Thursday, September 17, 2009

Does the Equation of Exchange Shed Any Light on the Crisis?

James Hamilton thinks the answer is no. In his reply to Scott Sumner's lead article at Cato Unbound, he questions Sumner's use of the identity MV = PY to explain the collapse of nominal spending over the past year. (In this equation M = money supply, V = velocity or the average number of times a unit of money is spent, P = price level, Y = real GDP, and PY = nominal GDP.) Hamilton contends the only meaningful use of the identity is to determine velocity (i.e. V=PY/M) and even then it is not totally reliable since it can vary based on the measure of money one uses. I believe, however, Hamilton under appreciates the insights this identity can shed on the crisis. It may not provide precise policy recommendations, but it does provide a starting point from which to think analytically and empirically about recent economic developments.

So what does this identity tells us about the crisis? To answer this question we first need to expand the identity a bit. To do so, note that the money supply is the product of the monetary base, B, times the money multiplier, m or

M = Bm.

Now substitute this into the equation of exchange to get the following:

BmV = PY.

Now we have an identity that says the sources of nominal spending are the monetary base, the money multiplier, and velocity. With this identity in hand we can asses the contribution of these three sources to the dramatic decline in nominal spending in the past year. Using MZM as the measure of money (see here for why MZM is preferred over M1 and M2) and monthly nominal GDP from Macroeconomic Advisers to construct velocity, the three series are graphed below in levels (click on figure to enlarge):

This figure indicates that declines in the money multiplier and velocity have both been pulling down nominal GDP. The decline in the money multiplier reflects (1) the problems in the banking system that have led to a decline in financial intermediation as well as (2) the interest the Fed is paying on excess bank reserves. The decline in the velocity is presumably the result of an increase in real money demand created by the uncertainty surrounding the recession. This figure also shows that the Federal Reserve has been significantly increasing the monetary base, which should, all else equal, put upward pressure on nominal spending. However, all else is not equal as the movements in the money multiplier and the monetary base appear to mostly offset each other. Therefore, it seems that on balance it has been the fall in velocity (i.e. the increase in real money demand) that has driven the collapse in nominal spending.

To get a better sense of what is happening with theses series note that log of the expanded equation of exchange can be stated as follows:

B+m+V = P+Y,

Now if we take first differences of the the quarterly log values of the series in the above identity we get a quarterly growth rate approximation. (Note, this approximation is not very good for large differences like the one for the monetary base in 2008:Q4.) Below is a table with the results in annualized values (Click to enlarge):

This table confirms what we saw in the levels: a sharp decline in velocity appears to be the main contributor to the collapse in nominal spending in late 2008 and early 2009 as changes in the monetary base and the money multiplier largely offset each other. It is striking that the largest run ups in the monetary base occurred in the same quarters (2008:Q3, 2008:Q4) as the largest drops in the money multiplier. If the Fed's payment on excess reserves were the main reason for the decline in the money multiplier and if the Fed used this new tool in order to allow for massive credit easing (i.e. buying up troubled assets and bringing down spreads) without inflation emerging, then the Fed's timing was impeccable. Unfortunately, though, it appears the Fed was so focused on preventing its credit easing program from destabilizing the money supply that it overlooked, or least underestimated, developments with real money demand (i.e. velocity). As a consequence, nominal spending crashed.

Now maybe this is self evident to some, but I find the above information from the equation of exchange useful as a starting point for discussing what went wrong and where to go from here.


  1. David,

    I would view things differently myself, but you have an interesting analysis.

    The most interesting story of all I think will be when we finally hear the Fed’s thorough explanation (if ever) on why they have been paying interest on reserves. My theory is that they are simply preparing the market for the eventuality that both the fed funds rate and interest rates on reserves will have to go up at some point, and very likely before they have withdrawn all of the accommodation that has created the “excessively excessive” excess reserve levels during the crisis. So it encourages that kind of adaptive expectation. That, plus, my theory is that they’ve actually never intended that excess reserves be “used” in the sense of a multiplier dynamic. They’re relying on bank capital restoration to move credit forward. Paying 25 basis points at the zero bound enforces this understanding.

    If you haven’t seen it, you may be interested in the attached interview of Paul Volcker, in particular, the following section, where he muses about interest on reserves:

    Volcker: Well, I asked you and others a question, I think, some months ago as to why the Federal Reserve is paying interest on excess reserves.

    Stern: You did. I don’t think anybody gave you a good answer.

    Volcker: No, the answer was “go ask Don Kohn” [vice chairman of the Board of Governors].

    Stern: Right.

    Volcker: But since then I’ve been better instructed. I’m told that the Web site still says the purpose is to maintain tight control over the federal funds rate, but the explanation I hear now is that we’ve got all these excess reserves in the system, and if we want to tighten up, the subtle way of doing that is to raise the interest rate on excess reserves.

    Stern: Yes, put a floor under interest rates.

    Volcker: But right now all the talk is, we’ve got to get more liquidity in the system and encourage banks to lend.

    Me: I would also point him to my theory. There is a fundamental distinction between liquidity and capital. (In fact, there was a lot of debate about that distinction in the early stages of the crisis, as it applied to the nature of the risk underlying the crisis itself.)

    The interview:

  2. Excellent discussion.

    Do you use monthly GDP figures often? Are you confident in them?

    If these things are obvious to macroecnoomists, they don't seem to be obvious to financial economists, much less politicians and other laymen.

  3. JKH: It would be great if we could get Bernanke to speak to this issue. Maybe the FT or WSJ could do a sit down interview with him on the motivation for the payment on excess reserves. David Wessel, in his recent primer on this issue,says the motivation now is simply to control interest rates.

    Bill: I have yet to use monthly GDP numbers in my own research. This is primarily because the data from Macroeconomic Advisers only goes back to 1993 (Maybe it goes back further if you are paying subscriber; I don't know). I have no reason to doubt their data and it appears to line up well with the quarterly GDP numbers. I typically opt for industrial production or the coincident indicator when I need a monthly real economic series.

  4. David,

    Makes sense.

    But I think the question some are wondering about is why the Fed is paying 25 basis points instead of 0 basis points, given that they have effectively reached the zero bound.

  5. Nice, David.

    That is a well-written and equally interesting article. In reference to Hamilton's article, do your results hold across the different measures of money? There's also a strong sense of lagged effects in your chart, since velocity started dropping in 2006 (coincidentally with home values in the aggregate).

    This makes total sense, though. My husband, being the good frugal German that he is, sold off all of the stocks in our portfolio and put the funds into high-interest saving (part of M2) and bonds in 2007. I, too, am guilty of mattress-stuffing.


  6. Rebecca:

    Good question. I need to check with the other monetary aggregates. I have just started defaulting to MZM lately after reading an article showing it has a comparatively stable money demand function for the post 1980s period. I too am guilty of mattress-stuffing! Makes for good stories in the classroom, though.

    Being a Texan you can appreciate this bit of news: the blistering heat finally let up here a few weeks ago. I forgot what cool weather was like after going thru this past summer.

  7. Darn good, David. Couldn't have said it as well.

  8. I read an analysis by John Maudlin about using MZM. I was attracted to this post because I was writing a lesson plan on the equation of exchange and wanted to see what you used for M. I can't match your work, but can learn from it. You have taught me much.

  9. Mike:

    What did John Maudlin say about MZM? And is there a link to it? Thanks.

  10. "This figure also shows that the Federal Reserve has been significantly increasing the monetary base, which should, all else equal, put upward pressure on nominal spending"

    As you say, both the transactions velocity of money & the banking system's expansion coefficient have fallen. But you use the wrong figures.

    And an increase in the monetary base is deflationary, i.e., it's almost all currency and an expansion of currency is deflationary unless offset by reserve bank credit. And there is no expansion coefficient for currency. And it could be that the non-bank public is just exchanging currency for demand deposits (the money supply stays the same).

    If one wanted to understand monetary flows (MVt) they would study monetary flows. I.e.,the only valid velocity figure is demand deposit turnover. Income velocity is a contrived (useless) figure.

    But even worse is that you assume that there is only 1 lag working thru the economy. There are many but there are two primary ones, one for real-growth and one for inflation. Your analysis doesn't take either into account.

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  12. excess reserves are misnamed. excess reserves are contingency reserves, they are contingent upon the remuneration rate and competing returns. But currently they are frozen

    and an increase in the remuneration rate functions just as raising reserve ratios would, both increase the volume of (unused) inter-bank demand deposits (reserves).

    only now is it appropriate to lable these reserves as "idle". there was never a "tax" on these balances. An expansion of reserves used to be associated with an increase in loans-deposits. In fact, as a system (given the expansion coefficent), one dollar of excess reserves equated with +$200 dollars of earning assets. That is not a tax. The system isn't the sum of it's parts.

  13. Perhaps the error of policy last year is analyzed most effectively in this post at nakedcapitalism:
    If the government had made transfer payments to debtor households, the crash in nominal income could have been ameliorated. The mistake was a feeble fiscal response. Of course, there are well known time lag problems with fiscal policy so it may not have been easy to achieve rapid boost of transfers in practice.

  14. Flow5: I am not sure if I follow you entirely, but I would note that the monetary base by definition is currency in circulation and bank reserves. In this particular crisis it has been the reserves component that has exploded, not currency in circulation. Also, an increase in monetary base is never deflationary. It may not be inflationary if offset by the money multiplier or real money demand (i.e. velocity), but by itself it never causes deflation.

    ECB: Thanks for the link. As much I have harped on the Fed for not stabilizing nominal spending I have to acknowledge that doing so probably would have been a tough political sell. As it was, there was plenty of inflation hawk rhetoric.

  15. "Also, an increase in monetary base is never deflationary"

    Wrong. An increase in currency decreases legal reserves and would cause bank credit contraction if not offset by open market operations of the buying type

    Yes you are correct, excess & required reserves have both exploded. It is the volume of reserves that primarily determines the expansion of money and the transmission of credit. This should be the FED's focus, however the FED doesn't watch member bank reserves (R. Anderson).

    But they should. For example: From Feb. 2006 until July 2008, the FOMC pursued a consistently tighter money policy. In fact, the rate-of-change in the volume of legal reserves (the proxy for inflation), fell for 29 consecutive months (out of a possible 29, or sufficient to wring inflation out of the economy).

    It’s only been in the last 10 successive months (since Sept. 2008), that it was necessary for the FOMC to switch from its “tight” monetary policy, to the extraordinarily “easy” monetary policy in force at present.

    Easy money is here defined as a growth rate of aggregate monetary demand (money times it's rate of turnover) in excess of the growth rate of product, and service output.

    Further I don't know how to target just nominal gdp. I don't think its possible. Only the rates-of-change in the proxies for real-growth & inflation added together would equal nominal gdp. Anyway, I can't trade using nominal gdp

  16. Flow5: You are confusing a change in the currency-to-deposit ratio with a change in the monetary base. Yes, an increase in currency holdings will drain reserves from the banking system and could be deflationary (if velocity does not offset it), but this is NOT a change in the monetary base. All this is is a change in the composition of the monetary base. Remember, the monetary base is equal to currency in circulation AND bank reserves, not just the former. As a result, if the Fed increases the monetary base it either adds to the reserves in the banking system and/or adds to the currency in circulation.

  17. I like this thread. Good reading.

  18. Forgive me for throwing another link at you, but Arnold Kling has some rude things to say about MV = PT (rude things I agree with and have said to you before - its an identity with no utility as an engine of analysis.)

  19. OK I grant you that. Nevertheless the monetary base is not a base for the expansion of the money supply. As far as MV=PT, the equation works. AAA corporates were 1/10 of a basis point off on Sept 81 as they were for the base year in 77. This equation has never, never, never, failed.

  20. The 87 selloff: monetary flows (MVt) dropped at the sharpest rate-of-change (2 months prior)since the Great Depression.

    By FEB 84 you could see the June 84stock market bottom.

    You could see the OCT 82 stock bottom for more than a year earlier.

    You could see the OCT 75 stock market bottom for more 3 months earlier.

    and the data was not perfect either

  21. Did you notice that the velocity of mackerel also declined? That is, the ratio of nominal GDP to mackerel crashed. The ratio of nominal GDP to lots of things crashed. That does not mean that by raising the denominator in one of those ratios you could have made a difference.

  22. If the supply of mackeral increases, there is immediate a surplus of mackeral. To get rid of the excess mackeral, sellers lower their prices.

    If mackeral served as the medium of account, so that all other prices in the economy were in terms of mackeral, then a fall in the price of Mackeral would be an increase in the price level, and so an increase in nominal income.

    More importantly, if mackeral served as the medium of exchange, then there would be no need to first sell the mackeral. The surplus mackerals at the existing price of mackeral can be exchange directly for any other good. People would take mackeral even if they had no interest in holding more mackeral, but wanted to sell them.

    The Kling theory is that the demand for mackeral always passively adjusts to the supply. And so, the increase in supply of Mackeral doesn't lead to a surplus of mackeral. Those with the extra mackeral just keep them. They keep whatever quantity of mackeral they happen to have.

    Treating the money as it is is just another good (or asset) and ignoring that it is both medium of account and medium of exchange explains why some economists excessively focused on finance don't understand monetary theory.

    Of course, it is hard to believe that any economist would believe that the impact of an increase in the supply of any good would be for the demand to passively adjust--rather than price fall, and quantity demanded rise to meet quantity supplied.

    While the equilibrium conditions may not be too different between money and other goods, or assets, the process by which an excess supply or demand for money impact the economy are unique.

  23. Arnold:

    As I note over at your blog, you're missing the importance of expectations. If the Fed were to change expectations of the future price level it would have an immediate and forceful effect on present spending. (For examplke, the Fed could set an explicit inflation target.)This means the Fed could affect velocity immediately if it really tried. The effect of exepectations is a key insight from modern macro.

  24. member banks don't loan out excess reserves. they are unencumbered in there lending operations

    income velocity, & the monetary base, are all contrived calculations. they are pure fantasy

    and a lag for the nominal gdp doesn't exist and thus it's not possible to make accurate projections using it

  25. The equation of exchange is an algebraic way of stating a truism; that the product of the unit prices, and quantities of goods and services exchanged, is equal (for the same time period), to the product of the volume, and transactions velocity of money. Velocity is the rate of speed at which money is being spent. It is self-evident from the equation that an increase in the volume, and or velocity of money, will cause a rise in unit prices, if the volume of transactions increases less, and vice versa.