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Wednesday, May 24, 2017

Does General Equilibrium Exist?

 One of the more entertaining parts of online (and academic) economic squabbling is the fights over the equilibrium concept used in mainstream economics. The reason for my amusement is that is is possible that the concept is not actually well-defined for macroeconomic models of interest. I must cautiously note that a proper definition might exist somewhere. However, I can say that it is possible to read a large portion of the modern mainstream literature and not find 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 by some research I am doing, some articles discussing equilibrium that I have recently read (Andrew Lainton, Peter Radford), and this question on equilibrium on the Economics Stack Exchange.

The questioner asks:
but is the model's equilibrium one that occurs naturally? Is real-life money supply equal to money demand? Is real-life investment equal to savings?
My question in short: why does the economy converge to these equilibriums?
This raises all kinds of questions. What is equilibrium? Is it a property of the real world? How is it possible to write a macro textbook and not answer these questions?

Seriously, It's a Set

Mainstream economics is highly mathematical. One of the perceived advantages of the mathematics is that argumentation is supposed to be clearer than literary economics. However, in order for mathematics to provide this advantage, concepts have to be formally defined. From the perspective of applied mathematics, this means defining concepts in terms of set theory (yes, there are weird corner cases in mathematical logic, such as self-referential sets, where we have to go beyond set theory).

Other than very highly stylised models, most macro models are specified by three things.
  1. A set of economic variables, which are elements in the set of "time series."
  2. "Constraints" on those variables: accounting identities, production functions, etc.
  3. A method to find a "solution" to the system. A solution is the set of variable values (1) that satisfy constraints (2). In general, there is an infinite number of potential solutions that satisfy constraints, so some method is needed to winnow down the choices.
Based on a textual analysis of economic writing, it seems clear that "general equilibrium" (whatever it is), is a technique for choosing a solution. By itself, this is innocuous. For example, we need to calculate the solution to the system of equations of a stock-flow consistent model; if we set up the equations correctly, there will be a unique solution. What is to stop us from labelling said solution the "general equilibrium"?  From an ideological perspective, that would be a big no-no (based on my reading of the literature). From a formal mathematical perspective, we would need a definition of "general equilibrium" that stops us from making that characterisation.

Economic Model Transitivity Fallacy

Before returning to general equilibrium, I want to explain (again) one of my complaints about the economic literature. Whenever I read Dynamic Stochastic General Equilibrium (DSGE) articles, there are commonly logical jumps in proofs, where assertions are made without any justification. The mathematical logic being used seems to rely on "Economic Model Transitivity."
  1. An "economic model" X has mathematical property A.
  2. Model Y is an "economic model."
  3. Therefore, Y has property A.
This obviously does not work from a formal mathematical perspective, unless we can validate that model Y has exactly the same characteristics as X, which allowed us to derive the result that A holds. The only way to be sure is to re-derive the proof that A holds for the new model Y.

In other words, we cannot just appeal to random theorems (or definitions) without citation; we need to explicitly list the conditions for the theorem (or definition), and then validate that the system meets those conditions.

What is General Equilibrium?

DSGE macro has its roots in optimal control theory.  However, the optimal control theory mathematics has largely been obscured by economists following a publishing convention that gets further and further away from its mathematical roots. It is entirely possible to read a few dozen DSGE journal articles, texts, or lecture notes, without finding a valid formal characterisation of how to find the solution for the macro model of interest. (When I refer to a macro model, it includes both households and firms attempting to optimise their utility/profits respectively, as well as a government sector. This creates a optimisation structure that is completely unlike an optimisation problem for one sector alone.) 

The implicit assumption is that the determination of "general equilibrium" was covered elsewhere. Walras? Arrow and Debreu? Intermediate microeconomics? The obvious question to ask: did that ultimate definition source solve the same macroeconomic system as the current journal article, or was the modern author relying on "Economic Model Transitivity"? That is, we can find definitions of "general equilibrium" that work for some models; the trick is to find a definition that matches macro DSGE models. At the time of writing, I still have not found anything satisfactory, but I want to underline that this is still a research in progress.*

Why Criticisms of Equilibrium do not Register

As a final note, I would suggest that this situation explains why heterodox complaints about the realism of equilibrium do not register among mainstream economists. In practice, the equilibrium assumption does not even get properly defined in papers that allegedly depend upon the assumption. Given the low level of attention to the concept, worrying about its realism is moot.

Footnote:

* The closest I have seen is in Section 7,3 ("Recursive competitive equilibrium") in "Recursive Economic Theory: by Lars Ljungqvist and Thomas J. Sargent. Although it appears quite formal, there were a couple of issues. One was a verbal formulation that was hard to translate into a statement about sets. (This could be viewed as a stylistic issue; in applied mathematics, it is normal to use verbal shortcuts. However, it is unclear how to resolve the ambiguity.) The second issue is more serious, as it does not take into account the differing objective functions of households and firms. I am in the process of reading the text, and I do not know whether this concern is addressed in a later section.

(c) Brian Romanchuk 2017

23 comments:

  1. Hmmm. I suspect the daily average use of milk is reasonably steady month-to-month. At one time, the cows produced seasonally so the supply side forced an annual variation in monthly averages. Cattle (for meat) have long had a expand-until-it-is-unprofitable negative feedback loop that is years in duration. I can't think of any industry that has been stable for more than a few years, and that stability at best a dynamic evolution of internal frothing.

    Maybe the communist system is more in equilibrium, such as Cuba.

    For my thinking, I don't see how you can have equilibrium when government can borrow from itself and not pay it back over a single lifetime.

    Rather than equilibrium, government borrowing creates an impulse event. A single shock to the economic system each time the borrowed money is spent.

    As you know, automobiles are powered by impulses. A single impulse, smoothed and repeated sequentially, can power a car smoothly. Repeated government borrowing, continued at increasing rates and spent daily, can power an economy.

    Ever expanding government debt is one kind of equilibrium. One method of powering an economy for many measuring periods. Is it equilibrium? Probably "yes" if we define "equilibrium" as the "steady state of government debt financed (choose your own phrase here)."

    Thanks for the thought-provoking post.

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    1. Roger,

      Your comments about impulse functions are thoughtful although I would like to point out that an internal combustion engine can have torque or speed regulation based on a negative feedback system despite being driven by impulses in each cylinder that are smoothed and balanced by the mass of the system. Oscillators are often designed to be similar to a swing set - energy is added in short periods of time - pushing the swing - and although the transient condition produces ringing and amplitude distortions - the steady state motion is characterized by what we call a stable sin wave.

      Financial markets generate positive feedback in the price of assets causing overshoot and collapse patterns. Government spending and taxes and monetary policy can introduce negative feedback of stabilizing efforts which are imperfectly effective.

      Brian - Using lay terms, if possible, what is the definition of equilibrium or a stationary system in stochastic models for optimum control systems? We know the default rate on loans is not a stationary value because it increases rapidly during a debt deflation. I imagine that is just one of many factors that would have a variable statistical mean value in a stochastic model of the economy.

      Joe

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    2. "Rather than equilibrium, government borrowing creates an impulse event."

      All borrowing creates an impulse event. There is nothing special about the government doing it - other than it is borrowing from itself.

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  2. "Is real-life money supply equal to money demand?"

    No.

    "Is real-life investment equal to savings?"

    Yes.

    "My question in short: why does the economy converge to these equilibriums?"

    It does not converge to the first. It confirms to the second because S=I is an identity constituted through our definition of savings and investment.

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    1. The identity S = I is derived from National Income and Product Accounts (NIPA) as follows:

      I = X - C (investment = national product X minus consumption C)
      S = Y - C (saving = national income Y minus consumption C)
      X = Y (total spending on national product = total income)
      S = I (saving equals spending for investment)

      When a government sector runs a net deficit the identity no longer holds and may be written in simplified form as follows:

      S = I + (G - T)

      where saving in the hybrid convention of NIPA with flow of funds accounting (FFA) now includes the amount of money or government bonds injected by the difference between government spending G and government tax revenue T.

      If one imagines the savings to be a fixed amount then it appears that government deficit spending "crowds out" non-government investment. However if one recognizes that financial instruments are generated and destroyed both in the non-government sectors and by finance activity of governments then it is clear that financial saving flows and stocks can depart significantly from levels of historical investment due to activities of government, foreign trade and investment flows, re-valuation of old assets with upward prices, and refinancing of old assets with new debt.

      The term "saving" does not appear to be limited to the NIPA definition especially when one investigates the generation and destruction of financial instruments for purposes other than "investment" as defined in the NIPA.

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    2. S = I refers to realized investment and savings and indeed, is true as a matter of accounting.

      But if the question refers to planned investment and savings then these are hardly ever equal.

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    3. @ Joe Leote

      A budget deficit or surplus should not cause disequilibrium between S and I as long as S and I are savings and investment for the economy as a whole, in contrast to private savings and investment.

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    4. @Joe Leote

      "S = I + (G - T)"

      I completely agree with your equation.

      One method for supporting the logic behind the equation is to apply it to the accounting for a locally issued gift certificate.

      Ownership of a gift certificate is identical to owning money so long as we are shopping in the issuing store. We can call the gift certificate "mercantile money".

      What happens if a merchant issues a gift certificate in exchange for electrical repair labor? New mercantile money has entered the economy.

      The merchant had better account for the claim he just issued on the goods in his store. (A gift certificate (or mercantile money) is a claim on goods, limited to the amount of value on the certificate.) The merchant should record "investment = value of the electrical repair" and, in a second entry for double accounting, "savings = value of the gift certificate issued".

      The savings value was created right out of the blue. Labor was performed and, at least for a while, was paid for by newly issued savings.

      Now who owns the savings? I think the merchant owns the savings until the gift certificate is finally satisfied. However, one could also make the argument that the electrical worker owns the savings. The worker is richer because he has the ability to claim goods at any time. The worker also has savings.

      So we see that both merchant and worker can claim to be richer because mercantile money was issued and electrical repairs were made.

      "S = I + (G - T)" allows us to see that savings are created by people working for government and receiving National Gift Certificates in payment.

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    5. Roger, Although I appreciate the logic of your conclusion it may not follow from the micro-accounting because, per my understanding, a repair can be capitalized to an asset account (perhaps qualifying as investment I in NIPA) or it can be recorded as an expense in the current year (which should correspond to spending for consumption C in NIPA). Since this is a thread about the existence of equilibrium perhaps it is not the proper context to extend such discussions?

      Regarding macro-economic stability, which is different from yet related to the concept of economic equilibrium, Hyman Minsky says the profit motive and financial system cause fluctuations in investments due to the volatility of expected profits linked to financial flows. A sufficiently large government could, in theory, enact policies where net spending (G-T) > 0 increases to avoid a significant loss of profits in the investment sector, which is a means to prevent debt deflation; and where a net surplus (G-T) < 0 is automatically caused when a credit fueled boom in private investment I would otherwise fuel significant inflation. In this model only a sufficiently large government (Big Govt in Minsky's lingo) can offset the instability introduced by the financing of assets and fluctuating profits in the investment sectors of an economy.

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    6. Yes, we might be wandering from the topic of equilibrium, but perhaps not. Is government debt funding investment or is it mostly funding savings?

      In modern times, government investment in infrastructure seems balanced with depreciation which would be counted as consumption. If that is the case, your equation would say that government debt funds mostly savings.

      In one sense, we could say that savings financed by government borrowing without corresponding investment is disequilibrium. On the other hand, if that is the way it is for year after year, then that IS equilibrium.

      Thanks for Brian for making us think.

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    7. You've definitely wandered from the topic of my article, which was the mainstream formal mathematical definition of equilibrium. 🙂 I am mainly focussed on that question, so I am not completely able to change gears to follow this conversation...

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  3. Chapter 6 has a comprehensive discussion of the different concepts deployed under the term "equilibrium" in economics:

    https://www.amazon.com/Reformation-Economics-Deconstruction-Reconstruction-Economic/dp/3319407562

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    1. Thanks for reminding me that the book is out...

      I believe it's safe to say that the definitions used have morphed over time. Does the book cover the formal defintions used in the post-1980 DSGE literature?

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    2. Yes. They really haven't morphed over time. All the mainstream models are based on the same notion of equilibrium. It's modified, i.e. with rigidities etc. But it's effectively the same.

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  4. Mainstream economics is built on a foundation of 3rd rate mathematics.

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  5. btw, this is well written:

    https://uneasymoney.com/2017/05/21/hayek-and-intertemporal-equilibrium/

    I'm not an expert in set theory, but I've seen enough of it to think it must be close to the big bang for the relevant conceptual foundation behind this sort of subject matter (macroeconomics).





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    1. Good article. The issue is that it is still approaching mathematical models as if there are tiny little people making decisions inside the model. We need to operationalise this as a formal statement about a set of solutions. What does "consistent with other agent's decisions" really mean? How can one agent change a decision without implying a change in the behaviour of its transaction counterparty? This is not a knock on Glasner; he's trying to decipher what's going on.

      It's safe to say that the majority of economists think that their intuition about equilibrium is sound, and that "somebody else" did the proper mathematical statement. "it's in the vast literature!" What was that about "mud moats"?

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    2. If an agent runs the credit department at a bank he or she decides unilaterally to offer credit on relaxed terms or offer credit on strict terms based on feedback from the balance sheets and expected future income of his or her customers. If all the credit departments feel fear they restrict credit and this changes the parameters of the DSGE or stock-flow model or whatever model one imposes to try to capture the effects of agent based decisions.

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  6. This link describes some basic principles of equilibrium as the concept applies in classical thermodynamics:

    http://web.mit.edu/16.unified/www/FALL/thermodynamics/notes/node11.html

    1. "The thermodynamic state of a system is defined by specifying values of a set of measurable properties sufficient to determine all other properties."

    2. "[P]roperties define a state only when a system is in equilibrium. If a process involves finite, unbalanced forces, the system can pass through non-equilibrium states, which we cannot treat."

    3. "An extremely useful idealization, however, is that only 'infinitesimal' unbalanced forces exist, so that the process can be viewed as taking place in a series of 'quasi-equilibrium' states."

    In economics we can cut down trees to burn (consumption) and/or to build a home (investment) where the investment produces future consumption (living in the house) until it is taken out of service (bulldozed) or depreciates in value (becomes unlivable via forces of natural decay). The amount of energy in a particular type of wood used as fuel does not depend on the price paid at any time. The value of living in a home has no monetary price until we invent credit and money customs in the legal and accounting systems. Therefore the price and savings system depends on the rules developed in law and accounting customs.

    Houses and other investment goods can be re-financed at higher prices via the credit markets. This activity should increase the float of savings without any increase in either current investments or future consumption flowing from past investments.

    Student loans are considered consumer loans and investments in education do not appear as human capital. The ability to repay loans is entirely dependent on the difference between income flows and cost of living in the future for families that incur debt today. The cash flow to service yesterday's debt is generated by the volume of debt rolled over and increased via today's credit deals. If the credit system does not generate the cash flows to validate yesterday's debts then either the government deficit spends to keep cash flowing or there is debt default which disturbs the price of assets and the investment incentive.

    The books are designed to balance whether one writes off a bad debt in bankruptcy or whether one collects the debt via actual cash flows. There does not appear to be a direct link between spending for investment and the accumulation of financial savings over time due to the distinction between real flows for consumption and investment and the independently determined financial arrangements under a system of law, accounting, and public policy making.

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    1. The physical definition of equilibrium was the original model for the economic concept, but the reality is that the analogy breaks. The question is what the mathematical definition used by the mainstream corresponds to, and if it is in fact mathematically coherent.

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    2. This 50 page paper has a bunch of math and claims to provide a proof that general equilibrium exists under certain conditions:

      General Equilibrium
      http://web.stanford.edu/~jdlevin/Econ%20202/General%20Equilibrium.pdf

      This 37 page paper is a criticism of general equilibrium with very little math:

      http://www.umass.edu/preferen/Class%20Material/Readings%20in%20Market%20Dynamics/Kirman%20HOPE%202006.pdf

      In this 8 page paper Perry Mehrling describes the mechanisms used by banks and other financial market makers which, when functioning properly, appear to validate more abstract theories of markets:

      http://www.perrymehrling.com/wp-content/uploads/2015/05/Lec-02-The-Natural-Hierarchy-of-Money.pdf

      He explains why interest rates can't adjust to generate equilibrium in financial markets under financial stress and this is a flaw in the more abstract IS/LM model.

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    3. The paper from Stanford that shows that equilibrium exists does so for "an exchange economy." As I noted, it seems possible to define GE for some models. The definition does not transfer to the models of interest, in which there are households and firms.

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  7. Thanks for the link to the Perry-Mehrling paper. An excellent description of the hierarchy of money.

    I should point out what might be an error to be noticed by a discerning reader. In part II, page 4, I think the author meant to say "Macroeconomic variables like interest rates and GDP are affected not by the outstanding gross quantity of inside credit, and (not "and", should be " but ) also by who is issuing it, who is holding it, and where that credit lies in the larger money-credit hierarchy"

    Of course, that edit is more than word-smithing, but with that change, the paper becomes (in my view) mechanically correct. Mechanical correctness, with full meshing of components, is important to me.

    In Part III, the author begins talking about the dynamic nature of the system.



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