Tuesday, December 1, 2015

Money is that which is conserved via a symmetry principle

Not sure where I got the link to this (so H/T to someone on the internet), but it was an entertaining jaunt through questions about "money" with supposedly more to come. I have my own theory about the origin of money that isn't really inconsistent with what Steve Roth (the author) says. He does say something that I'll take issue with, but in a constructive way I hope ...
Now to be fair: a definition of money will never be simple and straightforward. Physicists’ definition of “energy” certainly isn’t. But physicists don’t completely talk past each other when they use the word and its associated concepts.

Energy actually has a really beautiful and simple definition: that which is conserved due to the time-translation symmetry of the universe. Because the laws of physics don't change if you go backwards or forwards a billion years (time translation), there is something that is conserved by Noether's theorem. We call this energy, but once we have time and time-translation symmetry, we have this conjugate quantity and we should call it something. Energy is a fine name. The equivalent conserved quantity for the space translation invariance of the universe is momentum, again ... something that took us awhile to figure out was conserved.

Now if we just knew what time was ...

Anyway, to make this a constructive criticism let me posit that money is a unit of measure [and quantity] of that which is conserved because of the homogeneity of degree zero of the economic universe. Homogeneity of degree zero is a bit mathy, but it's a simple concept. If I double the demand for sheep and the supply of sheep, then the price of sheep will stay the same (is conserved) ... and prices are measured in money. That's how the tally marks in Roth's piece can be "money". If you doubled the amount of sheep owed and doubled the amount of tally marks, you haven't done anything accounting-wise to the price of a sheep.

But that's really it -- anything that becomes an intermediary between supply and demand can be money as long as it "works". What do we mean by "works"? Well, if changes in supply and changes in demand still carry the same amount of information with the intermediary in place (they are in information equilibrium), then we can say that intermediary "works" and therefore is money. In general, information equilibrium relationships are the bare minimum required to maintain this capability.

The tally marks and sheep above represent two things that are information equilibrium. If I change the tally marks, it is translated into a change in sheep. If I was to change the number of sheep in some sort of pattern (+1 sheep, -1 sheep, -1 sheep, -1 sheep, + 1 sheep) and I get the exact same pattern of change in the tally marks (+1 tally, -1 tally, -1 tally, -1 tally, + 1 tally), then I could in principle send a message (in Morse code, say). And if I can send a message, information must be flowing through a communication channel. We say the tally marks and sheep are in information equilibrium. I should be able to communicate a signal of changes in supply through the price mechanism, otherwise information is being lost along the way -- basic information theory.

But the real bonus is that something that can be used for a lot of products works the best to communicate that information without loss, especially if it isn't a product itself (doesn't have intrinsic value) and is completely fungible (high entropy). That's where my definition of money comes in:

Money is a thing that mediates transactions and has high information entropy

Tally marks fit this bill. So does our modern currency. But essentially, what money is doing is helping maintain homogeneity of degree zero, so that prices don't change if you double supply and double demand. And an information equilibrium relationship (linked above) is the most general form of the simplest condition required for this to be true. 


Update 12/6/2015

I thought I should express mathematically (and therefore more precisely) what the previous post means.

Homogeneity of degree zero (aka invariance under scale transformations or conformal symmetry) in the supply (S) and demand (D) defines the relationship (to leading order) 

P = dD/dS = k D/S

where P is the (abstract) price of whatever S is. This is the fundamental information equilibrium relationship and is denoted D ⇄ S. This is invariant under the transformation:

D → α D
S → α S


d(αD)/d(αS) = dD/dS
k (αD)/(αS) = k D/S

So P  → P under the symmetry transformation. That means P is a measure of whatever is conserved under the symmetry principle (per Noether's theorem) -- it is invariant under the transformation.

As an aside, if D = D(t), P = P(t) and S = S(t), the the symmetry transformation applies at all times simultaneously D(t) → α D(t), S(t) → α S(t); P(t)  → P(t) is invariant. This is really the idea that if you add a couple zeros to every price for all times, you haven't really done anything.

P is the (abstract) price in the information equilibrium model of economics, and it is related to money through the unit of account. But we can go further. Let's introduce something called M (the medium of exchange) and because of the chain rule (and multiplying by 1 = M/M)

dD/dS = k D/S
(dD/dM) (dM/dS) = k (M/M)(D/S)
(dD/dM) (dM/dS) = k (D/M)(M/S)

If D ⇄ M ⇄ S (D is in information equilibrium with M and M with S so that information gets from D to S via M), then we have a relationship

P'' = dD/dM = k'' D/M

And so we can re-write the equation above

(P/P'') P'' = (dD/dM) (dM/dS) = (k/k'') k'' (D/M)(M/S)
P' = dM/dS = k' M/S

Therefore we can capture the relationship P : D ⇄ S entirely with P' : M ⇄ S. Now dM/dS is the exchange rate for an infinitesimal quantity of money for an infinitesimal quantity of supply ...i.e. dM/dS = P' is the money price of a unit of S (as opposed to the abstract price P).

This equation is just another information equilibrium relationship and is invariant under the conformal symmetry transformation 

M → α M
S → α S

which leaves the money price P' invariant.

You can actually do the trick above to come out with P'' : D ⇄ M as the relationship that captures P : D ⇄ S entirely. Let's relax this to an information transfer relationship (non-ideal information transfer) P'' : D  M. Let's say D is an aggregate demand of several goods D₁, D₂, D₃, ... Dn. The budget constraint means you can't spend more than M (the total amount of medium of exchange) at a given time, so D₁ + D₂ + D₃ + ... + Dn ≤ M. Since this is a high dimensional system (n >> 1), the most likely (maximum entropy) point saturates the budget constraint D₁ + D₂ + D₃ + ... + Dn ≈ M. Therefore at any given time most medium of exchange will be allocated against all the investments, goods and services, etc ... and therefore the distribution of medium of exchange will be approximately equal to the distribution of goods and services. This means that medium of exchange minimizes the difference between the information content of D and M, i.e. minimizes I(D) − I(M), or more usefully

I(D) = β I(M)

for some β where 0 < β ≤ 1. Therefore (returning to the definition of the information equilibrium relationship)

(D/dD) log σd = β (M/dM) log σm

D/dD = (1/k''') M/dM 

dD/dM = k''' D/M

where k''' = (log σd)/(β log σm). I.e. the existence of a high entropy medium of exchange M allows us to write I(D) ≥ I(M) (where the conformal symmetry doesn't hold) as an effective information equilibrium relationship I(D) = I(M) (where the conformal symmetry does hold).

So to be more precise, the money as unit of account is that which is preserved by a symmetry principle (scale invariance or conformal symmetry). Money as medium of exchange makes the symmetry principle hold.


  1. "Energy actually has a really beautiful and simple definition "

    Well now "very simple" is in the eye of the beholder... This very simple definition requires some extraordinarily deep understandings before it seems "simple"... I'm working on that...

    You're even worse than me: I want to promulgate a new definition of money (simultaneously simpler and far more abstruse); you want to promulgate a new definition of energy _so that_ you can promulgate a new definition of money. Here's to our mutual ambitions... ;-)

    Oh and: one paper you linked to elsewhere, Kocherlakota's "Money is Memory" is a great Aha for me. So completely in keeping with my thinking above: "'Money' allowed humans to outsource some of that arduous mental recording onto tally sheets."

    1. Ha!

      Although I can't claim that is a new definition of energy since that is how it's taught in undergrad physics: symmetry <=> conservation law.

      Another interesting paper (for me at least, because it scooped me by 50 years) is Gary Becker's paper on irrational agents:


      Also as an aside, I took a shot at rewriting the principles of economics in Krugman and Wells in terms of information equilibirum:


  2. Here's something I'm struggling with in understanding your writeup above, would love to hear your thoughts:

    Supply and Demand aren't numbers. They're behavioral constructs and concepts: How much are people willing to buy and sell (and correlated, consume and produce) across a range of prices? They're only expressed via the whole S or D curve. When people say "demand for oil was $x in this period," I want to tear my hair out.

    This reality explains why S/D diagrams are always dimensionless. You can't look at the national accounts and point to any measure that tells you what S or D was, numerically, for anything.

    Also, the notions of flow supply vs stock supply, which is almost completely untheorized. (With the exception, to my knowledge, of Clower: http://www.jstor.org/pss/1907357.)

    More on this here:



    Any thinking help greatly appreciated on this.

    "Money is a thing that mediates transactions and has high information entropy"

    I think this may still be caught in the conceptual trap I describe: trying to simultaneously talk about the abstract entity and its embodiments.

    Closely related, I think: Am reading Cesar Hidalgo's Why Information Grows: The Evolution of Order, from Atoms to Economies. He speaks of information as "something that is not a thing." I think money is similar, and trying to speak about both it and "money things" simultaneously may be the root of much confusion. Love to hear your thoughts.

    1. Steve, I'm sure Jason will have a great answer for you, but, assuming you're not already familiar with it, you might try digesting the first 8 pages or so of his paper here that lays out the basics of the information transfer framework and explains single-good supply & demand curves in that framework:
      Beyond that intro there's AD and AS curves and a host of other subjects tackled in there. Also, if you're not familiar, check the right hand column for links to some other posts explaining the basics.

    2. ... by that last bit I meant there are links in the right hand column of this blog.

    3. Hi Steve,

      Thanks for reading! I broken up my responses to your comments into a couple of pieces

      (And thanks Tom)

      1. On supply versus quantity supplied

      The way I've constructed it, supply and demand are technically just information -- Hidalgo's "something that is not a thing" (I'm also reading his book, originally out of fear that I might have been scooped). One would set up a relationship where "supply equals demand" ... but in technical language you'd want to say:

      the information revealed by specifying a supplied widget from the distribution of supplied widgets (S) is equal to the information revealed by specifying a demanded widget from the distribution of demanded widgets (D)

      i.e. I(D) = I(S)

      It is somewhat of a shorthand to say "supply and demand are in information equilbrium" and there is a single distribution of widgets supplied (and widgets demanded) that makes this true. But this relationship also defines counterfactual dependencies of the price associated with the supply distribution on changes in the demand distribution or vice versa ... aka supply and demand diagrams.

      What is interesting about this framework is that you can have I(D) > I(S) ... e.g. some demands aren't met and there is information loss. But information equilibrium is a good assumption surprisingly often.

      The price is more like a flow meter measuring the information flowing between these distributions that keeps them in information equilibrium. When the distribution of supplied widgets are "out of whack" relative to the demand for them (in 'partial equilbrium' in econ terms), the price deviates from its usual'general equilibrium' (in econ terms). For example, if all the apples were in one store in Chicago, the aggregate price for apples would be huge because the distribution of demand for apples is much wider.

      I know that I have tended to gloss over the distinction between the 'supply schedule' ("the supply") and the 'quantity supplied'. Equivalently, I have glossed over the technical details in talking about "information" sometimes where being technically accurate doesn't aid clarity.

      However using uniform distributions for information sources as an initial approximation tends to make many of those distinctions unnecessary.

    4. 2. On money versus money things

      What I've found in messing around with this stuff is that what "money" is kind of depends on what you are trying to describe. The monetary base works well to describe short term interest rates, but not inflation. The monetary base without reserves does well with core inflation. MZM works well for long term interest rates. M2 does well with exchange rates (but mostly because it closely follows NGDP).


      As an aside, the information model looks like a quantity theory of money, but only in a high inflation limit. But that's really because the quantity theory for high inflation is basically a restatement of homogeneity of degree zero (the economic symmetry).

      But I'd say those are "money things" rather than abstract "money". As an analogy, energy has different forms (heat, kinetic, potential, etc), but e.g. the kinetic energy of a scuba tank doesn't change the thermal energy of the gas inside it.

      Abstract money is just something that comes between supply (S) and demand (D) -- written as an information equilibrium relationship:

      D ⇄ M ⇄ S

      i.e. I(D) = I(M) = I(S) ... a thing that has high information entropy: it's all the same, it's not really located anywhere (or is everywhere ... and everywhen). If it has low information entropy, then you have (a non-ideal information transfer relationship)

      D → M → S

      or I(D) > I(M) > I(S).

    5. 3. Stocks versus flows

      I completely agree that it's necessary to understand the distinction. A stock is a quantity; a flow is a quantity with a time scale. I've recently noticed that economics doesn't really pay attention to theory scales ...


      I will have a look at your links!

  3. Oh, I've got my reading (and thinking) cut out for me...

    1. Feel free to ask questions anytime -- none of the comment sections on this blog are closed, so if you come to a post that is confused add a comment and I'll get back you you.

  4. The most important point about money is that it is a man-made accounting construct. Nothing more. That’s true whether money is measured in tally sticks or Euros, and it’s true whether money is embodied in a piece of paper, a piece of metal or as 1s and 0s on a computer disk. It should therefore be defined in accounting terms i.e. in relation to the conservation laws which apply.

    As with other accounting constructs, we can think of the money we HAVE (an inventory of money at a point in time) and the money we CREATE / USE / DESTROY (in discrete events). Over a period of time, we have an opening inventory balance, followed by a number of events, resulting in a closing inventory balance.

    This is comparable with a science experiment. We might start with opening inventories of hydrogen and oxygen. We then construct an event which causes them to combine. We end with a closing inventory of water. No-one would take a chemist or a physicist seriously if they didn’t take accounting (and its constituent conservation laws) seriously. Accounting is a fundamental logical thinking discipline. It is not an accident that the word ‘accounting’ is just the word ‘counting’ with an added prefix. Physics and chemistry without accounting would be no different from magic.

    Accounting and conservation made sense in chemistry only once the basic entities that are conserved under specific changes were understood e.g. atoms, protons. Prior to this stage, chemistry, or rather alchemy, was concerned with phlogiston and other made-up substances with no clear conservation rules. This is the stage at which economics finds itself today. Economists who view accounting as something irrelevant to economics are like chemists who think that atoms and conservation laws are irrelevant to macro phenomena like explosions. Scientists use mathematics on top of a well-defined and well-observed accounting framework. Economists use mathematics instead of an accounting framework. This is why physicists build machines the size of a small European country to observe the Higgs boson while economists just wave their hands.

    Jason: “If I double the demand for sheep and the supply of sheep, then the price of sheep will stay the same (is conserved) ... and prices are measured in money”

    Price is not conserved under most conditions. See stock markets, supermarkets and just about any other markets. Price is not the same as money. Price is just a ratio of money per unit of goods. Goods and money are the more fundamental concepts.

    If I buy two sheep from you for €10 then both the sheep and the €10 are conserved.

    If I then sell the two sheep to a third party for €15 then both the sheep and the €15 are conserved.

    Nevertheless, I have made a profit of €5.

    You gained €10 from the first transaction and I lost the same €10. I gained €15 from the second transaction and the third party lost the same €15. Overall, you are +€10, I am +€5 and the third party is -€15. The pluses and minuses balance because money is conserved under exchange. Price is not conserved, and one consequence of the non-conservation of price is that I can make a profit. Profit arises BECAUSE price is not conserved.

    Of course, if money is conserved under exchange, we need separate processes to create and destroy money in order that money can exist and in order that the money supply can change. From the perspective of a non-physicist, the accounting for the money creation process is a bit like matter and anti-matter emerging from nothing in physics i.e. money (as asset) is always offset by an equal and opposite money (as liability).

    1. Hi Jamie,

      I think you've confused a symmetry transformation that is a property of equilibrium at an instant in time (doubling the supply of sheep and doubling the demand for sheep leaving the price unchanged) with a non-equilibrium process (selling at a higher price in short order means that the sheep market is not in equilibrium).

      If I created a second identical Earth, the price of everything on each Earth would be the same as the price of everything on the single Earth even if their markets were coupled together.

      Another way, price is an intensive property (this is well established in economics):


      Also the time symmetry of physics (and related energy conservation) does not mean every moment is the same (quite clearly, they are not). Energy changes forms from chemical to heat or kinetic to potential for example. You have energy entering the Earth system via the sun. But we don't say energy isn't conserved by the laws of physics because of this.

      In a similar way, the birth of new people adds new aggregate demand to the economic system. This does not mean economics looses its scale invariance (conformal symmetry or homogeneity of degree zero).

      Accounting does have this same symmetry and therefore will have a conserved quantity associated with it. If I double assets, double liabilities and double the bottom line, the accounting still balances (if I double all my debits and all my credits and my bank balance, it still all adds up). What I am saying is that the thing that is conserved is deeply connected with the unit of account.

      Of course, all of this only applies to systems that are in information equilibrium since the conformal symmetry (scale invariance) isn't true of the non-ideal equations where I(A) > I(B).

  5. Jamie, thanks for this. Very helpful to me. I haven't found time yet to ingest (much less metabolize) Jason's thinking, but because I've thought long and hard about accounting, I am able to get my conceptual teeth into this.

    "money we HAVE (an inventory of money at a point in time) and the money we CREATE / USE / DESTROY (in discrete events). Over a period of time, we have an opening inventory balance, followed by a number of events, resulting in a closing inventory balance. "

    Right. Modern accounting uses three interrelated constructs to represent this: balance sheets (HAVE), income statements, and flow-of-funds matrices. cf the Fed's Z.1, "Financial Accounts of the United States." More on this, and how those constructs are used in practice, below.

    Next: correct. The "money stock," however defined and measured, has an at-best iffy correlation to prices/inflation. "Double the number of sheep, and the number of dollars, and price is unchanged" doesn't play out very well in the lab.

    I'll ask you to provisionally adopt, for this paragraph, my def of money (short and imprecise form, "the [net] value of assets"). Curious cat that I am, I recently looked at correlations between household net worth and ∆NW, versus inflation indexes (levels and ∆s), with various lag times (positive and negative). Short story, all the correlations were below .20. Now sure, physical supply was changing along with the money "supply" (I hate that usage), so take this for what it's worth...

    (HH NW, btw, is at least a reasonable representation of private-sector NW, because households ultimately own all firms, at zero or more removes. Firms don't own households [yet]. I'll use HH here as convenient shorthand for private-sector.)

    (I remember seeing one rather ax-grinding monetarist paper years back suggesting that across the world, over very long periods, the M-to-price correlation actually approaches 1. But I don't remember much about it. I should look it up. But maybe you know this literature well?)

    Now: conservation. Looking at the whole economy, money is clearly created and destroyed, ab nihilo. When the sovereign currency issuer deficit-spends, that money goes onto the lefthand, asset side of HH balance sheets, with no change to the righthand side (RHS), and no necessary offsetting liability on the gov balance sheet. (Bond issuance is a separate issue, just a requirement gov imposes on itself, arguably just swapping one form/embodiment of money for another.) Voila: that discrete accounting event of def spending creates new assets, NW, "money."

    Private lending also puts newly created assets/money on HH balance sheets (LHS), though unlike gov def spending, it simultaneously creates matching RHS liabilities, for zero HH NW change.

    I tend to think of Treasury as a hole in the ground, which money comes out of (def spending) and goes back into (surplus). As Milton Friedman said, governments have both printing presses and furnaces. Basically the same mechanism for private lenders, but complicated some by the creation and destruction of associated liabilities: lenders' balance-sheet expansion and contraction.

    This is all straightforward MMT. But here's where I part with that school: existing-asset markets also create money, and in far greater amounts than gov def spending or private lending.

    Continued next...

    1. Quick one:

      You said:

      Next: correct. The "money stock," however defined and measured, has an at-best iffy correlation to prices/inflation. "Double the number of sheep, and the number of dollars, and price is unchanged" doesn't play out very well in the lab.

      I agree!

      But like how a thermodynamic macrostate breaks the underlying time symmetry of physics (the arrow of time), which allows useful energy to dissipate into heat, I think the conformal symmetry related to money is broken by the macroeconomy.

      In this ensemble of markets:


      The individual markets all obey the conformal symmetry (homogeneity of degree zero), but the macro state does not. The macrostate manifests as an apparent changing information transfer index ... and economic growth that is related to the second law of thermodynamics and the arrow of time:


      [I didn't state this stuff in terms of the symmetry principle in the preprint paper since I was trying to avoid being too weird to get published ....]

    2. "I was trying to avoid being too weird to get published" Lol... yes, that may have earned you a place on the "crackpot watch" website... which makes me wonder... I wonder if any of the crackpots identified there actually don't deserve to be there?

      Steve & Jamie, I would recommend the following link in the right hand column:

      "General information transfer model (Fielitz and Borchardt)"

      It's under the heading "Information transfer economics for beginners." There's some good examples in there of using the principle of information equilibrium for physics problems (like to find an expression for gravitational force and to find the ideal gas law). There are several versions of the paper with different examples. The math doesn't go beyond basic calculus and probability theory. I found it helpful background knowledge while digesting Jason's paper.

  6. Part II:

    Think of the money stock as the net value of all our assets (all our claims, at zero or more removes, on real goods). HH NW is a decent measure of that. For the US, about $80 trillion right now. That's the residual of assets minus liabilities -- what on a S-corp or C-corp's balance sheet would be called shareholder equity (a specially distinct "liability"), but with no shareholders; households own themselves, and their residual, net worth.

    That money stock is like a balloon. Problematically simplified, and confuted with real resources: it expands when production pours in, contracts with consumption (poof! goods are gone).

    But the balloon also expands when existing-asset markets go up. Animal spirits. When the stock or RE market goes up, there's more money. Magic. I'll outsource my explanation to myself:


    Short version: the existing asset markets are recognizing that the goods from previous periods' production are worth more than they thought when those goods were sold/bought, and delivering that income from production onto HH balance sheets via the mechanism of cap gains. Key heterodoxy here: cap gains (should) count as income from production.

    But what I really haven't touched on, which is the core of your comment: the paradox of monetary profits (in the this-period market for newly-produced goods). Where did that extra five pounds come from?

    I'll outsource that to Steve Keen (pdf):


    I haven't fully metabolized this, treat it rather as a black box. Money is created in the process of production and sale. More money comes out of that box than goes in. "Monetizing" the real input-to-output surplus from production.

    So I truly don't know how this relates to the notion of conservation. Would love to hear thoughts on that as I work my way into Jason't thinking.

    Just one more accounting discussion that I think is important here.

    Take a look at two things in the Z.1:


    • Table S.3.a, circa page 146: "Households and Nonprofit Institutions Serving Households." (They're treated together because neither has shareholders; they "own" themselves.)

    • The Flow of Funds Matrix on pages 1 and 2.

    The thing to notice: S.3.a's Revaluation and Other Changes in Volume accounts (roughly: cap gains) are absent from, exterior and invisible to, the FoF matrix.

    That closed-loop accounting matrix basically represents the National Income and Product Accounts (NIPA) construct going back to Kuznets' creation of those accounts in the '30s. Those accounts didn't and don't have balance sheets; they simply didn't have the measurement/estimation wherewithal to do that.

    So the FoF matrix basically represents this-period flow accounting. It imparts this period's (monetized) surplus from production, which expands the money balloon. (See Keen, above.)

    But it doesn't and can't resolve to net worth, because net worth is obviously affected by revaluation -- cap gains (in my thinking, newly "realized" income from previous periods' production).

    My point being: arguably the greatest engine of money creation and destruction -- the market for existing assets -- is invisible to the accounting construct that most people (including most economists) use to think about the economy: the NIPA/FoF construct.

    I'll leave that there. I hope this is useful to others, would love to hear feedback.

  7. This comment has been removed by the author.

  8. Jamie & Steve,

    I didn't see where Jason left a link to his latest post which addresses some of your questions. Here it is, in case you missed it.

  9. Jason, Steve, Tom,

    Thanks for your replies. I have read them and want to comment further as this is an interesting area. I am particularly interested in Steve's comments. I will give a fuller reply tomorrow (UK time).

  10. re: Tom's (removed) comment:

    "there appears to me to be a mismatch in focus between the way you're looking at it and the way he's looking at it."

    That is manifestly so. Vive la difference.

    Unfortunately I am some ways from being able to say anything useful about Jason's paradigm. Will work on it. (Pointing to more articles, btw, has limited value for me; the reading list is already very long -- will get to it as time and inclinations allow...)

    Meanwhile, wondering if anyone sees interesting ways that Jason's paradigm informs/challenges/whatevers the "accounting view" within which Tom and I seem to be more comfortable and competent.

    1. "Pointing to more articles, btw, has limited value for me; the reading list is already very long -- will get to it as time and inclinations allow"

      Lol... I know how you feel, especially at first. Jason is a fountainhead of new posts. Just so you know, he has yet another one up, this time looking at Steve Keen's work (on your recommendation).

    2. Just to say my link was about the profit mystery. The graph Jason discusses was just a stylized illustration to discuss that mystery.

      Still would love to hear discussion here:

      Where did that extra five pounds come from?

    3. One thing about Jason: he's one of the most patient and most willing-to-answer-questions macro bloggers out there. Like he wrote above: on any of his posts... going back to the very beginning. Don't be shy! (you'll see that Jason answered my questions this year -- two years after he originally wrote some of the posts!)

  11. Oops I meant Jamie and I. Jason feel free to edit that if you wish.

  12. Let's restate the core of the question of "where the €5 came from"; Jamie said:

    If I buy two sheep from you for €10 then both the sheep and the €10 are conserved.

    If I then sell the two sheep to a third party for €15 then both the sheep and the €15 are conserved.

    Nevertheless, I have made a profit of €5.

    ... Price is not conserved, and one consequence of the non-conservation of price is that I can make a profit. Profit arises BECAUSE price is not conserved.

    There are two possible scenarios here in terms of the information framework, but in a sense in both cases they come from something that does not exist at the micro level and can't be captured by accounting: entropy.

    If I add up all the energies of each atom in a gas, I don't actually get all the sources of potential energy (forces are potential energy gradients, or flows if you will). There is a term in the thermodynamic potential T*S (temperature times entropy). TS = 0 for each atom individually, and only collectively do they have a nonzero TS term. Entropy and information are intimately connected, so in the information framework, ecomomic output is all of the goods and services -- plus a bit from "economic entropy". That's where the €5 comes from and it can never be captured by accounting at the micro level.

    Here are the two specific mechanisms ...


    You are not in an (information) equilibrium (and markets aren't functioning well). Information equilibrium is the maximum entropy state, but we're not there. We have:

    I(D) > I(S)

    where D is demand for sheep and S is supply of sheep. This is non-ideal information transfer and information is lost in the market. In this case

    P1 < P2 < P*

    where P1 is the observed price €10 and, P2 is the observed price €15, and P* is the ideal price of say €40.

    In this case, there are sufficient sheep and demand (and Euros) to sustain this market at a price of €40, but something has gone wrong. I attribute this to "coordination", but bad coordination. Too many people are coordinating their plans to sell sheep (a market panic). In this case, any price P < P* can be realized (they're all part of the available state space). It takes less information to specify a market state where everyone is panicking than one where everyone's actions are unpredicatble.

    In this case, the symmetry/conservation law does not hold because I(D) > I(S); the symmetry/conservation law is a property of I(D) = I(S).

    So in this case, the €5 profit comes from pushing the market from I(D) > I(S) towards I(D) = I(S). When a market is ideal, all goods and services traded in it are worth more.

    Continued with II ...

    1. II.

      We are in information equilibrium:

      I(D) = I(S)

      In this case the €15 and the €5 profit represent economic growth and/or inflation.

      The general equilibrium solution to the information equilibrium condition

      P = dD/dS = k D/S


      log D ~ k log S

      log P ~ (k - 1) log S

      The k is called the information transfer index and represents a bit of that entropy piece. If k = 1, then there is no profit because

      log P ~ (k - 1) log S = 0

      So P ~ 1 (i.e. is constant). You always buy and sell sheep for €10. The existence of profit (in information equilibrium) depends on k > 1. For k > 1, the price of sheep increases with the supply because

      P ~ (k-1) (S/S0)ᵏ⁻¹

      so later transactions have higher prices. In the case that Jamie describes, if the total number of sheep when the first transaction happens is S0 = 10, then:

      10 € = (10 €) (k-1) (S/10)ᵏ⁻¹ if S = 10

      15 € = (10 €) (k-1) (S/10)ᵏ⁻¹ if S = 15 and k = 2

      15 € = (10 €) (k-1) (S/10)ᵏ⁻¹ if S = 90 and k = 1.5

      What has happened is that the extra information in the probability distribution of 15 sheep (or 90 sheep) relative to 10 sheep means there is more information in determining the allocation of a single sheep -- more information = higher price.

      Aside: there are 176 different allocations of 15 sheep versus 42 allocations of 10 sheep. Assuming a uniform distribution over the allocations, we need 7.5 bits to specify an allocation of 15 sheep versus 5.4 bits for 10 sheep.

      In this case (information equilibrium) the symmetry principle holds because:

      D → 2 D
      S → 2 S, then

      P = d(2D)/d(2S) = k (2D)/(2S)

      P = dD/dS = k D/S

      so the price is the same if you double demand and supply, Which means the measure of the information in the different allocations (the unit of account) has the same value.

      Note you have to be careful when doing this with the equations above ... since really:

      P = (k-1) (S/S0)ᵏ⁻¹

      and taking S → 2 S means S0 → 2 S0 so that

      P = (k-1) (2S/2S0)ᵏ⁻¹ = (k-1) (S/S0)ᵏ⁻¹

    2. Jason, you write:

      "10 € = (10 €) (k-1) (S/10)ᵏ⁻¹ if S = 10"

      k = 2 there though, else it doesn't work, right? If we only know that k > 1, then it reduces to

      10 € = (10 €) (k-1)

      so k must be 2.

    3. Very interesting comment BTW! It makes it more concrete.

    4. Jason, could you briefly tell us how you calculate 42 allocations with 10 sheep?

    5. Yes, k must be 2 in that case unless you change the coefficient. I forgot the (k-1) piece in the initial writing and went back and put it in ...

      42 is the number of integer partitions of 10

      10 = 10
      10 = 1 + 9
      10 = 2 + 8, 1 + 1 + 8



  13. Tom: "there appears to me to be a mismatch in focus between the way you're looking at it and the way he's looking at it."

    Steve: “That is manifestly so. Vive la difference”

    Apologies in advance for the length of these comments but I think they are important in identifying what is wrong with almost all economic discussions.

    I agree completely with Steve’s point in the above quote.

    My background is in working in business and government, helping to solve operational problems. One of the most powerful insights into any human system, even one as small as a large business or government department, is that there are many different perspectives on the system, and that useful insights often come from unexpected places.

    For example, the personnel manager in a business will view the business through a lens of its people and organisational structure. The finance manager will view it through a lens of its costs and revenues. The CEO will have a broad but often shallow data-driven view. The shop floor worker will have a narrow but deep physical view of one part of the factory. The factory manager will think of the factory in great detail. The head office planner will think of the factory in terms of a couple of equations in a mathematical model. A customer will think of the business in terms of its products.

    Which of these people has the “correct” way of understanding and analysing the business? Even to ask this question shows that it is an absurd proposition. The reality is that each person has a valid perspective and that businesses succeed by blending many perspectives rather than relying on any one of them.

    This is also true at a macro level. If an alien visited earth and studied our societies, he would observe that most of our successful institutions blend many perspectives, and that these institutions are better than any alternatives which rely on a single perspective.

    Markets succeed best when suppliers are forced to compete for the attention of customers. Apple may have more specialist knowledge of the internal workings of tablet computers than anyone else. However, it doesn’t get to decide which tablet computers we buy. Everyone else in society is allowed to pass judgement on Apple’s computers. Even economists are allowed to buy tablet computers and pass judgement on Apple despite the fact that most of them may know little or nothing about computer design.

    Governments succeed best when politicians are forced to compete for the attention of voters. Barack Obama doesn’t get to pronounce himself president. Everyone else in society is allowed to pass judgement. Even physicists are allowed to vote despite the fact that most of them may know little or nothing about almost everything from foreign policy to agriculture subsidies.

    Lawyers are not allowed to pass judgement on their own arguments. Juries pass judgement on the arguments. Even people like me are allowed to sit on juries despite the fact that most of us may know little or nothing about the law.

    I would argue that the use of multiple perspectives to understand complex human systems and problems is the single most important underlying principle of social design.

    When Jason says of me

    “I think you've confused a symmetry transformation that is a property of equilibrium at an instant in time … with a non-equilibrium process”

    all he is saying is that my perspective of the economy is different from his perspective. I might equally say that he has confused a non-equilibrium process with a symmetry transformation (if I knew what a symmetry transformation was).

  14. Steve: “Where did that extra five pounds come from?”

    There was no “extra” five pounds. It is an illusion. Money was conserved under exchange. No money was created and no money was destroyed.

    When stock market prices go up, many people, including famous economists and Noah Smith, say that “money poured into the stock market”. However, that is an illusion as money is conserved under exchange. If one person buys a share for €1 then another must sell the share for €1. This is true irrespective of whether prices are going up or going down, or how many people are involved.

    As Paul Krugman says “one man’s spending is another man’s income”. In the stock market, “one man’s purchase is another man’s sale”. These statements are about the conservation of money under exchange.

    If you observe the stock market in any detail, you will see lots of bid prices and lots of ask prices. An ask price is just a price at which one particular seller is willing to sell his shares. A bid price is just a price at which one particular buyer is willing to buy shares. Different people may have wildly different bid prices and ask prices at any one moment based on different valuations of the underlying asset. The thing that is called the “stock price” is just the last price at which one bid price and one ask price were matched. Nothing more.

    Most people think of price as something concrete which is just accepted by all market participants. However, multiple bid prices and ask prices show that this is not true.

    The same model is true in prediction markets. Different people have very different views of the probability of an event occurring e.g. Hillary Clinton to become president in November 2016 (current probability 54% according to Betfair). Prediction markets have the equivalent of bid prices and ask prices. A bid price is the implied probability someone is willing to accept in order to bet in favour of an outcome. An ask price is the implied probability someone is willing to accept for betting against an outcome. In Betfair, the “current price” is normally thought of as the highest current ask price e.g. Clinton at 54%, although the last matched price is also available for inspection. This is because most people bet that an event will occur rather than that it won’t occur, so they are looking to place a bid to match with an existing ask price.

    Outside stock markets and prediction markets, haggling is the best approximation of the bid / ask divide. In some societies, haggling occurs for many products and services. In western societies, haggling occurs mostly for high value assets e.g. houses, works of art. However, there is nothing to prevent you from haggling for any product e.g. an expensive hi-fi system or even a tin of beans.

    Profit arises from a difference between valuations of the underlying assets, goods or services. There is nothing mysterious about this.

    Jason (in linked post on valuation of businesses): “The basic idea is that the market capitalization and the "book value" of companies in stock indicies (sic) has wildly diverged”

    The valuation of a business is just a price, like any other price. Only fools and economists think that stock markets have an accurate understanding of the value of a business. Any serious person who considers buying a business will conduct extensive due diligence. This is the purchaser’s way of validating the bid price. Similarly, prospective house purchasers conduct due diligence by appointing surveyors to inspect the property on their behalf. Due diligence may result in a change to the bid price or to an end to the deal.

  15. Steve: “I'll ask you to provisionally adopt, for this paragraph, my def of money (short and imprecise form, ‘the [net] value of assets’)”

    Another observation from working in business and government is that the definitions of words, and the concepts behind them, are extremely significant in human communication, particularly in large collaborative teams where agreed terminology is vital. However, most people don’t think about this.

    Wittgenstein: “The limits of my language means the limits of my world”

    Economics is full of badly defined terminology. The majority of economic debates are based on badly understood terminology and badly articulated concepts. Irrespective of which definition of a term like “money” is “correct” (whatever “correct” means here), we can’t have a serious debate about concepts for which we have different definitions.

    Earlier this year, I read some posts on David Glasner’s blog about Keynes’ accounting identities. I like David’s blog. He has a different perspective from me. He writes about history and what specific economists said and did. I have little to add to his views so rarely comment. However, I did comment on his accounting identities posts.

    My management summary of David’s posts is that economists have been arguing about simple accounting identities for 80 years without making any progress. The interesting question for me is WHY does this happen?

    I asked David and his readers what economists mean SPECIFICALLY by the term “saving” in the identity

    Investment + Consumption = Consumption + Saving

    The answer I was given is that different economists have different definitions of “saving”. So we have an identity which is presumed to hold even if economists use their own definition of one of the terms!! Perhaps that explains 80 years of fruitless debate.

    I suspect that this happens because economists discuss concepts in the abstract and often seem to have no idea of how to tie the concepts explicitly to examples in the real world.

    Another example. Economists appear to think of all four terms in the Keynes’ identity above as though they represent “money”. In fact, you can use some simple examples in a toy economy to show that “investment” is not money. Investment is a valuation of something that is purchased with money.

    For example, if a car manufacturing business buys raw materials for five cars, labour for the manufacturing of five cars, and some overheads, and then manufactures five cars, then it has an investment in five cars. The valuation it places on the five cars is the thing that belongs in Keynes’ identity.

    This is, of course, why I = S. I represents a valuation of the cars. S represents the money that the car manufacturer spent on the car. The business’s spending became someone else’s income and, in the absence of any further spending, was added to their saving. (This assumes that the business values the cars based on the costs incurred in their manufacture, but that is standard practice).

    [I have a Google spreadsheet which shows how this works and explains some basic accounting concepts. If anyone is interested, I can publish the spreadsheet and provide a link].

    At least “saving” and “investment” are real things. Economists also use concepts which are just made-up. “Representative agents” don’t exist. However, the concept suggests that the economy is made up only of average people. Average people are not unequal to other average people. Average people don’t build up debt with other average people. This is my summary of a talk by Joseph Stiglitz I once listened to on YouTube. The use of the concept of “representative agents” in thinking about the economy basically precludes discussion of inequality and private-sector debt. Discussion of “representative agents” has a distinct right-wing bias.


  16. Of course, there is left-wing bias too. The MMT people make use of accounting concepts so you might think that I approve of MMT. However, MMT uses the term

    Government = Treasury + Central Bank

    where most other economists, and non-economists, use the term

    Government = Treasury

    with Central Bank seen as a separate part of Public Sector.

    This seems an inoffensive difference. However, it leads directly to MMT saying things like “government spends money into existence” which sounds very odd to other people’s ears. This puts many people off MMT, and it must be obvious to the MMT people that this is the case, so why do the MMT people persist with this? The only logical answer is that they do it because they can then show that fiscal policy should be used everywhere. (Monetary policy become irrelevant if you ignore the central bank).

    There are lots of other examples e.g. “monetary policy” is not a policy. It is just a vague collective term for a group of policies which are never fully articulated but which, according to some, can cure all known economic problems.

    That reminds me. Scott Sumner uses the terms “policy stance”, “tight money” and “loose money” all the time. What does he mean by these terms? I have no idea. Neither does Scott.


    I’d draw three conclusions from all of this. First, it’s impossible to represent economics as a serious subject if economists can’t even define their own basic terminology.

    Second, it may be that any set of definitions contains its own hidden political biases. When Marxists assume that the world is made up of “capitalists” and “workers”, there is a certain inevitability to their conclusions that all of the problems in the world can be attributed to this divide.

    Third, although I agree with a lot of the basic thinking in Steve’s posts, and although I think that we could have many interesting discussions, I disagree with his definition of “money”.

    Steve: ‘the [net] value of assets’

    I think this mixes up two different things: money and assets / goods valued in terms of money. An example. Suppose that I have

    One house which I value at €500,000
    One car which I value at €30,000
    100 shares which I value at €100,000
    €20,000 on deposit at the bank
    €100 cash in my wallet.

    I would say that I have

    Physical assets (house and car) with estimated valuation of €530,000
    Financial assets (shares) with estimated valuation of €100,000
    Money (deposit and cash) with valuation of €20,100.

    My net worth is the total of these figures.

    I think that it is important to distinguish money (as it is valued in terms of itself and it is used to value everything else) from things that are valued in terms of money. I also think that it is useful to distinguish between physical and financial assets.

    My main problem with this taxonomy is whether government and corporate bonds should be defined as money or as financial assets. I suspect that they should be defined as money. However, that would have huge implications. For example, governments would increase the money supply when they created new bonds but central banks would not change the money supply when they swapped bonds for money. They would merely change the terms of the government loan (central banks return interest on their bonds to government so they make the loan interest free).

    The most important point here is to acknowledge the need to define terminology careful, and to use examples like my net worth example to clarify the definitions.

    Apologies again for the length of these comments but there is endless repetition of the same debates on this and other blogs.

  17. Hi Jamie,

    Sorry about your comments getting stuck in the spam folder -- I fished them out. I have some responses to a few things ...

    You said:

    When Jason says of me

    “I think you've confused a symmetry transformation that is a property of equilibrium at an instant in time … with a non-equilibrium process”

    all he is saying is that my perspective of the economy is different from his perspective.

    Not really -- the differential equation

    P = dD/dS = k D/S

    obeys a symmetry transformation D → a D, S → a S that leaves the equation unchanged, therefore the price P is unchanged. There is no different perspective about this.

    If you sell a sheep at P = €10 and then at P = €15, this has no bearing on the symmetry principle of the equation (and therefore no bearing on the conservation law I was discussing). In that case dD/dS and k D/S have different values -- they can't have different values under the symmetry transformation D → a D, S → a S. Under that transformation

    dD/dS → d(aD)/d(aS) = dD/dS

    k D/S → k (aD)/(aS) = k D/S


    dD/dS = P → P

    There may be a difference in perspective about whether the equation is meaningful for economics (which I have no problem with), but the questions about profit and money being conserved under exchange are not germane to the symmetry principle/conservation law I was discussing in the post above.

    You said:

    When stock market prices go up, many people, including famous economists and Noah Smith, say that “money poured into the stock market”. However, that is an illusion as money is conserved under exchange. If one person buys a share for €1 then another must sell the share for €1.

    I am assuming that you mean the original share price was lower, like €0.50. If it was €1, then the market index wouldn't have gone up. If it was €0.50, then sale at €1 means €0.50 went into the valuation of the market (making the index go up) and €0.50 of cash (or whatever) went into the assets of the seller.

    The market in a sense 'created' €1 of value. That is €0.50 for the asset holder and €0.50 for the market -- actually more because there's more than a single share and that last bid-ask that was met now represents the current value of all existing shares. But this is what is meant by 'money pouring into the stock market'.

    The question at hand was where did this €1 come from (or valuations in general)? Your answer was:

    There was no “extra” five pounds [or here €0.50]. It is an illusion. Money was conserved under exchange. No money was created and no money was destroyed.

    I was under the impression that we were discussing where the extra value came from, not whether MZM went up. I agree that MZM, M1 or whatever measure didn't change. But the total valuation of assets changed, one person made €0.50 of capital gains that counts towards GDP.

    With the so-called representative agent, you can't really have that extra €0.50 of value because it comes from exchange. The representative agent can't make €0.50 by selling a €0.50 stock to himself for €1.

    And that is the problem with traditional economics ... profits and losses can't really arise without counterparties making mistakes, being 'irrational' ... or just existing at all.

    This is not an issue in the information framework. The information about a specific transaction (or any initial conditions) is lost as the economy returns to equilibrium. This is the exact same definition in thermodynamics: equilibrium means that the information about the initial conditions has been irrevocably lost (entropy increases).


    1. You said:

      Irrespective of which definition of a term like “money” is “correct” (whatever “correct” means here), we can’t have a serious debate about concepts for which we have different definitions.

      I agree! That's why I set up models and use empirical results to say which measure of money is "correct" (I personally don't care) ... and it turns out different measures have different uses. MB is connected to short term interest rates. M0 (MB without reserves) is connected to core inflation. MZM appears to be related to long term interest rates. M2 looks like a good measure for exchange rates.

      You said:

      Investment + Consumption = Consumption + Saving

      I always thought a) this is a debate that only exists because the people involved are nerds in the same way people debate about what in Star Wars is "canon", b) this is a debate that only exists because the terms saving and investment have colloquial meanings, c) that this equation really means

      [not consumption] + Consumption = Consumption + [not consumption]

      Where you call the [not consumption] on one side "investment" and on the other "savings" because reasons, and d) this is the problem when people use the equals sign instead of ≡ or ≈ or forget time averages 〈S〉 = 〈I〉.

      My usual response is to bang my head on my desk. There is zero insight that comes out of any discussion of S = I.

  18. Jason, that was quite an update you left there!


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