Thursday, March 31, 2016

Does saving make sense?

There were a couple of posts that came up recently that discuss how "saving" doesn't mean what you think it means (in economics). Here's Nick Rowe with a good macro class version. Here's Steve Roth's critical take on the term. (N.b. I'm not critiquing either post in this post).

I find Nick Rowe's (i.e. the traditional) definition of saving to be a perfectly acceptable definition:
... "saving" means literally anything you do with your income except buying newly-produced consumption goods or paying taxes.
Which is encapsulated in the identity Y = C + T + S. This says there are two economically interesting things you do with your income (consumption and pay taxes = C + T) and everything else (S = "saving" = boring!). Let's try to illustrate what is going on in the macroeconomy using the information equilibrium picture (e.g. here). Say taxes are blue, consumption is red and saving is everything else (yellow). The growth states of a macroeconomy might look like this:

This says e.g. consumption on item X went up in location Y, while taxes collected from activity A went down in state S, etc. Integrating up these diagrams from each quarter (year, month, etc) gives you the level of the contribution to Y on that color coded segment.

Does it make sense to break up Y into C, T and S in this case? Not really. Each type is uniformly distributed throughout the states. But what if there's a central government and all changes in taxes collected are correlated? Well, in that case we have a diagram that looks like this:

And in this case, it does make sense to break out T.

However, since S is "everything else" in the economics definition, it doesn't really make sense to break it out. It should be Y = C + T + R where R means residual. My diagram above makes a case that Y = X + T where T is taxes and X is everything else.

That's the idea behind breaking up income Y into a set of separate measures: you've identified some highly correlated components of your economic system -- so correlated, they appear to move together as a single unit to a decent approximation. And you've likely identified some sort of effect on the economy that goes along with that correlated component.

However, that also means that in most of those national income accounting identities, there will be a "residual". In Y = C + I + G + NX, it's I. In some post-financial crisis models, there's an argument to add a financial sector (gray):

This model is

Y = X + G + F

where G is government spending, F is the financial sector's contribution to GDP (whatever that is), and X is everything else.

Note these break-outs don't have to move exactly as a single block, and you could have something that looks like this at some period of time (imagine a transition from the first graph above to this graph -- a jump in consumption growth and a fall in "saving" growth, with Y growth staying the same):

In this case, it's a bit fuzzy to nail down exactly what is happening to what (this is probably the most realistic version -- with the large box versions being an approximation to this diagram). But overall, there are a few things that appear to operate as correlated units.

So the bottom line is that an accounting identity of the form:

Y = A + B + ... + R

Is a statement that someone (at some time) found A and B (and ...) seem to move as correlated units in a way that is useful to explain some effect or policy. And then there is everything else (R).

In our old friend the national income accounting identity, we have

Y = C + I + G + NX

where R = I. This identity happened because in the Great Depression consumption of newly produced goods seemed to be correlated (and falling), and while it was theoretically possible to try and export your way out of that mess, not everyone could (and most of your trading partners had the same problem). So boosting government spending was a potential solution.

This only makes sense if the real empirical distribution looks like that last one with G in blue and say C + NX in yellow and I (the residual) in red.

But if that isn't what the empirical distribution looks like, then boosting government spending (moving blue boxes) might just displace some red and yellow boxes with no net result -- e.g. due to Ricardian equivalence, or monetary offset. That is to say: declaring Y = C + I + G + NX does not mean the distribution usefully breaks up into C, I G and NX.


Update 1 April 2016

In case you were wondering if this is how a real economy looks, here's some data. I used the components of CPI instead of nominal output, but that should be a good proxy. They should be weighted by the relative size of each component, but this just counts up the number of CPI components with a given growth rate (and normalizes it).

As you can see, this shows a peaked distribution around the CPI growth rate (inflation rate), much like the graphs above (although those are nominal growth rate).


Update 10 December 2016: Which definition?

Steve Roth stops by in comments below to dispute my interpretation of the national income accounting identities. Let's consider two possible definitions of what we could mean by the "accounting identity" Y ≡ A + B. First a picture, and then some words:

  1. Y is defined as the sum of A and B which are independently defined.
  2. Y is independently defined. A is independently defined as a subset of Y. B is then "not A" and Y = A + Y\A (A plus the Y-complement of A).
In the second definition, B is the residual (what is left over after defining A), and there may be things in B (subsets of B) that are not (loosely speaking) well-defined except in terms of Y and A.

Now let's say (in the following, by consumption, I mean private consumption, and everything is relative to the "current year" and for "final use"):

Y = current-year produced goods and services
A = current-year produced goods and services that are consumed (C)
B = current-year produced goods and services that aren't consumed (I)

It seems obvious to me that we are looking at the second definition when it comes to the national income accounting identities Y ≡ C + I and  C + S (ignoring government and exports, but including them just creates three well-defined A's above, with I and S still remaining residuals).

It is critical to note that neither definition has any implications for the relative dynamics of Y, A, or B. It's a definition, not a model (or as Steve puts it, counterfactual). More on this here.

We can adapt the old example of shoes and socks used to discuss the axiom of choice in math to help intuition here.

The first definition could be A = left shoes and B = right shoes and Y = shoes. And that works for the other definition as well because the complement of right shoes in all shoes are the left shoes.

However if Y is instead the set of things you wear on your feet, then the Y-complement of right footware includes not only left shoes but socks such as tube socks that are neither left nor right. That is to say the second picture is the one we should have in our head, not the first.

When we look at Y = C + I (ignoring government and exports for simplicity), we have a situation where Y is defined and C is defined (Y that is consumed) leaving I to be the residual (Y that is not consumed, which could be anything and which economists confusingly dub "investment" since most normal people consider buying stocks or bonds "investing" rather than e.g. toilet paper bought for company bathrooms).

Steve additionally says that I isn't a residual, but S is (in Y = C + S), which is problematic for maintaining total income equal to total expenditure.

In our shoes and socks example, we have footware sold and footware bought. These should be the same.

However in one case Steve says the definition Y = C + I is more like definition 1 rather than definition 2 while the other (Y = C + S) is more like definition 2 than definition 1. This means in one case (items sold) we could potentially have things in the complement that aren't the "opposite" (e.g. the tube socks) that don't appear in the definition of the other (items bought).

We'd fail to maintain the income/expenditure identity. That is to say

C + S = Y = C + I + X

where X is unknown. To be explicit with the footware example, we have:
Left shoes bought + not left shoes bought = footware = left shoes sold + right shoes sold + other stuff sold (tube socks)
But previously we said as a matter of definition Y ≡ C + I. So X must be zero (i.e. the empty set).

Essentially, Steve must be changing the definitions of savings and investment to be different from economists' definitions. That's fine, but he can't the call me out for being wrong for not going along with his definitions. In those cases we usually say something like "I don't like those definitions" ‒ and I agree! That was the point of the my post above. I think my post gives a far more well defined way to proceed. We can even address some of those dynamics questions by linking the definitions of our partitions of Y to partitions that have sensible dynamics. Keynesians like to think that Y = C + I + G means that boosting G boosts Y, and if G is a correlated unit (as representing in the post above by the big blue box), then that is probably true. And if I is a residual, then there's really no telling what happens to I when G changes (∂I/∂G is not just model dependent, but not even well-defined).

The key issue is that when you define something by what it is not, you have to be careful about what you've put in your set.


  1. Text: "That's the idea behind breaking up income Y into a set of separate measures: you've identified some highly correlated components of your economic system -- so correlated, they appear to move together as a single unit to a decent approximation. And you've likely identified some sort of effect on the economy that goes along with that correlated component."

    Important point. :)

    Text: "However, that also means that in most of those national income accounting identities, there will be a "residual". In Y = C + I + G + NX, it's I."

    I think that you need to flesh that claim out. It is pretty clear that for economists, technology is a residual. But investment, outside of its common meaning of buying already existing stocks and bonds (which, IIUC, is classified as saving), is a fairly well defined economic activity. One thing that is included in investment, however, that seems to qualify as a residual is inventory.

    Text: "In our old friend the national income accounting identity, we have

    Y = C + I + G + NX

    where R = I. This identity happened because in the Great Depression consumption of newly produced goods seemed to be correlated (and falling), and while it was theoretically possible to try and export your way out of that mess, not everyone could (and most of your trading partners had the same problem). So boosting government spending was a potential solution."

    During the Great Depression it is unclear that we can treat either saving or investment as a residual. There was a vicious cycle. Every link in the cycle has to be significant. None of them can be a sink. In ordinary times it makes sense to treat saving as a residual, but if we treat saving as a residual, then the Paradox of Thrift does not arise. However, it does not arise in normal times, but it does in depressions.

    1. I don't need to "flesh that claim out" because it's not a claim. It is the definition of investment in the accounting identity.

      In the equation Y = C + I + G + NX, "investment" is anything that is not consumption of currently produced goods, government spending or net exports. It includes non-government buying of non-consumption goods (houses, production machinery), and anything else that contributes to Y isn't part of C, G or NX. It's a catch-all for the things not included in the other terms (i.e. it's all final goods and services that are not-G, not-C and not-NX), which have explicit definitions.

    2. I know that economists can be careless about labeling residual quantities. But I suspect that the original definition of I was not as a residual, but its meaning was diluted over time. If I cared enough, I expect that I could do a series of articles on changing definitions in economics. ;)

  2. Text: "That is to say: declaring Y = C + I + G + NX does not mean the distribution usefully breaks up into C, I G and NX."

    True. But as you say above, that division is something that was not just declared.

    Text: "But if that isn't what the empirical distribution looks like, then boosting government spending (moving blue boxes) might just displace some red and yellow boxes with no net result -- e.g. due to Ricardian equivalence, or monetary offset."

    Indeed. There are times when government spending will induce saving against taxes in the future, or (inclusive) a crowding out of private investment. Can we go further and say that in normal times we should expect government spending to induce some saving against future taxes and some crowding out of investment?

  3. I will say that if saving, S, and investment, I, are both residuals, then that makes the identity,

    S ≣ I

    easy to understand, since it just means

    Other ≣ Other


    1. Ain't that the truth.

      That's a lot of bars on that equal sign.

  4. Well, let me critique Rowe's post. He reads this blog, so maybe he will see this. (He has closed comments on the post.)

    Rowe received an email from a first year student that went something like this: "For a closed economy, we know that national saving equals national investment, and investing in newly-produced goods counts towards GDP. And "saving" means anything we do with our income other than paying taxes and spending on newly-produced consumption goods. So if I take $100 from my pay and stick it under the mattress, does that count towards GDP?"

    Rowe's first reaction -- and mine too, I admit --, was to say no. After all, the number of giraffes does not count towards the number of jellyfish.

    But upon reflection, I think that our shared reaction is wrong. "Counts towards" is not a technical term, and carries a connotation of "adds to". Obviously, sticking money under the mattress does not add to the GDP. But saving, which is what that is, counts toward the GDP because it is one of the things that adds **up** to the GDP. That "up" makes a difference. :)

    Now, it is not clear where the student's confusion lies, and as teachers we would want to find out. Rowe and the commenters touch on various possibilities. :)

    In his reflections in the text, Rowe goes astray, I believe. He thinks that the student should compare sticking the money under the mattress with some other action (or inaction). That is standard economics thinking. For instance, using the money to purchase a new good or service would add to the GDP, ceteris paribus, while sticking the money under the mattress would not. It is normal for economists to compare different possible scenarios, but, as Rowe mentioned later on, accounting does not work across possible scenarios. It works within scenarios.

    Let us compare the two scenarios and account for the GDP. Suppose that every other monetary transaction in the two scenarios is the same, except that in one the student saves $100 under his mattress, and in the other he pays $90 for massages for him and his girlfriend and pays a tax of $10. By comparison with the first scenario he has reduced his saving by $100, increased his consumption by $90, and increased his taxes by $10. All of this makes no net difference to Y. However, by comparison he has increased the income of the masseuse by $90, and, since the masseuse has not spent the $90, the saving of the masseuse by $90. This transaction has increased Y by $90 in the second scenario by comparison with the first.

    It should now be clear that, though buying the massages increased GDP by comparison with stuffing the money under the mattress, **both alternatives** count towards GDP, but towards different GDPs in different scenarios.

    So the answer to the student's question is yes. :) Whatever he does with the money is accounted for in each possible scenario, and counts towards the GDP for that scenario.

    1. How do we get the increase in Y of $90 in the second scenario by comparison with the first? C increased by $90, T increased by $10, and S decreased by $10, as the student's saving decreased by $100 and the masseuse's saving increased by $90. :)

  5. A partial critique of Steve Roth's post:

    First, a general comment. I think that Roth's argument could use a bit more mathiness. He does include an equation or two, but mainly relies upon a verbal argument which is vague and appears to be inconsistent. Some more math would add clarity. :)

    His main point is well taken. He defines aggregate savings as aggregate net worth, measured in money. Saving does not increase the quantity of money in the economy, nor does it increase the amount of non-monetary assets. It may add or subtract value by rearranging money or assets, but to a first approximation saving does not alter aggregate net worth.

    However, Roth wishes to go one step further by saying that there is no saving in aggregate. He does this by defining individual saving as income minus expenditures. OC, in aggregate the sum of (income minus expenditures) is zero. So there is no aggregate saving. With no aggregate saving there is no aggregate effect of saving on anything.

    The initial exchange between Roth and Ramanan is instructive in this regard. Ramanan says that Roth's definition of individual saving is incorrect, that it is disposable income minus consumption expenditure. Roth **agrees**, but states that "the Personal Saving construct does not provide useful way to think about how the economy works."

    Roth's intuition about how the economy works leads him to deny the equation,

    Y = C + S + T

    defining S to be identically zero, yielding the equation

    Y = C + T

    OC, that equation is wrong.

    Roth's overall point does not depend upon aggregate saving being zero, and setting it to zero messes up the math. Here is a case where a little mathiness would have helped. :)

    1. Hmmm. Actually, saving by buying existing assets may increase the entropy of the economy, thereby decreasing aggregate net worth.

  6. Hey Jason, just came across this. I've got a bunch of issues with it.

    1. In the national accounting, investment (aka capital formation) is NOT a residual. It's measured using empirical methods.

    2. Saving IS a (the) residual.

    3. "'saving' means literally anything you do with your income except..." No. Saving is quite explicitly NOT doing anything with your income. Saving is not-spending. That's why it's the residual.

    4. Investment IS spending — spending on newly produced goods that will not be (fully) consumed within the accounting period.

    I think there are others -- definitely the whole cap gains and balancing-to-assets-and-net-worth issue in my latest evonomics post, but that's all I have right off.



    1. Hi Steve,

      Let's assume closed economy and there's no government. Then

      Y = C + I
      Y = C + S

      If those are accounting identities, then one of Y, C, or I (or one of Y, C, or S) is implicitly defined in terms the other two. That is what I meant by residual.

      If we take Y and C to be the defined (think of a Venn diagram with Y as the big set and C as a proper subset), then it's true that I is also well-defined (and independently measureable). However I's definition must be the Y-complement of C otherwise there is no "identity".

      Let's say (simply, for sake of argument) that Y = total spending on all currently produced goods and services. And we define C (simply, for sake of argument) as those that are "consumed". Then I is defined by "all currently produced goods and services that aren't consumed". That is a perfectly well-defined set of things. But that does not mean it is not a residual -- it's definition depends on the definitions of Y and C.

      Likewise, S is also a residual because income/expenditure method says Y = Y so C + I = C + S, therefore S = I (in our simple economy). C is income from consumption or expenditure on consumption. These are equal by definition (purchases = sales). Therefore both I and S are residuals (their definition is implicit in the definitions of Y and C.

      Regarding 3), yeah, I think I got that one backwards. However, it doesn't change the meaning.

      If C + S has S as residual, then C + I must have I as the residual (where residual means defined in terms of Y and the other component -- i.e. the Y-complement of C).

      Here's a way to think of it. Let's say we look at all mass in the universe. We measure some of it in stars, dust, planets, etc. But there's more. We call it dark matter. It is all the stuff that has mass, but isn't stars. That is a perfectly fine definition (and we can measure the mass of all the dark matter, empirically, to be about 85% of all the mass of the universe). But it's still a residual -- not ordinary baryonic matter.

      (Notwithstanding Verlinde's latest paper.)

  7. Hey Jason:

    I've started a somewhat lengthy reply, but realize that much of my issue with this thinking is explained in my latest on Evonomics, the deep conceptual problem of "monetary circuit" theory, and what I call "the conservation of money fallacy."

    It's excessively lengthy, but I will timorously suggest we might move this conversation faster if you read it first? It seems up your alley.

    Briefly: the universe presumably has a fixed amount of visible/dark matter/energy. The economy does not have a fixed amount of "money."

    Related but not quite same issue, economists' widespread and egregious misconception: because Y = I + C, if C is higher then I must inevitably be lower. You see this thinking constantly. But that's only true in accounting retrospect, and it assumes Y is unchanged. In economic prospect, higher C can (almost certainly does) cause higher I (and Y). We're beyond accounting identities, into economic reaction functions. It's a "which counterfactual?" issue.

    Those reaction functions, btw, can be presented in many ways, like the slices/boxes in your presentation. What are the (heterogenous) individual reaction functions to higher C, I, or Y? (Think: agent-based simulations.) What are the sectoral reaction functions? (Much MMT thinking.) How do different economic classes react? (Sliced by income? By wealth?)

    Happy to go on at more length, but hoping this moves us forward...



    1. Hi Steve,

      I read your Evonomics post, but it has nothing to do with what I'm saying (or even what you seem to be saying). Neither does conservation of money, monetary circuits, nor the dynamics of boosting government spending. My point has to do with basic logic (set theory); I updated the post above with some pictures that might help.

      The dark matter analogy was in reference to the set theory argument I was making that is not intertemporal, so whether it grows or not (or the amount of "money" grows or not) is not germane to the argument. The economy in a given year does have a fixed amount of final goods and services purchased -- which is what the "accounting identities" are about.

      I also realized that in #3 in your comment above was me quoting Nick Rowe, and as he's a macro professor I'll stick with his version.

      In your original comment you implied I was incorrect with regard to a generic economic consensus framework, but it now seems you were calling me incorrect with regard to your own framework.

  8. Jason,

    In NIPA/FOFA/SNA2008, each sector saving is defined as follows:

    saving = capital_formation + income - expense.

    capital_formation = acquired non-financial assets from your financial expense (i.e. "non-consumed" expense). Thus, there are many ways to count capital_formation in NIPA/FOFA for different purpose

    NIPA is used for measuring production/income purpose. The capital formation counts all acquired non-financial assets from investment(I) expense including 4 kinds of financial expense: equipment, construction, IP, and inventory.

    FOFA is used mainly for measuring wealth/networth purpose. The capital formation counts expense on equipment, real estate (construction+land), IP, inventory and durable consumer goods such as cars, etc.from consumption(C) expense

    No matter how you counts capital formation, we have this accounting identity for each sector: saving - capital_formation = income - expense.

    Sum of all sector (saving - capital_formation) = 0

    In NIPA, S - I = 0 if S includes saving from foreign sector and I here is capital_formation from I expense. S - I = Balance of Current Account(BCA) if S only counts saving from domestic sectors. In foreign sector, saving = 0(no capital_formation) + income - expense = - BCA, where, BCA (Balance of Current Account) = expense from imports/other expense - income from export/other income from a resident view.

    Note, I am not defending SFC model, What I am describing is NIPA/FOFA/SNA2008 underlined data schema(in temporal database sense) about variable relationships in terms of accounting identities.

    As you may notice, SFC is a just weak model with many assumptions. Due to these assumptions, all SFC balance equations are not true accounting identities. For example, government deficit = T - G in SFC model. It should be: government deficit = T - G - NonG(non-discretionary expense) = government Saving - government I(capital formation). This is due to SFC model did not cover all types of expense in SFC matrix. Thus, you cannot use variable names from SFC models to access Fed data to get your expected SFC equations. I think that Steve and many other people just misunderstand. the semantics about NIPA/FOFA/SNA2008.

    1. Peiya,

      Thank you for the detailed explanation, however the issue I am trying to grapple with above is much more abstract than specific accounting definitions.

      My question is whether "saving" is a well-defined concept in any partition of national income. For example, is "capital formation" a well defined concept? Is there a handle I can turn to crank up capital formation? Does that handle then reduce some piece of GDP?

      My main conclusion is that only "G", government spending, appears to be a well-defined partition. Everything else is a "residual" that has no specific behavior without resorting to a model.

      There is a case to be made that there might be a financial sector that is also a well-defined "handle" (as I also mention above).

    2. Jason,

      You can totally forget the residual concept from SFC.
      Precisely speaking, each sector "residual" has two parts: financial residual (income - expense) and non-financial residual = non-consumed non-financial assets (so called capital formation). In other words, each sector saving has two parts: nonfinancial assets and financial part in forms of financial instruments(assets/liabilities).

      All sectors in national income partition has already defined their own "saving" in terms of its capital formation, income and expense uniformly. Thus, we have government saving, households saving, business saving, financial sector, etc. from each sector partition in national income.

      GDP is defined as sum of all sector "discretionary" expense (i.e. C + I + G + NX)

      GDI is defined as sum of all sector "earned" income (i.e. wages + NetOperatingSurplus + ConsumptionOfFixedCapital + Tariffs - Subsides)

      Of course, we can derive a GDP calculation equation involved with capital formation/saving. GDP = C + G + S + NX - BCA.
      Domestic saving S here only constains nonfinancial capital formation from I expense and financial saving parts from domestic sectors are zeroed out.

      Capital formation is more useful in calculating sector wealth/net worth. We have two methods: market price or replacement cost for reevaluating financial/nonfinancial assets in net worth calculation.

      Real estate in nonfinancial assets and all financial assets/liabilities are based on market price in reevaluation. All other non-financial assets are based on replacement costs currently.

      The relationships are:

      sector saving(t) = capital_formation(t) + financial asset(t)- liabilities(t)

      sector NW(t) = NFA(t) + FA(t) - FL(t)

      Conceptually, NW(t) is calculated as follows. Details are in FOFA R and S tables.

      NFA(t) = Reevaluation NFA (t-1) + capital_formation(t)
      FA(t) = Reevaluation FA (t-1) + financial assets(t)
      FL(t) = Reevaluation FL (t-1) + financial liabilities(t).

      We have separate stock and flow accounting identities:

      (flows consistency)
      saving(t) - capital_formation(t) = sum of financial asset(t)- liabilities(t) in all financial instruments.

      Sum of all sector [saving(t) - capital_formation(t)] = 0

      (stocks consistency)
      NW(t) - NFA(t) = sum of FA(t) - FL(t) in all financial instruments

      Sum of all sector NW(t) - NFA(t) = 0

      There is no stock/flow accounting identity unless we assume no reevaluation as in SFC matrix. More importantly, flow variables such as financial saving(i.e. financial asset(t)- liabilities(t)) are not calculated from delta stock variables, but stock variables are defined in terms of flows and reevaluation of stocks up to end of previous period(i.e. t-1).

  9. It should be this GDP = C + "GC" + S + NX - BCA
    G = GC (government consumption) + GI(gov investment)

    I forget to mention one point that for these equation derivation (S - I = 0 or S - I= BCA), there no need to go through market assumptions by using demand/supply equilibrium or SFC model matrix. They are based on total sector (income - expense) = 0

  10. Jason,

    Nick's concept "saving as residual" is based on a wrong and widely-misused accounting identity.

    C + I + G + NX = Y = C + T + S or (S-I)+(M-X) = (G-T)

    ... "saving" means literally anything you do with your income except buying newly-produced consumption goods or paying taxes.

    Correct one is this: C + GC + NX = C + GC + S + NX - BCA. It simply replace all investment I + GI with S - BCA. This one

    (Private S - Private I) + (Public S - Public I) = BCA

    It is sector S-I balance accounting identity

    We need to look at saving from finance not production aspects. Saving(= capital_formation + income - expense ) is a concept of "assets allocation" to build up net worth. How we allocate our income to different financial/non-financial asset classes? Each asset class has different risk-adjusted returns during assets reevaluation.

    Fluid vs Finance Dynamics:

    Fluid dynamics: water flows from higher places to lower places.

    Finance dynamics: money flows from lower yield assets into higher yield assets.


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