Basic Math for the Math Challenged

Since Richard Posner has decided to exhibit his math skills again, I thought it useful to work through some math to see how one can obtain back-of-the-envelope estimates for the stimulus package. I’ll use Mr. Posner’s numbers to illustrate.


Stipulate that $89 billion in stimulus funds were expended (this is Economy.com‘s estimate), in combined tax rebates, transfers to support state spending on goods and services and transfers, and direct Federal spending. This is higher than the $60 billion cited by the IMF (page 6), but lower than the $100 billion cited by Dr. Romer in her study (where she used IRS information on tax cuts; as Donald Marron has pointed out, Recovery.gov reports only expenditures on goods/services and transfers).


Assume 40% of the $89 billion was transfered to the states, of which most supports spending on goods and services.


Note that the GDP deflator is about 10% higher now than in 2005.


Calculate real government spending on goods and services by end-2009Q2:


(89 × 0.40)/1.10 = 32.4 Ch.2005$


Divide this stimulus by 2009Q2 GDP, not at annual rates. 2009Q2 GDP is about 3223 billion Ch.2005$.


32.4/3223 = 0.01


The resulting percentage increment to GDP assuming the multiplier for spending on goods and services is 1.0 is 0.01.


Annualize the implied increment to 2009Q2 GDP:


(1.01)4 = 1.04


In other words, the $89 billion results in 4 percentage points increase in growth, assuming zero spending out of tax cuts. Assume 0.5 multiplier for spending on goods and services (kinda wierdly low, but plausible when discussing impact multipliers), and one still gets 2 percentage points increase in growth.


Update 8/22 2:40pm Pacific: I should note that such math errors are not restricted to Mr. Posner. See also this example, also referring to Dr. Romer’s testimony (and my rejoinder of 8/13).

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29 thoughts on “Basic Math for the Math Challenged

  1. Michael

    How does this all meaningfully compare to the 609B of mortgageback securities the Fed has purchased so far? That is money that would otherwise not be in the system. Does that imply a flat or near-flat GDP number reveals that we would have had a 13.68% fall in GDP otherwise assuming the same .5 multiplier (609/89 * 2%) had the Fed not put this 1.25T MBS purchase program into place?
    Looks like the ‘stimulus’ is a decoy. The real substantial ‘fiscal’ policy is disguised as monetary, and no one is looking.

  2. Anonymous

    Menzie, I was introduced to your blog by Morris Davis and I’ve been an avid reader for almost 2 years. So please understand these are serious questions that I really want answered.
    Now can you factor in the present value of the payback of the debt? Can you explain why it is a good idea to borrow 89B at 2.5% interest to increase GDP by 38B? Also, if you can then take it one step further and explain why we don’t just stimulate ourselves every year?
    Thanks.

  3. Buzzcut

    Hmm. Do we really know where the money to the states went to? I’ve heard a lot went to Medicaid spending. I would NOT assume that much went to goods spending. Most probably went to pay salaries of health care workers. Knowing Illinois, much went to pay past due bills.
    So maybe a 0.5 multiplier is not “weirdly low”.

  4. aaron

    I must agree with Anonymous. Once interest is considered, I’m almost certain the result is negative over time. It seems like propping up the number in one year and sacrificing the growth rate indefinitely. It’d be nice to find out I’m wrong.

  5. aaron

    If it went to pay down debt at a higher interest, I think that probably the best thing it could have been spent on is paying down debt at higher rates. The only other things that might be better are improving efficiency on roads (dispelling the myth of slow being efficient, seperating shipping and car traffic)and reduce the burden of doing business (speeding up bureaucracy, reducing the time and manpower to comply with regulations, making processes simpler and clearer, making forms more readable and intuitive, making laws less complicated).
    I’m doubtful that much of any of the spending produces a return on the investment, especially when the interest in considered.

  6. spencer

    Buzzcut–so what if it went to health care spending.
    Care to explain why a dollar paid to a health care worker will have a different multiplier than a dollar paid to a mechanic or a retail sales clerk.
    Remember, if it is spent on goods a large share of the spending will end up in the hands of a Chinese workers–international leakages will be very large.
    So actually, spending on goods as your seems to favor might have a smaller multiplier.

  7. Scott

    @michael
    Sweet God, no! Not . . . a 2.5% interest rate! At that usurious interest, if the inflation returns to a long run average of around 3%, that would mean a negative real interest!
    That’s the point of stimulus that most people here seem to miss. That if you believe stimulus is a credible way to stop the economic bleed and bring some sense of normalcy to the market, than of course it’s worth it. The long run cost of servicing the debt is worth it, when you consider the alternative of deflation, catastrophic unemployment, and broad insolvency or cutting of services from the states.
    It seem like a lot people here could benefit on reading up on the output gap, deflationary spirals, and required spending policy when the public has a debilitating excess savings rate.

  8. JoshK

    I just don’t get the connection between increased GDP and this stimulus spending. There’s really no control here. It’s just as plausible to make the government-crowding-out argument. With no counter factual it seems very self serving to try to construct a multiplier.

  9. Menzie Chinn

    Anonymous: See this post for some background on the multiplier. The Keynesian multiplier being greater than zero relies upon some underutilized resources in the economy. In the neoclassical synthesis, then, the multiplier is smaller as one approaches full employment. There are other multipliers, even in RBC models. For instance, in a prototypical one, higher government spending induces greater output via work effort because the government spending is on goods “thrown into the ocean”.

    Buzzcut: Well, spencer beat me to the punch. Government spending to pay health care workers is not a transfer in a NIPA sense (Posner was confused by the terminology of Federal transfers to the states). That’s a purchase of goods and services. So, the multiplier should be one.

    JoshK: You are indeed free to make any assumptions you wish. But rather than just parroting the phrase “crowding out”, it might be useful for you to consider under what conditions full crowding out might occur. Consulting a textbook could be helpful in this regard.

  10. aaron

    Scott, 2.5% doesn’t seem like much, but it’s more than the return on the asset that that money will be used to create. Also, 2.5% is the current short term rate, the borrowing also affects the long term rate for the broader economy, pushing up interest rates.
    http://www.ustreas.gov/offices/domestic-finance/debt-management/interest-rate/yield_historical.shtml
    And the long term rate is where all the risk in the system is. It never went away, we just shielder certain people and organizations from it.
    Also remember, bad practices lead to mispriced assets for 2005-2008. It was a boon for some sellers, but for a mortgage originated for a new purchase during that period, that means the effective interest rate for a buyer is greatly higher than the nominal rate of the time. That 6% mortgage need to get down to 4% for them to get to a reasonable rate by recent standards.

  11. AJStrata

    Ok, not that’s pretty pathetic. $89 billion includes unemployment – which is not ‘growth’. It includes covering state employee salaries – not growth. Includes tax rebates which probably did not come close to offsetting losses in savings, earnings and wealth.
    You tried to show how one could have thrown a cup of water overboard while the Titanic was sinking and declared salvation.
    When you factor in reality this was too little and too late.

  12. DickF

    So we only bought $89billion of our own products and got 2% growth in GDP. Just imagine, if we had bought $445billion of our own products – only a fraction of the total allocated to the bailout – we could have GDP growth as high as China. Ain’t math great!
    Wow, what a country!

  13. Buzzcut

    Spencer, who has a greater capacity utilization, health care or retail? Isn’t skewing the money to areas that AREN’T in recession (state and local governments added workers so far during the recession, health care has done relatively well) counterproductive?
    Now, a “Cash for RVs” program would do a lot more good. ;) Highest unemployment rate in the US is Elkhart, Indiana, home of the RV and boat manufacturing industrues.

  14. Menzie Chinn

    Buzzcut: Your argument makes sense if health care workers consume only health care services; you’re focusing only on first round impacts. Anyway, July 2009 unemployment in health care/education is 6.1%, still consequential. See discussion here.

    AJStrata: I didn’t claim any miracles. I merely pointed out how to do math correctly. If this offends, I apologize.

  15. Anon

    So explain to me still, how an 89B (regardless of interest expense) is a good investment if we only get a 39B return. It seems even if we got a multiplier of 2, we’d still only be at 80B and that is still a negative return. It seems like we’re just delaying the pain.

  16. John Smith

    Why doesn’t the government just create several trading companies and get them to sell things to each other in a massive circle-jerk?
    All those transactions would count towards GDP and at the end of the day no goods would have to actually be moved (it can all be done on paper) so it would be real cheap. In fact the goods wouldn’t even have to actually exist.
    I can see no downside to my scheme and it seems to fit in well with current government ethics. Definitely on a par with allowing banks to value assets way above market prices, the Fed/Treasury circle-jerk, FHA taxpayer-backed, no-money-down loan guarantees to credit deadbeats, cash for clunkers, Government Motors, etc, etc etc.

  17. Leigh Caldwell

    Anon: $89 billion of stimulus is not meant to get a “return” as if it were some kind of private investment. It is not meaningful to compare the $89 billion with the $39 billion because one represents a transfer, and the other a pure gain in GDP.
    There is a meaningful comparison to be made, however, which is to measure the actual economic or efficiency cost of that $89 billion with the economic gain of $39 billion. I have outlined that argument at http://www.knowingandmaking.com/2009/08/what-is-return-on-fiscal-stimulus.html and the result is a lot more flattering than you imply.

  18. ray

    do stimulus flows that end up in banks get counted as part of multiplier effect? it seems that money saved would then increase due to leverage.

  19. antoniusb

    I am a little confused on the math. I don’t see how the annualization procedure to the GDP growth rate applies when you are using quarterly GDP totals.
    In other words, the measured effect of the stimulus on GDP is claimed to be $32.4 billion. Let us set up a static model and hold all figures constant for four quarters. The effect on an annual basis would be $130 billion.
    Annual GDP would be ~$13 trillion so the impact of the stimulus is 1% on an annual basis and would only account for a little less than one quarter of the difference in GDP growth rates between Q1 and Q2.
    It seems to me that you can scale either the stimulus impact or the growth rate but not both. Someone please let me know where I have gone wrong.

  20. Cabbie

    Here is some very simple math. If you continue to spend more than you earn, you will eventually go bankrupt. Don’t matter if you are an individual or a government. Future has some pretty dark clouds for the USA and her closest trading partners. Will probably have to go and start another war somewhere !!!

  21. Tom

    Any analysis of the impact of deficit spending on consumption must account for the fact that it comes out of the pool of money available for private investment.

  22. GNP

    Annualizing the stimulus percentage increase makes sense.

    Why is the stimulus deflated back to 2005 dollars? Strikes me as unnecessary for the point being made. The calculated values are not any more “real” despite the extra effort. What am I missing?

  23. Menzie Chinn

    GNP: The numbers were converted to Ch.2005$ because the growth rates were expressed in real terms. That explains the use of deflators (here, and in general).

  24. GNP

    Duh…. Of course! Thanks Menzie! hehe

    I understand the use of deflators well, actually better than most. Am now back on track. Gotta wonder if Posner’s problem is that he lacks friendly but critical proof-readers.

    And, yes please do continue to hold these guy’s feet to the fire despite the pathetic attempts at diversion and no matter how inspiration some of the earlier work might have been.

  25. kurtteej

    All very nice classroom scenarios. The reality is that the transfers to the states were mostly given to bankrupt states to pay off bills that were previously accounted for (paying off deficit spending) and therefore added nothing to overall growth. The multiplier also does not take into consideration the fact that banks are sitting on cash, people are sitting on cash, companies are not hiring and won’t be any time soon. Classroom theory has to be factored against certain realities that distorted numbers will never be able to accurately account for.
    Remember, we lost jobs last month and mysteriously the unemployment rate dropped. That doesn’t even work in classroom theory and can only be explained away in politispeak.

  26. Owe Jessen

    I wonder – whats the use of the American obsession to use annualized growth rates? Especially for one-off events, they just inflate numbers.

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