Does Full General Equilibrium Exist?

 One of the to a greater extent than entertaining parts of online (and academic) economical squabbling is the fights over the equilibrium concept used inwards mainstream economics. The argue for my entertainment is that is is possible that the concept is non genuinely well-defined for macroeconomic models of interest. I must cautiously complaint that a proper Definition powerfulness be somewhere. However, I tin say that it is possible to read a large percentage of the modern mainstream literature as well as non observe a formal Definition that survives scrutiny. Such an omission is curious, given the alleged importance of the concept.

Does the Real World Converge to Equilibrium?

This rant was triggered yesteryear to a greater extent than or less inquiry I am doing, to a greater extent than or less articles discussing equilibrium that I get got of late read (Andrew Lainton, Peter Radford), as well as this query on equilibrium on the Economics Stack Exchange.

The questioner asks:

but is the model's equilibrium ane that occurs naturally? Is real-life coin render equal to coin demand? Is real-life investment equal to savings?
My query inwards short: why does the economic scheme converge to these equilibriums?

This raises all kinds of questions. What is equilibrium? Is it a belongings of the existent world? How is it possible to write a macro textbook as well as non response these questions?

Seriously, It's a Set

Mainstream economic science is highly mathematical. One of the perceived advantages of the mathematics is that argumentation is supposed to live clearer than literary economics. However, inwards club for mathematics to provide this advantage, concepts get got to live formally defined. From the perspective of applied mathematics, this agency defining concepts inwards damage of laid theory (yes, at that spot are weird corner cases inwards mathematical logic, such equally self-referential sets, where nosotros get got to become beyond laid theory).

Other than really highly stylised models, most macro models are specified yesteryear iii things.

1. A laid of economical variables, which are elements inwards the laid of "time series."
2. "Constraints" on those variables: accounting identities, production functions, etc.
3. A method to observe a "solution" to the system. Influenza A virus subtype H5N1 solution is the laid of variable values (1) that satisfy constraints (2). In general, at that spot is an interplanetary space number of potential solutions that satisfy constraints, therefore to a greater extent than or less method is needed to winnow downwards the choices.

Based on a textual analysis of economical writing, it seems clear that "general equilibrium" (whatever it is), is a technique for choosing a solution. By itself, this is innocuous. For example, nosotros demand to calculate the solution to the organisation of equations of a stock-flow consistent model; if nosotros fix the equations correctly, at that spot volition live a unique solution. What is to halt us from labelling said solution the "general equilibrium"?  From an ideological perspective, that would live a large no-no (based on my reading of the literature). From a formal mathematical perspective, nosotros would demand a Definition of "general equilibrium" that stops us from making that characterisation.

Economic Model Transitivity Fallacy

Before returning to full general equilibrium, I desire to explicate (again) ane of my complaints nearly the economical literature. Whenever I read Dynamic Stochastic General Equilibrium (DSGE) articles, at that spot are normally logical jumps inwards proofs, where assertions are made without whatever justification. The mathematical logic beingness used seems to rely on "Economic Model Transitivity."

1. An "economic model" X has mathematical belongings A.
2. Model Y is an "economic model."
3. Therefore, Y has belongings A.

This land does non piece of work from a formal mathematical perspective, unless nosotros tin validate that model Y has precisely the same characteristics equally X, which allowed us to derive the result that Influenza A virus subtype H5N1 holds. The only way to live certain is to re-derive the proof that Influenza A virus subtype H5N1 holds for the novel model Y.

In other words, nosotros cannot only appeal to random theorems (or definitions) without citation; nosotros demand to explicitly listing the weather condition for the theorem (or definition), as well as and therefore validate that the organisation meets those conditions.

What is General Equilibrium?

DSGE macro has its roots inwards optimal command theory.  However, the optimal command theory mathematics has largely been obscured yesteryear economists next a publishing convention that gets farther as well as farther away from its mathematical roots. It is solely possible to read a few dozen DSGE mag articles, texts, or lecture notes, without finding a valid formal characterisation of how to observe the solution for the macro model of interest. (When I refer to a macro model, it includes both households as well as firms attempting to optimise their utility/profits respectively, also equally a authorities sector. This creates a optimisation construction that is completely dissimilar an optimisation work for ane sector alone.) 

The implicit supposition is that the conclusion of "general equilibrium" was covered elsewhere. Walras? Arrow as well as Debreu? Intermediate microeconomics? The obvious query to ask: did that ultimate Definition source solve the same macroeconomic organisation equally the electrical flow mag article, or was the modern writer relying on "Economic Model Transitivity"? That is, nosotros tin observe definitions of "general equilibrium" that piece of work for to a greater extent than or less models; the play tricks is to observe a Definition that matches macro DSGE models. At the fourth dimension of writing, I yet get got non constitute anything satisfactory, but I desire to underline that this is yet a inquiry inwards progress.*

Why Criticisms of Equilibrium create non Register

As a lastly note, I would advise that this province of affairs explains why heterodox complaints nearly the realism of equilibrium create non register with mainstream economists. In practice, the equilibrium supposition does non fifty-fifty larn properly defined inwards papers that allegedly depend upon the assumption. Given the depression marking of attending to the concept, worrying nearly its realism is moot.

Footnote:

* The closest I get got seen is inwards Section 7,3 ("Recursive competitive equilibrium") inwards "Recursive Economic Theory: yesteryear Lars Ljungqvist as well as Thomas J. Sargent. Although it appears quite formal, at that spot were a couplet of issues. One was a verbal formulation that was difficult to interpret into a declaration nearly sets. (This could live viewed equally a stylistic issue; inwards applied mathematics, it is normal to utilisation verbal shortcuts. However, it is unclear how to resolve the ambiguity.) The minute number is to a greater extent than serious, equally it does non accept into occupation organisation human relationship the differing objective functions of households as well as firms. I am inwards the procedure of reading the text, as well as I create non know whether this concern is addressed inwards a afterwards section.

(c) Brian Romanchuk 2017