaussian          2.0 x 10^-93
  expected waiting time, Gaussian          2.0 x 10^90 years
  age of the universe                      1.4 x 10^10 years
  P(that move or worse), cubic tail        2.5 x 10^-5
  expected waiting time, cubic tail        161 years

The cubic tail is not a device chosen to be flattering: exponent 3 is what Gabaix, Gopikrishnan, Plerou and Stanley (2003) measure on index returns. The Gaussian model does not say the crash was unlikely. It says it could not happen, and it happened on a Monday. One hundred and sixty-one years is a number a treasurer can plan against. Ten to the ninetieth is a number that quietly removes the event from the risk register.

Now the Balenciaga cut, and it goes against this chapter's own title.

Everyone reaches for the metabolism. The economy breathes, digests, circulates. It is a beautiful figure and it is wrong, measurably, with a sign error.

Biological metabolism scales sublinearly with mass: Kleiber's law, B ∝ M^0.75, confirmed across twenty-seven orders of magnitude. Double an organism's mass and its metabolic rate per gram falls 15.9 percent. Large animals are slow. That is what the exponent means.

Bettencourt, Lobo, Helbing, Kühnert and West (2007) ran the same analysis on cities. Infrastructure scales at β ≈ 0.85 — sublinear, as the metaphor predicts. But wages, GDP and patents scale at β ≈ 1.15, which is superlinear. Double a city and output per head rises 11.0 percent. Socioeconomic activity does the opposite of metabolism, and the exponents sit on opposite sides of one.

  metabolic rate,  β = 0.75   per unit on doubling   0.841   (−15.9 %)
  infrastructure,  β = 0.85   per unit on doubling   0.901   ( −9.9 %)
  wages, GDP,      β = 1.15   per unit on doubling   1.110   (+11.0 %)

So the economy is not an organism. An organism has one genome, one objective, one bounded life and a target adult size. An economy has none of those. What it is, is an ecosystem — many lineages, no shared objective, no target size, no death by design. Everything that is useful in the living-systems frame survives that correction; everything that was sentimental does not. And the same paper names the cost: superlinear growth on a finite resource reaches a singularity in finite time unless innovation resets the curve, which is a harder claim about growth than any degrowth argument makes.

The honest negatives, in full, because this is where they belong.

One. Diversity does not buy stability. May (1972) proved that a randomly assembled community of S species with connectance C and interaction strength σ is almost surely unstable once σ√(SC) > 1. More species and more connections make it more fragile, not less. At S = 20, C = 0.3, σ = 0.3 the criterion is 0.735 and the system is stable; at S = 50 it is 1.162 and it is not. Anyone arguing "diverse, connected, therefore resilient" is contradicting the only place that claim has been made precisely — and Haldane and May carried exactly this result into banking, where it explained why a densely interconnected financial network amplifies rather than absorbs.

Two. Ulanowicz's window is a shape, not a constant. Robustness R(a) = −a·ln a peaks at a = 1/e = 0.3679, where the ordered share of throughput is roughly a third. It is elegant and it is used, but 1/e is a property of the function, not a measured constant of economies. No dataset has shown that a national economy at 0.368 outperforms one at 0.30. Use it as a shape.

Three. Agent-based models have a degrees-of-freedom problem. A model with thirty free parameters can be fitted to almost any history and has thereby explained none of it. Fagiolo and Roventini (2017) treat the calibration and validation problem seriously and do not pretend it is solved. Until an ABM is judged out-of-sample, on data it was not tuned to, it is a story with a compiler.

Four, and the one a hostile economist will reach for: the neoclassical instrument is still the better one for five namable questions.

Those five are not concessions. They are the boundary of the claim, and a claim without a boundary is not a claim.

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DREAM

What becomes ordinary

In the economics that has absorbed this, the first question asked of any model is what is the distribution, and the second is what happens at the tail. The average is reported because it is conventional, not because anyone decides on it. A risk committee that is shown a mean and a standard deviation asks for the exponent, and it is in the pack, because somebody built the report once and now it runs monthly.

Firms are understood as adapting populations rather than optimising points. A strategy is judged by how it performs across the range of futures the system can produce, not by its expected value under one. Nobody finds this exotic; it is how an engineer has always sized a bridge, and it is how a treasurer sizes a buffer.

Policy is tested in simulation before it is tested on people. When a regulator proposes a leverage limit, a capital surcharge, a loan-to-income cap, there is a model with heterogeneous households and firms in it, and the model is run a thousand times, and what is reported is the distribution of outcomes with the bad tail shown at full size. The model is published with its code. When it is wrong it is corrected in public, which is what makes it worth having.

Industrial strategy is chosen by looking at what a country's capability network already nearly supports, rather than by naming a sector that is fashionable elsewhere. Ministries have the map. Firms have it too, and use it to decide what to make next, because proximity in that network is a better predictor of success than an ambition is.

And the interaction is the object of study, not the friction. The structure of who supplies whom, who lends to whom, who learns from whom, is measured and maintained as an asset in its own right — because it is one. A supply network is not a cost to be minimised. It is the organ through which a firm senses an economy, and organisations that treat it as such find out about the world two quarters before the ones that treat it as procurement.

None of this needs a new paradigm to be announced. It needs four reports that do not currently exist and one committee willing to read them.

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DESIGN

The structure that gets there

Three layers, and they are built in order, because each one makes the next affordable.

Layer one: measure the distribution, not the average. This is a reporting change and it is nearly free.

For any series your organisation already keeps — daily revenue, claim size, outage duration, order size, supplier concentration — compute four things beside the mean: the standard deviation, the 99th percentile, the expected shortfall beyond it, and a tail exponent estimated by the Hill estimator or by a rank-frequency regression. Four numbers. If the tail exponent is below 2, the variance of that series does not exist and every tool your organisation uses that assumes it does is producing a number with no referent. That single finding, delivered once, changes how a risk function behaves permanently.

Layer two: model the interaction. Build the smallest agent-based model that could possibly answer one real question, and hold it to a hard rule.

The rule is the one that separates this from theatre: the model must be calibrated on one period and judged on another it has never seen. Pick the question first — will this pricing change cascade through our distributors, does a loan-to-income cap bind on the households we care about, what happens to lead times if our second-largest supplier stops. Write the agents with the fewest attributes that make the question answerable. Then withhold three years of data and do not touch it until the model is frozen.

Report three quantities every time: the median outcome, the tenth percentile, and the fraction of runs in which something breaks. Report the last one even when it is zero, because a model that has never produced a bad run has not been tested.

Layer three: govern for the window. Ulanowicz's shape, used properly.

Efficiency and resilience are the same axis read from opposite ends. A firm that maximises efficiency — single supplier, zero inventory, one channel, one product — drives its ordered share of throughput toward one, where robustness is near zero. A firm that diversifies without limit pays coordination costs it cannot carry. The governance move is to name the axis and pick a position on it deliberately, then write the position into a covenant so that the quarterly pressure toward efficiency has something to push against.

In practice that is three numbers in the board pack: concentration of revenue by customer, concentration of supply by input, and concentration of capability by person. Each with a floor and a ceiling, each reviewed at the same meeting as margin. The floor is the part nobody writes down, and the floor is the whole mechanism — because margin will argue for the ceiling every quarter without help, and nothing in the standing agenda argues for the floor.

Sequence. Layer one in a quarter, by one analyst. Layer two in two quarters, once layer one has shown the tail is real and the appetite exists. Layer three at the board, on the evidence of the first two. Attempt them in the other order and you are asking a board to adopt a philosophy; attempt them in this order and you are asking it to respond to its own numbers.

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DESTINY

How it holds when nobody is pushing

What sustains this is not conviction. It is that the distributional report is cheaper to keep running than to switch off, and that once a tail exponent is in the standing pack, removing it requires somebody to explain why.

Three failure modes, named so they can be seen coming.

The model becomes an oracle. An agent-based model that is never wrong in public is being used wrongly. Its value is that it produces a distribution of futures; the moment somebody quotes its median as a forecast, it has become a worse version of the thing it replaced. The guard is a published back-test against withheld data, refreshed annually, with the misses shown.

The metaphor outruns the mathematics. "The economy is a living system" is a sentence that can be made to justify almost anything — deregulation, because ecosystems self-organise; intervention, because ecosystems need keystone species. Both inferences are unlicensed. The analogy earns its keep exactly where a specific result carries across with its assumptions attached, as May's criterion did, and nowhere else. When you cannot name the theorem, you are using a figure of speech.

The complexity becomes the product. Sophistication is seductive and a model that only its author can run will die with its author. Every model in this programme carries a one-page description that a person outside the team can read and a second owner who has run it alone.

And the deeper thing that makes it hold: this framing is falsifiable, so it can be improved. The exponent can be measured and it can come back at 3.4 instead of 3. The out-of-sample test can fail. That is not fragility — it is the property that lets a body of knowledge compound rather than merely persist, and it is the reason this chapter spends as much of itself on what the frame cannot do as on what it can.

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DELIGHT

What it feels like

There is a moment, the first time you plot your own data on log-log axes and it comes out straight, when something reorganises. The scatter you had been apologising for as noise turns out to have a slope. It was never noise. It was structure, in a coordinate system you had not been using.

And then the better pleasure, which is the loss of a particular kind of anxiety. Once you know that the extreme event is not an aberration but a property of the distribution, you stop being ambushed by it. You size for it. The wondering whether this quarter is the one where something breaks is replaced by a number, and the number is in the buffer, and the buffer is committed.

There is also something quietly lovely about the fact that nobody set the price on the slate in the plate. Nobody at Santa Fe wrote volatility clustering into the artificial market and it came anyway. You are not looking at a machine somebody built badly. You are looking at a thing that is doing what things of its kind do, and that is a far more interesting object to work inside.

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OPERATIONALIZE THIS

At the level of finance

The arithmetic above ends in one place a treasurer can act on: a liquidity facility sized by the tail rather than by the variance.

The structure: a committed revolving credit facility, undrawn, sized on expected shortfall under a fat-tailed distribution, with a covenant holiday triggered by a published dispersion measure.

Most corporate liquidity buffers are sized on a 99 percent Value at Risk computed under a normal distribution, because that is what the spreadsheet does. Here is what that costs, computed:

  at 99 %       VaR, Gaussian   2.33 σ     VaR, cubic tail   2.62 σ   (1.13x)
  at 99 %       ES,  Gaussian   2.67 σ     ES,  cubic tail   4.04 σ   (1.52x)

  the buffer multiple   k  =  ES(cubic, 99 %) / VaR(Gaussian, 99 %)  =  1.74

A facility sized on Gaussian 99 percent VaR holds 42 percent less than the average loss in the very tail it was built for. That is the entire proposal in one line, and it is an arithmetic statement, not a philosophical one.

The mechanics.

• Size. k × your current Gaussian-VaR-based buffer, where k is computed on your own series, not borrowed from this page. Compute the tail exponent first; if it comes back above 4, k will be close to 1 and you should not do this, which is what makes the test worth running.
• Form. Committed and undrawn. You are buying the option to draw, not the cash. The cost is the commitment fee, typically a fraction of the margin, and that fee is the premium on the only insurance that pays in the state where insurance matters.
• Trigger. A covenant holiday on leverage and interest cover, keyed to a published, external, non-discretionary dispersion measure — a volatility index, a sector dispersion measure, a named credit spread crossing a stated level. External and published, so it cannot be argued about in the month it fires.
• Term. Three years, with a hard review at eighteen months against the re-estimated exponent. The exponent moves; the facility should be told.
• Balance-sheet treatment. Undrawn, it is a disclosed commitment, not a liability; the commitment fee runs through the income statement. Drawn, it is debt on normal terms. Take this to your auditors at the start — the conversation is about disclosure of commitments, which they have every year.

The counterparty. Your existing relationship banks, at renewal, as a re-sizing of a facility that already exists. This is materially easier than a new facility and it is the reason to time the work to the renewal calendar. A syndicate of three rather than one, because the whole argument is about concentration.

The number that decides it.

        k × (annual commitment fee on the increment)
   ------------------------------------------------------   <  1
    (probability of the tail state) × (cost of the tail
     state: distressed financing spread + forced asset
     sales + lost options)

If the left side is below one, the facility is cheaper than the event it insures. Most organisations have never computed the right side, and the act of computing it is worth more than the facility.

The first ninety days.

Read alongside the other seventy-six instruments in this edition, this one has a particular place: it is the first that is sized by a shape of distribution rather than by a level. Everything else in the catalogue can be sized once. This one is re-sized whenever the exponent moves, which makes it the only instrument here that is genuinely alive.

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APPRECIATIVE QUESTIONS

Twelve, for a room

Discovery — what is already working

1. When has someone here correctly predicted a market move that the official model missed? What were they paying attention to that the model was not?
2. Where in this organisation do we already treat something as a population rather than an average — and what does that let us see?
3. Which of our relationships with suppliers, customers or competitors behaves most like an ecology rather than a transaction? What makes it work?

Dream — what becomes possible

1. If every number in our board pack arrived with its distribution attached, which decision would change first?
2. Imagine we could run our next major policy a thousand times before we ran it once. What would we most want to learn from the thousand?
3. If we understood our supply network as an organ of perception rather than a cost centre, what would we start noticing two quarters earlier?

Design — what we build

1. Which single series should we compute a tail exponent on first, and who already has the data?
2. What is the smallest question an agent-based model could answer here that would genuinely change a decision?
3. What floor — on diversity, on redundancy, on slack — would we be willing to write into a covenant, so that next quarter's efficiency pressure has something to push against?

Destiny — how it holds

1. What would we want to see published every year so that this way of working could be checked by someone who did not build it?
2. Who is the second person who can run our model alone, and what would it take to make that true by March?
3. What is the first sign we would see that our metaphor had outrun our mathematics, and who in this room is most likely to notice it?

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Fagiolo, G. and Roventini, A. (2017). "Macroeconomic Policy in DSGE and Agent-Based Models Redux." Journal of Artificial Societies and Social Simulation, 20(1), 1.

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Note on figures. The Scarf orbit, the Gaussian and cubic tail probabilities, the expected-shortfall buffer multiple k, the Ulanowicz robustness curve, the scaling factors on doubling, the Pólya urn distribution, May's criterion and the network-shock decay rates are all computed in lib/verify/II_02.py and reproducible with python3 lib/verify.py II.02. Published constants — the Pareto exponent 1.059, the tail exponent 3, the scaling exponents 0.75, 0.85 and 1.15 — are attributed at the line that prints them.