Sunday, 17 February 2013

Fiscal Multipliers Explained (and Proved)

The Panic of 1837, blaming the Treasury Policies of Andrew Jackson
The IMF spokesperson, Mr Jeffrey Rice, said the following in a Press Briefing in Washington about 10 days ago:
(...) there's been a lot of discussion of this fiscal multiplier, which is probably something very few people had heard of until some months ago.(...)
I have shown the inaccuracy of Mr Rice's statements in a previous post and have no intention of doing the same again. It appears that the publication of Oliver Blanchard and Daniel Leigh's IMF paper about the problems of forecasting multipliers has caused more stir than it should, because it essentially does not say much. Quoting from their conclusions (page 19): "Our results suggest that actual fiscal multipliers have been larger than forecasters assumed. But what did forecasters assume? Answering this question is not easy, since forecasters use models in which fiscal multipliers are implicit and depend on the composition of the fiscal adjustment and other economic conditions."

What has not been noted in the existing literature of comments about the paper is that it does not really estimate multipliers or anything similar. It just points that the estimations concerning these multipliers, when the IMF staff were making them in 2008, can be proven wrong, ex post. (For those interested the IMF staff assumed a multiplier value of merely 0.5. We will see later why this does not appear to be very valid)

Thus, what we may infer from their comments is that the previous models of estimating multipliers had been wrong, yet they are not proposing how these errata can be fixed. In fact, they are not actually even saying how a multiplier actually works or how it affects the economy. Since many economists and politicians do not appear to understand the rationale of fiscal multipliers, I have decided to play the game on their own terms.

Let us consider the following model:

In each country the yearly output is measured by GDP=G+C+I+NX where G stands for government spending, C for Private Consumption, I for Private Investment and NX for Net Exports (Exports - Imports). We assume then that G is increased by X billion. The amount of G here is unimportant to the result we are about to reach.

First of all, we have to understand where the X billion chunk of Government Spending goes to. Wikipedia states that Government spending is the sum of government expenditures on final goods and services. It includes salaries of public servants, purchase of weapons for the military, and any investment expenditure by a government. Thus, we may safely conclude that this amount goes right into the consumers' pockets in forms of salaries, or other payments towards companies or individuals. 

Then, what do these consumers do with that X billion "given" to them by their government? As usual, they have two options: either spend it or save it. There is also the option of repaying their loans but this falls in "saving" part for reasons which are not the purpose of this article. We may assume that this X billion amount is saved or consumed in any fraction we can think of. Here we will note it as s*X billion where is s signifies the amount saved in banks or other institutions (hiding under the mattress is not an option here!) and (1-s)*X billion the amount consumed.

As it may have become visible through this analysis, Private Consumption will be increased by (1-s)*$ billion. Then, if we believe neoclassical economics, i.e. David Ricardo or Alfred Marshall, all savings are used in the form of loans to entrepreneurs by banks and thus Investment is increased by s*X.

What can be seen here is that through the standard measures of GDP, Government Spending is calculated twice. Once as spending by the government in the G part of the equation and then partly as an increase in Private Consumption (in C) and an increase in Private Investment (I). The total increase of C and I equalsX.

Now imagine if Government Spending was to be decreased by X. The total contraction of GDP would equal 2X in the above example. (In the real world this amount might be slightly exaggerated or underrated, since other issues may also arise. For example the economy's rate of growth would cause the effect to be less but a further decrease in consumption as a result of people's fear about the future would contract GDP even more.)

For the skeptics, let us assume that neoclassical economists are all wrong and that the fraction of G which ends up in banks does not find its way to the entrepreneurs. Then the an X increase in G would mean a (1-s)*X+X increase in GDP, which is still greater than X. The decrease would go the other way around.

Are we forgetting something here though? If the reader was careful during the above analysis the fact that in the definition of government spending transfer payments, such as social security or unemployment benefits are not included should have rang a bell. Now, think about it again: transfer payments are being employed the same way as government spending, i.e. given to people for them to consume or save. Now, have a look at the following data compiled in an older post:

Now I will not even suggest that all of this money goes unnoticed in measuring Government Spending for GDP (although probably most of it does). Yet, even if 40% of the above amounts goes unnoticed and as already mentioned, people consume or save that amount, then an X cut in the benefits would mean a 0.4*X*(1-s) reduction in consumption and a 0.4*X*s reduction in investment. In total, even though these amounts are not measured in GDP their effect appears to be of great importance.

The above are in accordance to empirical findings by the IMF (have a look at this article, pages 41-43) which now estimate that fiscal multipliers are in the range of 0.9 to 1.7. (This admittance that the multipliers were actually higher than they had expected, did not actually gain much publicity, only in the form of articles like Is the IMF short for I Must Fail?) Although the 0.9 number does appear to be rather too little based on the previous analysis, it may occur if the change in spending is little and the country experiences strong growth (although it would be tremendously difficult to achieve such low values for rapid changes in the level of spending) the 1.7 one appears to be more in the line of the above simple model.

Conclusion: Government spending is affecting GDP more than we believe. This means that we either have to alter our understanding about it or change the way we measure GDP. The former is much easier. I think...

Note: The above calculations would only be important if the amount slashed off the budget is significant. A country's GDP will surely reflect a 1-2% reduction in spending, yet nobody is going to notice a 0.0001% reduction in G when the natural growth rate of an economy is about 1%. On aggregate that is...


  1. And you only cover the technical direct references here - I would add that government spending also influences private spending - there are many indiscriminate connections between the two, particularly considering how fragile economy is to rumors and speculation.

    1. Yes indeed, the psychological factors which can be affected by a cut in Government Spending cannot be easily evaluated that is why I have refrained from doing so.

      It appears that the Fiscal Multipliers are indeed much larger than we can think of...

  2. In some manner I'm actually glad is not quantifiable - working with defined operators can sometimes prove more dangerous on a wider implementation scale, while a certain amount of risk and danger means that there are fewer players willing to take that chance. However, those who do assume the risk, also end up sometimes doing more damage that 1000 players on smaller scales.

  3. In GDP=C+I+NX+G, where's tax?

    1. Direct taxes are not accounted for in the GDP equation. They are removed later yielding what is known as disposable income (GDP - direct taxes). VAT (or indirect taxes in general) are included in the prices and depending on which method is being used (e.g. expenditure approach and income approach add indirect taxes less subsidies on products to get GDP while the expenditure approach already accounts for that in C)

  4. GDP=C+I+NX+G, has already calculated GDP, and you can't calculate GDP without knowing what tax is. So tax must be there in GDP=C+I+NX+G somewhere.

    1. As said in the previous comment, it depends on the method and the tax

  5. You said first we have to understand that the €X billion of G goes into the pockets of consumers. But before that, it came out of the pockets of consumers as tax, so there's no gain.
    Governments spend tax. Tax is not in C+I+NX, it's in G.
    GDP=C+I+NX+G + €1 shows that GDP increases only €1. The €1 does not get multiplied.

    1. It depends on where the money to fund G comes from: is it local banks? Is it foreign entities? Is it an increase in money supply? In all of the above cases an increase in G means that the consumers are earning more so the extra €1 is in fact multiplied. Perhaps the specific (emphasis: specific) multiplier of G is not as high as stated here (always open to new ideas) but the fact is that G does have important effects on GDP.

      The situation is not as clear when it comes to increasing G, but it makes much more sense when it comes to decreasing it. I'm not arguing for a positive multiplier here but for a negative one.

  6. Are you still trying to tell me that G isn't funded by tax?
    That money borrowed by the govt for G, isn't repaid by taxing citizens?
    That inflation isn't a tax on citizens' purchasing power?

    G is funded by Joe Public's taxes.
    If pols don't spend €1 in G, Joe gets to spend it in C+I+NX.
    So there's no positive multiplier from pols spending Joe's money,
    and no negative multiplier from them not spending Joe's money.

    GDP=C+I+NX+G + €1 shows that GDP increases only €1, no matter who spends the €1, or what they spend it on.

    There's no multiplier. And that's why we see all these huge national debts from G.

    1. Please read what I am writing: not entirely funded by taxes. If there is an increase in the money supply then there surely is a multiplier. Nothing more nothing less. Inflation has nothing to do with this sorry. And no, not all money borrowed is repaid by taxation if the sovereign can print money.

      If Gov just taxes and does not spend then we also have a reduction of purchasing power. If it does, then the same euro/dollar/whatever is spend once by the government (counted as 1) and again by less from the receiver (i.e. consumption+savings-tax=1) and so on. This is like the geometric series 1+d+d^2+...., with d<0. As simple as that. Not going to go into any ideological wars here sorry.

  7. It's not ideological, it's math. GDP=C+I+NX+G + €1 shows that GDP increases only €1, no matter who spends the €1, or what they spend it on.

    1. Then you are assuming that banks do not create money either. No multipliers at all. You haven't done your IS-LM / AS-AD / IS-MP homework have you?

  8. I'm not assuming that banks don't create money. I'm saying that the Keynesian "multipliers" are mathematical nonsense.

    1. If you believe that people actually consume more when they have more income and they consume less when they have less then multipliers exist. Prominent example of this line of thinking are Franco Modigliani on Life-Cycle hypothesis and Milton Friedman on the Permanent Income Hypothesis. Thus, we are not talking about anything "Keynesian" here but of something evident in other schools of thought as well.

      P.S. I do not identify myself as Keynesian and neither am I a follower of any other particular school of thought

  9. What you describe, is the Keynesian fiscal "multiplier".
    And GDP = C+I+NX+G +€1 proves that it's mathematical nonsense.
    If GDP = C+I+NX+G, is 4 = 1+1+1+1, and you add €1, you get 5 = 1+1+1+1+1.
    GDP increases €1, whether it's €1 of C, I, NX, G, or divided into some combination of those.
    If it's €1 of G, then ΔGDP = ΔG = €1.
    If the added €1 is tax, the result is 4 = 1+1+1+1-1+1, for no net gain.
    You can't just ignore the results from that equation.

    1. First of all, economics is not mathematics.

      Then, think of it this way: At period 1 (and by period I do mean a very small time increment not a quarter or a year) after the increase in G what you say is correct, namely GDP increases by 1. But what matters with regards to the multiplier, happens in period 2 when the effect of this increase is still out there.

      Think of it in AS/AD terms: at t, AD (i.e. GDP) increases (i.e. shifts to the left) because of an increase in G. As producers see this, at t+1, AS increases (shifts to the right) as a result of the shift in AD meaning that output is again higher than before at their equilibrium point. The effect of the shift in AD is thus passed on to the AS curve and in addition, during this time, we have higher output than before.

      I cannot explain this any clearer. Again I repeat: what you say holds at period 1. At period 2, is where the things get interesting. The combined effect is what has been called the multiplier effect.

      Let me further make this point: the multiplier is not stable but has different values along the business cycle. Thus, even if you subtract at e.g. period 10 what you used at period 1, the effect will not be as large if private consumption and investment are rising. The effect is more intense during recessions though (see the Great Depression) and almost zero when the economy is booming (see Thatcher's policies).

      Hope I have clarified this.

  10. The fiscal "multiplier" was math from the beginning. And GDP = C+I+NX+G +€1, proves that it's nonsense. ΔGDP = Δ(C+I+NX+G), no matter who spends the money or what they spend it on. That goes for every step. There's nothing "still out there".

    1. Please read what I wrote. What you say holds in period t, the multiplier is what happens from t+1 onwards. So please, if you want to say about maths, check the sum of an infinite series of small numbers (which is what the multiplier mathematically is). What you say does not hold for every step exactly because of what I wrote above.

      I consider this discussion over, there is no point arguing when you do not even bother to read what I write.

  11. I read what you wrote, but there's no multiplier.
    1+b+b^2+b^3+...= 1/(1-b) = the "multiplier".
    GDP = Yt
    Yd = disposable income

    1) Yt = C+I+NX+G
    2) Yt = bYd+Co+I+NX+G
    3) Yt = b(Yt-T)+Co+I+NX+G
    4) Yt = bYt-bT+Co+I+NX+G
    5) Yt -bYt = -bT+Co+I+NX+G
    6) Yt(1-b) = -bT+Co+I+NX+G
    7) Yt =(1/(1-b)) (-bT+Co+I+NX+G)

    Add a €1 increment of G in equation 7. Both sides must give the same result.

    7) Yt +1 =(1/(1-b)) (-bT+Co+I+NX+G) +1

    Yt = GDP increases €1.

    The increment can't be multiplied. The Keynesian "multiplier" adds the €1 inside the parentheses, illegally placing addition before parentheses and multiplication, and producing a different result on each side, and a different result from equation 1.

    1) Yt +1 = C+I+NX+G +1

    The infinite series idea also doesn't account for the "saved" fraction of income, which is Yt(1-b) = -bT+Co+I+NX+G. If (-bT+Co+I+NX+G) disappeared, it wouldn't be in the equations, and there would be nothing to "multiply", including G.

    Keynesians also don't account for:
    1+(1-b)+(1-b)^2+(1-b)^3+... = 1/b

    If you're really looking for truth and real life, check the math.

    1. Why in equation (2) do you have that bYd+Co=C (what do bYd and Co mean?)

      Still, you do not add it outside the equation! You add it in G, meaning that (7) becomes Yt +1 =(1/(1-b)) (-bT+Co+I+NX+G+1) which is what the multiplier is.

      What you say does not make any sense. You add it in the parenthesis because everything on the right (even though you should specify your terms so I can fully see it your way) adds up to what's on the left. I do not increase GDP by 1. I just increase G or C or anything else by 1. (Yes, it might hold for everything on the right under your assumptions)

      Saying that -bT+Co+I+NX+G would "disappear" makes no sense. Where would it go???

      Now you see what I mean by maths?

  12. C = bYd+Co is the Keynesian "consumption function". Look it up.
    bYd is the spent fraction of Yd
    Co [Csubzero] is "autononomous" consumption, supposedly independent of income.

    Some numbers with b = 0.8, and an added $1 of G.
    Keynesians say equation 7 returns $100:

    1) Yt = C + I + NX + G
    100 = 80 + 5 + 5 + 10
    100 +1 = 80 + 5 + 5 + 10 +1
    101 = 101

    7) Yt = (1/(1-b)) ( -bT + a+I+NX + G)
    100 = (1/ 0.2) ( -8 +8+5+5 + 10)
    100 +1 = (1/ 0.2) ( -8 +8+5+5 + 10) +1
    100 +1 = (5) ( 20 ) +1
    100 +1 = 100 +1
    101 = 101

    The Keynesian method:
    7) Yt = (1/(1-b)) ( -bT + a+I+NX + G)
    100 = (1/ 0.2) ( -8 +8+5+5 + 10)
    100 +1 = (1/ 0.2) ( -8 +8+5+5 + 10 +1)
    100 +1 = (5) ( 21 )
    101 = 105

    2) Yt = bYd+Co+I+NX+G
    b = Yt-(Co+I+NX+G) / Yd
    You can't increment G without changing the value of b.
    Keynesians keep the value of b fixed at 0.8, by using illegal math.

    If you follow the order of operations
    or just use Yt = C+I+NX+G
    you don't have the illegal math problem.

    Yt(1-b) = -bT+Co+I+NX+G doesn't disappear.
    You're the guy saying that it does, with:
    1+b+b^2+b^3+... = 1/(1-b)
    which only accounts for the spent fraction bYt.

    If you account for both bYt and (1-b)Yt,
    the b in the series is replaced with b + (1-b) = 1
    and each step is just another $1 added.

    1. Why use that? Use the usual IS/LM function... Don't know if that even makes sense or if labeling it Keynesian is correct... Unless you mean that Co=c*Yt meaning it's a fixed percentage of income which I can accept. Otherwise this does not make sense to me.

      You say:
      "b = Yt-(Co+I+NX+G) / Yd
      You can't increment G without changing the value of b."

      Yes you can, if you increase Yt or Yd...

      I didn't understand what you claim I say that it disappears.

      If you account for both bYt and (1-b)Yt then you cannot use the term for taxes. It is because you spend out of new income that the multiplier exists

  13. The consumption function is not my idea. Keynesians thought it up. Look it up.
    Co isn't cYt, or bYt. It's Co = C - bYd.

    Take equation 7:
    7) Yt =(1/(1-b)) (-bT+Co+I+NX+G)
    divide both sides by (-bT+Co+I+NX+G):
    Yt/(-bT+Co+I+NX+G) =(1/(1-b))
    That's the "multiplier". So equation 7 actually says:
    7) Yt =(Yt/(-bT+Co+I+NX+G)) (-bT+Co+I+NX+G)

    Keynesians would add $1 to the G in the multiplicand (-bT+Co+I+NX+G),
    while leaving the value of the G in the denominator of the multiplier (Yt/(-bT+Co+I+NX+G)) unchanged. You can't increment G, without incrementing G.

    It's illegal to increment either of the G's, but if you do add the $1 to both G's, the value of the "multiplier" changes so that Yt will increase only $1.

    You can't increment G, without changing the value of b, and changing the value of the "multiplier". Keynesians want the "multiplier" to have a fixed value greater than 1.

    If you are using
    1+b+b^2+b^3+... = 1/(1-b)
    you are only accounting for the bYt, in Yt = bYt + (1-b)Yt.
    That means that you don't account for the spending in (1-b)Yt = -bT+Co+I+NX+G.
    The"new income" you have talked about is G.
    If you don't account for (1-b)Yt = -bT+Co+I+NX+G, then you don't account for G,
    and don't account for new government spending, which is nonsense.

    Taxes are accounted for in G.
    1) Yt = C+I+NX+G
    2) Yt = bYd+Co+I+NX+G

    In equation 3:
    3) Yt = b(Yt-T)+Co+I+NX+G

    Keynesians substitute Yt -T for Yd, which means that:
    Yt -T = Yd
    Yt = Yd + T

    So we've got:
    Yt = C+I+NX+G
    Yt = Yd + T

    Yd = C+I+NX
    T = G

    Keynesians use T = G in their Tax Cut Multiplier, and Balanced Budget Multiplier scams.

    T = G is their problem, not mine.

    1. I'm not a Keynesian as already said and not a follower of any other school of thought. If I do not like a function then I don't care about looking it up.

      I fail to see your mathematics making sense. All you are left with is Yt=Yt which is exactly what you have to be left with. Again, you are not employing time in your equation. By saying Yt you are implying that only what happens at t matters and as I've already said (more than once) the multiplier is what happens in the future.

      I also fail to understand your obsession with Keynesianism. Why on earth does T=G? Final note: what you are saying holds only CONTEMPORANEOUSLY! Not if you add more periods as then it would be the integral or the summation of all future streams of payments.

  14. The fiscal "multiplier" is pure Keynesianism.
    Keynes invented it.
    You're using it.
    And it's mathematical nonsense.
    The 1+b+b^2+b^3+... = 1/(1-b) story, which you use,
    is a Keynesian cover story
    which is contradicted by the Keynesian equations.

    If you don't like the Keynesian consumption function, then you won't like the fiscal "multiplier", because the consumption function is part of the fiscal "multiplier"equations which include T and G.

    The equations are not mine.
    They're Keynesian equations.
    Math is math, and I'm just pointing out what the Keynesian "multiplier" equations actually say.

    Both sides are dollars, euros, whatever.
    Both sides are always equal, so the equations always say Yt = Yt.
    And both sides must give the same result.
    If you add a $1 increment of anything,
    Yt increases $1 ... on both sides: Yt +1 = Yt +1.

    I am accounting for time.
    I also account for the "saved" fraction of income.
    1+b+b^2+b^3+... = 1/(1-b)
    does not account for the "saved" fraction: (1-b)Yt = -bT+Co+I+NX+G.
    When you do account for (1-b)Yt, you must replace b with b + (1-b) = 1, and get 1+1+1+1+1 ... as far into the future as you want to go.

    Yt is total income, which is GDP. The little "t" is a label, which indicates total, the same as the little "d" indicates disposable in Yd, the same as the "o" in Co, indicates exogenous or autonomous.

    T = G:
    1) Because that's what the "multiplier" equations say.
    2) Because Keynesians use T = G in the tax cut "multiplier" scam, in which they say government spending is better than tax cuts. They use T = G in the balanced budget "multiplier" scam in which they say that if T is increased $1, and G is increased $1, then Yt increases $1.

    If you don't like T = G, complain to Keynesians.

    Add any number of periods or payments you want,
    and it's always 1+1+1+1+1 ..., as far into the future as you want to go.

    But then if you actually think:

    7) Yt = (1/(1-b)) ( -bT + a+I+NX + G)
    100 = (1/ 0.2) ( -8 +8+5+5 + 10)
    100 +1 = (1/ 0.2) ( -8 +8+5+5 + 10 +1)
    100 +1 = (5) ( 21 )
    101 = 105

    is a correct result,
    and that addition comes first in the math order of operations,
    then you shouldn't be complaining about math not making sense.

    Does 80 + 20 + 1 = 101, or 105?
    Does 5 x 20 + 1 = 101, or 105?

    If you're looking for truth and real world,
    check your assumptions,
    and check the math.

    I'm finished.

    1. Sorry I do not find it easy talking to people who are creating straw men and cannot see the errors in their own math.

      Your error is that you do not increase Y unless you increase the other 4 parts. The right hand side defines Yt and NOT vice versa. It's like saying that increasing sugar crops means increasing the sun and not vice versa. You do not increase GDP and increase everything else. You increase everything else and by that you increase GDP.

      All I had to say and what I have been saying for a very, very long while but you have not been paying attention.