Tuesday, May 2, 2017

Mathiness in modern monetary theory

Simon Wren-Lewis sends us via Twitter to Medium for an exquisite example of my personal definition of mathiness: using math to obscure rather than enlighten.

Here's the article in a nutshell:
Any proposed government policy is challenged with the same question: “how are you going to pay for it”. 
The answer is: “by spending the money”.
Which may sound counter intuitive, but we can show how by using a bit of mathematics. 
[a series of mathematical definitions] 
And that is why you pay for government expenditure by spending the money [1]. The outlay will be matched by taxation and excess saving to the penny after n transactions. 
Expressing it using mathematics allows you to see what changing taxation rates attempts to do. It is trying to increase and decrease the magnitude of n — the number of transactions induced by the outlay. It has nothing to do with the monetary amount.
I emphasized a sentence that I will go back to in the end. But first let's delve into those mathematical definitions, shall we? And yes, almost every equation in the article is a definition. The first set of equations are definitions of initial conditions. The second is a definition of the relationship between $f$ and $T$ and $S$. The third set of equations define $T$. The fourth defines $S$. The fifth defines $r$. The sixth defines the domain of $f$, $T$, and $S$. Only the seventh isn't a definition. It's just a direct consequence of the previous six as we shall see.

The main equation defined is this:

\text{(1) }\; f(t) \equiv f(0) - \sum_{i}^{t} \left( T_{i} + S_{i}\right)

It's put up on the top of the blog post as if it's $S = k \log W$ on Boltzmann's grave. Already we've started some obfuscation because $f(0)$ is previously set to be $X$, but let's move on. What does this equation say? As yet, not much. For each $i < t$, we take a bite out of $f(0)$ that we arbitrarily separate into $T$ and $S$ which we call taxes and saving because those are things that exist in the real world and so their use may lend some weight to what is really just a definition that:

K(t) \equiv M - N(t)

In fact we can rearrange these terms and say:

f(t) \equiv & f(0) - \sum_{i}^{t} T_{i} -  \sum_{i}^{t} S_{i}\\
f(t) \equiv & M - T(t) -  S(t)\\
K(t) \equiv & M - N(t)

As you can probably tell, this is about national income accounting identities. In fact, that is Simon Wren-Lewis's point. But let's push forward. The article defines $T$ in terms of a tax rate $0 \leq r < 1$ on $f(t-1)$. However, instead of defining $S$ analogously in terms of a savings rate $0 \leq s < 1$ on $f(t-1)$, the article obfuscates this as a "constraint"

f(t-1) - T_{t} - S_{t} \geq 0

Let's rewrite this with a bit more clarity using a savings rate, substituting the definition of $T$ in terms of a tax rate $r$:

f(t-1) - r_{t} f(t-1) - S_{t} & \geq 0\\
(1- r_{t}) f(t-1) - S_{t} & \geq 0\\
s_{t} (1- r_{t}) f(t-1) & \equiv S_{t} \; \text{given}\; 0 \leq s_{t} < 1

Let's put both the re-definition of $T_{i}$ and this re-definition of $S_{i}$ in equation (1), where we can now solve the recursion and obtain

f(t) \equiv f(0) \prod_{i}^{t} \left(1-r_{i} \right) \left(1-s_{i} \right)

This equation isn't derived in the Medium article (and it really doesn't simplify the recursive equation without defining the savings rate). Note that both $s_{i}$ and $r_{i}$ are positive numbers less than 1. There's an additional definition that says that $r_{t}$ can't be zero for all times. Therefore the product of (one minus) those numbers is another number $0 < a_{i} < 1$ (my real analysis class did come in handy!) so what we really have is:

\text{(2) }\; f(t) \equiv f(0) \prod_{i}^{t} a_{i}

And as we all know, if you multiply a number by a number that is less than one, it gets smaller. If you do that a bunch of times, it gets smaller still.

In fact, that is the content of all of the mathematical definitions in the Medium post. You can call it the polite cheese theorem. If you put out a piece of cheese at a party, and if people take a non-zero fraction of it each half hour, those pieces will get smaller and smaller but eventually there is nothing left (i.e. somebody takes the last bit of cheese when it is small enough). Which is to say for $t \gg 1$ (with dimensionless time) $X \equiv T + S$ because $f(t) = 0$ with $t \gg 1$. 

But that's just an accounting identity and the article just obfuscated that fact by writing it in terms of a recursive function. Anyway, I wrote it all up in Mathematica in footnote [2]. 

Now back to that emphasized sentence above:
Expressing it using mathematics allows you to see what changing taxation rates attempts to do.
No. No it doesn't. If I write $Y = C + S + T$ per the accounting identities, then a change in $T$ by $\delta T$ means [3]

\delta Y =  \left( \frac{\partial C}{\partial T}+ \frac{\partial S}{\partial T} + 1 \right) \delta T

Does consumption rise or fall with increased taxation rates? Does saving rise or fall with increased taxation rate? Whatever the answer to those questions are, they are either models or empirical regularities. The math just helps you figure out the possibilities; it doesn't specify which occurs (for that you need data). The Medium article claims that all that changes is how fast $f(t)$ falls (i.e. the number of transactions before it reaches zero). However that's just the consequence of the assumptions leading to equation (2). And those assumptions represent assumptions about $\partial C/\partial T$ (and to a lesser extent $\partial S/\partial T$). Let's rearrange equation (3) and use $G = T + S$ [4]:

\delta Y = &  \frac{\partial C}{\partial T}\delta T + \frac{\partial S}{\partial T}\delta T  + \delta T \\
\delta Y = &  \frac{\partial C}{\partial T}\delta T + \frac{\partial G}{\partial T}\delta T \\
\delta Y = &  \frac{\partial C}{\partial T}\delta T + \delta G

And there's where we see the obfuscation of original prior. In the medium article, $f(0) = X$ is first called the "initial government outlay". It's $\delta G$. However, later $f(t-1)$ is called "disposable income". That is to say it's $\delta Y - \delta T$. However those two statements are impossible to reconcile with the accounting identities unless $X$ is the initial net government outlay, meaning it is $\delta G - \delta T$. In that case we can reconcile the statements, but only if $\partial C/\partial T = 0$ because we've assumed 

\delta Y - \delta T & = \delta G - \delta T\\
\delta Y & = \delta G

This was a long journey to essentially arrive at the prior behind MMT: government spending is private income, and government spending does not offset private consumption. It was obfuscated by several equations that I clipped out of the quote at the top of this post. And you can see how that prior leads right to the "counterintuitive" statement at the beginning of the quote:
Any proposed government policy is challenged with the same question: “how are you going to pay for it”. 
The answer is: “by spending the money”.
Which may sound counter intuitive, but we can show how by using a bit of mathematics.
No, you don't need the mathematics. If government spending is private income, then (assuming there is only a private and a public sector) private spending is government "income" (i.e. paying the government outlay back by private spending).

Now is this true? For me, it's hard to imagine that $\partial C/\partial T = 0$ or $\delta Y = \delta G$ exactly. The latter is probably a good approximation (effective theory) at the zero lower bound or for low inflation (it's a similar result to the IS-LM model). For small taxation changes, we can probably assume $\partial C/\partial T \approx 0$. Overall, I have no real problem with it. It's probably not a completely wrong collection of assumptions.

What I do have a problem with, however, is the unnecessary mathiness. I think it's there to cover up the founding principle of MMT that government spending is private income. Why? I don't know. Maybe they don't think people will accept that government spending is their income (which could easily be construed as saying we're all on welfare)? Noah Smith called MMT a kind of halfway house for Austrian school devotees, so maybe there's some residual shame about interventionism? Maybe MMT people don't really care about empirical data, and so there's just an effluence of theory? Maybe MMT people don't want to say they're making unfounded assumptions just like mainstream economists (or anyone, really) and so hide them "chameleon model"-style a la Paul Pfleiderer.

Whatever the reason (I like the last one), all the stock-flow analysis, complex accounting, and details of how the monetary system works serve mainly to obscure the primary point that government spending is private income for us as a society. It's really just a consequence of the fact that your spending is my income and vice versa. That understanding is used to motivate a case against austerity: government cutting spending is equivalent to cutting private income. From there, MMT people tell us austerity is bad and fiscal stimulus is good. This advice is not terribly different from what Keynesian economics says. And again, I have no real problem with it.

I'm sure I will get some comments that say I've completely misunderstood MMT and that it's really about something else. But please don't forget to tell us all what that "something else" is. But the statement here that "money is a tax credit" plus accounting really does say, basically, that government spending is our income.

But with all the definitions and equations, it ends up looking and feeling like this:

There seems to be a substitution of mathematics for understanding. In fact, the Medium article seems to think the derivation it goes through is necessary to derive its conclusion. But how can a series of definitions lead to anything that isn't itself effectively a definition?

Let me give you an analogy. Through a series of definitions (which I have done as an undergrad math major in that same real analysis course mentioned above), I can come to the statement

\frac{df(x)}{dx} = 0

implies $x$ optimizes $f(x)$ (minimum or maximum). There's a bunch of set theory (Dedekind cuts) and some other theorems that can be proven along the way (e.g. the mean value theorem). This really tells us nothing about the real world unless we make some connection to it however. For example, I could call $f(x)$ tax revenue and $x$ the tax rate ‒ and adding some other definitions ($f(x) > 0$ except $f(0) = f(1) = 0$) and say that the Laffer curve is something you can clearly see if you just express it in terms of mathematics.

The thing is that the Laffer curve is really just a consequence of those particular definitions. The question of whether or not it's a useful consequence of those definitions depends on comparing the "Laffer theory" to data.

Likewise, whether or not "private spending pays off government spending" is a useful consequence of the definitions in the Medium article critically depend on whether or not the MMT definitions used result in a good empirical description of a macroeconomy.

Without comparing models to data, physics would just be a bunch of mathematical philosophy. And without comparing macroeconomic models to data, economics is just a bunch of mathematical philosophy.


Update 5 May 2017:

Here's a graphical depiction of the different ways an identity $G = B + R$ can change depending on assumptions. These would be good pictures to use to try and figure out which one someone has in their head. For example, Neil has the top-right picture in his head. The crowding out picture is the bottom-right. You could call the picture on the bottom-left a "multiplier" picture.

Update 6 May 2017: Fixed the bottom left quadrant of the picture to match the top right quadrant.



This is basically equivalent to what is done in the Medium article.

[2] Here you go:

[3] If someone dares to say something about discrete versus continuous variables I will smack you down with some algebraic topology [pdf].

[4] I think people who reason from accounting identities seem to make the same mistakes that undergrad physics students make when reasoning from thermodynamic potentials. Actually, in the information equilibrium ensemble picture this becomes a more explicit analogy.


  1. I'll be first. I think a more accurate statement of MMT's core assertion is:

    Government *deficit* (or net) spending (by a sovereign-currency-issuing government) delivers private "surplus."

    More precise, but equally obvious. But only in the undefined vernacular. "Surplus" isn't a word used anywhere in the national accounts — not in the NIPAS, the FOFAs, or the IMAs.

    And this raises a point that conjoins with yours: every set of accounting statements is simply a set of definitions — defining the labels of each measure in those accounting statements in relation to each other. It's a set of mutually coherent definitions. eg "Income" has these accounting relationships to these other measures. People reify and deify accounting measures, but they're just defined terms among many other related defined terms, part of an economic model.

    I much prefer: "gov def spending adds assets to the private sector balance sheet." And since it adds no liabilities (except some vague future off-balance-sheet possible need to pay more taxes), it increases private sector net worth. (Unlike bank lending, which adds assets the the left side of private sector balance sheets, but also adds equal liabilities so no new net worth>)

    I don't think this makes sense: "private spending pays off government spending." The unstated assumption on gov spending is that it can only pay money to the private sector. Sort of, basically true. But most private spending is within the private sector, not to gov. Only taxes and etcs remove assets from the private sector balance sheet, disappearing them back into gov's asset-creating-and-destroying-hole-in-the-ground. So only those payments reduce private-sector assets/net worth.

    Not a very organized comment but there you go. Stopping now.

    1. Yes, I was sloppy about net spending above except at one point.

      However I do believe the Medium article (and per Neil below) is trying to say that "private spending pays off government spending" (by essentially creating output that is taxed).

      I show that in my reply to Neil's comment below (but again, pointing out the model assumptions that are going into it).

      I just thought of a funny way to think about the assumptions in physics terms. It's an iso-consumption government expansion. Whether or not you can do this is an empirical matter. I think it appears to be approximately a valid process when inflation is low and/or the at the zero lower bound.

  2. “The outlay will be matched by taxation and excess saving to the penny after n transactions.”

    Problem with that idea is that if the economy is already at capacity (the point at which further demand will cause excess inflation) then the extra demand stemming from “the outlay” before it disappears in the form of tax and saving will cause excess inflation. Ergo that extra demand is not allowable. Ergo in that circumstance, and as per conventional wisdom, the question “how are you going to pay for it” becomes very relevant. To be exact, $X of extra public spending will have to be balanced by around $X of tax.

    A second flaw in the above idea is that (ignoring tax and inflation for the sake of simplicity and concentrating just on saving), if the private sector’s stock of saving is what it wants at current rates of interest, then additional public spending will push savings above the latter desired level, which will result in the private sector trying to spend the surplus away (hot potato effect). Again, the result is excess inflation.

    1. This may or may not be true.

      However it represents a different set of assumptions about fiscal policy. In this case

      δY = (∂C/∂G)δG + δG

      we are taking ∂C/∂G = -1 when the economy is at capacity so that under a change G → G + δG

      δY = 0

      Again, this may or may not be true. It represents assumptions about how macroeconomies work that should really be left to empirical data.

  3. "Problem with that idea is that if the economy is already at capacity"

    It isn't Ralph. Stop banging on about it. It makes you sound like you have dementia.

    If you are at capacity you get no more G to start with because there is nothing free to buy at a price you're prepared to pay.

    1. Neil, I think it's you that has dementia (if you want an exchange of insults). My "at capacity" point was what's known as an A-S-S-U-M-P-T-I-O-N. Assumptions (however unrealistic) are often made in discussion in economics and almost every other subject. They help simplify issues and help everyone concentrate on points under discussion.

      As to whether the economy really is at capacity, clearly that's a moot point as anyone with half a brain realizes. For example the Bank of England is clearly contemplating raising interest rates, and nearly voted to do so at their last (as of mid June 2017) Monetary Policy Committee meeting, but didn't actually raise them.

  4. "As you can probably tell, this is about national income accounting identities"

    Actually it isn't really. You and Simon have assumed that and got the wrong end of the stick. You didn't bother to ask first. You just ploughed on with your own assumptions. It's not as if I'm that difficult to get hold of.

    This is a piece of *additional* government spending. Say £100. That is used to buy something. The next person then has £100 less some tax - which is at a rate because taxes are generally expressed in terms of rates. That person then decides what to save (as an amount because people don't save in percentages, they save in chunks). Then they spend.

    The result of all that is the maths I've described - a recursive function that causes government spending to be matched by an amount of taxation and an amount of saving. To the penny, each time every time. And because we have a floating rate non-convertible currency it applies *wherever people are on the planet*. None of this nonsense about 'leakages' to strange foreign climes. There is no such thing. It's just fiscal savings in your denomination held somewhere.

    So if I want 10,000 more police I can pay for it fiscally by simply spending the money and that will cause a transaction chain that will generate a balancing amount of taxation and additional non-government saving.

    If I increase the tax rate, then the number of transactions in the sequence will likely fall unless people start saving less for some feedback reason.

    It's a simple formulation of the accounting transactions that occur at central bank level. HM Treasury accounts put funds in commercial bank reserve accounts, and the transactions at the commercial bank level put some of it back due to the effects of taxation and savings asset swaps. The rest stays in the reserve accounts. There is no limiting control function there. There is nobody to say 'no' who can make that 'no' stick.

    That's it. That's all it says. It doesn't say anything about where there 10,000 people will come from, what they are doing now that is so much more important or anything else about the private circuit response, or whether they are available at a price that is worth paying.

    It just deals with the journalist response 'how will you pay for it'. And the answer is 'by spending the money'. Because the central bank layer is a completely closed system with no overriding control function.

    Once that is out of the way you can get down to the political discussion obfuscated by the money illusion - should a nation's resources be allocated first by elected people in parliament, or Simon's unelected Central Bankers from their ivory towers on behalf of corporations?

    I'll also point out that your 'empirical data' is tainted by bad policy. It is the description of a man in a strait jacket and collected on that basis. That empirical data cannot help describe what happens when you take the jacket off. We don't know because we've never tried it.

    So going around shouting 'empirical data' is just a political trick to try and keep the straitjacket on. To work with something you've never tried you need a simulation, and a way of trying out scenarios with it with real people. We call that a 'multiplayer computer game'.

    Next time why not ask first? It's considered polite and I don't bite. Not too much anyway.

    1. "Actually it isn't really [about accounting identities]. This is a piece of *additional* government spending."

      But it is. Additional government spending means

      G → G + δG

      Which means that (using Y = C + G for simplicity)

      Y + δY = C + δC + G + δG
      Y + δY = C + (∂C/∂G)δG + G + (∂G/∂G)δG
      Y + δY = C + (∂C/∂G)δG + G + δG
      δY = (∂C/∂G)δG + δG

      so when you say:

      " ... can pay for it fiscally by simply spending the money and that will cause a transaction chain that will generate a balancing amount of taxation and additional non-government saving."

      This is an assumption that (using your model) the number of consumption transactions that occur don't change -- i.e. ∂C/∂G ≈ 0. I have no problem with this assumption (it's the traditional Keynesian picture), but it is assumed. Assuming ∂C/∂G ≈ 0 we have:

      δY ≈ δG

      which also means via your definition of taxes

      T + δT ≈ r (Y + δY) r (Y + δG)
      δT ≈ r δG

      i.e. your point about the spending leading to taxes being collected. No recursive function necessary.


      "That person then decides what to save (as an amount because people don't save in percentages, they save in chunks)."

      This is silly. If I save £20, £25, £20, and £5 that is mathematically equivalent to saying I save

      20% of £100
      50% of £50
      50% of £40
      25% of £20

      I didn't say the percentages were constant (listed as $s_{i}$ -- different for each round of transactions).


      "Next time why not ask first? It's considered polite ..."

      Discussion in a public forum shouldn't proceed by private communications. What I would "ask" is exactly as stated in the blog post I wrote above. When you write things in public, they can and will be criticized without your permission -- hopefully in the interest of intellectual progress. I have a lot of expertise in mathematics much of which I obtained via public funds. I try to use that in the public interest when I can.

    2. "This is an assumption that (using your model) the number of consumption transactions that occur don't change"

      There is no such assumption. Government spending is there to increase the number of consumption transactions because there is something available worth buying at a price government is prepared to pay. I hire an unemployed person and they can now buy stuff which causes it to be produced. That creates a consumption transaction that didn't previously exist and wouldn't previously have existed, some savings if there is anything left and some taxation. Rinse and repeat. In the aggregates: G goes up, C goes up a bit more and so does Y to balance. How many extra transactions you get depends on how much drag you get from savings and taxation. That's the fiscal multiplier.

      Savings drag can be forced by production constraints. I'd like a Tesla Powerwall 2 but they are on pre-order. In the meantime the cash for it stays in the bank. So you get a Time shift response, not a price response. The queue at the barbers is the same concept.

      In a competitive market a firm that quantity responds will outcompete one that price responds. Price responses should be seen as an indication of a lack of competition in a market segment and dealt with accordingly.

      What you're aiming for is a set of quantity response, time shifters.

      "I didn't say the percentages were constant "

      Ok, I see what you've done, but 'solving' the general case sort of misses the point of why it was left in the form it is.

      What you are doing is the equivalent of taking a poem and saying "all it says is that she is going to the shops". It's not what is said. The value is in the way it is said.

      It's far easier to step through my version with a few lego models and a pile of monopoly money because I'm summing, not multiplying and keeping the savings in amounts. Usually when I'm demonstrating I keep the tax in amounts too.

      And that is its purpose - to describe using relatively low level maths that whatever government spends it gets back in taxation and saving while causing a sequence of transactions. All in nominal terms. That's it. You pay for government spending by spending the money.

      P.S. The two articles I link to at the bottom of my article derive the standard fiscal multiplier, which may be more where you are thinking.

      All this just goes to show that when you use maths, all you actually do is obfuscate and reduce your audience. That's why I don't generally use it.

    3. "There is no such assumption."

      Yes there is.

      You compute the numbers of transactions based on the people in on the government spending largess chain of transactions assuming 1) those people have no other transactions they might otherwise engage in (and either slowing down or speeding those up) and assuming 2) there aren't any other people who don't get in on the largess chain of transaction that adjust their spending (slowing down or speeding up).

      This may or may not be a good approximation, but it is definitely an assumption.

      The "crowding out" picture assumes in this picture that people slow down their transactions when government does some deficit spending, lowering output. Instead of B buying two apples from A, A buying 2 carrots from C, and C buying 2 bananas from B each at a value of £1 (for a total output of £6), they instead buy 1 of each from each other (total output £3) because they only produce 1 of each because they think others might only be able to afford 1 of each because of future taxes. Or maybe the δG made some apples so everyone figured that A wouldn't be able to buy as much, meaning B and C wouldn't be able to either. Note the net accounting here is zero (in one case £1 circulates, in the other £2 circulates).

      I don't necessarily believe this story -- especially at low inflation -- but your model assumes this *cannot* happen. And that is your assumption, the assumption that when G → G + δG we have

      δY ≈ δG

      The above "story" about apples, bananas, and carrots is basically a story for ∂C/∂G < 0 (i.e. consumption falls for an increase in government spending).

      The people that claim "crowding out" happens are not morons, so it is highly unlikely some "simple logic" or "simple math" shows that they are morons. And sure enough that "simple math" simply assumes "crowding out" doesn't exist. Maybe it doesn't, but that seems like you need a much more complex model and comparison with empirical data to figure that out.

    4. Many economic theories are based on wrong interpretation on accounting identities and underlying data semantics. These theories work only under special conditions. A more complete and correct case analysis for output and inflation impact from government deficit spending should be like something below.

      Assume Y = PQ = C + I + G (public and private sectors only)
      δG = Public I and C spending change(G = Gov I + Gov C)
      δNonG = Public Non G spending change = 0 (often δNonG ≈ δG )
      δNonIC = Private non I and C spending change

      (1) δG + δNonG = δC + δI + δNonIC (accounting identity)
      δY = δC + δI + δG

      Some special cases:
      (1a)δY =δG if δC=0, δI=0 (nominal output change is δG)
      (2b)δY =2(δC+δI)=2δG if δNonIC=0(nominal output change is 2δG)

      (2)δY = δPQ = PδQ + QδP + δPδQ (accounting identity)

      Some special cases:
      (2a) δY = PδQ if δP = 0 (real output change δQ)
      (2b) δY = QδP if δQ = 0 (inflation change δP)

    5. Peiya,

      I'm not sure what you mean by δNonG or δNonIC. I don't know how there can be any "public non-G spending change" as "public" means "G" in the traditional definition of the income accounting identity.

    6. Jason,

      Traditional definition of the "income accounting identity" (C+I+G = C + S + T or S-I = G-T) is widely-misused with implicit assumption NonG = 0 . NonG = government nonIC spending = government transfer/interest/rental/others payments to private sector. This implicit assumption is not a realistic assumption when addressing government deficit spending issues based on this equation. In US government spending, NonG ≈ G,

      Derivation steps:
      Total IC spending(GDP=C+I+G) = total earned income(i.e GDI)
      private earned income = C+S.
      government earned income = GC + GS = T - nonG
      government total income = T (i.e. tax)
      government IC spending = G = GI + GC
      government nonIC(nonG)spending= Transfer/Interest/others payments
      total earned income(GDI)= C+S + T - nonG

      Definitions of S, GS and G
      (a)S(Private sector saving) = I + private total income - private total spending.
      (b) GS(Public sector saving) = GI(government investment) +
      T(government total income) - government total spending.
      (c) G = GI(government investment) + GC(government Consumption)

      Two kinds of income and spending are in NIPA detailed NIPA Primer

      For earned income(GDI) and IC spending(GDP),
      GDP = I + C + G
      GDI = Wages + NetOperatingSurplus + CFC + Tariffs-Subsides

      For each sector, there are also unearned income/nonIC spending such as tax receipts/payments, interest receipts/payments, transfer receipts/payments, rental receipts/payments, etc. These unearned income/nonIC spending are not counted as productive GDI/GDP.

      δNonG is used for government nonIC spending change since G is already used for government I + Gov C in GDP formula.

      δNonIC is used for private sector nonIC spending change since I and C are already used for private I and C in GDP formula.

      Top-level NIPA accounting identities:
      (1) total income = GDI + total unearned income(all sectors)
      (2) total spending = GDP + total nonIC spending(all sectors)
      (3) GDI = GDP
      (4) total unearned income=total nonIC spending(all sectors)

      Basically, these 4 identities describe economic principles:
      (a) my expense is your income, (b) my IC spending is your earned income, and (c) my nonIC spending is your unearned income.

      Now you can see what is wrong in this ad-hoc formula: C+I+ G = C + S + T or S-I = G-T

      For comparison purpose, correct variations are as follows:
      (1)C+I+G = C+ S + T - NonG
      (2)S-I = GI - GS
      (3)C+I+G=Wages+ NetOperatingSurplus + CFC + Tariffs-Subsides
      (4)private total income - private total spending(I+C+NonIC)
      = government total spending(G+NonG) - T
      (5) GDP = Total Income - Total NonIC Spending

      Typical production money flows from deficit spending to private sector are:

      (1) NonG --> private unearned income --> private IC spending or private nonIC spending or private saving.

      (2) Government IC spending(G) --> private earned income --> private IC spending or private nonIC spending or private saving.

      Note that private sector income can come from private sector itself. For example, private sector wages are from private corporation spending.

    7. Peiya,

      You said:

      Traditional definition of the "income accounting identity" (C+I+G = C + S + T or S-I = G-T) is widely-misused with implicit assumption NonG = 0.

      For the purposes of this blog, we'll stick to the traditional definition unless there is e.g. a model of empirical data that warrants a change of definition. Changing definitions of accounting identities and saying "Many economic theories are based on wrong interpretation on accounting identities" is a bit disingenuous.

      Imagine if I said you were wrong because I define accounting identities as statistical equilibrium potentials? I could say that there is no entropic force associated with your "nonG" term, therefore you have a wrong interpretation of the accounting identities.

      But I don't say that. And you shouldn't say that about the "traditional" definition of accounting identities unless you have a really good reason backed up with some peer-reviewed research or at least open presentations of that research.

      You must always try to "bend over backwards" to consider the fact that you might be wrong. Or at least note when you are considering some definition that is non-standard that it is in fact non-standard. In my link above, I admit the approach is speculative. I say "At least if [the equation presented] is a valid way to build an economy." I recognize that it is a non-standard definition of the accounting identities.

      Saying people misunderstand a definition and then presenting a non-standard version of that definition is not maintaining the necessary integrity for intellectual discussion and progress.

    8. All this stuff is a bit off topic neither necessary for the specific issue.
      However, the identities do not describe much economic principle, they rather show some (biased) point of view about describing economic system and capitalism.
      With statistics always there are many problems, but one can take net taxation and simplify. More than enough in this case.

    9. "For the purposes of this blog, we'll stick to the traditional definition unless there is e.g. a model of empirical data that warrants a change of definition"

      Here is the chart to illustrate C+I+G+X-M ≠ C+S+T.
      Red line = C + I + G + X - M
      Green line = C + S + T
      Dot black line = Y = GDP = Red Line
      Blue line = X - M

      1. Dot Black line = Red line ≠ Green Line
      2. Image version

      I read out NIPA semantics, put out correct identities
      and verify them with empirical data. I put a blue line to show the difference between red line and green line is not due to X-M. It is due to NonG part. NonG is increasing due to government non-discretionary spending.

      I can formally prove

      Models with this tradition definition ⊨ ¬ O(Observations).

      NIPA data is a temporal model!!

    10. You left out net government saving and used gross private saving instead of net private saving.

      Those are at least two errors. You also neglected the statistical discrepancy:


      which accounts for the measurement errors associated with compiling data in different ways.

      Look, the BEA isn't a bunch of morons. You should probably work at it until you figure it out and get the accounting identities to work out right.

      You might learn something!

      Here's a good resource:


    11. Let NIPA data speak themselves. By the way, I never define NIPA data terms and just read out data semantics from BEA NIPA guide https://www.bea.gov/national/pdf/nipa_primer.pdf. Many people just read in their own interpretation without checking NIPA economic data.

      You think what you think is not what NIPA data sayings. For proof of that, I add government saving(GS) even I think S here is only private saving. I also add statistical discrepancy(SD) data series into your income accounting identity.

      It gets worse since you miss out NIPA data semantics.

      Blue line(GDI+SD)=red line=black line(GDP)≠ C+S+T+GS+SD


      Note that meta logic theories for proving empirical invalidity.

      Models with this tradition definition ⊨ ¬ O(Observations).

      As a consequence, it creates further empirical invalidity.

      ¬O => ¬O1 V ¬O2 V ¬O3 V ...V¬On
      if O is involved with O1,..,On in another accounting identity

    12. You should really start with the prior that the BEA and the accounting identities are correct and that you are making a mistake.

      You should work on it until you get them to match.

    13. Jason,

      I can symbolically verify equation's invalidity. That's not my purpose. The paper idea about government deficit spending and private paying can be either good or bad depending on

      (1) how government is spending by using G, NonG, or policies, etc.
      (2) how private sector is spending by using I, C, NonIC, stock buybacks, production reduction or increasing goods prices for excess money from government.

      We can drag a horse to a river, but we cannot enforce the horse to drink the river water. We can pump out NGDP artificially, but we cannot be sure it's higher in price level(P) or quantity (Q) since supply side(producer) determines it for their own best profits.

      Have a good weekend!

    14. You cannot "symbolically verify" a definition's "invalidity".

      If I define

      Y ≡ C + G
      S ≡ Y - C - T


      S = C + G - C - T
      G = S + T

      and so

      Y = C + S + T

      There is literally no way around this. You cannot prove this wrong because it is just a series of manipulations of a definition. You cannot "verify" its "invalidity". It is a definition.

      You obviously have no idea what you are talking about.

    15. Let me prove "it is you not me obviously have no idea what you are talking about formal systems".

      Y ≡ C + G
      S* ≡ Y - C - T
      GS* ≡ S* - PS
      Y = C + S* + T

      Just few formula is not a complete formal system. In order to become a formal system, you need to define either model theory(semantic) or proof theory(syntactic) for these symbols/formula in this formal language. Otherwise, it is just a formal language and no interpretation on symbols and formula. For example, you don't mean that G here stands for Gravity, right?

      Let me put a simple model theory for you based on your intention.

      1. G,C,Y and T here means NIPA time-series data for government investment and consumption (G=GI+GC), private sector consumption, GDP assumed private I=0, and government current receipts respectively.

      2. S* means NIPA time-series Y - C - T. I use S star for your attention since it is your definition for total sector saving. Symbol S is used for NIPA total sector saving.

      3. PS means NIPA time-series for private sector saving

      4. GS* means government saving in terms of S* - PS since S* must be equal to GS* + PS

      This formal data system violates economic data consistency principles: total income = total spending, and total saving = total investment
      1. S* - GI = GC - T ≠ 0

      Note NIPA data S-GI= 0 assumed private I = 0

      2. Y = C + S* + T ≠ C+ PS + T

      That means total spending Y is not equal to total income( government income T plus private sector income C+PS)

    16. Peiya,

      Asserting your data consistency conditions 1&2 implies:

      S* = GI
      S* = PS
      PS = GI

      Why should S* = GI? Total public + private saving is not equal to public investment. Total saving is total investment.

      Why should S* = PS? Total saving is not equal to private saving. Total saving is private and public saving.

      Why should PS = GI? Private saving is public investment? That makes no sense. Total saving is total investment.

      Again, I think you've convinced yourself you are infallible and the BEA is wrong. But really you should consider that lots of people look at this stuff all the time. It would be astounding if hundreds of people have been making a mistake for 80 years and you -- a blog commenter -- found it and remain the only person who understands it.

      If you can find me a person you've convinced that you are right, then I might consider it. But for now I'm going to side with the BEA.

      Sorry, but you haven't demonstrated any credibility with these comments and you keep making mistakes in them.

      You should really consider the possibility that maybe you are deeply confused. You are starting to sound like Egmont/AXEC and you will be banned for lack of scientific integrity. You do not "lean over backwards" to show how you could be wrong but instead just assert you are right.

      This is not just because I disagree; it is because you lack introspection. You don't ask questions or yourself.

      The first question you should ask yourself is: Why would I be the first notice a mistake the BEA has been making for years?

      If you can provide me with a good answer to that question, you won't be banned.

    17. It seems that you did not get my points that BEA NIPA temporal database is correct!!. That's our ground truth. Our defined views on NIPA database sometimes get data inconsistency with underlying NIPA database. Our math applications work with our defined NIPA data views only. Here we have 4 levels:

      BEA NIPA temporal database--> our defined database views --> our math apps -> narrative explanations.

      If final narratives seem limited, we can track back to see which level we made assumptions.

      Our NIPA database view:

      Y ≡ C + G
      S* ≡ Y - C - T

      If we want to define S* in terms of Y, C, and T for total saving, then we need to subtract nonG to make our view data consistent with underlying NIPA temporal database.

      Since it is tedious to get NonG (multiple NIPA time-series),
      we can just use government investment(GI)/saving(GS) time-series to define it.

      Assuming private I = 0 since Y = C + G
      If S* is for total saving, S* = Y-C-T-nonG = GI
      If S* is for private saving, S* = Y-C-T-nonG = GI - GS.

      Hope to get more clear what issues we addressed.

    18. "Our defined views on NIPA database sometimes get data inconsistency with underlying NIPA database."

      Why would you have your own personal "defined views"?

      Why would you have your own personal "defined views" that are inconsistent with the data?

      From what I can work out, you are basically saying you have some opinion about the BEA data that is inconsistent with the BEA data that requires you to invent new accounting identities in order for you to say that it's the BEA that's inconsistent (per your links to FRED above), not your opinion.

      That's messed up.

      Say the global climate database says the Earth is warming, but your defined database view is that it isn't. Therefore you redefine the laws of physics to add in a temperature correction called "nonCO2" that makes the data no longer shows warming?

  5. Ralph does characterize another core MMT tenet: The practical limit on gov def spending is inflation.

    But — maybe I'm hearing him wrong — he seems to think we're at max-capacity now even though:

    1. We've 25 years of declining/low inflation.

    2. All measures of inflation expectations are low.

    3. Labor-force participation rate is at historic lows.

    4. Hours worked/capita are well down from their peak.

    Seems like right now, people and businesses would be happy to just sell more stuff at current prices.

    Takeaway for government? Spend!

    1. I agree that low inflation and low interest rates traditionally (i.e. in the Keynesian tradition) mean borrowing is a better way to finance government spending than collecting taxes.

      I do personally question whether fiscal policy or monetary policy has anything to do with inflation (my thinking has come around to high inflation in the 70s being a demographic phenomenon like Steve Randy Waldman/Interfuidity). But given this seems to be a minority view, I have no problem with the traditional Keynesian view that deficit spending is fine given macro conditions.

    2. I agree there was something odd about the 1970s. My sense (at least as far as the UK was concerned) was that trade unions were in a more than normally belligerent mood and that that sparked off a wage price spiral. That idea is supported by the unusually high proportion of GDP going to wages in the 1970s.

  6. Well, I don't know if you are interested, but I have another kind of math model based on MMT. But I don't think it represents the community, and most MMTers would classify it as "mathiness", but maybe it could generate some discussion...

    It don't know if links are blocked, but here it is:


    1. The next step would be to compare it to some data. A lot of data is available here for the US (as well as other countries):


      As a side note, the equations of the form

      (∂x/∂t) (t/x) = τ

      are information equilibrium conditions x → t with information transfer index τ.

  7. "This advice is not terribly different from what Keynesian economics says. And again, I have no real problem with it."

    "Without comparing models to data, physics would just be a bunch of mathematical philosophy. And without comparing macroeconomic models to data, economics is just a bunch of mathematical philosophy."

    I stand with your claims though it is a bit like shooting fish in a barrel in this case. Even the great JMK would get bothered spotting that a useless math description mostly risks to hijack his thinking and logic. It is a fact that many critics of the insane and ideological neoclassicism and brainwashing mainstream are neither confident in their grammar nor in themselves (as such being paradoxically very anti-JMK)so as to believe that a tautological mathematics may give soundness and authority to their explanation. It is not even true that "The value is in the way it is said" as in a "poem" or literary masterpiece as for instance Frederick Nietzsche explained about science, though in his case, of Neil Wilson, it is indeed his narrative and syntax to be little flawed.
    However, his flaw narrative shouldn't out-weigh his point of view and logic, which are fundamentally correct.
    The main point in studying the economic phenomena and capitalism lies in the selection of behaviors relations and causalities.

    The neoclassicist inmates have taken over running asylum and world: Their dominant insane ideological model consists in some crazy and irrelevant assumptions which then are magically confirmed by junk mathematics. All the mathematical apparatus has only one "scientific" scope, proving and confirming the ideological presumptions fabricating a fantasy land lacking any connection and isomorphism to real world.

  8. Jason,

    I agree that Neil’s post is an example of mathiness. However, I think that your reply is also mathiness.

    Let’s stand back a little. The problem with these conversations is that everyone asks and answers different questions at the same time. As a result, there is never a meeting of minds. I’m never sure whether you want to engage with the PK / MMT guys or whether you just want to criticise them. If you want to engage, you need to separate out the various questions and answers. The questions are something like:

    Q1: What is the point of accounting identities?
    Q2: How do accounting identities work in the simplified world of economic theory and why are they still the source of such controversy?
    Q3: How do accounting identities work in the real world i.e. NIPA?
    Q4: How can accounting identities be used in forecasting? Is it the stock-flow consistency or the modelling assumptions which do the work?
    Q5: What are the implications for Keynesian policy prescriptions e.g. Keynes said that governments should run a deficit in recessions but run a surplus in booms; modern mainstream Keynesians seem to sorta kinda agree with this but have fluctuating definitions of when we are in a recession / boom; MMTers say the government should run a deficit until the point at which inflation occurs; you say inflation is not caused by government deficits, so under what conditions do you think the government should run a deficit?

    I think that the questions need to be answered in roughly the order shown. You are mostly interested in Q4 but, as far as I can see, Neil is interested in Q1, Q2 and Q5. My perception is that you and Neil parted company at Q1. Neil thinks that the point of accounting identities is to tell a logically consistent story which can be understood by non-economists. I am on Neil’s side. You might argue that telling a story is not science but Einstein told stories about people on trains and people on platforms watching each other. If stories are good enough for Einstein, they’re good enough for me.

    Regarding the technicalities of how accounting identities work in the simplified world of academic theory, I will re-iterate a comment I have made several times previously. If an accounting identity holds in ANY economy, we can carry out a thought experiment on how the identity holds in an economy with just ONE single transaction. We can then catalogue the identity behaviour for each type of transaction. The economy is just a sum of these building blocks in various combinations.

    Imagine that the single transaction is that I pay €20 in tax to the government. Note that I lose €20 from my saving (that’s the S term in the identity) and the government gains the €20 (that’s the T term). Hence, T + S = 0.

    Now imagine that the government spends the €20 by buying my labour. This time the government loses the money and I gain it. T is -€20 while S is €20. Again, T + S = 0.

    Hence, if we imagine an economy made up of millions of tax transactions and millions of government spending transactions, the sum of the T + S terms will indeed be zero as in Neil’s argument. However, that is because all of the T + S terms are zero. We don’t need complicated mathematics to understand this. This is just a consequence of what should be the first law of economics - money is conserved under exchange.

    If the government wants to run a deficit, we can create a new type of transaction(s) where the government obtains the extra money before it spends it. Neil and I would disagree about how that occurs. In the real world, the government creates bonds and sells them to the private sector. In MMT-world something else happens. Nevertheless, in either case, we can imagine an economy where the only thing that happens is that the government obtains the money for deficit spending and we would need to say explicitly how that process impacts the accounting identity.

  9. This comment has been removed by a blog administrator.

    1. I am reposting maiko11's comment, removing the dollar signs that interfere with my set up of mathjax:

      JS is quite right and the issue is not accounting identities, maybe by nature he is just harsher on confused post keynesians and more compliant with asylum inmates.
      Neil's story beside using a not needed math is flawed and hijacks JMK, independently of his good intention and correct perception of capitalism working.
      YOur considerations on accounting identities are a bit off topic and confused, but even worse you write a sort of non sense story.
      It doesn't seem you master them well yet.
      The €20 you pay in taxes maybe reduce your consume not your saving. Nevertheless, when government buys your labor force accounting identity is going to change. The €20 you get even though you do not spend in consume and save them have changed a bit the situation.

    2. Maiko11,

      I am happy to discuss this but you need to use logic rather than making sweeping statements.

      Neil’s model has two sectors – public and private. Logically, there are four type of exchange that can occur in such a model:

      Someone in private sector buys something from someone else in the private sector
      Someone in the public sector buys something from someone else in the public sector
      The public sector taxes the private sector
      The public sector buys something from the private sector.

      There are no other logical possibilities. I discussed the latter two in my previous post. The first two are trivial but I will discuss them here for completeness.

      If I sell a bicycle to Jason for €100 then Jason’s €100 moves from Jason to me. Meanwhile, the bicycle moves from me to Jason. There is still one bicycle and €100. At a macro sector level, nothing has changed. The macro-economy doesn’t care if Jason or I have the bicycle or the €100.

      This would be different if we had a three-sector model – with businesses and households split and with me as a business and Jason as a household. If that were the case, the bicycle transaction would cross a sector boundary and would be relevant at a sector level. However, that is not the case here.

      The argument is similar if someone in the public sector buys something from someone else in the public sector.

      Even if we had a three sector model (or any other number of sectors), the public sector can collect tax only if it comes from the private sector.

      If you disagree with this, please present a logical argument. I am not interested in name calling.

    3. Dear Jamie thanks for your attention and i must voice that i agree with your feelings about economy and capitalism. I just do no think necessary trying to explain JMK in a confused or wrong way. And wasting time in irrelevant entertainments.

      The main problem raised by JMK (in his short run perspective) is that in capitalism and specially in an advanced capitalism savings are in excess, causing big troubles. Of course at first a big war is always a solution.
      MMT shares that vision therefore considers public sector spending an offsetting necessity.

      About your accounting identities handling you write "Now imagine that the government spends the €20 by buying my labour. This time the government loses the money and I gain it. T is -€20 while S is €20. Again, T + S = 0."
      If the government buys your labor you get a higher income, gdp is greater and government revenues should increase too. And all this will be recorded in accounting identities.

  10. Jason,

    Have you seen this post from 2009? It is one of my all-time favourites. It is Brad DeLong (pretending to be a governor of the Bank of England from the 1920s) pointing out that Eugene Fama doesn't understand accounting identities.

    It was this post which first sparked my interest in accounting identities. How could a famous economist (now a Nobel winner) fail to understand a simple identity which is taught in entry-level courses? This could happen only in economics.

    Note that the problem is that Fama doesn't understand the meaning of at least one of the terms.


    1. DeLong is a joke and as much incompetent as Fama.
      The problem is not that Fama doesn't understand accounting identities which are just a convention, (he understands them very well) but that Fama according to asylum inmates of whom he is a honor member gives a fictional interpretation apparently similar to the "Treasury View".
      If two individual spend a week counting people passing in front of a door their statistic should be equal and an identity disregarding some minor errors.
      If one of them additionally assumes that all those people are going to Miami that's usually just a sign of insanity.

    2. Maiko,

      You know that just ad hominem calling people "jokes", "incompetent", or "asylum inmates" or referring to things as "fictional" or "nonsense" without any supporting evidence or references is just childish name-calling.

      If this continues, you will be banned from my blog as your comments do not display academic integrity.

    3. Okay, well i am aware i am spending time on someone else blog instead of doing something for myself like going to gym and i know you can do here whatever you like. I should just blame myself.

      About DeLOng assertion that Fama cant understand accounting identities it is false and, speaking of integrity, misleading. I do not trust him.
      Accounting identities are just a convention, being identities the two sides express the same thing or phenomenon statistically measured in different ways.
      So everybody knows what I (investment) includes.
      Fama as all zealots and fanatics assumes that everything is going well according to Say law and so on.
      Hence what he says is logically coherent according to his vision or model. I repeat that it is not a problem of understanding accounting identities as DeLOng claims. Maybe he is the one not understanding well the issue or he is just a bit intellectually dishonest.


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