# Arrow of time and irreversibility in Economics

In physics there is the concept of arrow of time, which states that time only evolves in one direction (forward), and that this should be visible in the material world. As the Wikipedia article states (emphasis mine):

Physical processes at the microscopic level are believed to be either entirely or mostly time-symmetric: if the direction of time were to reverse, the theoretical statements that describe them would remain true. Yet at the macroscopic level it often appears that this is not the case: there is an obvious direction (or flow) of time.

The emphasis above wants to highlight an important question, which I think it is pertinent to economics. Namely, if the direction of time were to be reversed, would the theoretical statements described by these models still hold? This is, are dynamic economic models time-asymmetrical or time-symmetrical?

For example, consider a Solow model. Is there an arrow of time in this model? To put it differently, draw a Solow diagram of the accumulation of capital. Draw the path of the economy from A to B. Does this imply necessarily a unidirectional time evolution?

This relates to the issue of irreversibility and determinism. If we use deterministic models (as the standard Solow), we can reconstruct "back" everything from "the future". However, complex systems, where uncertainty and stochastic shocks prevails are by definition irreversible.

As this article on the Chemistry Nobel prize winner Ilya Prigogine states:

Prigogine contends that determinism is no longer a viable scientific belief: "The more we know about our universe, the more difficult it becomes to believe in determinism". This is a major departure from the approach of Newton, Einstein and Schrödinger, all of whom expressed their theories in terms of deterministic equations. According to Prigogine, determinism loses its explanatory power in the face of irreversibility and instability.

Any hints of how this discussion translates into economics?

• Not sure I follow. If I understand correctly you would consider a detirministic dynamic system $x_{t+1} = f(x_t)$ reversible if $f$ is bijective because I can reconstruct the past from the present? – Giskard Sep 19 '17 at 13:08
• But then why are stochastic systems different? I can already not forecast the future from the present with certainty. And it seems I could always have some sort of probabilistic belief about the past based on present conditions using some prior belief and Bayes or maximum likelihood methods. – Giskard Sep 19 '17 at 13:10
• @denesp Not sure I follow either. Hence the question. Deterministic to me is that you can tell any state of the system, regardless of what arrow of time you assume. Irreversible means the arrow of time can only go in one way. – luchonacho Sep 19 '17 at 13:48
• in your example related to the Solow model, if you are referring to capital accumulation towards a steady state-given the initial conditions-eventually you end up at the steady state. If you are talking about going from one steady state to another one then I think that in this case there is definitely no time involved; there is no description in the Solow model on how we get from one steady state to another; anything goes – user14471 Sep 19 '17 at 18:37

The question and the answer by user14471 seem to relate to the issue of ergodicity in economic systems and models. Economic systems (in reality) cannot conceivably be ergodic, while some economic models are ergodic (those that do not attempt to reflect any of the non-ergodic properties).

Necessary concepts to answer this question: Ergodicity, microstates, macrostates

Ergodicity is the property of a system to spend approximately equal amounts of time in each of its microstate (if you observe it over sufficiently long times). A microstate is simply the state of a system if you consider every property of every element of the system. Not all microstates are distinguishable. Undistinguishable microstates constitute a macrostate. That is, all macrostates are distinguishable but some have more microstates than others. An ergodic system will be more likely to assume those macrostates with more associated microstates (that is, the macrostates with higher entropy).

Examples

Consider a non-economic example: gas in a container. For the definition of a microstate, the position of every molecule is important, for a macrostate only the distribution of the gas matters. While there is a macrostate in which all the molecules are crammed together on one side of the container, this one is extremely unlikely; a macrostate with an even distribution of molecules across the container is much more likely. Since the gas can, however theoretically assume each of the microstates in the future, independent of which other microstates have been assumed in the past, the system is ergodic.

Now consider an economic example: the distribution of dollars across the population. A microstate considers which dollar is owned by which person, a macrostate only considers the distribution. While microstates are not distinguishable or even measurable (how do you distinguish the different dollars in your bank account?), the number of microstates per macrostate is important. Theoretically, all dollars could be owned by one person (while everyone else has nothing). This is an extremely unlikely macrostate, while a more even distribution across the population is much more likely. Actually the likelihood (and entropy) will be maximized by a Gaussian distribution. (However, if you measure the distribution of wealth, you will find that it is not Gaussian but heavy-tailed.)

How ergodicity, irreversibility, and the arrow of time, are related

Ergodicity can be phrased differently as: every microstate is reachable from every other microstate. It is reversible, it does not have an implicit arrow of time. It is easy to see that this is not the case if the system has an attractor (a stable equilibrium) that captures trajectories (development paths) which will then not be able to leave the attractor again.

Ergodic and non-ergodic models in economics

Armed with these concepts, we can now return to the question of reversibility in economics.

Models of simple exchange of goods etc. are ergodic. Depending on how and according to what prices and preferences the goods are exchanged, every microstate of the system is reachable and possible. What is more, every transaction can be reversed.

In the case of models of growth, developent, or technological change, ergodicity will typically not hold any more (unless you can have degrowth such that the system reverses its growth trajectory exactly). These systems typically have attractors (Solow-Swan models for instance have an attractor once you remove the neutral technical change term) they have production functions that would generally not be considered reversible, and they may not allow for recessions and reversion of technological change. More complex models of technological change and development from evolutionary economics or so will definitely

That said, economic models have a certain tendency to assume ergodicity for whatever is not part of the model. Macro-models (including Solow-Swan models) assume that the structure of the microlevel (that is not modeled) will not interfere with how the model works, that agents are exchangeable (representative agents) and that transactions are neutral. For RBC and DSGE models this is made more explicit by assuming an unbiased i.i.d. stream of shocks acting on a largely homogeneous population of agents. Some varieties (HANK etc.) try to address this, but the heterogeneity those models allow for is extremely limited. Agent-based models address this from a different angle and are able to achieve the non-ergodicity property observed in real economic systems (see last paragraph below), but they have their own problems, as was discussed in a different question.

Ergodicity in real economic systems

In real economic systems, it is obvious that development paths are not reversible. You may suffer a recession, you may even experience an extensive collapse and lose advanced technologies, but you will not be able to dismantle and sell off the associated capital goods (and the human capital) in a way mirroring how they were acquired. Further, humans react vastly different to gaining and losing wealth (loss aversion).

And beyond this, the very distribution of wealth - a heavy tailed, Pareto distribution, not a Gaussian - will show you that the system does not achieve (or even come close to) the entropy maximizing macrostate with the most associated microstates. This is a structural property typically found in complex systems with self-organizing properties (the issues Prigogine wrote about as mentioned in the OP). You will find similar distribution also among firm sizes, regional agglomerations (city sizes), and a ton of other things that are more or less related to economic systems.

Any hints of how this discussion translates into economics?

I think the short answer is that it doesn't.

There is an old paper by Thorstein Veblen where he tries to explicate why economics is not an evolutionary science; what he means by 'evolutionary' is in a sense the opposite of what he calls the 'taxonomical' view of his contemporary economics (late 1800's) namely a theory based on cumulative causation and not on explanation with recourse to immutable 'natural' laws (in this paper he doesn't seem to deal with the 'mechanics' of selection, mutation etc but is more dedicated to the idea of change that happens over (long?) periods of time due to a sequence of causes).

This 'evolutionary' flavor of economics never came to fruition in the sense that it never made it in the curricula for undergraduate studies nor did it ever influence policy in a distinguishable manner. I'm using this evolutionary analogy because I think that in many ways time irreversibility, path dependence etc is about evolutionary processes. Economics has always been for the major part about guiding policy; to the extent that policy makers are to some degree accountable for their actions, they cannot recourse for advice to a theory that permits in principle 'one thousand roses to bloom'.

'Non-linearity', 'time-irreversibility', 'complex systems' applications in economics are ideas that have their origins in attempts to tackle problems in an interdisciplinary manner. Their distinctive characteristic-if we're allowed to single one out-is their fundamental pluralism in methods and open-endedness in results.

Those characteristics are a no-go for economics for the simple reason that economics is politics disguised in formulas and politics is about conflict resolution.

In general if someone is interested in those themes they could look into this or this or this for that matter.

• Thanks!. I'm not sure if by "it doesn't" you mean "such discussion has not have much traction in economics", or "that discussion has no role in economics". – luchonacho Sep 20 '17 at 9:25