Ensayos de Economía

The system is made up of numerous heterogeneous units that interact. This is the "microeconomic base" of the system.

In an economic system viewed under complexity, these units can be human beings, companies, or the central government itself —which is considered here simply as one of the agents (units) of the system—.

The interaction of these units is governed by simple, albeit non-linear, adaptive rules. These are rules of the form "if these conditions are given... then this happens: ...". It is these rules that form the "mesoeconomic basis" of the system.

In an economic system viewed under complexity, these are not high-level rules of cognitive introspection, but, rather, rules of inductive behavior. The reason for this is that, according to modern psychology, human beings in situations that are complicated or undefined are only moderately good at deductive logic and make only moderate use of it. However, we are outstanding in seeing or recognizing patterns that confer evolutionary benefits of survival and well-being. And for that, we construct internal temporal models or hypotheses (schemata) to work with (Arthur, 1994; Sauce & Matzel, 2017).

Human beings make localized deductions based on our current hypotheses and make decisions accordingly. As feedback comes in "from the outside", we strengthen or weaken our beliefs in our current hypotheses and discard those that no longer work, replacing them with new ones.

In other words, when we cannot reason completely, or the problem is not well defined, we resort to simple models to fill in the gaps in our understanding. This is called inductive behavior.

For example, in chess players typically study the configuration of the board at a given time and recall their opponent's moves in past games (de Groot, 1965). They then use "adaptive rules" to form hypotheses or internal models about the opponent's possible responses to their possible moves. But there is no total rationality here. Typically, there is adaptation and non-Bayesian evolution; that is, the "ability" to survive and win. However, modeling all this requires different tools, especially computational ones. That is computer programs that can "learn" and recognize themselves, creating hypotheses, adapting, discarding, and mutating².

But also, for years, results from behavioral experiments, field experiments, intra-, and extra-species studies, archaeological and anthropological data, models of cultural evolution, and innovations in classical and evolutionary game theory have been integrated to show us a deep insight into the role of cooperative prosocial human behaviors (norms and institutions) in the creation and evolution of economies (Dhami, 2016).

The dynamics of the simple adaptive rules of the micro-units show, most commonly (but not exclusively), positive feedbacks.

These are iterative amplification (or reinforcement) mechanisms (each step is reinforced or augmented by the next) that occur in (almost) all complex systems: positive feedbacks amplify the current state of the system's behavior. Thus, for example, a behavior can become contagious and disperse in the system, but it can also disappear.

One of the best-known positive feedbacks is the "conformity effect", which is characterized by the property that a certain behavior of a micro-unit becomes more likely if "nearby" units have the same behavior. In turn, "network effects" ("network externalities" or network effect) are a particular case of conformity effect in which closeness is established by the distance (in ties) of one micro-unit to another one in a network (Figure 1).

Particularly, in an economic system viewed under complexity, financial panics and the formation of ghettos or similar forms of segregation are possible examples of the operation of these network effects. That is also the case in a market, where a good shows network effects if the value for a new buyer of adopting the good is increasing with the number of buyers who have already adopted it. Thus, the more customers adopt the good, the more valuable it becomes to potential adopters. However, these positive feedbacks can also work in reverse: if the adoption does not reach the "critical mass" of buyers, the good may eventually disappear from the market.

Another very explanatory example in which positive feedbacks play a central role is Pólya's stochastic urn processes (Pólya, 1930). Here a process with conformity effects is formulated. Let us consider an urn of infinite capacity, to which are added balls of two colors: red and black. Suppose we start with a red and a black ball inside the urn, adding one ball at a time, indefinitely, according to the following rule: we randomly choose a ball in the urn; if it is red, we add it to the urn, in addition to the extracted ball, another red ball; and if it is black, we add, similarly, an additional black ball.

Then we can ask ourselves: does the proportion of red and black balls stabilize around a single ratio? That is, does the law of large numbers apply? Pólya (1930) showed that the proportion of red balls does tend to a limit X with probability 1, but that this is a random variable uniformly distributed between 0 and 1. X is a random variable uniformly distributed between 0 and 1. That is, structure (regularity) does emerge, but the realization of this structure is random: the regularity that emerges is not an equilibrium (steady state), but a random variable!

In the more general case, when the urn starts from an arbitrary number of red and black balls, the proportions, once again, tend to a distribution X which this time is a two-parameter beta function:

which is one of the best-known regularity-generating functions in the theory of complex systems. In fact, for a certain wide range of parameters (fixed y and relatively large x), the beta function generates stochastic power laws of the form p(x)=Cx^-α with C,α>0 fixed, which, by taking logarithms on both sides of the equation, is in the form of a decreasing straight line in log-log scale:

These Pólya urn models are very naturally used to represent phenomena in statistical mechanics, evolutionary biology, population processes and cases of emergence and formation of industrial clusters, and even disease contagion processes (Banerjee et al., 1999; Eggenberger & Pólya, 1923; Jhonson & Kotz, 1977).

A final example of the emergence of positive feedbacks is the scale-free network (Barabási et al., 1999; Jackson, 2019) in which the micro-units, as they into the system, "adhere" to the units (nodes) that, at that moment, have more units already connected. These networks will be characterized, as an emerging regularity, by some very popular large connection centers (called hubs) and others (the vast majority) with very few adherents, following (by number of adherents) a stochastic power law (see Figure 1). Note that, once again, a random variable appears as a macro regularity, through positive feedbacks.

Figure 1.: Scale-free network*

* The nodes most highlighted in black are the "larger" hubs because they have more connected adherents. The size distribution of the nodes follows a power law of the form [2] above. Source: Monsalve and Ávila (n. d.).

Other examples of such positive feedbacks abound. These include herding behaviors, mobs, peer effects, etc. (Bikhchandani & Sharma, 2001; Jackson, 2019). It should be noted that these positive feedbacks are just one of the drivers that make the dynamics of a complex system always "multi-level" and that, at the systemic level, "the whole is more than the sum of its parts". After all, these amplifications are at the heart of the information transmission processes between micro-units.

The non-linear mesoeconomic dynamics of complex systems co-create behavioral "regularities" in the long run (self-organization).

Not only do the micro-units contribute to the formation of macro regularities, but these, in turn, contribute to the behavior of the micro-units (retroactive behavior of the system), in "eternal disequilibrium". Arthur (2009) and Arthur et al. (2020) point out this characteristic of complex systems in an important way, in the case of the problem of innovation and technological development of an economy. First, they claim that the "skeleton" of an economy is made up of its technologies, and that the rest of the economy, i.e., trade activities, flows of goods and services, and investment, only "wrap around" that skeleton, although integrally dependent on it. Thus, the economy does much more than readjust to technological changes: it is also from it, in fact, that technologies emerge.

Let us observe the systemic causality proposed by Arthur: technology creates the structure of the rest of the economy (which adapts to new changes), and this, in turn, leads to the creation of other technological innovations. This systemic causality (technology creating the economy-economy creating technology) is not widely seen in mainstream economic literature. This phenomenon is known as structural change in the economy. And since these changes do not occur on a day-to-day basis, it is economic historians who are responsible for studying it. Here we do not see the introduction of new technologies as the origin of small adjustments and growth, but of changes in the structural composition of the economy itself.

In practice, a new technological change (which has been made possible by a combination of previous technologies and has outperformed its rivals in competition within the economy) is a call for the creation of new industries. However, it should be cautioned, as Arthur also points out, that the process of technological change does not follow a strictly Darwinian type of evolution (natural selection and mutation), but a "combinatorial evolution" in the manner of cellular automata (Mitchell, 2009).

Thus, this process of technological change requires, each time, a new industrial organization; which, in turn, may cause new technical and social problems; and from there niches of opportunities are created for new technologies (institutions) to solve this within the economy. However, there is nothing inevitable, nothing predetermined in all the above. In fact, very different combinations, and arrangements (technological and institutional) can solve the technical and social problems posed by the new technology.

Which combination, ordering and path is chosen within the network of previous technologies is, in part, a question that depends on small historical changes. For example, the order in which problems are studied, the predilections and actions of the people in charge of the studies and decisions. In other words, according to Arthur, it is not possible to predict future results, but we can understand the mechanisms by which technological innovations are obtained. Thus, technology determines the structure of the economy, but which technology emerges is not determined a priori.

Today, economics is a discipline that is often criticized because, unlike the "hard sciences" such as physics or biology, it cannot be described by a set of facts that do not change over time ("economic laws"). But after the above discussion, it is postulated that this is not a flaw, but is proper and natural to it: economics does not operate through "economic laws" in the manner of classical mechanics. After all, it is not a simple system; the arguments seem to show it, rather, as a complex system in constant evolution, and the structures it forms are also constantly changing. This means that our interpretations of the economy must also change constantly over time.

In complex systems, micro-units behavior shows limited predictability of future behavior of the system...

Determinism, which is a doctrine of a necessary relationship between all events and phenomena, and conditioning to chance events, reached a development in natural science and materialistic philosophy with Bacon, Galileo, Descartes, Newton, Laplace, Spinoza, and the French materialists of the eighteenth century. Therefore, and in keeping with the level of the natural sciences at that time, determinism had a mechanistic and abstract character. Hence, an absolute value is assigned to the form of causality. This is described according to the rigorously dynamic laws of classical mechanics, which leads to the identification of causality with necessity and to the denial of the objective character of chance.

The French mathematician Pierre Simon de Laplace (hence mechanical determinism is also known as Laplace's determinism) is the one who formulated this point of view with the greatest prominence. In 1799, Laplace began his five-volume traité de la mécanique céleste with the statement that, if the velocity and position of all the particles of the Universe were known at an instant, their past and future could be predicted.

For more than 100 years Laplace's assertion seemed correct and, therefore, it was concluded that freedom would not exists, since everything was determined. However, the progress of science has refuted Laplace's determinism in all areas. Henri Poincaré's chaos theory (1903) and Werner Heisenberg's uncertainty principle (1927) were, at the time, two pillars in refuting the hypothesis of Laplacean determinism.

The origin of chaos theory arose in 1903 when Poincaré studied the old problem (the three-body problem) of whether the Earth-Moon-Sun system would be stable forever, and, surprisingly, found that it depended sensibly on the initial conditions. Specifically, the three-body problem consists in determining, at any instant, the positions and velocities of three bodies of any mass, subject to mutual gravitational attraction and starting from some given positions and velocities. Although it had already been rigorously studied by Newton, Euler, and Lagrange since 1687, Poincaré was the first to note the existence of an infinite number of periodic solutions with "high sensitivity" to the given initial positions and velocities.

Specifically, Poincaré noted that small variations in the initial conditions could imply large (in fact, exponential) differences in future behavior, thus making prediction impossible due to the problem of numerical approximation in those initial conditions. Note that this happens although these systems are strictly speaking deterministic (they are not random); that is, their behavior can be completely determined by knowing their initial conditions. From then on, examples of this "chaotic" behavior began to be observed not only in this problem of classical mechanics, but also in demographic, climatic, epidemiological, etc. problems.

On the other hand, the establishment of the indeterminacy (or uncertainty) principle by Werner Heisenberg (1927), in quantum mechanics, also revealed the inconsistency of Laplacian determinism by showing that quantum particles do not follow definite trajectories: it is not always possible to know exactly the value of all the physical quantities that describe the particle's state of motion at any moment, but only a statistical distribution. Therefore, it is not always possible to assign a trajectory to a particle, although it is possible to say that there is a certain probability that the particle is in a certain region of space at a certain time.

Chaos theory and Heisenberg's uncertainty principle (among others), led, then, to the conclusion that science is not a predictive function through universal laws, but, very deeply, a function of understanding through relationships, patterns, or regularities of the objects of study.

In the case of an economy, and despite Poincaré and Heinseberg, the notion of uncertainty in a complex system is not preponderantly established, today, around the notion of chaos, as it is in other sciences. This is because since the 1990s, the tests developed for its detection in time series do not favor the hypothesis that the economic variables are chaotic (Faggini & Parziale, 2012), even though, curiously, the dynamics of Walras' tâtonnement and the dynamics of the spider's web (among many others), which are the basic dynamics with which microeconomics theoretically explains the formation of equilibrium prices, are chaotic (Kaizoji, 2010; Saari, 1995). That is why, currently, uncertainty in complex economic systems is being established, instead, around the notion of "fundamental uncertainty" à la Shackle (1938).

For Shackle (1938), each agent must form subjective beliefs about his and others' futures, and they will do the same. So, there is no such thing as an "optimal trajectory". For example, different entrepreneurs who are starting high-tech companies may not know how well their individual technologies will do, how the government will regulate them or what their competitors will be, and so on. This happens because they are subject to fundamental uncertainty and, thus, the problem they face is not well defined, in the manner indicated by the dominant neoclassical school.

Formally, Shackle objected to the representation of individual beliefs by an additive measure of probability a la Savage (1954), as with expected utility or expected benefit. The reason is that probabilities do not apply to decisions that are not repeated under the same conditions, nor it is possible to establish, a priori, the complete list of "states of nature." That is, in Shackle's world, decision-makers cannot divide an event into small "micro-events". Then, fundamental uncertainty ensures that even if agents have intelligent or sensible behaviors, they cannot have rational deductive behaviors because deductive rationality is simply not well defined.

Thus, the well-known "choice under rationality" is not well defined either: after all, there can be no logical solution to a problem that is not well defined. In other words, there is no such thing as rational choice. Moreover, as we said before, people normally act all the time in situations that are not well defined, forming hypotheses —or internal models— about the situation they are in, and continually updating them. In fact, people are adopting and discarding their hypotheses, strategies, and actions as they explore. That is, they always act by induction rather than deduction.

Something of extreme importance to mention here, and which has been seen in complex systems, is that it seems that the inductive shaping of expectations is a very strong contributor to the elimination of indeterminacy at the macro level, as shown by Arthur (1994) in the El Farol model of expected attendance at a restaurant, among many other examples in the same direction (Cara et al., 2000; Savit et al., 1999; Swain & Fagan, 2019). Although there is still much research to be done on this point, everything leads us to believe that proceeding by induction is at the heart of the behavior of an economy that is presented to us in permanent disruptive movement, as agents explore, learn, and adapt. An example of this can be seen in Ávila (2022), which presents an approach to the problem of the Invisible Hand posed by Adam Smith (1776/1969) that, inspired by the game of the Minority (Challet & Zhang, 1997; Savit et al., 1997), by adding adaptive and/or evolutionary behaviors. In this paper, problems of segregation, poverty traps, bubbles —and therefore crises—, cyclicality, out-of-equilibrium dynamics, incomplete information, among many others, are observed; and these are standard characteristics of observable markets.

A complex system satisfies the non-ergodicity condition.

In words, the time averages of the behavioral trajectories of the micro-units of the system exist but cannot be obtained by means of the long-term averages in the aggregate. Thus, the statistical behaviors of any macro regularity of the system are not determined by the statistical behavior of the micro-units.

The definition of ergodicity is inspired by the famous ergodic theorem —George Birkhoff (1917, 1931)— which was a key piece in the study of statistical mechanics of systems (Gibbs, 1902). As this is a very general theorem (it requires measurement theory to be fully understood), it is common to assume that a system is ergodic if the average outcome (and its variance) of a large group of units coincides with the time-averaged outcome (and its variance) of a single unit over time. If these two outcomes coincide in mean and variance, the system is ergodic. For example, flipping a coin (heads or tails) is an ergodic stochastic process, because if a very large number N of people flip a coin, the average result (and its variance) will be the same as if only one person flipped the same coin N times.

Therefore, in a non-ergodic system, the means (and variances) of the individual states do not determine the mean (and variances) of the joint distribution of the system. Thus, if one wants to determine whether a stochastic process is (a priori) ergodic (or not), a first (and fundamental) test is to observe whether the following two outcomes are (or are not) equal: 1. Average (and variance) of single-unit outcomes over time; 2. Average (and variance) of the results of a very large number of units at a single point in time. This shows the difference between individual (micro) average behavior and aggregate (macro) average behavior. Consequently, in a non-ergodic system the law of large numbers is not satisfied. After all, the law of large numbers is an application of Birkhoff's ergodic theorem.

In the same sense, and as the famous meteorologist Edward Lorenz (1993) would have pointed out, individual trajectories are analogous to day-to-day weather, while long-term averages are analogous to seasonal weather. Thus, in the case of weather, long-term averages reveal the full picture of the dynamic process with much more information than an individual trajectory could ever provide.

Going a little further, it can be shown from Birkhoff's ergodic theorem that a non-ergodic system has three additional fundamental characteristics:

• Path-dependence. This is immediate because the ergodicity condition implies that the time-averaged trajectory of any statistic is independent of the initial condition of the system. It is shown that the dynamic behavior of the system depends sensitively on the initial condition at each stage.

• Any shock to some part of the system at a certain stage affects the system in the long run. This is for the same reason of dependence on the average time path under the initial condition.

• The process from micro-units to long-term macro regularities is irreversible. This means that from a macro regularity, it is not always possible to predict exactly which were the micro-units that formed it, nor its dynamics.

When the system is non-ergodic, the interaction in any economy will be, then, by hierarchical levels, or, in colloquial terms, in the form of "terraces". That is, at each level, the stochastic characteristics of the interactions are completely different. This, as we know, is known as the "hierarchical structure" of the complex system.

A famous example of positive feedbacks and path-dependence arising from a non-ergodic process in the high-tech industry is the QWERTY (patented by Christopher Scholes, 1868) versus the DVORAK (Dvorak et al., 1936) on the keyboards of old typewriters and today's computers. This was studied by Paul David as early as 1985, who claimed that the adoption of the QWERTY keyboard is a consequence of decentralized and uncoordinated decisions in an environment where there are strong network effects. The latter, remember, means that the use of one standard or another will depend on how many others have adopted it as well, and this could lead to everyone choosing a single standard (which may not be the best).

These network externalities led to the QWERTY keyboard being "locked in" as a technological standard, even though there were more efficient and ergonomic keyboards such as the DVORAK. So, due to network effects, the market dominance of the QWERTY keyboard prevailed, even though it was only one of several long-term standards that could have emerged. And of course: under a different sequence of shock realizations at the beginning of the process, the DVORAK keyboard would have emerged as the standard keyboard.

Phenomena such as QWERTY are very common. This is also the case with the most widely used languages in the world and their preponderance (English, for example), which, despite not being the most grammatically rich, are typically used today in almost all Western countries and in many Eastern ones as well. It is also the well-studied case of the size and distance between rails of trains: the standard used today was not the best option.

On the other hand, let us emphasize that the relationship between ergodicity and chaos is not completely established. However, curiously enough, many of the chaotic systems to which the economic literature has resorted are ergodic, which has facilitated the analysis and the obtaining of average results, at the cost of having this highly demanding hypothesis. Therefore, chaos, in general, can be attenuated (or, in some cases, regulated) if the system is ergodic. An example of this is the stochastic cobweb dynamics in the partial equilibrium model, which, in addition to being chaotic, Huang & Day (2001) proved that it is ergodic.

However, it is common to find multiple examples in economics in which the dynamics with positive feedbacks are non-ergodic, meaning that the expected value of changes in utility does not reflect the average growth over time. A simple example is that of a bet with a correct coin: Heads, gains 50% of the current wealth; and Tails, loses 40% of the current wealth. W and Stamp loses 40% of it. Counter-intuitively, while this gamble has an expected value of 1.05 times the current wealth per flip, then, the expected value of the change in utility is 1.05 times the current wealth per flip:

also has a time average of, approximately, 0.95 times the current wealth, as this is shown (Peters & Gell-Mann, 2016) to be equal to:

Therefore, maximizing the expected value is wrong, and this person could eventually go bankrupt.

Ultimately, when making economic decisions about wealth, people often want to know how fast their personal fortunes will grow under different scenarios. But this requires determining what happens (over time) to wealth. Mistakenly assuming that wealth behavior is ergodic, replacing average temporal wealth with the expected value, can bankrupt individuals. It is now known that wealth is not ergodic (see, for example Peters & Gell-Mann, 2016), so the average behavior of a population cannot be deduced from averaging the temporal behavior of an individual.

This example shows that under non-ergodicity, temporal behavior implies that agents will have to adapt their preferences according to the dynamics and history (path-dependent) of the situations they face. Non-ergodicity is a criticism at the heart of expected utility theory (Poitras, 2013). Indeed, it is at the very origin of the typical textbook criticisms of Allais, Ellsberg, and the St. Petersburg Paradox. But, very importantly, it is at the heart of why the non-existence (except in very particular cases) of the representative agent, to which micro-founded macroeconomics appeals (Gorman, 1961; Jackson & Yariv, 2019).

A complex system adapts and evolves in response to the environment (system boundary) that surrounds it (Figure 2).

For Complexity, an economy must be seen as a socioeconomic system that develops multiple endogenous control mechanisms that make it function, but these mechanisms are continuously evolving (evolving) integrated with other complex boundary systems (system boundaries) such as, for example, the biosphere. After all, the biosphere is also a prototype of a complex system that includes the economies of all species and where the regularities of higher levels emerge from localized interactions and processes of adaptation, reaction and innovation carried out at lower levels (Capra & Luisi, 2014).

Figure 2.: Diagram of the universal operation of a complex system

Source: Monsalve & Ávila (n.d.).

In fact, terrestrial life has survived multiple crises over millions of years, and the pandemic we are facing today is one of them: we are being threatened by a virus that has been generated in the biosphere. To survive, human organisms have, among others, a mechanism known as "herd immunity". When a disease spreads among the population, an individual can protect himself by means of a vaccine that immunizes him. But this is not all because immunization of the individual means that he or she will never transmit the disease to others, effectively reducing the possibility of the disease proliferating in the population. Because of this, a disease can be eradicated, even if the entire population has not been vaccinated. This herd immunity effect is, of course, positive feedback from those we have already discussed, only now present in the interaction between complex systems (Solé & Elena, 2018).

Another problem we face, which is one of interaction between complex systems in the biosphere, is the threatening climate change, which manifests itself as a physical phenomenon, and originated (as a start-up) by the human economy in interaction with its frontiers. In 2018, William Nordhaus received the Nobel Prize for his work on global warming, where he claimed that allowing the Earth to warm by 3.2ºC would provide the optimal trade-off between climate change and "damage" to the economy. But most scientists who are experts on the subject think that this level of warming would be disastrous. It is now known that the problems with Nordhaus's analysis stem from an overestimation of the cost of combating climate change and an underestimation of the damage it would cause, as well as the use of an oversimplified model based on unrealistic assumptions.

Since some years ago, Complexity theory has been presenting a new perspective on this problem. For example, Farmer et al. (2020) have analyzed the costs of the transition to green energy and obtain results opposite to Nordhaus. For example, the data show that energy costs over the last century and a half (fossil fuel era) are surprisingly constant, and the forecasts show that renewables would lower energy costs substantially. So regardless of climate change, economically it is much better to convert to renewables, and doing so quickly would benefit us even more.

In fact, the authors show that it is possible to change most of the energy system within the next 20 years, save a lot of money and keep global warming below 2ºC. Arriving at these results was the authors' commitment to empirically grounded model building with the criteria of a complex economy (starting from the bottom up). Nordhaus, on the contrary, starts from aggregate models that rely on unrealistic assumptions such as rationality and utility maximization.

Complexity theory offers us, then, the possibility of using very modern tools to understand the economy (of human beings) and to suggest effective solutions. We are now close to the possibility of collecting data on economic activity at a remarkable level of detail and relying on powerful computers (quantum computing) to process them.