Equilibrium Together With Steady State Inwards Sfc Models

The usage of the term "equilibrium" has move out almost meaningless inwards economics, but this does non halt it from widespread use. It is associated amongst the economist Walras inwards classical too neo-classical ("mainstream") economics. Equilibrium has the political undertone that the economic scheme has reached an optimal state, too everyone gets their "just desserts." Post-Keynesians tend to hold upward allergic to the term, likely because of that political slant. If nosotros ignore the politics, the classical notion of equilibrium tin hold upward related to the one-period solution of a stock-flow consistent (SFC) model. Meanwhile, SFC models innovate the notion of a "steady state,"  which could hold upward idea of every bit a "long-term equilibrium." However, since SFC models let for involuntary unemployment, these equilibria notions exercise non imply whatsoever cast of "just desserts."

Dynamical Systems definition of Equilibrium

Within dynamical systems theory, equilibrium is a well-defined. If nosotros receive got a discrete fourth dimension dynamical arrangement defined by:

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

where the nation x(t) is a vector of all of the underlying fourth dimension serial values that define the system, too z is the initial status of the system. (For those of you lot who are unfamiliar amongst vectors, it agency that nosotros stack upward all of the private fourth dimension serial inwards a column, too care for the entire column every bit a unmarried mathematical entity.)

The indicate y is an equilibrium if:

y = f(y).

It is straightforward that the fourth dimension series x(t) = y is the solution for the arrangement when the initial status is x(0) = y. That is, all of the components of the nation variable are fixed at the same levels for all times. (Within mathematics, this is also referred to every bit a "fixed point," too thence 1 runs into "fixed indicate theorems" inwards systems theory.)

An equilibrium indicate may hold upward stable or unstable, which are also well-defined mathematical concepts (but the definitions are less intuitive, thence I volition skip the formal definitions). If an equilibrium is stable, nation variables volition receive got a vogue to converge towards the equilibrium every bit fourth dimension passes.

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

The telephone commutation indicate to banking firm notation that this definition of equilibrium is a fourth dimension serial property; it makes no feel to verbalize over it without a notion of time. Classical economic science was developed inwards an era amongst extremely express access to fourth dimension serial data, too thence the notion of fourth dimension was solely vaguely modelled ("the brusk run" versus "the long run"). I volition render to the classical definition at the halt of this article.

The One-Period Solution Of H5N1 SFC Model

I instantly render to the model SIM (from the text Monetary Economics by Godley too Lavoie), the solution of which I described inwards "Finding the Solution In a Simple SFC model." (I had some questions almost how this solution relates to equilibrium past times a reader "Bill"; this article is an extended response.)

It must hold upward underlined that model SIM is the simplest possible SFC model; it has a unmarried truthful nation variable. (That is, the solution at a fourth dimension indicate solely relies on the previous value of solely 1 variable -- menage sector coin holdings.) In lodge to move out on the model at a unmarried nation variable, the describe of piece of work concern sector has to receive got the superhuman mightiness to forecast revenue, every bit it always has no profit. If nosotros let for describe of piece of work concern sector forecast errors, nosotros thence take away to rail the describe of piece of work concern sector holdings of money, which creates a instant nation variable. We tin scale upward SFC models to let for added nation variables -- but it is no longer slow (or possible, depending upon the model) to discovery the solution past times hand.

Since I wanted to explicate how to discovery a solution without relying on numerical solutions, I referred to this uncomplicated model. The difficulty is that it is hard to relate this superhuman describe of piece of work concern sector to existent footing entities. However, 1 simply needs to focus on what households are doing, too ignore the describe of piece of work concern sector for now.

As I noted inwards that article, the accounting relationships inside the model (which must hold) exercise non pivot downward the model solution. The menage sector has a determination to brand -- how much of its income volition hold upward saved or consumed? Once nosotros create the saving charge per unit of measurement (and the grade of authorities spending, which is laid exterior the model), the laid of equations thence fixes a unmarried solution.

This unmarried flow solution looks similar what many people recall of every bit "equilibrium" inwards a modern Dynamic Stochastic General Equilibrium (DSGE) model. However, it is solely the solution for a laid of equations at a fourth dimension point; in that place is correspondence to the definition of an equilibrium inwards dynamical systems. We should solely refer to it every bit the "model solution" at the fourth dimension point. (If in that place is no solution, it implies that nation variables cease to exist, which is non a sensible belongings of a mathematical model of a existent footing system.) This is discussed farther below.

What Is Happening During The Period Does Not Matter

The best respond to the query "What sequence of transactions is occurring during the period?" is that it does non matter. You receive got laid a arrangement of equations, too you lot receive got institute a solution. The exact underlying transactions are non modelled, too in that place is non plenty information to model them. We solely receive got access to the aggregated values at the halt of the accounting period.

And nosotros receive got to hold upward realistic -- in that place is no direct a opportunity that nosotros tin model all of the underlying transactions inside an economy. We tin create virtual economies where everything is modelled, similar a video game (or "agent-based models"), but fifty-fifty those models are non going to hold upward easily described mathematically. Although an interesting area, in that place is no guarantee that nosotros tin calibrate them against real-world economical data.

I am currently reading Capitalism: Competition, Conflict, Crises by Anwar Shaikh, which has a give-and-take of "emergent properties" inwards economical models. (The mass is novel too really interesting; I hold back to review it shortly.) Although Shaikh is somewhat dismissive of some of post-Keynesian economics, 1 could adapt his logic on emergent properties to explicate why the micro demeanor does non matter.

We could imagine whatsoever number of plausible behavioural rules that could hold upward followed past times private households at the microeconomic grade inside model SIM. (Once again, the describe of piece of work concern sector is assumed to hold upward superhuman, too non worth thinking about.) We could thence aggregate their demeanor to larn the total model solution.

Once nosotros solve the novel model, nosotros volition larn a solution that is consistent amongst a SFC model amongst some behavioural parameters. Therefore, at a unmarried fourth dimension point, nosotros cannot tell the departure betwixt the models.

Over a longer period, nosotros run the simulations too thence compare them. Over an interplanetary space fourth dimension horizon, too amongst an interplanetary space number of model runs, nosotros could presumably tell the departure betwixt the master copy SFC model too the novel model. However, if nosotros are restricted to a unmarried finite fourth dimension history, too amongst the solution starting nigh a "steady state" value (see below), the model solutions are probable going to hold upward indistinguishable. That is, the uncomplicated consumption constituent inside a SFC model volition guess the demeanor of a hypothetical to a greater extent than complex competitor, which is reasonable since savings rates are fairly stable inwards the existent world. In other words, the SFC model acts similar a starting fourth dimension lodge approximation of the to a greater extent than complex model.

In lodge to generate a big modify inwards menage savings, nosotros would probable take away to impose a structural modify inwards menage behaviour, which is beyond the compass of model SIM. (Please banking firm notation that this is purely a guess on my part, but it looks similar a reasonable guess based on the stability of existent footing data.)

If the model flow is 1 quarter or 1 year, in that place are a keen many payroll cycles inside a fourth dimension step. There allows whatsoever number of behavioural patterns to converge towards the same macro result. In the context of the illustration I discussed inwards the previous article, it is unclear how rapidly the describe of piece of work concern sector would ramp upward hiring inwards answer to the increment inwards authorities spending inside the model accounting period. (There is a multiplier greater than one, thence production increases past times to a greater extent than than the authorities consumption increase.)

Finally, I would banking firm notation that Godley too Lavoie verbalize over the determination of the solution inwards Section 3.3.2 "Mechanisms adjusting render too demand." They contrast a number of approaches, including classical too Keynesian. They approach the theme inwards the commons way for economists, but it may non deal non-economists' agreement of the exact sequences of transactions inside the accounting period. Their explanation is to a greater extent than easily understood inwards the context of how Walras framed the number (discussed below).

Model Steady State

Within model SIM, it is possible to demonstrate that if nosotros move out on authorities purchases constant, all variables (such every bit total production) volition converge towards fixed values. (There are some mild conditions; for example, if the taxation charge per unit of measurement is zero, variables volition grow without bound.) The model is linear, too thence if nosotros double authorities spending, thence all other variables volition double inwards steady state.

If the model starts amongst variables at the steady nation values (and authorities spending is fixed), the variables are unchanged for all time. This makes the steady nation aspect similar an equilibrium, using the dynamical systems definition of the term. However, it must hold upward kept inwards hear that this equilibrium depends upon authorities spending beingness fixed; if nosotros modify authorities spending, the equilibrium changes.

Having variables converge to fixed values plant inwards an economic scheme that is non growing. However, nosotros to a greater extent than oft than non desire model models to exhibit steady growth rates. In this case, nosotros take away to define "steady state" inwards terms of the ratios of variables. For example, the ratio of variables scaled past times gross domestic product 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 of import to move out on the ii concepts separate.

Walras' Equilibrium

I volition instantly give a simplified overview of how Walras defined equilibrium. For those who are interested inwards to a greater extent than details, Anwar Shaikh has an first-class longer give-and-take inwards Chapter iii of Capitalism.

• We write downward the render too demand curves for all markets, which map prices to quantities supplied/demanded. (Prices drive classical economics; if nosotros receive got all the prices, nosotros tin create upward one's hear all of the other variables from them too the render too demand curves.)
• We thence take away to discovery the laid of prices that brand render equal demand inwards all markets ("market clearing").
• All entities are assumed to hold upward toll takers, a supernatural "auctioneer" sets the prices. It keeps announcing novel sets of prices until all markets clear. (The procedure of adjusting prices is known every bit tâtonnement -- groping).
• The in conclusion prices that pick out render too demand into residual are the "Walras equilibrium" prices.

If nosotros receive got a fourth dimension serial of the laid of prices, the notion of "Walras equilibrium" would check the dynamical arrangement definition. However, encounter a huge job -- how exercise nosotros create upward one's hear that fourth dimension sequence of auctioneer prices?

This is where mathematical sloppiness hits. The render too demand curves are constraints for the solution. According to the assumptions almost render too demand curves, in that place is solely supposed to hold upward 1 solution to the arrangement of equations -- the equilibrium laid of prices. We cannot discovery whatsoever other prices inside the model, since constraints would hold upward violated. We tin verbalize almost "non-equilibrium prices," but that is nonsense inside the context of a mathematical model. You either receive got a solution, multiple solutions, or none at all. (Multiple solutions is extremely awkward; nosotros commonly recall of in that place solely beingness a unmarried value for economical fourth dimension serial at whatsoever given time.)

From a mathematical perspective, the solely interpretation of "equilibrium" that stands upward is that it is essentially the same thing every bit the unmarried flow SFC model solution. My reading of the DSGE literature is that is how it is to a greater extent than oft than non interpreted; nevertheless in that place is soundless an insistence on giving it the mystical elevate of "equilibrium" instead of "solution."

Finally, nosotros tin clit a parallel betwixt the procedure of tâtonnement and the the procedure of the nation variables inwards a SFC model converging towards equilibrium over multiple periods. However, inwards this case, the prices are beingness determined via mathematical constraints at each fourth dimension period. If nosotros equate "Walras equilibrium" amongst a unmarried flow solution existing, the arrangement was genuinely at "Walras equilibrium" during the whole tâtonnement process. This agency that this interpretation is quite dissimilar from the way these concepts are usually explained.

(c) Brian Romanchuk 2015