Thursday, September 8, 2016

Central Bank Liabilities Part II: Currency, Negative Nominal Interest Rates, etc.

Central banks are indeed banks. As financial intermediaries, their social role depends on what a central bank can do that some private financial intermediary cannot, or cannot do as well. Fundamentally, financial intermediation is asset transformation - how the intermediary structures itself so that its liabilities have attractive properties as compared to its assets. In our changing world, central banks have to adapt their liabilities to evolving technologies and macroeconomic developments. My first installment on this topic was about new types of central bank liabilities - the idea that a central bank's reach could expand beyond supplying old-fashioned currency, with a focus on liquid overnight liabilities that reach into the upper levels of financial interaction, beyond retail payments.

This installment will deal specifically with currency. Are paper money and coins going the way of the dinosuars? Should they? If we were to eliminate or significantly curb the circulation of Federal Reserve notes and coins in the United States, should the Fed issue new types of liabilities to replace them? What types of liabilities might work, and how should they be managed? You can find plenty of writing on this topic recently. Ken Rogoff and Larry Summers make a case that we should elminate large-denomination currency. Agarwal and Kimball, Kocherlakota, Goodfriend (new version), and Goodfriend (old version) discuss various schemes for either eliminating currency or making it more costly to hold. In contrast to Rogoff and Summers, who think eliminating $100 Federal Reserve notes, for example, will put a big dent in undesirable and illegal activities, the latter group of authors is interested in ways to effectively implement negative nominal interest rates.

What does standard monetary theory tell us about how much currency we should have? The basic logic comes from Friedman's "Optimum Quantity of Money" essay - what's come to be known as the "Friedman rule." If the nominal interest rate on assets other than cash is greater than zero, then people economize too much on cash balances. Cash balances are costless to create, and economic forces will create the optimal amount of real cash balances if the central bank acts to equate the rate of return on cash to the rate of return on other safe assets. There are different ways to do this. One approach is for the central bank do whatever it takes to make the nominal interest rate on safe assets zero forever. Another approach, if feasible, is for the central bank to pay interest on cash at the nominal interest rate on safe assets. There are other approaches, but we'll stop there.

How do we connect the Friedman rule with what people are saying about currency? Those people are saying we have too much currency, but the Friedman rule says that a zero nominal interest rate is just right, and otherwise (with nominal interest rates above zero) we have too little currency. Of course, there's a lot going on in the world that is important for the problem at hand, and is absent in standard monetary models. First, currency is certainly not costless to create. Real resources are used up in designing currency so as to thwart counterfeiters, to print and distribute it (think about guards and armored trucks), and to maintain it (taking in and shredding worn-out notes). Second, currency can be stolen. A virtue of currency is its anonymity - people will accept my Federal Reserve notes without worrying about my personal characteristics - but this also makes currency easier to steal than some other assets. Third, the use of currency implies social costs. Again, a virtue of currency is its anonymity, in that the existence of currency allows us to conduct transactions in private. In some ways, we might think this is consistent with democratic notions of individual freedom - we can conduct whatever transactions we want out of the reach of surveillance. But this anonymity also lowers the cost of carrying on domestic illegal activity, and international activity that might be harmful to domestic interests, which is what Rogoff and Summers are on to. The illegal domestic activity could be various aspects of the drug trade and organized crime, while harmful international activity might involve, for example, the international market in military weapons.

Currency is a remarkably resilient payments technology. As I pointed out in my last post, the ratio of U.S. currency to GDP is as high as it was in the 1950s, and has risen since the beginning of the financial crisis. Though about 80% of the stock of Federal Reserve notes consists of $100 bills, currency still shows up in surveys of legal payments as an important transactions medium. As we might expect, this Boston Fed publication indicates that currency is used much more intensively by poor people than by the rich. The advantages of currency are obvious. It's easy to carry around (except if you want to buy a house or a car with it, for example); it comes in small denominations, making it convenient for small transactions; and it provides immediate settlement without the use of electricity (unless the other party in the transaction needs to open the cash register).

So, any scheme to eliminate currency altogether faces significant potential costs. Unless an alternative low-tech payments medium can be provided, a lot of poor people (and possibly some rich ones as well) are going to be worse off. But Rogoff and Summers have a point - it's hard to see what the benefits of $100 bills are. In fact, there would not be much loss in convenience for legitimate exchange from eliminating $50 notes as well. The U.S. government, however, would lose part of the revenue from the inflation tax. How large is that? U.S. currency outstanding is about $4500 per U.S. resident, so a 2% inflation rate generates about $90 per person in seignorage revenue. If eliminating $100 and $50 bills reduced the stock of currency outstanding by about 70% (assuming that some of the existing demand migrates to other denominations), the loss would be about $63 per person, which perhaps is a small price to pay for putting a serious dent in criminal activity and terrorism. Thus, what Rogoff and Summers are proposing seems like a no-brainer.

My best guess is that no one would be talking about currency much if it weren't for the negative interest rate enthusiasts, which gets us into a very different set of issues. Here's their (Kocherlakota/Agerwal/Kimball/Goodfriend) argument:

1. Take as given that the most important role (at the extreme, the only role) for monetary policy is correcting the distortions arising from sticky wages and prices. That's a key assumption - none of these authors thinks it is necessary to defend that.
2. The real interest rate is low, for reasons unconnected to monetary policy, and is expected to remain low into the indefinite future. It's hard to quarrel with this, but the reasons for the low real interest rate could matter.
3. (1) coupled with (2) implies that, in pursuit of inflation-targeting and output-gap targeting, the central bank will in the future be encountering the zero lower bound (if indeed it is a bound) with greater frequency.
4. Once constrained by the zero lower bound, there are losses in economic welfare due to output gaps and departures from inflation targets.
5. Such welfare losses can be avoided, if only nominal interest rates can go negative - this relaxes the zero lower bound constraint.
6. There are no costs to negative nominal interest rates, only benefits.

The hardline NK (New Keynesian) view, shared by Agarwal/Kimball/Kocherlakota/Goodfriend (AKKG) is that things that get in the way of the downward descent of nominal interest rates are an "encumbrance" that we are better off without. In this respect, currency just gets in the way. As the argument goes, a negative nominal interest rate on reserve balances will tend to make all safe short-term rates of interest negative. But for banks, there's a problem. The bank's assets are close substitutes for assets earning negative rates of interest, but its deposit liabilities are substitutes for zero-interest currency. So banks are squeezed - if they lower deposit rates, they lose small depositors; if they don't their profit margin goes down.

What to do about that? AKKG argue that there should be programs put in place to make currency less desirable, thus giving bank depositors a poor alternative to flee to. In Goodfriend's 2000 paper he suggests a scheme for taxing currency - equip every Federal Reserve note with a strip. Whenever the note returns to a bank, the bank charges the accumulated tax since the note was last turned in, and deducts this from your deposit. Alternatively, Agarwal and Kimball suggest a scheme for a changeable exchange rate between currency and reserves. The current arrangement between financial institutions holding reserves and the Fed, is that reserve balances can be converted one-for-one into currency, and vice-versa (presumably with service fees added for armored trucks and such). Under the Agerwal/Kimball arrangement, during periods of negative interest rates the rate at which reserves can be converted to currency would decline over time to match the negative interest rate on reserves.

The Goodfriend scheme seems problematic as, if the vintages of Fed notes ("vintage" meaning the time since the tax was last paid) are known, then notes should trade at different prices. This of course messes up the whole currency system, according to which the denominations are set up so that it's easy to make change - seems hard to do that if one note marked with a 10 trades at 97% of the value of another note marked with a 10, for example. All of these schemes will not make things any better for banks, who are in part selling a convenience to small depositors - the right to exchange their deposits one-for-one with currency, which these people find useful for making transactions. And people and firms are quite ingenious when it comes to getting around regulations. If a negative interest regime is in place for a long time (something of course the advocates don't imagine, but no one in Japan imagined they would have interest rates close to zero for over 20 years either), this creates profit opportunities for various types of cash hoarding operations. Hoarding cash is an activity subject to increasing returns. A financial intermediary could, in principle, set itself up as a simple currency warehouse, supplying safekeeping services, and make a profit in such an environment. Of course, AKKG, or people like them, could think up many ingenious ways to thwart such arbitrage schemes.

But would all this be worth it? In the process we would put sand in the gears of financial institutions that are serving a useful social function, and for what? We're going to have better monetary policy, apparently. This part of the story gets a bit complicated. Academic Keynesian economics is a well-defined thing - it's clear what it is, and you can evaluate it and determine for yourself whether or not it's useful. Keynesian economics in practice can sometimes be a completely different beast, saturated with myth rather than science. One myth is that, in reducing nominal interest rates, we can have our cake and eat it too. Conventional wisdom seems to be that reducing nominal interest rates increases inflation and reduces unemployment. That's just Phillips curve logic, right? Wrong. Even in basic New Keynesian (NK) models with Phillips curves, lower nominal interest rates make inflation go down - that's basic neo-Fisherism. The Fisher effect is ubiquitous in models, and it's there in the data too. Here's an example. Data for Switzerland on the overnight nominal interest rate and inflation look like this:
The Swiss went to negative interest rates at the beginning of 2015. Since then inflation has also been below zero. Of course, monetary policy isn't the only factor affecting the inflation rate. There is substantial variation in inflation in the chart, but you can see the trend. Negative nominal interest rates for 21 months in Switzerland are hardly making inflation explode.

As is well-known, a belief that low nominal interest rates cause high inflation inevitably leads to self-perpetuating low nominal interest rates and low inflation. That's part of the force behind the negative nominal interest rate lobby. Surely, they think, pushing interest rates into negative territory will cure the low-inflation problem. Well, sorry, adding another hole to your shoe doesn't fix your shoe.

But, the other motivation for negative nominal rates is stabilization policy. It may be true that a lower bound of zero gives the central bank less room to move when real interest rates are persistently low, but some people seem to find it difficult to reconcile themselves to the realities of the stabilization policy they claim to believe in. If we really think that moving a short-term nominal interest rate around has large real effects on aggregate economic activity, we have to recognize the need to make intertemporal tradeoffs. That is, we may think that lowering nominal interest rates confers some benefit, but we can't always be moving nominal interest rates down. Sometime they have to go up. Seemingly, a key principle of monetary stabilization is that it's less costly at the margin to increase nominal interest rates in good times than in bad times. Also, that it's of less benefit, at the margin, to lower nominal interest rates in good times than in bad times. Therefore, optimal monetary stabilization is about increasing interest rates in relatively good times, and reducing them in relatively bad times, while on average hitting an inflation target. Every time interest rates go up, central bankers are asking people to bear some short-term pain, in exchange for larger future short-term gains from lowering interest rates. Larry Summers, for example, seems to have a hard time recognizing this. According to him, we're in a long period of secular stagnation, which implies that we're in a relatively good state compared to the future. Further, Summers is very concerned with the fact that interest rates can't go down much if a recession happens. So, one might think he would be in favor of interest rate increases now, but he's not. Go figure.

Just as wrongheaded inflation control can lead to perpetually low nominal interest rates, so can wrongheaded stabilization policy. People who are single-minded about stabilization always see inefficiency when they look at the economy. Things are never quite right, so there's always a reason to lower interest rates. Given these tendencies among policymakers, perhaps a zero lower bound is a good thing. It's a constraint that prevents well-intentioned interventionists from defeating themselves, in terms of their own goals.

Where does this leave currency? Rogoff and Summers seem to have good arguments for getting rid of large-denomination currency. Getting rid of all currency, or making its use more costly seem like too high a cost, given the likely benefits of negative interest rate policy - about which proponents seem pretty confused. Central banks should, however, be thinking about electronic alternatives to currency. Blockchain technologies are potentially useful, and could be tied to the decentralized transfer of central bank liabilities. There are private alternatives to currency, for example Green Dot provides stored-value cards that are not tied to checking accounts. Would it be a good idea for central banks to issue stored value cards? In any case, we're not there yet with these alternative payments technologies so, for the time being, currency is a convenient low-tech payments instrument that we should keep.

Friday, September 2, 2016

Central Bank Liabilities: Part I

In this first installment, I'll discuss issues related to Greenwood, Hansen, and Stein's Jackson Hole paper, and leave discussion of the future of currency for installment #2.

An important model for modern central banking was the Bank of England, founded in 1694, which subsequently developed a symbiotic relationship with the British crown. The crown needed to finance spending, particularly on wars, and the Bank was looking to make a profit. The crown granted the Bank an ever-expanding monopoly on the issue of circulating currency, culminating in Peel's Bank Act of 1844, under which the Bank became the sole supplier of circulating currency in the UK (save for some grandfathered Scottish private banks, which are still issuing currency in 2016). Given the Bank's monopoly on currency issue, it could lend to the crown at a low interest rate, and still make a profit. The Bank, through experimentation or accident, discovered crisis intervention. During financial panics the Bank could, through judicious use of information available to it, engage in lending to banks with liquidity problems (banks experience a lot of bank note redemptions). In so doing, the Bank would expand its note issue to fund lending to banks that it deemed illiquid but solvent and, on the flip side of that, give those holding bank notes a safe asset to run to. The Bank did this, driven by its motive to make profits on superior information.

It wasn't like some economist was promoting the idea that a monopoly on currency issue and financial crisis intervention were the natural province of a central bank, or that this central bank should be have some form of public governance. Those ideas came much later. Indeed, the Bank of England remained a private institution until 1946. However, the Federal Reserve System was founded as a public institution, albeit with semi-private governance, in terms of how the regional Federal Reserve Banks are run. The work that led up to the Federal Reserve Act of 1913 included, for example, the National Monetary Commission, created by Congress to study the U.S. financial system, and systems elsewhere in the world, and to come up with recommendations for reform. The Commission produced volumes of useful stuff, which you can find on Fraser. The basic ideas in the Federal Reserve Act are that the role of a central bank in the United States was to: (i) through its monopoly power, provide a safe currency; (ii) make the currency "elastic," i.e. make the supply of currency respond to aggregate economic activity, and to financial panics, by way of discount window lending.

Basically, the framers of the Federal Reserve Act had in mind an institution that would play the same role as the Bank of England, but the motivation was different. A currency monopoly for the central bank was seen as socially beneficial, as experiments with private currency issue in the U.S. had not turned out well. And crisis intervention was seen as a means for preventing the disruption to the aggregate economy that had occurred during the banking panics of the National Banking era (1863-1913).

Fast forward to 2016. In some ways things haven't changed. U.S. Federal Reserve notes are basically the same stuff (with some anti-counterfeiting features added) they were in 1914, and currency is still an important Fed liability. Here's the stock of currency relative to U.S. GDP:
The ratio of currency to GDP has risen to close to 8% from less than 6% before the financial crisis, and is at the same level as in the mid-1950s. Retail payments using currency have fallen, but perhaps not as much as one might think. For example, surveys by the Boston Fed show a declining use of currency, but currency is still important in consumer payments, accounting for 26.3% of payments in 2013, as opposed to 31.1% for debit cards. But, the quantity of Federal Reserve notes per U.S. resident is about $4,500 currently. I don't know about you, but I'm not holding my fair share of that. In this chart you can see that, by value, about 80% of currency outstanding is in $100 notes, and studies indicate that about half of the stock of U.S. currency is held outside U.S. borders. There are some legitimate questions about the role played by currency, and whether its use should be curtailed, but more about that in the next installment.

The key point is that, if we took large-denomination currency - which is likely serving no useful social function - out of the mix, only a small part of the existing Fed portfolio would be funded by what was once thought to be the primary central bank liability. As it is, even with all those $100 bills included, interest-bearing Fed liabilities are greater than the quantity of currency outstanding:
The key Fed liabilities are, in order of magnitude (from small to large):

1. Reverse repurchase agreements, or ON-RRPs, used in "temporary open market operations." This is overnight lending to the Fed, with securities in the Fed's asset portfolio used as collateral. ON-RRPs are roughly reserves by another name - overnight interest-bearing liabilities of the Fed that can be held by a wider array of financial institutions than are permitted to have reserve accounts with the Fed. In particular, money market mutual funds cannot hold reserves, but they are key participants in the ON-RRP market. ON-RRPs play an important role in post-liftoff Fed implementation of monetary policy. The idea is that the ON-RRP rate, currently 0.25%, puts a floor under the fed funds rate, so that fed funds will trade between the interest rate on excess reserves (IOER), currently at 0.5%, and the ON-RRP rate.

2. Reverse repurchase agreeements held by foreign government-related institutions. This is somewhat mysterious, and explained here. Just as there is a foreign demand for Treasury securities, there is a foreign demand for very short-term liabilities of the Fed.

3. Currency. We know what this is about.

4. Reserve balances. As is well-known, this quantity has grown substantially, as this stuff financed the Fed's large-scale asset purchases post-financial crisis.

We're now in a world in which interest-bearing Fed liabilities have become very important. Is this a temporary change, or is it permanent? Should it be permanent? This gets us to Greenwood/Hansen/Stein (GHS). Basically, their answer to the last question is yes. GHS first argue that short-term safe assets are useful in financial markets and that, by looking at market interest rates, we can find evidence of "moneyness" for these assets. If assets are used in facilitating some kind of exchange, or they are in wide use as collateral, they bear a liquidity premium. That is, people are willing to hold these assets at lower rates of return than seem consistent with the actual payoffs on these assets. For example, Gomme, Ravikumar, and Rupert calculate real rates of return on capital in the U.S. since 2007 of from 5%-7% (after tax, for all capital). But here are some short term nominal rates of return:
Before December 17, 2015 (liftoff date), the IOER was 0.25% and the ON-RRP rate was usually 0.05% (with some experimentation). After that date, IOER was set at 0.5% and ON-RRP at 0.25%. In the chart, you can see that, after liftoff, the fed funds rate has typical fallen in the range 0.35%-0.40%, except at month-end and quarter-end (for technical reasons). Also, four-week T-bills trade in the same ballpark as ON-RRP, with some variation.

What can we conclude? (i) Reserves are a less liquid asset than ON-RRP or short-term Treasury debt. Reserves can be traded among a smaller set of financial institutions than these other assets, and you have to pay banks more to take reserves than to take T-bills or ON-RRP. (ii) Short-term Treasury debt and ON-RRP seem to have roughly the same liquidity properties.

So, once the case has been made that short-term safe assets bear liquidity premia, reflecting their usefulness in financial markets, who is going to supply these assets? There are three options: (i) the private sector; (ii) the Treasury; (iii) the Fed. GHS argue that the private sector does not do a very good job of this. Why? First, there are some "externalities" involved, according to GHS. This is somewhat vague, but GHS seem to have in mind that private sector production of short-maturity assets involves intermediating across maturities, which is risky. And maturity transformation makes these private financial intermediaries sensitive to market stresses, according to them. Second, regulatory changes, for example the Supplementary Leverage Ratio requirement, gums up the private sector's ability to produce safe short-maturity assets. GHS also argue that the Treasury does not issue enough short term debt, because it is worried about the risk of auction failure if it has to roll over a lot of short-term debt.

The heart of the argument is that the Fed has advantages over both the private sector and the Treasury in issuing short-term debt. But what kind of short-term debt should the Fed issue? Currently, the choice is between reserves and ON-RRP, though in principle we can think about the possibility the Fed could issue Fed bills - short term circulating debt that looks exactly like Treasury bills. The Swiss National Bank can issue such securities, for example. But, if wer're constrained to considering only reserves and ON-RRP, it seems clear that ON-RRP is much more successful in serving the liquidity needs of financial markets than are reserves. After all, you have to give the institutions that are permitted to hold reserves a 25 basis point inducement to get them to do so.

So, it seems that, in general, ON-RRP is a better instrument for the purpose than reserves. As GHS point out, if the Fed were to expand its ON-RRP program and shrink reserves, this would amount to savings for taxpayers, given the level of short-term market interest rates. This is becasue the Fed could have the same effect on market interest rates, but be paying lower interest on its liabilties, thus handing over larger transfers to the Treasury. What expands the ON-RRP program? One approach would be to set the ON-RRP rate equal to IOER.

But, how large should the Fed's ON-RRP balances be, and what's the optimal size for the Fed's balance sheet? Prior to the financial crisis, the size of the Fed's balance sheet was essentially determined by the demand for currency (in real terms), given the level of market interest rates, as the quantity of reserves was very small, and almost all of the Fed's asset portfolio was financed with currency. If we take the GHS proposal at face value, there should be a similar natural limit for ON-RRP. If the Fed sets an ON-RRP rate, and conducts a fixed-rate full-allotment auction, with IOER equal to the ON-RRP rate, what could happen? (i) the Fed reaches the upper bound on securities that it can use as collateral in ON-RRPs - it gets more bids than it can satisfy. (ii) the Fed does not reach the upper bound on available collateral, and there are some reserve balances outstanding. In case (i) it seems the ON-RRP program is too small, and in case (ii) it's the right size, but the Fed's balance sheet is too large. The bottom line, if we accept GHS's hypothesis that a Fed supply of short-term interest-bearing liquidity is a good thing, is that the market can determine how much of this stuff it needs. I should make it clear that this is my conclusion, not theirs. But I think this is the logical implication of their analysis.


1. How does this relate to quantitative easing (QE)? GHS seem to think this is a different issue - that QE is aimed at some short-run problem, and the role for ON-RRP is long-run. I don't think so. What GHS is putting forward is indeed a theory of how QE works. They are assuming that the Fed has a special role in maturity transformation, and when the Fed does this it matters. And this matters all the time, not just in unusual circumstances, so the implication is that QE should be an ongoing thing. But it seems there is a role for long-maturity debt in financial markets too - presumably the Fed shouldn't be sucking up all the long-maturity assets in existence, and GHS don't seem to be thinking about that.

2. I've got doubts about whether there are limitations on the private sector's ability to intermediate across maturities. Turning long Treasury debt into overnight debt is risky, but there are ways to manage such risk. It's not clear that the Fed and Treasury are any better at bearing such risks than are private financial institutions.

3. If the problem with private sector liquidity transformation is what can happen in crises, why shouldn't the Fed's intervention be confined to crisis times? For example, expand the ON-RRP program in a crisis, and contract it when the crisis is over.

4. A concern of GHS is a possible flight to safety during a panic. The argument is that, during times of financial stress, financial market participants could abandon private sector liquidity for ON-RRPs, and that could be bad. The "cure" for this is caps on the ON-RRP program. I've always found this idea puzzling. In the ON-RRP auction, the Fed can either set the quantity, or the price, but they can't set both. If there's a binding cap on the ON-RRP program, the price is too low, i.e. the ON-RRP rate is set too high. Presumably, in a crisis the correct policy is to lower the ON-RRP rate.

5. GHS's explanation for why the Treasury did not, or could not, issue more short-term debt did not make sense to me. This seemed more like a call for the Treasury to do a better job of auctioning their securities. Further, if a Treasury auction "fails" on a given day, I'm not sure why that's the end of the world. For example, the Treasury has a reserve account with the Fed, and the balance looks like this:
You can see that the balance in this account fluctuates a lot - it's an important buffer for the Treasury in managing cash inflows and outflows. Further, the average size has increased substantially, to around the $300 billion range. If rollover risk is greater with more short debt outstanding, why can't the Treasury have a larger average balance in that account?

In general I found this paper very useful. It's the first coherent story I've seen about a legitimate role for a central bank in what we would typically call "debt management." But much more research and thought needs to go into these questions.

Friday, August 26, 2016

We are All Neo-Fisherites

I was looking at this piece by Mark Thoma on increasing the inflation target. In some discussions of this issue, people seem to have a hard time getting to the core of the argument, but Mark does not. He has a good discussion of the Fisher effect, and his concluding paragraph is:
The Fed did not have enough room to cut interest rates before hitting the zero lower bound when the recession hit. Raising the target inflation rate, which would increase average interest rates and give the Fed more space for rate cuts, is something the Fed ought to seriously consider.
He's not quite as blunt as I might like, but he's saying that, if a central bank wants to hit a higher inflation target, it has to set nominal interest rates higher, on average. So, in the course of transitioning to a higher inflation target, the central bank must, at some time, have to raise nominal interest rates in order to produce higher inflation. But then, it must be true that, if the central bank has an inflation target of x%, and inflation is persistently y%, where y < x, then the central bank must raise its nominal interest rate target.

Monday, August 22, 2016

Danger!! Crazy Neo-Fisherians on the Loose!!

Not sure how I missed this Narayana post, but better late than never. I may not be Jacques Derrida, but a little deconstruction can be good fun. Here goes.

Opening paragraph:
Some economists argue that the Federal Reserve should take a highly unconventional approach to ending a long period of below-target inflation: Instead of keeping interest rates low to spur economic activity and push up prices, it should raise rates.
Clicking on "argue" takes you to my St. Louis Fed Regional Economist piece on neo-Fisherism. This is about as low-tech an elucidation of these ideas as I've been able to muster - it's got one equation, two figures, and 3,000 and some words. In any case, apparently I'm "some economists." John Cochrane is also well out of the Fisherian closet, and we have certainly received some sympathy from others (to whom the idea is obvious - as it should be), but neo-Fisherism is hardly a movement. As you can see, particularly in Narayana's post, there's plenty of hostile resistance.

The last sentence in the quote offers you a false choice. That choice is either a world of low interest rates, which obviously spurs economic activity and pushes up prices or the alternative: increases in interest rates which, the reader would naturally assume, would give us the opposite - less economic activity and lower inflation. The basic neo-Fisherian idea is that this is not the choice we're faced with. Let's put aside the issue of how monetary policy affects real economic activity, and focus on inflation. Neo-Fisherism says that conventional central banking wisdom is wrong. A lower nominal interest rate pushes inflation down, and no one should be surprised if an extended period of low nominal interest rates produces low inflation. Indeed, that's consistent with what we're seeing in the world right now.

Let's move on. Two paragraphs later, we have this claim:
Neo-Fisherites believe that modern economies are self-stabilizing.
I've been staring at that sentence for several minutes now, and I'm still not sure what it means, so I don't think I could "believe" it. But let's give this a try. In conventional Econ 101 macroeconomics, students are typically told that, in the short run, wages and prices are sticky, and there is a role for short-run "stabilization" policy, which corrects for the short-run inefficiencies caused by stickiness. The Econ 101 story is that, in the long run, prices and wages are perfectly flexible, and the inefficiencies go away. So, the standard story people are giving undergraduates is that modern economies are indeed self-stabilizing, but "in the long run we're all dead" as Keynes said. What's this have to do with neo-Fisherism? Nothing. Indeed, most conventional models have neo-Fisherian properties, whether those models have a role for short-term government intervention or not. In this post I worked through the neo-Fisherian characteristics of Narayana's favorite model, which is certainly not "self-stabilizing" in the short run. Bottom line: Basic neo-Fisherism is agnostic about the role for government intervention. It just says: Here's how to control inflation. You've been doing it wrong.

Narayana says that neo-Fisherism leads in the direction of "unusual policy recommendations."
Suppose, for example, the long-run equilibrium real rate is 2 percent. Neo-Fisherites would predict that if the Fed holds nominal rates at 0.5 percent for too long, people's inflation forecasts will eventually have to turn negative -- to minus 1.5 -- to get the real rate back to 2. Conversely, if the Fed raises its rate target to 4 percent and keeps it there, inflation expectations will rise to 2 percent. Because such expectations tend to be self-fulfilling, the result will be precisely the amount of inflation that the Fed is seeking to generate.
That doesn't describe "unusual policy recommendations" but is actually the prediction of a host of standard monetary models. For example, there is a class of representative agent monetary models (money in the utility function, cash in advance, for example) in which, if the subjective discount rate is .02, in a stationary environment, then the long run real interest rate is indeed 2%. In those models, it is certainly the case that a sustained nominal interest rate of 0.5%, supported by open market operations, transfers, whatever, must ultimately induce a deflation equal to -1.5%. Indeed, those models also tell us that deflation at -2% would be optimal - that's the Friedman rule. So, Narayana's thought experiment is not controversial in macroeconomics - that's the prediction of baseline monetary models. Things can get more interesting with fundamental models of money that build up a role for asset exchange from first priciples - e.g. overlapping generations models from back in the day, or Lagos-Wright constructs. New Keynesians seem to like taking the money out of models altogether, in "cashless" frameworks. NK models, and fundamental models of money typically have many equilibria, which presents some other problems. Multiple equilibria can also be a feature of cash-in-advance models. But, as I discuss in this post and this one, multiple equilibria need not be a serious problem for monetary policy, as we can design policies that give us unique equilibria - with Fisherian properties.

But, from Narayana's point of view, standard macroeconomics is not standard - it's crazy and dangerous. His claim:
Traditional economic models, by contrast, predict the opposite. If the central bank raises rates and credibly commits to keeping them high, people and businesses become less willing to borrow money to invest and spend. This undermines demand for goods and services, putting downward pressure on employment and prices. As a result, the economy can plunge into a deflationary spiral of the kind that afflicted the U.S. in the early 1930s.
What "traditional models" could he be talking about? This can't be some textbook IS/LM/Phillips curve construct, as he's discussing a dynamic process, and the textbook model is static. The only "tradition" I know of is a persistent narrative that you can find if you Google "deflationary spiral." Here's what Wikipedia says:
The Great Depression was regarded by some as a deflationary spiral. A deflationary spiral is the modern macroeconomic version of the general glut controversy of the 19th century. Another related idea is Irving Fisher's theory that excess debt can cause a continuing deflation. Whether deflationary spirals can actually occur is controversial, with its possibility being disputed by freshwater economists (including the Chicago school of economics) and Austrian School economists.
The closest thing to actual economic theory supporting the idea is Irving Fisher's debt-deflation paper from 1933. That's just another narrative - you won't find an equation or any data in Fisher's paper.

The best I can come up with in terms of a genuine theory of deflationary spirals, is what can happen in Narayana's NK model if inflation expectations are sufficiently sticky, and initial inflation expectations are sufficiently low - basically, people have to start off expecting a lot of deflation. Further, in order to support a "deflationary spiral," i.e. sustained deflation, conventional asset pricing tells us that there has to be sustained negative growth in output. But if there's a lower bound on output, which is natural in this type of environment, then the deflationary spiral isn't an equilibrium. Conclusion: Narayana has things turned around. Traditional macroeconomics gives us a long run Fisher effect. Deflationary spirals are not part of any "traditional" (i.e. serious) macroeconomic theory.

On the empirical front, the "deflationary spiral ... that afflicted the U.S. in the 1930s" looks like this:
There's a body of macroeconomic history that ascribes that deflationary episode to the workings of the gold standard. Indeed, the deflation stops at about the time the U.S. goes off the gold standard. Not sure why we're using a gold standard episode to think about how monetary policy works in the current context. In modern economies, I have no knowledge of an instance of anything consistent with Narayana's "traditional model" in which increases in nominal interest rates by the central bank cause a "deflationary spiral" (if you know of one, please let me know). But, when "deflation" enters the conversation, some people will mention Japan. Here's the CPI level for Japan for the last 20 years:
This is one of my favorite examples. We wouldn't really call that a "deflationary spiral" as the magnitude of the deflation isn't high at any time, and it's not sustained. Over 20 years, average inflation is about zero. Further, since mid-1995, the Bank of Japan's nominal policy interest rate has been close to zero, and recently the BOJ has thrown everything but the kitchen sink (except, of course, a higher policy rate) at this economy in an attempt to generate inflation at 2% per year - to no avail. Note in particular that the blip in inflation in 2014 can be attributed almost entirely to the direct effect of an increase in the consumption tax of 3 percentage points.

So, that's an instance in which a form of "traditional" macroeconomics doesn't work. That traditional macroeconomics is textbook IS-LM/Phillips curve with fixed inflation expectations. In that world, a low nominal interest rate makes output go up, and inflation goes up through a Phillips curve effect. A standard claim in world central banking circles is that a low nominal interest rate, sustained for a long enough time, will surely make inflation go up. I don't know about you, but if I want to catch a bus, and I go down to the bus stop and find someone who has been waiting for the bus for twenty years, my best guess is that sitting down in the bus shelter with that person has little chance of making a bus appear any time soon.

Next, in Narayana's post, he shows a time series plot of the fed funds rate and a breakeven rate. To be thorough, I'll include other breakeven rates, and focus on the post-2010 period, as the earlier information isn't relevant:
Narayana says:
The Fed held the nominal interest rate near zero from late 2008 until late 2015 -- a policy that, according to Neo-Fisherites, should have driven inflation expectations into negative territory. Yet they stayed roughly the same for most of that period. They started to slide downward only after the Fed began to tighten policy in May 2013 by signaling that it would pull back on the bond purchases known as quantitative easing. Also, the recent modest increase in the nominal rate has not led to a commensurate increase in inflation expectations.
"According to Neo-Fisherites?" No way! Any good neo-Fisherite understands something about low real interest rates, and what might cause them to be low. Here's me thinking about it in 2010. If there are forces pushing down the real interest rate, we'll tend to get more inflation with a low nominal interest rate than we might have expected if we were thinking the long run real rate was 2%, for example. So, by 2013, yours truly neo-Fisherite was certainly not surprised to be seeing the breakeven rates in the last chart.

But how should we interpret the movements in the breakeven rates in the chart? On one hand, breakeven rates have to be taken with a grain of salt as measures of inflation expectations. They can reflect changes in the relative liquidity premia on nominal Treasury bonds and TIPS; they're measuring breakeven rates for CPI inflation, not the Fed's preferred PCE inflation measure; when inflation falls below zero, the inflation compensation on TIPS is zero; there is risk to worry about. On the other hand, what else can we do? There are alternative market-based measures of inflation expectations, but it's not clear they are any better than what I've shown in the chart.

So, suppose we take the breakeven measures in the chart seriously. The 5-year and 10-year breakevens can be interpreted as predictions of average inflation over the next 5 years, and the next 10 years, respectively. The five year/five year forward rate can be interpreted as the average inflation rate anticipated over a five-year period that is 5 to 10 years from today. Given that the interest rate Narayana is focused on here is the overnight fed funds rate, what matters for these market inflation expectation measures is the course of monetary policy for up to the next 10 years - in principle, the structure of the Fed's policy rule over that whole period. There are plenty of other things that matter as well - world events, shocks to the economy, and how those events and shocks matter for the Fed's policy rule. Narayana seems to think that the Fed "tightened" in May 2013, but I remember that episode - the "taper tantrum" - as a prelude to a period in which the public perception of the future course of interest rate hikes was constantly being revised down. A downward path for long-term inflation expectations seems to me consistent with a neo-Fisherian view of the world, with the market putting increasing weight on the possibility that nominal interest rates and inflation will remain persistently low.

Narayana finishes off in true hyperbolic fashion by raising the twin specters of the Great Depression and Great Recession:
I, too, once believed that the horrific events of the early 1930s, when economic output fell by a quarter and prices by even more, could not recur in a modern capitalist economy like the U.S. Then 2008 happened, and we all learned where a religious belief in the self-correcting nature of markets can lead us. If we want stability, we have to choose the right policies. Raising rates in the face of low inflation is not one of them.
I hope you understand by now that I think: (i) economics is about science, not religion; (ii) neo-Fisherism has nothing to do with the "self-correcting nature of markets." Do I think that Narayana's policy prescriptions are crazy and dangerous? Absolutely not. If he's right, which I think he's not, then good for him. If he's wrong, and his policies get implemented, what harm gets done? Inflation stays low, and central banks may proceed to demonstrate, through experimentation, that unconventional policies don't do much. Or maybe we find some that actually work. Who knows? We would really be in danger if the people who think of high inflation as a cure-all figure out how to produce it. But I don't think that will happen.

Sunday, July 31, 2016

Multiple Equilibria, Installment #2

The goal in this post is to provide some more illumination with respect to Narayana's note, and my previous post. As well, if I could eliminate Nick Rowe's confusion, that would be great.

The problem at hand is one of multiple equilibria. Sometimes multiple equilibrium models are used in an attempt to explain real-world phenomena. That's Roger Farmer's approach - maybe we're stuck in bad, suboptimal states because of self-filfilling low expectations. Sometimes policy rules can lead to multiple equilibria in models we study. That's considered problematic as, to analyze policy in a coherent fashion, we would like to have a unique mapping from policy rules to outcomes, so that the optimal policy problem we're solving is well-specified. That's the problem that comes up in New Keynesian models, but it's certainly not unique to that class of models, as we'll show in the example below.

For people who work in monetary economics, multiple equilibria are ubiquitous. In any model that builds up a role for valued fiat money from first principles, there is always an equilibrium in which money is not valued - if people believe that money will not have value at any date in the future, it will never have value. Fiat money has no intrinsic payoffs, so if people believe that others will not accept it in exchange, they will not accept it either - valued money is supported as an equilibrium because everyone has the self-confirming faith that it will always be valued. So in models of fiat money, there is an equilibrium in which money is not valued, and typically many equilibria in which it is.

One old workhorse of monetary economics is Samuelson's overlapping generations model. The specific example I'm going to use comes from Costas Azariadis's 1981 paper. Time is indexed by t = 1,2,3,..., and at t = 0 there are some old people endowed with M(0) units of money. In each period there are N two-period-lived people who work when they are young and consume when they are old. Each has preferences

(1) U[c(t+1),n(t)] = u[c(t+1)] - v[n(t)],

where c is consumption and n is labor supply. One unit of labor input produces one unit of consumption good. In equilibrium, the young work, purchase money from the old in exchange for goods, and then sell the money for goods when they're old. The government can inject money each period through lump sum transfers to the old. The money stock in period t is M(t). Assume preferences have standard properties: u is strictly concave and v is strictly convex, etc.

In equilibrium, everyone optimizes, and markets clear. There can be plenty of equilibria, including sunspot equilibria and cycles (see Azariadis's paper), but we'll focus on the deterministic ones. In general, we summarize equilibria as sequences {n(t)} that solve the difference equation

(2) [M(t)/M(t+1)]n(t+1)u'[n(t+1)] - n(t)v'[n(t)]=0,


(3) p(t) = n(t)/M(t),

where p(t) is the price of money - the inverse of the price level.

Here's an example. Let M(t)=1 for all time, and assume u has constant relative risk aversion a, with v just n to the power b. Here, a > 0 and b > 1. Then, if we write the difference equation (2) in logs (don't know how to deal with exponents in html), we get

(4) ln[n(t+1)] = [b/(1-a)]ln[n(t)]

So, if a < 1, then (4) looks like this:
And if a > 1, it looks like this:

In either case, there are two steady states: (i) n = 0, where money has no value forever, and nothing gets produced. You can't see that in the second picture, but it's an equilibrium nevertheless. (ii) n = 1. The second steady state is the quantity-theoretic equilibrium. The money growth rate is zero, the inflation rate is zero, the growth rate in output is zero, and the velocity of money is constant forever. But there are also other equilibria, depending on parameters.

First, suppose a < 1. In the first chart, there are many equilibria with 0 < n(0) < 1 which all converge in the limit to n = 0. These are hyperinflationary equilibria for which the inflation rate increases over time without bound. There are also many equilibria with n(0) > 1 for which n(t) grows over time without bound. These are hyperdeflationary equilibria, for which the inflation rate falls over time without bound.

So, those are all the equilibria for that case (I think there are no cyclical or sunspot equilibria either - see Costas's paper). What would Narayana's note say about this? He's interested in the limiting equilibria of finite horizon economies. If we looked for such equilibria here the search is not difficult. Suppose we fix the horizon at length T, where T is finite. Then p(t)=0 for all t. No one would want to hold money in any period, because it has no value in the final period. So, the only finite-horizon equlibrium is n = 0 for any T, so if I take the limit I get n = 0. So, Narayana's claim that a limiting equilibrium of the finite horizon economy is an equilibrium in the infinite horizon economy is correct, but we only found one equilibrium by this approach - the one where money has no value.

We could take a broader view, however. Take the infinite horizon economy, fix p(T), solve the difference equation (4) backward, then let T go to infinity. In this case, the difference equation is stable backward. So, this picks out two equilibria, n = 0 and n = 1. That's an equilibrium selection device which, if we took it seriously, would permit us to ignore all the non-steady-state equilibria that converge to n = 0 in the limit. But that approach shouldn't fill us with confidence. By conventional criteria, in this case n = 0 is "stable" and n = 1 is "unstable."

Next, consider the case 1 < a < 1 + b. In this case, the slope of the difference equation in the second figure is not too steep at n = 1. In addition to the two steady states, there are now many equilibria with n(0) > 0 that converge in the limit to n = 1. Again, literally following Narayana's advice gives one equilibrium, n = 0, but if we following our other limiting approach, the difference equation is unstable backward, and there are three limiting equilibria: (i) n = 0; (ii) n = 1; (iii) a two cycle {...,0,inf,0,inf,0,inf,...}. So that's an example for which Narayana's claim is not correct, as that's not an equilibrium of the infinite horizon economy, since n = 0 is a steady state.

One problem with the model I've specified is that it permits, under some conditions, hyperdeflations in which output grows without bound. A simple fix for that is to put an upper bound on labor supply, keeping preferences as we've specifed them. That will kill off all the hyperdeflationary equilibria, as well as the limiting two-cycle we get by the Narayana method. Then, the Narayana method, taken literally, gives us one equilibrium: n = 0. The Narayana method, taken liberally, gives us two equilibria: n = 0 and n = 1. Note that Narayana's NK model is misspecified in a similar way (see my previous blog post). Given his Phillips curve, he finds equilibria for which i = inf and i = -inf. But in the first such equilibrium, output is rising at an infinite rate, and in the second it is falling at an infinite rate. An upper bound on labor supply would put an upper bound on output, and kill the first equilibrium. As well, in Narayana's model, the Phillips curve is derived by assuming that a fraction of firms charge last period's average price. So, if i = -inf the sticky price firms sell no output, but the flexible price firms have to sell some output. This puts a lower bound on output, which kills off the i = -inf equilibrium. Thus, by the liberal Narayana method, there is only one equilibrium in his NK model - the Fisherian one.

What about the literal Narayana method in his NK model? Here we have a problem. In spite of the fact that this is a cashless model, nominal bonds are traded as claims to money. But in a finite horizon model, the value of money must be zero in the final period, and thus in all periods. So the price of nominal bonds is zero. Thus, we can't even start discussing the usual NK approach, which is assuming that the central bank can set the price of a nominal bond. The central bank is stuck with a price of zero.

Of course, we can wave our hands at this point, and claim that, in a finite horizon monetary model, the price of money is pegged in the last period through fiscal intervention. But that would be a different model, and we might ask why the fiscal authority doesn't do that intervention in every period - then we're done. The central bank should abandon its assigned job and hand it over to the fiscal authority.

Here's something interesting. In line with my previous blog post, there is an optimal monetary policy in this model that kills off indeterminacy. It looks like this:

M(t+1)/M(t) = {n(t+1)u'[n(t+1)]}/{n*v'[n(t)]}

where n* solves u'(n*)=v'(n*). In equilibrium, the money supply is constant, and the policy rule specifies out-of-equilibrium actions that eliminate the indeterminacy.

Question: Does Narayana have a point? Answer: Nah.

Monday, July 18, 2016

More Neo-Fisher

What follows is an attempt to make sense of Narayana's note on Neo-Fisherism. That discussion will lead into comments on a paper by George Evans and Bruce McGough.

Start with basics. What are Neo-Fisherite ideas anyway? Narayana says the absence of shocks, the equilibrium inflation rate should be constant if the nominal interest rate is pegged forever. The Fisher equation then implies that the inflation rate should move one for one with the nominal interest rate. This logic is sometimes referred to as “neo-Fisherian”.
I would actually call these New Keynesian (NK) claims. For example, in "Interest and Prices," Mike Woodford takes pains to address the concern, which came out of the previous macro literature, that nominal interest rate pegs are unstable. Woodford's claim is that a Taylor rule that conforms to the Taylor principle (a greater than one-for-one increase in the nominal interest rate in response to an increase in inflation) will imply determinacy. That is, if there are no shocks, then the nominal interest rate is pegged at a constant forever, and the inflation rate is a constant - the inflation target. Further, in the basic NK model, if Woodford's claim is correct then, in the absence of shocks, if the central bank wants to increase its inflation target, then the nominal interest rate should increase one-for-one with the increase in the inflation target, and actual inflation will respond accordingly. Under basic NK logic, this behavior is supported by promises to increase the nominal interest rate in response to higher inflation - and this inflation never materializes in equilibrium.

But, whatever we think Neo-Fisherite or New Keynesian ideas are, Narayana is making a particular argument in his note, and we want to get to the bottom of it. I don't think the analogy part is particularly helpful though. There are two problems considered in Narayana's note. One is an asset pricing problem, and the other has to do with the properties of a particular NK model. As far as I can tell, the extent of the commonality is that solving each problem can involve geometric series. Otherwise, understanding one problem won't help you much with the other.

The asset pricing problem looks like a trick question you might give to unwitting PhD students on a prelim exam. The equilibrium one-period real interest rate is negative and constant forever, and we're asked to price an asset that pays out a constant real amount each period forever. Question: Solve for the steady state price of the asset. Answer: Dummy, there is no steady state price for the asset. Since a rational economic agent in this world values future payoffs more than current payoffs, if we compute the present value of the payoffs, it will be infinite.

Well, so what? On to the second problem. Narayana uses a version of the standard NK model. We're in a world with certainty - no shocks. I'll change the notation so I don't have to use Greek letters. From standard asset pricing, and assuming constant relative risk aversion utility, we can take logs and get
Here, y is the output gap (the difference between actual output and efficient output), i is the inflation rate, R is the nominal interest rate, and r is the subjective discount rate (or the "natural real interest rate"). The second equation is a Phillips curve
This is the only difference from standard NK, as the Phillips curve doesn't have a term in anticipated inflation. This makes the solution easy, but I don't think it otherwise changes the basic mechanics.

In general, we can solve to get the difference equation
Then, an equilibrium involves finding a sequence of inflation rates that solves the difference equation (3) given some sequence of nominal interest rates, or some policy rule governing the central bank's choice of the nominal interest rate each period.

So, suppose that the nominal interest rate is a constant R forever, and suppose that, in period T the inflation rate is i(T). Then, we can solve the difference equation (3) forward to get
Similarly, we can solve (3) backward to get
So, for any real number i(T) equations (4) and (5) describe an equilibrium. Thus, there is a whole continuum of equilibria, indexed by i(T). In equation (4), the second term on the right-hand side converges to zero as n goes to infinity, for any i(T). Thus, all equilibria converge in the limit to an inflation rate of R-r. That's the long-run Fisher relation. In equation (5), the second term does not converge as n goes to infinity, i.e. as time runs backward to minus infinity. If i(T) < R - r, then inflation runs off to minus infinity as time runs backward, and if i(T) > R - r, then inflation runs off to infinity as time runs backward. This is typical of course - we have a difference equation that's stable if we solve it forward, and it's unstable if we solve it backward. Note that one equilibrium is i(t) = R - r in every period.

What Narayana does is to take equation (5), and let T go to infinity, so he's only looking at the backward solution. As should be clear, I hope, that's not describing all the equilibria. By any conventional notion of what we mean by convergence and stability, the nominal interest rate peg is stable, and all the equilibria converge in the limit to R - r. The Fisher relation holds in the long run. As a practical implication of this, I've heard many people argue that, if the central bank holds its nominal interest rate at zero, then surely inflation will eventually rise to the 2% inflation target. Well, they can't be thinking about this model then. In any equilibrium with R = 0 forever and with inflation initially lower than some inflation target i*, inflation either falls to -r in the limit, or rises to -r in the limit. If -r < i*, the central bank will never achieve its target by staying at zero.

But, with a nominal interest rate pegged at some value forever, we have an indeterminacy problem - there exists a plethora of equilibria. This makes it hard to make statements about what happens when the interest rate goes up or down. For example, it's certainly correct that, if we set T=0 in equation (4), and think of time running from zero to infinity, solving the difference equation (3) forward, then given i(0), the inflation rate will be higher along the whole equilibrium path, if R rises. But i(0) is not predetermined - it's not an initial condition, it's endogenous and the first step in only one equilibrium path. Who is to say that economic agents don't treat R as a signal and jump to another equilibrium path? We might also be tempted to set i(0) = R*-r, then solve for the equilibrium path given R = R**, and think of that as describing the effects of an increase in the nominal interest rate from R* to R**, since an inflation rate of R* - r is the long run inflation rate when R = R*. Though that's suggestive, it's not precise, due to the indeterminacy problem.

So what to do about that? If we follow the usual NK approach, we would specify a Taylor rule
In equation (6), the Taylor principle is d > 1, and Mike Woodford says that gives us determinacy. But what he means by that is local determinacy - that is, determinacy in a neighborhood of the inflation target i*. But this model is simple enough that it's easy to look at global determinacy - or indeterminacy, in this case. From equation (3) and (6), we get
And the picture looks like this:
D is the difference equation from (7). Note that the kink in the difference equation is where the nominal interest rate hits the zero lower bound (for low inflation rates). A is the desired steady state where the central bank hits its inflation target, and B is the undesired steady state in which the inflation rate is - r and the nominal interest rate is zero. A is an equilibrium, but it's unstable - there are many equilibria that converge in the limit to B. We won't discuss equilibria in which inflation increases without bound, as the model needs to be fixed a bit so that those make sense, but that's possible in a slightly modified model. These are well-known results - the Taylor principle has "perils," i.e. it yields indeterminacy, and there are many equilibria in which the central bank falls short of its inflation target forever - not great.

So, we might look for other policy rules that are better behaved. Here's one:
That rule implies a difference equation that looks like this:
The equilibrium is
The first part of the rule, (8), acts to offset effects of future inflation on current inflation, thus killing off equilibrium paths that will imply current inflation above target. (8) is only an off-equilibrium threat. The second part of the rule, (9), acts to bring inflation back to target next period. The equilibrium result is that inflation can be lower than the target in period 0, but the central bank hits its target in every future period. Further, note that the rule is neo-Fisherian, in more than one way. First, the central bank reacts to low inflation by increasing the nominal interest rate above its long-run level, temporarily. Second, the equilibrium satisfies the properties in the quote at the beginning of this post. After period 0, the nominal interest rate is constant forever, and inflation is constant. If the inflation target increases, then the nominal interest rate increases one-for-one in periods 1,2,3,... Narayana says those are Neo-Fisherian properties, and I stated above that I thought these were claims made of standard NK models under the Taylor principle. Seemingly, these are deemed by some people to be good properties of a monetary policy rule.

What Narayana seems to be getting at is that stickiness in expectations matters. In the example he gives in his note, fixed expectations in the infinite future can have very large effects today. You can see that in equation (5), for example, if we fix i(T) and solve backward. Indeed, it seems that conventional central banking wisdom comes from considering expectations as fixed, as is common practice in some undergraduate IS-LM/Phillips curve constructs. Take equation (1), fix all future variables, and an increase in the current nominal interest rate makes output and inflation go down. Indeed, sticky expectations is what George Evans and Bruce McGough have in mind. Here's their claim:
Following the Great Recession, many countries have experienced repeated periods with realized and expected inflation below target levels set by policymakers. Should policy respond to this by keeping interest rates near zero for a longer period or, in line with neo-Fisherian reasoning, by increasing the interest rate to the steady-state level corresponding to the target inflation rate? We have shown that neo-Fisherian policies, in which interest rates are set according to a peg, impart unavoidable instability. In contrast, a temporary peg at low interest rates, followed by later imposition of the Taylor rule around the target inflation rate, provides a natural return to normalcy, restoring inflation to its target and the economy to its steady state.
We can actually check this out in Narayana's model. Following Evans-McGough (E-M), we'll assume a form of adaptive expectations. Let e(t+1) denote the expected rate of inflation in period t+1 possessed by economic agents in period t. Assume that
So, h determines the degree of stickiness in inflation expectations - there is less expectational inertia as h increases. Using (1), (2), and (11) we can solve for current inflation and expected inflation for next period given the current nominal interest rate and expected inflation as of last period:
How this dynamic system behaves depends on parameters. To see some possibilities, consider extreme cases. If h=0, this is the fixed expectation case - expectations are so sticky that economic agents never learn. Letting e denote fixed inflation expecations,
That's the undergrad IS-LM/P-curve model. If you want inflation to go up, reduce the nominal interest rate. The other extreme is h = 1 which is essentially rear-mirror myopia - economic agents expect inflation next period to be what it was this period. This gives
That's extreme Neo-Fisherism. If you want inflation to go up by 1%, increase the nominal interest rate by 1%.

The question is, what happens for intermediate values of h? There are three cases: sticky expectations
medium-sticky expectations:
Not-so-sticky expecations:
The sticky expectations case gives the results that E-M are looking for. If the central banker follows a Taylor rule then, if inflation expectations are sufficiently low, the central banker goes to the zero lower bound, inflation increases, the Taylor rule eventually kicks in, and inflation converges in the limit to the inflation target i*. But, with medium-sticky or not-so-sticky expectations, from (12) increases in the nominal interest rate increase inflation. Further, if expectations are not-so-sticky there are Taylor rule perils. If d > 1, then there always exist equilibria converging to the zero lower bound with i = -r in the limit. In those equilibria the central bank undershoots its inflation target forever.

Under no circumstances is the standard Taylor rule with d > 1 well-behaved. At best, if inflation is initially below target, the inflation target is only achieved in the limit, and at worst the central banker gets stuck at the zero lower bound forever. But, there are other rules. Here's one:
Under this rule, the central banker hits the inflation target every period, provided initial inflation expectations are not too far below the inflation target. In the worst case, the central banker spends a finite number of periods at the zero lower bound when inflation expectations are too low. But, if inflation expectations are medium-sticky or not-so-sticky, the period at the zero lower bound exhibits inflation above the inflation target - i.e. a period at the zero lower bound can serve to bring inflation down.

The critical value for inflation expectations is
That is, under the rule (19), the central banker goes to the zero lower bound if inflation expections fall below e*. Note that e* is decreasing in h and goes to minus infinity as h goes to 1. As expectations become less sticky, the zero lower bound kicks in only for extreme anticipated deflations.

In their paper, E-M say
As we have shown, the adaptive learning viewpoint argues forcefully against the neo-Fisherian view and in support of the standard view.
As I hope I've made clear, that's overstated. I take the "standard view" to be (i) staying at the zero lower bound will eventually make inflation go up; (ii) a standard Taylor rule is the best the central bank can do. In Narayana's model, under adaptive learning, (i) is only correct under some parameter configurations - actual inflation and expectation inflation both have to be sufficiently sticky. Further, (ii) is never correct.

Tuesday, June 21, 2016

Attitude Adjustment

For this post, note the disclaimer at the top of the page. I'm just speaking for myself here, and my views do not necessarily reflect those of the St. Louis Fed, the Federal Reserve System, or the Board of Governors.

This is a reply to Narayana's recent Bloomberg post, which is a comment on this St. Louis Fed memo.

First, Narayana says that Jim Bullard thinks that
... the economy is so weak that a mere quarter-percentage-point increase would be enough for the foreseeable future.
I don't think the memo actually characterizes the economy as "weak" - it's not a pessimistic view of the world as, for example, Larry Summers or Robert Gordon might see it. As I noted in this post, one would not characterize the labor market as "weak." It's in fact tight, by conventional measures that we can trust. The view in the St. Louis Fed memo is that growth in real GDP, at 2% per annum, is likely to remain lower than the pre-financial crisis trend for the foreseeable future - i.e. "weaker" than we've been accustomed to. But "so weak" is language that is too pessimistic. And there remains the possibility that this will turn around.

Second, Narayana says:
Bullard’s rationale focuses on productivity...
That's not correct. The memo mentions low productivity growth, but a key part of the argument is in terms of low real rates of interest. According to conventional asset pricing and growth theory, low productivity growth leads to low consumption growth, which leads to low real rates of interest. But that effect alone does not seem to be strong enough to explain the fall in real interest rates in the world that has occurred for about the last 30 years or so. There is another effect that we could characterize as a liquidity premium effect, which could arise, for example, from a shortage of safe assets. I've studied that in some of my own work, for example in this paper with David Andolfatto. In recent history, the financial crisis, sovereign debt problems, and changes in banking regulation have contributed to the safe asset shortage, which increases the prices of safe assets, and lowers their yields. This problem is particularly acute for U.S. government debt. A key point is that a low return on government debt need not coexist with low returns on capital - see the work by Gomme, Ravikumar, and Rupert cited in the memo.

Third, Narayana thinks that:
Bullard uses a somewhat obscure measure of inflation developed by the Dallas Fed, rather than the Fed’s preferred measure, which is well below 2 percent and is expected to remain there for the next two to three years.
"Obscure," of course, is in the eye of the beholder. Let's look at some inflation measures:
The first measure is raw pce inflation - that's the Fed's preferred measure, as specified here. The second is pce inflation, after stripping out food and energy prices - that's a standard "core" measure. The third is the Dallas Fed's trimmed mean measure. Trimmed mean inflation doesn't take a stand on what prices are most volatile, in that it strips out the most volatile prices as determined by the data - it "trims" and then takes the mean. Then we calculate the rate of growth of the resulting index. One can of course argue about the wisdom of stripping volatile prices out of inflation measures - there are smart people who come down on different sides of this issue. One could, for example, make a case that core measures of inflation give us some notion of where raw pce inflation is going. For example, in mid-2014, before oil prices fell dramatically, all three measures in the chart were about the same, i.e. about 1.7%. So, by Fisherian logic, if the real interest rate persists at its level in mid-2014, then an increase in the nominal interest rate of 50 basis points would make inflation about right - perhaps even above target. Personally, I think we don't use Fisherian logic enough.

Finally, Narayana says:
...the risk of excess inflation is relatively manageable.
That's a point made in the memo. The forecast reflects a view that Phillips curve effects are unimportant, and thus an excessive burst in inflation is not anticipated.

Here's a question for Narayana: Why, if a goal is to have "capacity to lower rates" in the event of "say, global financial instability," does he want rates reduced now?