Krugman’s getting tougher on the Fed

Make it eight posts today.  I couldn’t resist giving kudos to Krugman for this outstanding post.  I take pride in the fact that I have been making some of these points all year.  But he has some sharp observations that I did not make:

As far as I can tell, what’s going on in monetary policy debate is a policy in search of a justification. Many central bankers just hate, absolutely hate, being in the position of being so accommodating; yet economic analysis offers no justification for tightening. So they’re inventing new policy doctrines on the fly to justify doing what they want to do.

Krugman’s like a good attorney.  You may not like him.  You may not agree with him.  But you want him on your side during a debate.  Right now he’s on my side—telling the Fed to get more expansionary.  (Recall that hawkish Fed talk is equivalent to a contractionary policy, in both his model and mine.)

PS.  Krugman and I both like to bring up 1937-38.  The revised 3rd quarter NGDP numbers are out—making it almost certain that 2009 will see the biggest drop in NGDP since 1938.  The 4.3% NGDP increase was revised down to 3.3%.  The normal rate is 5%.  How’s that fiscal stimulus doing?  But give the US economy credit.  Even with Fed hawks doing all they can to inhibit AD, and even with the government restraining SRAS with 40% minimum wage increases and a tripling of the duration of unemployment benefits, the economy eked out 2.8% growth, as the SRAS curve shifted a bit to the right with sharp wage cuts.  But there has to be a better way to run a modern economy.  This is pathetic.

HT: rob



29 Responses to “Krugman’s getting tougher on the Fed”

  1. Gravatar of MIke Sandifer MIke Sandifer
    27. November 2009 at 18:45

    I think I understand the moral hazard argument when it comes to extending unemployment benefits, but what else can be done? Projections have unemployment being very high even three or more years out in some cases. How would failing to extend benefits be helpful on the net?

    Of course, if there was anything like the kind of monetary stimulus necessary for job recovery, or indeed had there been enough to prevent such high unemployment, this would be a moot point.

  2. Gravatar of Doc Merlin Doc Merlin
    27. November 2009 at 23:13

    Mike Sandifer:
    “Of course, if there was anything like the kind of monetary stimulus necessary for job recovery.”

    Dude, the AMB doubled in the course of a few months; the USDX has lost ~16 points since march; the overnight rate dropped to near zero. While monetary tightness may have precipitated this mess (as Scott claims) I am thoroughly unconvinced that what we have now is tightness.

  3. Gravatar of jean jean
    28. November 2009 at 02:19

    I suspect that now that the bill on health care has gone through the Senate, Krugman no longer fears that the health care reform will be repelled because of the “deficit digging” argument. So now, he can say what he really thinks.

    But maybe I am a bit too paranoid.
    Good news anyway.

  4. Gravatar of Doc Merlin Doc Merlin
    28. November 2009 at 04:09

    Hrm, I need to clarify:
    WRT monetary tightness, MB is what the fed can do directly, M1,M2,M3 are created by the economy and with regulation. The fed opened the floodgates

    Thats a ~1.1T USD jump.

    But this hasn’t really done anything, in large part because the fed is paying interests on reserves so excess reserves have become huge.

    Thats a ~1T USD jump

    So, we still have about 100B USD of extra money being passed around. This is probably what is causing the rather large drop in the USD.

    This paper suggests the reason the Fed is paying interest on reserves is because it doesn’t want the banks to spend the money.

    This is a WTF moment for me. I don’t understand it from a macro point of view. This makes no sense to me. They talk about how this recession is one of the worst ever and then they try to keep banks from loaning money? Dropping the over night rate to 0 thats expansionistic, buying up all the assets they can and shoveling money out the door, also expansionistic. Increasing the AMB by over 100%, also expansionistic. Paying interest on reserves, thats short-term tight money and long term expansionistic. I am scratching my head, what game is the Fed playing at. Is it trying to scare China into floating the Yuan? Is it trying to open the door for much higher reserve % banking? Is it just trying to create more power for itself by giving itself this option? Has it been captured and is it just become a way to funnel money to well connected banks? I have no idea.

  5. Gravatar of ssumner ssumner
    28. November 2009 at 06:36

    Mike, I was making a positive argument, not a normative one (although I completely understand that it seemed I was making a normative one.) Obviously I think the solution is greater AD, that would help the unemployed much more than simply extending benefits. What to do if we don’t boost AD? That is a tough question. One compromise might be a significant lump sum to the unemployed, so that they have some money to live on but there is no disincentive to find work. (Yes, even lump sums have some disincentive effects, but not as much.)

    But from a purely postive perspective, the Chicago School folks are right about Un. Comp., it does raise the unemployment rate. But Krugman’s right that AD is a much bigger problem than AS. So right now I have more sympathy for the left.

    Doc Merlin, You may think money is easy, but it is nowhere near as easy as it needs to be. It should be easy enough to raise NGDP growth expectations high enough to get back on track. We are nowhere near that level. It’s up to the Japanese to make the dollar stronger, as I pointed out in another recent post. If Japan runs 2% deflation, then a tight policy of 1% inflation in the US can make the dollar look weak.

    jean, My conspiracy theory is that deep down he knows how powerful monetary policy is (even at low rates) and he is getting sacred about the next election. He raised the spector of “Sarah Palin’s second term as president” in a recent post.

    Doc Merlin, I mostly agree with your second comment. But I think interest on reserves is contractionary both short and long term. The long term you refer to is that once rates rise above zero, it can be made far less contractionary by setting the rate below the T-bill yield.

  6. Gravatar of David Pearson David Pearson
    28. November 2009 at 07:42


    I know you think the gold price increase is mostly Indian retail borrowing (btw, notice that the Indian Central Bank is also buying?), so I call your attention to 5-yr TIPS.

    What is the security telling us about AD and stimulus. Well, the real TIPS yield says you are absolutely right: at .14% it is the lowest ever (except for the brief post-Lehman spike-down). Clearly, this shows that expectations for future REAL growth are in the cellar. What kind of economy produces risk-free real rates of return of .14%? Certainly not one that is growing.

    So a .14% real rate should also imply some kind of deflation threat. Let’s check the 5-yr TIPS inflation premium. The nominal 5-yr is at 2.04%, which makes the inflation premium 1.9% — close to its recent high.

    Now, 1.9% may still be too low in your book. Fine, but that’s an answer to the wrong question. The right question is, “how will the TIPS inflation premium move in reaction to Fed attempts to stimulate AD?” The evidence thus far is that one should expect the real rate to crash while the TIPS inflation premium widens. This is certainly problematic for those that believe that nominal and real AD march in lock-step. Throw in the rise in the gold price (now, are those Indian RETAIL investors recently buying 200 tons?), and what you get is a plausible scenario where monetary policy is only effective in stimulating nominal demand.

  7. Gravatar of MLB MLB
    28. November 2009 at 08:19

    Money too tight and $1100 gold? You’ve got to be kidding.

  8. Gravatar of David Pearson David Pearson
    28. November 2009 at 08:20

    Correction — that low in the TIPS real yield actually occurred in early March of 2008, not post Lehman.

    BTW, I neglected to mention that the 2yr to 10yr Treasury yield spread, at around 250, is still close to multi-year highs, and needless to say, the premium of 10yrs over Bills is sky-high. So, the bond market seems to be saying:

    -ZIRP will continue for years
    -It will produce no growth
    -It will produce moderate inflation
    -And Treasury investors therefore require a higher term premium.

    Something is quite different between the Japanese experience (1% 10-yr JGB yields during ZIRP) and the U.S. one. That “difference” is that inflation expectations are keeping the cost of borrowing higher for us. Let’s hope when the Fed adopts your prescriptions — and in the case of an inflation target, it surely will — we don’t get a higher cost of borrowing combined with little or no growth.

  9. Gravatar of StatsGuy StatsGuy
    28. November 2009 at 08:28

    Next time you decide to go on vacation and store up EIGHT posts in one day, have some pity on us!

    Currently, the Fed’s message is: “We’re going to keep rates down, but when inflation peeks we’re going to crush it… perhaps even when it looks like it _might_ peak.”

    The issue with the exit strategy is that the Fed has discovered that everyone everyone else knows that the Fed doesn’t (and never did) have control over velocity given its current toolset. When everyone _thought_ it had control, it sort of did have control – vis-a-vis Nick Rowe’s post-modernist interpretation of monetary policy.

    Now that the cat is out of the bag, there is a presumption that IF/WHEN velocity picks up, it could move so fast that the Fed won’t be able to draw money out of the economy fast enough. Hence, there is a pick-up in LONG TERM rates even as sub-5-year rates stay quite low. The yield curve has sharpened around that 5 year cutoff (which, btw, is consistent with Krugman’s Rudebush/Taylor projections).

    We have a lack of credibility on BOTH ENDS of the yield curve: low credibility that the Fed will keep rates “low” enough long enough to get the economy going, and low credibility that it will cut rates when it needs to. And, IMHO, the fundamental reason we’re likely to not raise rates long term is because of a huge fiscal debt we’re incurring trying to prop up the under-utilizing economy with consumption-related expenditures (primarily mandatory transfers and “automatic stabilizers”) and our failure to aggressively close the savings/investment gap. In other words, the Fed’s short term failure is making its long-term failure more likely.

    Here is the clincher:

    IF the Fed was really doing its job, the yield curve would move in the _opposite_ direction as we’d see short term inflation (to restore the NGDP trajectory) coupled with long term moderation. In other words, create more permanent money now, so that we don’t have to run massive fiscal deficits (MOST of which is driven by lost revenue).

    The current policy trajectory back-loads inflation. At least, that is what the bond markets are telling us. It is, I fear, a continuation of the policies of Bush Jr.

  10. Gravatar of ssumner ssumner
    28. November 2009 at 11:56

    David, Those are good points, but here is my response:

    1. I don’t favor targeting inflation, but if we insist on inflation I’d like to see two year inflation expectations up to 2%. Right now they are closer to one percent. And it is precisely the next two years that I worry about–I think the economy will be much weaker than it should be. After that, I agree that inflation may well return to the low twos.

    2. I prefer to target NGDP. Because of the low real rates you mention, I think we can agree that real growth may be low, or at least low for a recovery. This leads to my third point.

    3. I favor level targeting not growth rate targeting. I suppuse one could argue it is too late for this recession. Nevertheless, if the Fed commits to at least partly return to the old trend line (obviously it would be too inflationary to go all the way back) then it could help anchor expectations in the next recession. We need to convince markets that we won’t move up or down onto a very different trend line, because when expectations shift that much it is very destabilizng for the economy.

    Regarding the AS/AD distinction, I still think AD is a big problem. Of course if you are right about nominal growth not helping, then we also should root for fiscal policy to fail. If fiscal policy succeeded in boosting NGDP growth, then it would hurt the economy if we face a vertical AS curve. I do not believe we face a vertical SRAS curve, although I agree that there are supply problems in the economy.

    Since you are mentioning asset prices, recall that US stocks did very poorly during the high inflation 1970s. But US stocks have recently rallied as the dollar has fallen. That tells me the stock market thinks a lower dollar will at least somewhat boost growth.

    MLB, We have also seen gold prices rise sharply in yen terms over the past 5 years. Should the Japanese be worried about high inflation in yen terms?

    David#2, My hunch is that the bond market expects about 1% inflation for 2 years, and 2.3% thereafter. Isn’t that consistent with what you see in asset markets? I think NGDP growth will also be about 5% after 2012, it’s the next two years when I think we need faster NGDP growth to partly catch up to the previous tredn line (again, I don’t favor a complete catch up, maybe 7% NGDP growth for 2 years, then 5% thereafter (or even less if we can figure out how to do monetary policy in a low interest rate environment.)

    Statsguy, I agree with your comment, with the proviso that the lack of long term confidence is not that great a problem. The expected value of long term inflation is a bit over 2%. But because of fat tails I still think the “mode,” if that’s the right term, is around 2%. I don’t see any major credibility problem in the long term. But I do agree with the market view that short term inflation will be well below target, and long term it will be a bit above.

  11. Gravatar of TGGP TGGP
    28. November 2009 at 14:31

    The economist who I hear most often harping on supply is Casey Mulligan.

    John Taylor complains about Krugman using a modified version of his “Taylor rule” here.

  12. Gravatar of ssumner ssumner
    28. November 2009 at 15:29

    TGGP, I think Casey Mulligan grossly underestimates the extent to which our current problem is too little AD. Indeed I hope to do a post on that if I get time.

    The Taylor Rule doesn’t much interest me, as I favor targeting NGDP expectations, not interest rates. I do think that policy needs to be much more expansionary, so in the short run I’m with Krugman I guess.

  13. Gravatar of StatsGuy StatsGuy
    28. November 2009 at 19:28

    “Since you are mentioning asset prices, recall that US stocks did very poorly during the high inflation 1970s. But US stocks have recently rallied as the dollar has fallen. That tells me the stock market thinks a lower dollar will at least somewhat boost growth.”

    Yes, but primarily that’s the carry trade. If you want evidence, compare the year-to-date performance of stocks that derive most of their revenue domestically (for example, the telecoms) to stocks that derive revenues internationally.

    BTW, in 1970, not only were international capital markets much smaller, but all of Europe, Latin America, etc. was also aggressively inflating to achieve higher employment and faster catch-up growth. Where could one park wealth, except in gold (which spiked)?

  14. Gravatar of Jim Jim
    29. November 2009 at 02:10

    I love the idea of a negative interest rate. One argument against might be this: Japan ran negative interest rates for years. Wall St. and Hedge funds borrowed the money for nothin’ and plowed it into bonds, and investments all over the world, getting rich on nothing investments and generally also making stupid risky investment.

    It could be argued that it was this global interest rate that confounded Greenspan when he raised short term rates and long term bonds actually went down, decoupling the Fed for the first time in history. We’re living in global world.

    Somewhere along the way Japan actually became the largest currency in the world. None of it really helped Japan.

    What say you?

  15. Gravatar of ssumner ssumner
    29. November 2009 at 08:00

    Statsguy, Are you telling me domestic stocks haven’t risen? That can’t be right, the overall market is up 60%. If you are saying they have risen less than international stocks, then I am not surprised–the weak dollar obviously helps multinationals. So I stand by my argument, unless you have data that domestic stocks haven’t risen at all.

    Jim, I don’t agree with your facts. Short and long rates often move in the opposite direction. Why is that a big deal?

    I am not sure what you mean by negative interest rates in Japan. And as far as Japan having the world’s largest currency, if that is true, I presume it reflects their deflationary policies which encourage the hoarding of currency.

  16. Gravatar of jean jean
    29. November 2009 at 08:41

    @MLB: Do you mean that the only real wealth is gold?

  17. Gravatar of Tom P Tom P
    29. November 2009 at 10:08

    If only you posted 8 times every day! This is consistently my favorite blog.

    Keep up the good work!

  18. Gravatar of David Pearson David Pearson
    29. November 2009 at 20:09


    The stock market and bond market’s real growth expectations are in contradiction.

    Statistically, the consensus expectation for 2010 S&P earnings growth implies real economic growth of around 4%. So if the stock market is a good guide to where your NGDP futures would trade, then you would be opening a bottle of bubbly and toasting to the Fed’s management of monetary policy.

    The bond market, of course, predicts .14% real growth, on average, over the next five years (using real yields as a proxy for growth expectations). Were the NGDP futures to reflect the bond market’s view, you would be condemning the Fed for risking a double-dip — or worse.

    Where would NGDP futures trade today if they existed? Would they match expectations of the stock market, the bond market, or an average of the two? And if markets are efficient, then why do the two deep, liquid markets (stock and bond) imply such different conclusions about the future state of the economy?

  19. Gravatar of Doc Merlin Doc Merlin
    29. November 2009 at 21:24

    @David, this odd sort of Scitzoid behavior exists all over the place right now.
    Gold is very expensive and predict inflation, but the TIPS T-Bill spread is kinda small. Stocks are rising heavily but bonds are doing poorly.

    Some possible reasons:
    1. Falling dollar (wrt other national currencies as shown by the USDX index) means that gold buying is up in national banks.
    2. The creation of the GLD ETF has drastically increased gold demand by lowering the barrier to entry in the gold market.
    3. The Chrystler and GM takeovers by the US gov and the UAW mean that the bond market is riskier than was once thought.
    4. Gold production isn’t doing so great
    5. Much higher barriers to entry on the supply side of the stock market, due to SarOx. The IPO as a model has died in the US, which means that established companies have fewer competitors when it comes to stock sales.
    6. There is no six.
    7. A single number unfortunately doesn’t actually tell you what is going on. Right now we are having very strong deflation and inflation, depending on what market you look at.

    More examples:
    Oil for example is way up from the low, as is gasoline. Natural gas has crashed, hard. Flash rom chips/drives which had been falling in price at a nice steady clip of ~50% per mb for a very long time, recently rose in price in the US. Black friday sales were down 8% this year. Detroit sold its football stadium that cost over 50M to build for about half a million dollars, but Arlington, TX just finished the new Cowboys’ stadium that is selling out its seats for games.

    P.S.: WRT the futures targeting, as Scott and Bill and I have said, you have to use a convertibility mechanism otherwise you suffer from circularity.

  20. Gravatar of David Pearson David Pearson
    30. November 2009 at 08:14


    My point is that futures targeting is subject to inconsistencies in the “signal” being sent by the market.

    Imagine a situation where stocks are undergoing a major correction but NGDP futures are surging. Meanwhile, oil and gold are going up, and TIPS inflation premiums are rising. The Fed removes liquidity to bring NGDP futures back to target, and stocks decline further and credit spreads widen.

    Implausible? No. Quite probable if the market expects NGDP to rise but real growth to fall. In such a case, the Fed will face intense market and political pressure to raise its NGDP target in order to prevent more draining of reserves. It will argue that market expectations embedded in NGDP futures trump whatever the stock market and credit spreads are signaling, and the market, reading the Fed’s response as blind intransigence, falls further. Eventually, the decline in related markets is enough to spook NGDP futures

    I believe I’ve never heard NGDP futures targeting proponents challenge their own thesis with detailed counterfactuals. Its a shame because the scores of econometrics-driven economists that argued for low credit losses in the economy had a similar deficiency (never testing their own assumptions or conclusions). I urge Scott to play out several scenarios, such as the one above, in his blog, and to tell us exactly how the benefits of NGDP targeting outweigh the risk of the relatively inflexible policy prescription going off the rails. The reason is that that I strongly believe that Scott and his colleagues WILL be listened to by the Fed, at least in the form of an inflation targeting regime. So let’s anticipate the result this time, rather than react to it.

  21. Gravatar of StatsGuy StatsGuy
    30. November 2009 at 09:32


    “So I stand by my argument, unless you have data that domestic stocks haven’t risen at all.”

    _Primarily_ the carry trade… Not _entirely_ the carry trade. I am unaware of an index that tracks stocks which have 100% domestic revenue exposure, but if you pick a handful of bellwether companies (outside finance) that have high domestic exposure you can see the trend.


    Waste Management

    They are essentially flat on the year… Going into October, they were negative (with trailing PEs ~ 15 and forward PEs ~ 12), but there has been some exodus from the dollar/risk trade in the past 1.5 months.

    Still, I’m not arguing against you – asset price appreciation is of course going to generate a demand effect which is going to be priced rapidly into domestic stocks too. I’m merely arguing that today, stocks are more aggressively used as an inflation hedge than in the 1970s for several reasons. So even if “bad” inflation were coming, one would not necessarily expect to see this moving stock market aggregates lower this time. The stock market is simply inconclusive.

  22. Gravatar of MikeM MikeM
    30. November 2009 at 11:18

    “…, the economy eked out 2.8% growth, as the SRAS curve shifted a bit to the right with sharp wage cuts.”

    This is contrary to the data that I just read, via Becker, that wages actually increased by 2% this recession so workers who have a job have been better off than in past recessions. Explaining why this is so may be the most interesting question no one is answering.

  23. Gravatar of Doc Merlin Doc Merlin
    30. November 2009 at 14:05


    Well, we did have a rather large minimum wage hike and also massive subsidy/bailout to many firms.

  24. Gravatar of ssumner ssumner
    30. November 2009 at 14:32

    Thanks Tom.

    I’m not so sure about that. I believe they both expect around 3% real growth. Keep in mind:

    1. Stocks are still far below the level of two years ago.
    2. When output is severely depressed, there is relatively little investment even when RGDP is growing. We saw the same thing in the 1930s. Fast growth from a low level did not raise rates. So the bond market also may be expecting 3% RGDP growth.

    Doc Merlin, Those are good points, but it just shows how complex the world is. Lots of things may be affecting gold prices. Natural gas is being affecting by dramatic breakthroughs in the technology of extraction (relative to oil.) Etc, etc.

    David#2, NGDP futures aren’t like other targets. Generally when you target M2, or interest rates, or exchange rates, you do so in the hope that it tends to stabilize the variables that you really care about. I am saying expected NGDP growth is the variable that I really care about. So as long as it is on target, I don’t care whether stocks or bonds or commodities or foreign exchange go to zero or infinity. If those things matter, then we shouldn’t be targeting NGDP. We should target a weighted average of NGDP and whatever other variables matter.

    BTW, of course they “matter” in some sense, I mean to they matter for monetary policy above and beyond their impact on NGDP. I doubt it.

    Statsguy, I agree with your intuition that things are different from the 1970s. My view is that in the 1930s stocks responded positively to inflation because we needed more inflation. In the 1970s stocks responded negatively to inflation because we needed less inflation. Now stocks are again responding positively because just as in the 1930s we need more inflation. And the correlary is that if inflation got to be more than what we need, stocks would again respond negatively

    There is also a levels/rates distinction. Stocks (in nominal terms) always respond positively to higher price levels, and usually respond negatively to higher rates of expected inflation.

    Mike, I can’t figure out Becker’s post. 12 month wage growth in the private sector has slowed from 2.4% to 1.2%, the slowest rate in I don’t know how long. And that means lots of workers are getting pay cuts, which used to be a rarity. He also says that we can’t reduce real wages by increasing inflation because right now deflation is the bigger concern. Huh? A commenter in my most recent post was equally as puzzled as I was.

    Doc Merlin, That further supports my point. I wonder what has happened to the average wage of non-minimum workers?

  25. Gravatar of Doc Merlin Doc Merlin
    1. December 2009 at 13:26

    “Doc Merlin, That further supports my point. I wonder what has happened to the average wage of non-minimum workers?”

    Scot, the average wage of the top few quintiles probably went down. But, any time that that unemployment goes up in the lowest quintile more than in the higher quintiles the average wage will increase… just from the statistical effects. Thus, it is not surprising that the average wage goes up during a recession.

  26. Gravatar of David Pearson David Pearson
    1. December 2009 at 17:15


    Look specifically at 5-yr TIPS: how is a .10% real yield consistent with bond market expectations of 3% real growth? I’d like to understand the theory that equates such low required real returns with such robust growth expectations…

    Also, you misunderstand my comment on stocks. To achieve the level of earnings growth expected in 2010, the economy would have to deliver around 4% real growth in the year. This has nothing to do with where stocks were relative to the past — it is a relationship between expected margins, revenue growth, and the correlation between real GDP and revenue growth.

    The two sets of expectations — bond and stock — are in direct conflict, and this calls into question how an NGDP futures market will be a reliable guide to future expectations.

  27. Gravatar of Scott Sumner Scott Sumner
    3. December 2009 at 05:46

    Doc merlin, Yes, I had forgotten about composition bias.

    David, I don’t see 3% real growth as robust when we are in a deep recession. Suppose the growth pushes factory utilization from 70% to 73%. Why would that cause companies to build new factories and office buildings? Now suppose things were better, suppose we were at a more normal 82% factory utilization rate; then 3% growth would lead to new business investment. So I still think you are overlooking the “levels vs rates” issue. Real rates tend to be low in deep downturns, even when growth is normal. So I don’t see the expectations as necessarily being in conflict, although it is obviously possible.

    BTW, if it is possible to objectively measure growth expectations in the stock market (without recourse to market prices) why can’t we do the same in the bond market. I assume you did not get the stock forecast of 4% growth from stock prices, but rather from some survey. Are those surveys also done for bonds?

  28. Gravatar of Doc Merlin Doc Merlin
    3. December 2009 at 23:04

    “I assume you did not get the stock forecast of 4% growth from stock prices, but rather from some survey. Are those surveys also done for bonds?”

    Nah, actually you just calculate it with a few (sometimes terribly wrong) assumptions. The models assume

    A really simple pricing model that assumes constant growth forever, for example is:

    P = E1 / (R – G)
    P is price
    G is the expected decimal growth rate
    R is the decimal discount rate (That is your expected rate of return, this is not to be confused with the macroeconomic variable of the same name.)
    E1 is the earnings over the next twelve months.

    This page here explains some simple earnings based models.

    The Black-Scholes formula is a more complex model that also is used to price (mostly for options but if you can price options you can instantly price a a security as well) and there are many others:

  29. Gravatar of ssumner ssumner
    6. December 2009 at 10:57

    Doc Merlin. Those formula’s don’t really prove anything about market efficiency, as they depend on the (dubious) accuracy of their assumptions.

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