## Dynamical Systems Definition of Equilibrium

Within dynamical systems theory, equilibrium is a well-defined. If we have a discrete time dynamical system defined by:

*x(t+1) = f(x(t)), x(0) = z.*

where the state

*x(t)*is a vector of all of the underlying time series values that define the system, and

*z*is the initial condition of the system. (For those of you who are unfamiliar with vectors, it means that we stack up all of the individual time series in a column, and treat the entire column as a single mathematical entity.)

The point

*y*is an

*equilibrium*if:

*y = f(y).*

It is straightforward that the time series

*x(t) = y*is the solution for the system when the initial condition is

*x(0) = y*. That is, all of the components of the state variable are fixed at the same levels for all times. (Within mathematics, this is also referred to as a "fixed point," and so one runs into "fixed point theorems" in systems theory.)

An equilibrium point may be stable or unstable, which are also well-defined mathematical concepts (but the definitions are less intuitive, so I will skip the formal definitions). If an equilibrium is stable, state variables will have a tendency to converge towards the equilibrium as time passes.

Physics also has a related notion of equilibrium, but the definition may drag in physical analogies which are inappropriate in a purely mathematical setting.

The key point to note that this definition of equilibrium is a time series property; it makes no sense to discuss it without a notion of time. Classical economics was developed in an era with extremely limited access to time series data, and so the notion of time was only vaguely modelled ("the short run" versus "the long run"). I will return to the classical definition at the end of this article.

## The One-Period Solution Of A SFC Model

I now return to the model SIM (from the text

*Monetary Economics*by Godley and Lavoie), the solution of which I described in "Finding the Solution In a Simple SFC model." (I had some questions about how this solution relates to equilibrium by a reader "Bill"; this article is an extended response.)

It must be underlined that model SIM is the simplest possible SFC model; it has a single true state variable. (That is, the solution at a time point only relies on the previous value of only one variable -- household sector money holdings.) In order to keep the model at a single state variable, the business sector has to have the superhuman ability to forecast revenue, as it

*always*has no profit. If we allow for business sector forecast errors, we then need to track the business sector holdings of money, which creates a second state variable. We can scale up SFC models to allow for added state variables -- but it is no longer easy (or possible, depending upon the model) to find the solution by hand.

Since I wanted to explain how to find a solution without relying on numerical solutions, I referred to this simple model. The difficulty is that it is hard to relate this superhuman business sector to real world entities. However, one just needs to focus on what households are doing, and ignore the business sector for now.

As I noted in that article, the accounting relationships within the model (which

*must*hold) do not pin down the model solution. The household sector has a decision to make -- how much of its income will be saved or consumed? Once we fix the saving rate (and the level of government spending, which is set outside the model), the set of equations then fixes a single solution.

This single period solution looks like what many people think of as "equilibrium" in a modern Dynamic Stochastic General Equilibrium (DSGE) model. However, it is only the solution for a set of equations at a time point; there is correspondence to the definition of an equilibrium in dynamical systems. We should only refer to it as the "model solution" at the time point. (If there is no solution, it implies that state variables cease to exist, which is not a sensible property of a mathematical model of a real world system.) This is discussed further below.

## What Is Happening During The Period Does Not Matter

The best answer to the question "What sequence of transactions is occurring during the period?" is that it does not matter. You have set a system of equations, and you have found a solution. The exact underlying transactions are not modelled, and there is not enough information to model them. We only have access to the aggregated values at the end of the accounting period.

And we have to be realistic -- there is no chance that we can model all of the underlying transactions within an economy. We can create virtual economies where everything is modelled, like a video game (or "agent-based models"), but even those models are not going to be easily described mathematically. Although an interesting area, there is no guarantee that we can calibrate them against real-world economic data.

I am currently reading

*Capitalism: Competition, Conflict, Crises*by Anwar Shaikh, which has a discussion of "emergent properties" in economic models. (The book is new and very interesting; I expect to review it shortly.) Although Shaikh is somewhat dismissive of some of post-Keynesian economics, one could adapt his logic on emergent properties to explain why the micro behaviour does not matter.

We could imagine any number of plausible behavioural rules that could be followed by individual households at the microeconomic level within model SIM. (Once again, the business sector is assumed to be superhuman, and not worth thinking about.) We could then aggregate their behaviour to get the full model solution.

Once we solve the new model, we will get a solution that is consistent with a SFC model with some behavioural parameters. Therefore, at a single time point, we cannot tell the difference between the models.

Over a longer period, we run the simulations and then compare them. Over an infinite time horizon, and with an infinite number of model runs, we could presumably tell the difference between the original SFC model and the new model. However, if we are restricted to a single finite time history, and with the solution starting near a "steady state" value (see below), the model solutions are likely going to be indistinguishable. That is, the simple consumption function within a SFC model will approximate the behaviour of a hypothetical more complex competitor, which is reasonable since savings rates are fairly stable in the real world. In other words, the SFC model acts like a first order approximation of the more complex model.

In order to generate a big change in household savings, we would likely need to impose a structural change in household behaviour, which is beyond the scope of model SIM. (Please note that this is purely a guess on my part, but it looks like a reasonable guess based on the stability of real world data.)

If the model period is one quarter or one year, there are a great many payroll cycles within a time step. There allows any number of behavioural patterns to converge towards the same macro result. In the context of the example I discussed in the previous article, it is unclear how quickly the business sector would ramp up hiring in response to the increase in government spending within the model accounting period. (There is a multiplier greater than one, so production increases by more than the government consumption increase.)

Finally, I would note that Godley and Lavoie discuss the determination of the solution in Section 3.3.2 "Mechanisms adjusting supply and demand." They contrast a number of approaches, including classical and Keynesian. They approach the topic in the usual way for economists, but it may not help non-economists' understanding of the exact sequences of transactions within the accounting period. Their explanation is more easily understood in the context of how Walras framed the issue (discussed below).

## Model Steady State

Within model SIM, it is possible to show that if we keep government purchases constant, all variables (such as total production) will converge towards fixed values. (There are some mild conditions; for example, if the tax rate is zero, variables will grow without bound.) The model is linear, and so if we double government spending, then all other variables will double in steady state.If the model starts with variables at the steady state values (and government spending is fixed), the variables are unchanged for all time. This makes the steady state look like an equilibrium, using the dynamical systems definition of the term. However, it must be kept in mind that this equilibrium depends upon government spending being fixed; if we change government spending, the equilibrium changes.

Having variables converge to fixed values works in an economy that is not growing. However, we generally want model models to exhibit steady growth rates. In this case, we need to define "steady state" in terms of the ratios of variables. For example, the ratio of variables scaled by GDP tend towards fixed values. However, such a definition moves the "steady state" away from the dynamical systems definition of equilibrium. As a result. it is important to keep the two concepts separate.

## Walras' Equilibrium

I will now give a simplified overview of how Walras defined equilibrium. For those who are interested in more details, Anwar Shaikh has an excellent longer discussion in Chapter 3 of*Capitalism.*

- We write down the supply and demand curves for all markets, which map prices to quantities supplied/demanded. (Prices drive classical economics; if we have all the prices, we can determine all of the other variables from them and the supply and demand curves.)
- We then need to find the set of prices that make supply equal demand in all markets ("market clearing").
- All entities are assumed to be price takers, a supernatural "auctioneer" sets the prices. It keeps announcing new sets of prices until all markets clear. (The process of adjusting prices is known as
*tâtonnement*-- groping). - The final prices that bring supply and demand into balance are the "Walras equilibrium" prices.

If we have a time series of the set of prices, the notion of "Walras equilibrium" would match the dynamical system definition. However, run into a huge problem -- how do we determine that time sequence of auctioneer prices?

This is where mathematical sloppiness hits. The supply and demand curves are constraints for the solution. According to the assumptions about supply and demand curves, there is only supposed to be one solution to the system of equations -- the equilibrium set of prices. We cannot observe any other prices within the model, since constraints would be violated. We can talk about "non-equilibrium prices," but that is nonsense within the context of a mathematical model. You either have a solution, multiple solutions, or none at all. (Multiple solutions is extremely awkward; we normally think of there only being a single value for economic time series at any given time.)

From a mathematical perspective, the only interpretation of "equilibrium" that stands up is that it is essentially the same thing as the single period SFC model solution. My reading of the DSGE literature is that is how it is generally interpreted; however there is still an insistence on giving it the mystical name of "equilibrium" instead of "solution."

Finally, we can draw a parallel between the process of

*tâtonnement*and the the process of the state variables in a SFC model converging towards equilibrium over multiple periods. However, in this case, the prices are being determined via mathematical constraints at each time period. If we equate "Walras equilibrium" with a single period solution existing, the system was actually at "Walras equilibrium" during the whole

*tâtonnement*process. This means that this interpretation is quite different from the way these concepts are usually explained.

(c) Brian Romanchuk 2015

Brian, Roger Sparks and I have been having an extended conversation about SIM, his variant of SIM, and various physical analogies. You can easily construct a circuit (for example) that matches SIM at the sample times. I propose one here to model SIM's state and one other variable (Y, T, YD or C... your choice). The circuit could be elaborated to model the full set. And no need to build it... a website like this can let you try it without getting your hands dirty. ;)

ReplyDeleteI emailed Anwar about his views of one of Jason's pages on Jason's approach based on Gary Becker's ideas from the early 1960s. Anwar was a student of Gary Becker. He referred me to one of his books (chapter 3) (I don't know if that's the one you're reading... Did it cost $38.50? ... :)

... I first suggested Roger think of a simple spring and dashpot analogy, and he liked that and ran with it, but it's way easier for me to think in terms of simple circuits. And I realize that SIM is about as simple as it gets! Why are we doing that?... I can't speak for Roger, but extreme nerdiness might be an appropriate explanation for me perhaps.

DeleteThe problem with the circuit analogy is that it fails as soon as we experience nonlinear/time varying effects. For example, in the model PC, the growth of Treasury bills is the stock times the interest rate. If the interest rate is not constant, we either have a linear time-varying system (if interest rates are externally specified), or the model is nonlinear if the interest rate depends upon the state of the system (for example, a Taylor Rule). This is why I would not spend time on those analogies; it only works because all first order linear time invariant systems are specified by the same simple equation:

Deletex(t+1) = (1+r)x(t) + u(t).

As for Anwar's book, it is "Capitalism"; it's new, but I think the hardcover is more expensive than that (that might be the cost of the ebook, which is what I bought).

... non-linearities... true, but you can also build circuits with non-linearities. ;) ... also, you can linearize a non-linear system about a particular operating point (nominal solution), which can also be helpful for analysis in some cases. And of course you already know all that.

DeleteI'm not recommending you DO this of course!

... time varying effects (some of them) can also be OK. The system will even remain linear in some cases (assuming it started out that way). For example, a tunable R, L or C element.

Delete... I just checked my email: yes, it was called "Capitalism" ... $38.50 (with the attached postcard :). $30 as an ebook. Have you read chapter 3 yet? Does it talk about irrational agents (or random agents)?

DeleteThe book is 1000 pages - it talks about everything...

DeleteChapter 3 is on microfoundations, and how the same macro behaviour can emerge from different micro behaviour rules.

"and how the same macro behaviour can emerge from different micro behaviour rules."

DeleteThat sounds a lot like what Jason talks about all the time. Jason likes to say you can think of there being an emergent "macro agent" which isn't dependent on the particulars of the micro-agents. This is something that David Glasner often speaks of too (using different words). Glasner talks of "macro foundations" rather than micro foundations. In fact it was David that pointed Jason to Gary Becker's work from the early 1960 on this... and it's the very thing I've heard Anwar talk about (briefly) in an interview on youtube somewhere, mentioning as well that he had been a student of Becker's. Becker has a paper from 1962 called "Irrational Agents" or something like that, which Jason unknowing replicated in a series of posts... and that's when Glasner let him know about Becker's earlier work.

Here's the reference:

Delete[1] Becker, Gary S. Irrational Behavior and Economic Theory. Journal of Political Economy Vol. 70, No. 1 (Feb., 1962), pp. 1-13

That from Jason's post covering the same material.

BTW, I found Anwar from another youtube lecture I saw Marc Lavoie do. He had a couple of slides on Anwarw's work.

DeleteBrian, you might get a kick out of this: follow this link to an online circuit model, and then close the switch by clicking on it. The three plots across the bottom are (left to right), Y, T and H. The explanation is in my post I link to above.

DeleteThis one is better. I adjusted the time step and speed to make sense.

DeleteOK, one final iteration of this circuit (click on the switch at top-center to activate... it'll take a second or two to get going), this time using op amps, and modeling one possible continuous time "equivalent" to SIM with "instantaneous flows" which should obey G&L's accounting identities over any time interval. 1 default SIM period is 1-second, and one SIM unit of money (e.g. a dollar) is represented by a coulomb of charge. Thus to recover SIM flows (G, Y, T, YD and C) over each sample period, the charge is integrated in a 1-farad (1-F) capacitor (only done for Y, but the same circuitry would apply for the others).

DeleteHere's an annotated screen shot describing what you're looking at, and there's more information in the post here.

"I had some questions about how this solution relates to equilibrium by a reader "Bill"; this article is an extended response."

ReplyDeleteYou have my gratitude, Brian.

Muchisimas gracias. :)

No problemo.

DeleteEconomic stock-flow models would have stocks which are populations such as people, non-financial goods (e.g., factories, houses, durable goods), and financial instruments (e.g., money, debt, equity). Flows would be the "state transition rules" that account for changes the population levels over a period of time. The concept of stable versus unstable homeostasis is a better fit for economic activity and economic history than the concept of equilibrium states. I think the Post-Keynesian models which assume agents who make decisions under uncertainty with margins of safety and who use buffer stock inventories is the better model for recognizing the fact of unstable homeostasis.

ReplyDelete