Tuesday, May 14, 2019

Accounting identities and conservation laws

David Glasner brought me into a twitter thread that resulted in a disagreement between Noah Smith and myself which is unfortunate because I think we're saying the same thing regarding "accounting identities" in economics being arbitrary definitions. Noah just seems to think that conservation laws aren't also (direct consequences of) definitions. First, let me get all relativistic (in both senses).

Newton's laws are basically a series of definitions that say "I am defining a quantity called momentum (1st and 2nd laws) that is conserved by definition (3rd law)" [1].  This turned out to be one of the most useful definitions in science — though it was counterintuitive at the time. Imagine people reading the 1st law about objects tending to stay in motion when everything in their life usually ground to a halt due to friction.

This definition of momentum led to being able to predict the orbits of comets and the paths of projectiles pretty well. Emmy Noether eventually discovered the reason: it's because the universe has an approximate translational symmetry such that laws of physics at a point x is the same as the laws of physics at a point x + dx. It gets a few things wrong, like the orbit of Mercury — that's because the actual symmetries are Lorentz invariance and general covariance. But by "actual" here, we mean that they dynamics that result from the definition of momentum that arises from assuming Lorentz invariance (4-momentum) gives the results we measure. We also often arbitrarily separate the momentum conserved due to rotational symmetries (angular momentum) from the momentum conserved due to translational ones.

But the universe doesn't care about momentum or its conservation — we humans defined it based on a way we decided to decompose the universe that we found useful. And in the end, it comes down to the definition of what a derivative is. That x + dx is directly related to the momentum operator d/dx and the better definitions of momentum that are conserved have covariant derivative momentum operators. So, it's really our human definition of calculus.

As we found issues with the definition of momentum, we've expanded it and made it more nuanced — because the purpose of the definition of momentum is to get empirically accurate theories, not hold on to Newton's definition.

Now not being an economist I may get this wrong but Simon Kuznets' original definition of GNP/GDP as the market value of final goods and services produced in a quarter (or year, or other time period) was so defined because people thought it would be useful as a measure of production related to the current level of employment. If a Starbucks barista made you a coffee this morning, it'll be in this quarter's GDP. If I sell you my vintage 1980s Dougram collection, it doesn't employ anyone in the current period so it's not in GDP.

Just like how momentum was defined in order to try and produce a useful theory of motion ("physics"), GDP was defined in order to try and produce a useful theory of employment ("macro"). It's just a definition:
GDP = (your cup of Starbucks Coffee) + (my cup of Starbucks Coffee) + ... (other Non-Coffee final goods and services produced in 2019 Q2)
We can also arbitrarily group (partition) them:
GDP = C + NC
That arbitrary grouping is probably not useful. But this one has stuck around for a long time:
GDP = C + I + G + NX
We call this arbitrary grouping a national income accounting identity. Now just because it has stuck around doesn't make it right, but it does capture one useful aspect of modern economies — G tends to move all at once with changes in government fiscal policy and can move in the same or opposite direction relative to e.g. C. Empirical data appears to show that changing G can be used to offset the collapse in C during a recession, for example. In a long-ago blog post, I discussed how it might be useful to think of an additional financial sector (which redefines C, I, and NX a bit) so that:
GDP = C + I + F + G + NX
That's another arbitrary grouping that's purely a definition. But like our evolving definition of momentum that's been found wanting on occasion, we can evolve our definition of the national income accounting identity to pick out a financial as well as a government sector. Why? Because the financial sector is also large and may move in the same direction all at once like in 2008. If we think of the distribution of growth rates of various companies and government entities in terms of their contribution to GDP, the whole collection will have some average growth rate based on the average of that distribution:

GDP growth will be the ensemble average. Like partitioning down to the individual coffee level, this may or may not be useful. However, if we group the financial sector into one big box (gray) and the government sector into another (blue), we instead have maybe something like this:

If there's a financial crisis, then maybe the whole financial sector shrinks:

and the new ensemble average growth rate results in a GDP that declined in that quarter. Shifting the government sector up could potentially offset that a bit. Again, maybe this arbitrary grouping is useful and maybe it isn't — a lot depends on the interactions of the various pieces (I talk about that a bit more here).

The main point is that definition of GDP and the accounting identity partition of it are both completely arbitrary, but like the arbitrary definition of momentum (which is based on our calculus-based approach to physics) they might be useful.

It's true per Noah's original tweet in the thread that we don't want to put too much weight on what is essentially an arbitrary definition that might not be useful. And you definitely want to be careful about reasoning from the identity alone — also because calculus:

This graphic lays out the possibilities for the interaction of those sectors (including the little boxes) in the distribution pictures above (but this graphic is for levels, while the distribution is for growth rates). The picture I showed in the distributions is the upper right where the change in the financial box doesn't do anything to the other boxes.

Macroeconomics is a nascent science compared to physics — Newton's definition of momentum is from 1687 while Kuznets' definition of GDP is from 1934. So yes, by all means recognize that the definition of GDP and its various accounting identities are arbitrary definitions. GDP seems to be a useful macro definition for studying employment and fiscal policy does seem to have an effect on the economy warranting a separate G term. Maybe there are better — more useful — definitions. But don't go too far in the other direction and think that the enormously successful definition of momentum as a conserved quantity makes it not arbitrary. It's still just a label we humans applied to the universe.



[1] Actually Newton's Lex II was a bit vague:
Lex II: Mutationem motus proportionalem esse vi motrici impressae, et fieri secundum lineam rectam qua vis illa imprimitur.
A somewhat direct translation is:
Second Law: The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.
The modern understanding is:
Second Law: The change of momentum of a body is proportional to the impulse impressed on the body, and happens along the straight line on which that impulse is impressed.
Where momentum and impulse now have very specific definitions as opposed to "motive force" and "motion". This is best interpreted mathematically as

I ≡ Δp

where I is impulse and p is the momentum vector. The instantaneous force is (via the fundamental theorem of calculus, therefore no assumptions of relationships in the world)

I = ∫ dt F

F ≡ dp/dt

where p is the momentum vector. The alteration of "motion" (i.e. momentum) is Δp (or infinitesimal dp), and the rest of the definition says that the vector (and impulse vector I) is parallel to the vector. Newton would have written in his own notes something like f = ẋ using his fluxions (i.e. f = dx/dt).


  1. When I started reading economics, I asked myself a couple of questions: “Are accounting identities useful” and “Why are accounting identities so controversial”? What I found was a totally dysfunctional debate.

    Let us go back to basics. GDP was created as a measure of PRODUCTION. It is a measure of the creation and dispersal of new things in the economy, as opposed to the trading of existing things.

    I could write the most basic identity:

    Total economy = GDP + non-GDP.

    Note that this trivial identity is missing from all mainstream economic discourse. Consider the monetarist claim that it is useful to think of GDP as the amount of money (M) times its velocity (V).

    I could rewrite my basic identity as:

    Total economy = (V * M) + non-GDP.

    That raises the question of what monetarists think happens in the non-GDP part of the economy. Does it use money? Does that money have velocity? Why is the velocity component of the GDP term important when no attempt is made to describe the non-GDP term in a similar way?

    This is a real problem because some monetarists think that increasing the money supply will cause inflation. However, that intuition assumes that new money will always inflate the GDP term rather than the non-GDP term. This is important for policy purposes as a policy such as QE merely replaces government bonds (in the non-GDP term) with money. However, the people who bought the bonds initially did so because they had no GDP-related spending requirement for that money, so they will use the new money to speculate in asset markets. If instead, new money was created and passed to poor people who would spend it on food and clothes etc, we would see a completely different outturn.

    My point here is that economics fails even at this extremely basic level. I might add another issue. Apparently, we need to control “inflation” in the GDP term as that is always “bad” whereas we do not need to control “inflation” in the non-GDP term as that type of inflation is “good”!

    My underlying question is why does no-one make this obvious point? I have no idea but, from my perspective, it is not a good look.

    Another example. We know from science that materials and energy are conserved under change. We could write a trivial identity:

    Material & energy inputs = Material & energy outputs

    This would be true at any scope. At the scope of all GDP products, we could write:

    GDP inputs = Useful GDP outputs + Waste GDP outputs

    How do we get from that identity to economics? We assign a financial value to the Useful GDP outputs term. We then ignore the other two terms. We then promote policies to increase the Useful GDP outputs term. What could possibly go wrong?

    Why is that science? Why is applying mathematical analysis to the Useful GDP outputs term science? You cannot make something science just by using mathematics.

    If climate change has significant adverse impacts later in this century, future generations will ask what happened? They will figure out the argument I have just made as it is a trivial argument. They may then conclude that our academic economists were amongst the most stupid people who ever lived.

    Compared with these examples the Keynesian accounting identity is a work of genius. This comment is already too long but your logic here is obscuring a simple but useful identity. The key point is that in GDP

    Income = Expenditure

    You can break that down into however many sectors you want. The traditional sectors exist because they helped answer the questions that the originators of the identity wanted to answer. If you change the question, you can change the sectors required to answer that question e.g. add in a Finance sector.

    One of the things that has gone wrong in economics is that important questions have got lost amongst useless methods. Another is that economists have failed to link economics back to basic science. Another is that basic conservation principles, like those in school level physics and chemistry, have been mangled into hopeless discourse.

    1. There is some definite ways to make sense of

      total economy = GDP + non-GDP

      where the typical non-GDP components are e.g. household production (cleaning your own house, writing blog posts) or the buying and selling of not-current-quarter produced goods and services (selling my Dougram collection as I mention above).

      But in that definition, the operation of "+" is ill-defined unless we talk about either side as "actions". Some "actions" have GDP-related correspondences like household production (cooking your own dinner vs getting take out) but others do not (taking a shower). In that sense it's hard to define "+" and make this a meaningful accounting identity. Although I didn't go into it in the post above, a lot of accounting identities are of the form

      A = X + Y + residual

      Which may or may not be useful. GDP has "I" which represents goods produced in the current quarter but not consumed or exported. Of course that definition means "I" gets into all kinds of debates about what economists mean by investment versus what a normal human means by investment.

      My overall point was not that some definitions are good to answer certain questions and others are not, but rather at a deep network level there are parts of GDP that act as connected components — correlations that are meaningful and actually empirically measureable.

      We could treat the ocean as just a bunch of milliliters of water combined, but there are definite underlying structures (the eastern pacific associated with El Nino/La Nina, the gulf stream). These emergent structures that depend on a particular coarse graining are what I'm talking about. It's not just defining things as we want to study them, but rather discovering the components in the macroeconomic system that are relevant to understanding the behavior of macro systems.

  2. Jason: “the operation of "+" is ill-defined unless we talk about either side as "actions"”

    That is a very Jason-like comment! In fact, the economy itself is both ill-defined in scope and somewhat arbitrary when we look at the detail. That is one of the biggest problems in studying it.

    There is always a sense that:

    Total economy = total measured economy + total unmeasured economy

    so, if I were being more precise, I should write

    Total measured economy = measured GDP + measured non-GDP

    where the measured economy is made up of recorded transactions of various types (the things we do e.g. mining, production, exchange) and recorded valuations of assets & inventories (the things we have e.g. raw materials, finished goods in GDP; property, shares, bonds in non-GDP; money in both).

    I agree that we need to be careful when things move from measured to unmeasured or vice versa. My favourite example relates to the purchase of a book in the 21st Century versus the 20th Century.

    In the 20th Century, I would have to drive to town, walk to the bookshop, browse the shelves, standing in line to pay etc. In the 21st Century, I order the book from Amazon in 30 seconds and wait for it to be delivered. There is a major time saving for me. However, my time saving is part of the unmeasured economy, so economists are unaware of it.

    I recall a post of yours where you quoted a famous economist saying that he could not see much evidence of productivity improvements from the internet! That is, in part, because much of the improvement is in unmeasured time savings for the consumer.

    Jason: “a lot of accounting identities are of the form A = X + Y + residual”

    That is not correct with respect to Keynesian accounting identities. As I said, the essence of these identities is

    Income = Expenditure

    Paul Krugman likes to say: “one person’s expenditure is another person’s income”.

    If your Keynesian identity uses four sectors, you break both Income and Expenditure into four terms – one for each sector. There is no residual. The term GDP does not appear anywhere in a Keynesian accounting identity even though the scope of the identity is GDP.

    I am not a fan of MMT even though I agree with them that accounting is important in economics. However, they have a historical chart of the sectoral balances for the US (where a sectoral balance for a sector is Income - Expenditure for that sector). It is a striking chart as it is symmetrical, so it forces the realisation that the sectoral balances do indeed add to 0.

    There are real issues surrounding the Keynesian identities. However, they are concerned with ambiguity in the definitions of terms, and the fact that most economists do not understand basic accounting.

    Jason: “that definition means "I" gets into all kinds of debates about what economists mean by investment versus what a normal human means by investment”

    The debates regarding the definitions of Keynesian accounting identity terms concern what accountants mean (and what is consistent with what accountant mean) versus what normal humans mean. This is one of the rare occasions where economists are in the normal human group.

    Jason: “That coarse graining is something I talk about here”

    This is interesting. I will comment separately on that post.

  3. By the way there is another macro-accounting identity called the Kalecki Profit Equation that is similar in nature to the traditional Keynesian identities. However, it is different in important detail.

    Even if you dismiss my arguments about consistency with accounting, and even if you dismiss my non-economist view, there is still an issue within economics as the two identities contradict each other. There are underlying questions that are never debated openly by economists concerning why the two identities are different and which of them is a more accurate representation of the macroeconomy.

    My money is on Kalecki. However, this is yet another area where focusing back on specific underlying questions would improve the quality of economic discourse.


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