This is not rocket science

I'd like to offer a few thoughts on this piece by Brad DeLong.

He seems to think that some people in the profession are confused about things like the natural rate of interest and its relation to the market rate. What's ailing the economy is so painfully obvious. Why are these dopes trying to make things harder than they are? This is not rocket science after all.

Well, I freely admit to being a bit confused. Let me explain why.

First, the "natural rate of interest" is a term that seems to mean different things to different people. According to DeLong, the natural rate of interest is the (real) interest rate consistent with "full employment." Well, that's all fine and good, except that he does not define what he means by "full employment." (I certainly hope he does not define it as the level of employment consistent with the natural rate of interest!)

In my view, the natural rate of interest and the full employment level of employment are theoretical objects. They may or may not exist in reality. Economists use these terms to help them interpret the world. Nobody knows for sure just by looking at the data where the economy sits in relation to the natural rate of interest or full employment (however these terms are defined theoretically).

One might of course try to estimate their values by sample averages in the data. But this approach is not without problems. The employment-population ratio in the U.S. shows secular variation. Moreover, it varies across different demographic groups. Take a look at this, for example. While there are more sophisticated ways of estimating these things, the whole exercise is predicated on the assumption that these objects actually exist. (They most certainly do, if only in the minds of some economists).

In any case, with that little bit out of the way, I'd like to see whether I can make sense of the various claims that DeLong makes in his column.

1. The current natural interest rate is much lower than it is normally--the natural rate is too low--and that is a problem.

One way to understand this statement is with the classic "loanable funds" market diagram. There is a supply for loanable funds S(r,Y), increasing in the real interest rate r, and increasing in the level of real income Y. The the demand for loanable funds, or investment demand, I(r) is a decreasing function of r. Let Y* denote the full-employment level of GDP. Then, in a closed economy, the natural rate of interest r* satisfies S(r*,Y*) = I(r*).

Now, imagine that investment demand collapses for some reason. Moreover, assume that the zero lower bound (ZLB) is not binding. What happens to the full-employment level of Y? I am not sure what DeLong is assuming here. I think he assumes that it remains unchanged at Y*, but that actual Y may move along some "short run" (sticky price) AS curve. So, are we currently in the short-run or long-run? He does not say. I say that after four years, the sticky price frictions are probably no longer relevant, especially since U.S. CPI is close to its long-run trend. If this is the case, then he must be assuming that Y* is currently at its pre-crisis level (or would be, if the ZLB was not a problem).

But what is the problem here? If the interest rate is free to move, then the economy achieves full employment and GDP remains at "potential" Y*. (This, despite the collapse in investment, which lowers the future capital stock, future real GDP, etc.? How can Y* remain unchanged years after lower-than-normal investment?) I am led to infer from this discussion that he views the real problem as stemming from the ZLB, which he tackles in his next statement.

2. The current market interest rate is higher than the natural rate--the market rate is too high--and that is a problem.

Alright, suppose that r0 cannot be attained because of the ZLB on nominal interest rates (despite the fact that yields on longer maturities are not at the ZLB, but anyway, ignore this too.) Then the market interest rate r1 is too high; i.e. r1  > r0. If this situation  persists, then according to this simple model, the level of output must decline to some Q satisfying S(r1,Q) = I(r1); where Q < Y*. This is also a problem; see also Krugman.

If this latter problem is indeed the problem, then there are simple fixes. First, the Fed could raise its inflation target. Inflation serves as a tax on saving because it lowers the real rate of interest (given the ZLB). At the same time, it stimulates spending. The fiscal authority could achieve the same result by just taxing saving directly. I wonder, however, how many people really believe this to be *the* problem.

*The* problem seems more related to whatever caused the collapse in investment demand in the first place. I'm sure that most economists, including DeLong, would agree with this. But the key question, in my view, is precisely what factors (institutional arrangements and exogenous shocks) caused the collapse and whether the desired policy response should be made contingent on specific causes. My experience in working with economic models is that the optimal policy response generally depends a great deal what one assumes about the underlying shock. I believe that this lesson likely holds for real economies too and I wish that economists would spend more time talking about this.

3. Increasing G--printing more Treasury bonds, selling them, and buying goods and services--(a) increases the supply of safe assets, (b) lowers the proper value of safe assets via supply and demand, thus (c ) raises the "natural" rate of interest, and (d) could fix our problems if the policy raises the natural rate of interest so much that it is no longer lower than the market rate of interest.

Listen folks, I am not against increasing government spending (see for example, here). But we have to be clear about what economists mean by "increasing G." What they mean is an increase in spending in any form. This is the proverbial "digging up holes and filling them up again" prescription. Or Krugman's famous "let's build a bunch of stuff and export it to Mars" prescription. Of course, they do not mean that this is the way resources should be used. The statement is simply that even if the resources were to be used in a wasteful manner, it would work. Well, forgive me for not being so comfortable with that proposition.

Now, suppose we instead take Steve Williamson's view (more correctly, my interpretation of his view) that the collapse in investment spending is related to the evaporation of private label liquidity products (like the MBS products that used to circulate in the repo market). In fact, Steve has an interesting paper where he lays out the details of a specific model here: Liquidity, Monetary Policy, and the Financial Crisis. Brad may want to read this himself, once he works his way through Wicksell.

Unfortunately, there is no simple way to summarize Williamson's model in a short blog post. As in reality, what happens depends on a specific set of circumstances. In one case he considers, the shock that afflicts the banking sector drives up the demand for relatively safe securities, which drives the real interest rate down. Hence, the statement that "the interest rate is too low" (relative to the rate that would prevail if the economy worked perfectly). There is no talk of "natural" rate of interest or "full employment" in Willamson's paper. In Descartes-like fashion, we could say that Willamson had no use for that language.

In his model (as in reality), U.S. Treasuries are substitutes for the now missing private collateral objects. In his "scarce interest-bearing asset" case, a one time open market operation by the Fed (purchase of UST) has a perverse effect: it diminishes the supply of good collateral and therefore raises its price; i.e., it lowers the real interest rate--but in a way that is bad for the economy because it further depresses investment (that is, the lower interest rate is a reflection of a downward shift in the investment demand schedule.) Williamson calls this the "illiquidity effect."

[Note: I am not arguing that this is in fact what is happening. It may or may not constitute one of many forces that are currently in operation. It is a property that emerges from a standard economic model with realistic frictions. It is standard practice to use theory to help guide us what to look for in the data. For example, it was theory that motivated modern day NIPA design, not the other way around!]

The policy prescription in Williamson's model is for the Treasury to expand its supply of debt, both to meet the hightened demand and to cover for the shortfall in (private) supply. What is the Treasury to do with its proceeds? In principle, the Treasury could buy up private securities (this is like a banking operation--an open market operation, but with Treasuries instead of money). Or taxes could be cut. Or, yes, an increase in G too (but whether and how much to do this should be based on standard cost-benefit calculations.)

Unlike the "this is not rocket science model" that DeLong likes to work with, there are many interesting cases that emerge in the Williamson model. Macroeconomists should go read it. His paper is by no means the last word on the subject. Indeed, it may turn out one day to be all wrong. But that is the nature of research. There are still a lot of things we do not understand about the way the macroeconomy works.  Well, for those of us who remain confused, in any case.

Monetary policy in a liquidity trap

Krugman has an interesting article today, Monetary Policy in a Liquidity Trap. I (sort of) agree with much of it. But I believe that a few comments are in order.

Consider this statement:

So, at this point America and Japan (and core Europe) are all in liquidity traps: private demand is so weak that even at a zero short-term interest rate spending falls far short of what would be needed for full employment. And interest rates can’t go below zero (except trivially for very short periods), because investors always have the option of simply holding cash.

This statement is, in varying degrees: [1] interpretative, [2] assertive, [3] misleading, and [4] wrong.

First, the quoted passage above suggests that a liquidity trap is the byproduct of "insufficient private demand," with the implication, of course, that more "public demand" is needed to rectify the situation. This may or may not be true. Regardless, the statement is [1], [2], and [3] above. Beware of economists making bald assertions.

Second, the statement is wrong in suggesting that our current liquidity trap is associated with zero nominal interest rates. Liquidity trap phenomena are much more general than this. And if you really want to further your understanding on this matter, please go read this piece by Steve Williamson: Liquidity Traps, Money, Inflation, and Bond Yields. As Steve says: this is not your grandma's liquidity trap.

In grandma's liquidity trap, the real interest rate is too high because of the zero lower bound. Steve argues that in our current liquidity trap, the real interest rate is too low, reflecting the huge world appetite for relatively safe assets like U.S. treasuries.

If this latter view is correct, then "corrective" measures like expanding G or increasing the inflation target are not addressing the fundamental economic problem: low real interest rates as the byproduct of real economic/political/financial factors.

Given these "real" problems, Steve's view is that the Fed is largely irrelevant. But he does assign hope to the Treasury: increase the supply of its securities to meet the world demand for them. I've been making similar arguments for some time now; for example, here.

Apart from all this, it will be interesting to see how the experiment in Japan plays out. Most of the massive purchases announced by the BOJ are for JGBs -- I'm really skeptical what sort of effect this should have (since the operation constitutes swaps of two assets that are close to perfect substitutes--although some purchases will take the form of higher risk assets--see Noah Smith on this). But what I think really does not matter--it is what market participants think--and the program does appear to be having some effect in financial markets.

Thank you, Japan, for this interesting experiment. Domo arigato, gozaimasu!

Poor Germany

Well, here's an eyebrow raiser: Germans Among Poorest in Europe: ECB Study. 

The paper is available here: The Eurosystem Household Finance and Consumption Survey. The cross country comparison of net wealth can be found in Table 4.1 on page 76.

Median net wealth in Germany for 2010 was 51K eur. Compare this to median wealth in Greece (101K), Italy (173K) and Spain (182K).

This just doesn't sound right to me, but I haven't gone through the report in detail. Evidently, the differences are driven primarily by real estate wealth. Thankfully (?), Germany escaped the housing price "bubble" that afflicted many European countries; see figure below.

Moreover, as I noted here, the German growth experience over the past 20 years has been nothing to write home about.

Germany: low growth, no asset price bubbles, low wealth, but...stable. Das ist gut?



April 16, 2013: Update here from VOX, who emphasize that the wealth distribution in Germany distorts the picture presented here. 

Krugman on Taylor

I had five minutes to kill so I read this: Cogan, Taylor and the Confidence Fairy, by Paul Krugman.

Let's see, what do we have here? Ah, yes...I see Krugman accusing Taylor of being dishonest. Paulo, Paulo...tsk tsk.

To be more specific, Krugman accuses Cogan and Taylor of dishonesty concerning the current state of macroeconomic research. The claim they made was that expectations did not play and explicit role in "old style" Keynesian models. Of course, Keynes and most of the other great economists of the 19th and early 20th centuries assigned a critical role to expectations. But if by "old style" Keynesian models we mean Hicks' IS-LM representation of (a part of) Keynes' theory, then Cogan and Taylor are largely correct. For example, do you see expectations entering explicitly anywhere in this exposition?

By the way, I find it a bit of a hoot that it is Krugman complaining that others in the field are not keeping abreast with the literature. Steve Williamson has addressed this on more than one occasion; see here, for example. But anyway, this is getting rather tiresome.

In fact, it appears that Krugman did not even read the Cogan and Taylor paper. Fortunately, Noah Smith helps lift the lazyman's load: John Taylor's Austerity Model. (Oh gosh, it appears that their results have nothing to do with the "confidence fairy." Oh well, "confidence fairy" looks so good in the title of a NY Times Op-Ed piece.) Noah summarizes his assessment of their work as follows:

Upshot: If you have no Zero Lower Bound, and if the Fed partially counteracts the demand-side effects of fiscal policy, and if people have forward-looking expectations, and if you don't cut government purchases much, and if taxes are very distortionary, then austerity works. This is not really a new result, but it rarely gets shown so explicitly, so it's good that John Taylor and his co-authors went ahead and did it.

Now there's something that economists can debate.

Unfortunately, Noah stumbles a bit on the partisan divide with this concluding statement:

So John Taylor is not committing some major fallacy. He's just using a standard mainstream New Keynesian DSGE model to stump for the Republicans.

I may very well be wrong but I do not ever recall Noah saying something like (say, in regard to the Eggertsson and Krugman model):

So Paul Krugman is not committing some major fallacy. He's just using a standard mainstream New Keynesian DSGE model to stump for the Democrats.

I mean, come on ... let's all stop this nonsense. Debate the substance of the argument and the evidence supporting it. Doing anything else distracts from the task at hand.

Wonkish (and important) note: Those"powerful fiscal effects at the zero lower bound" that Noah alludes to are quite possibly an artifact of the manner in which the NK DSGE model is linearized around steady state (linearization not taking into account the zero lower bound, with zero lower bound imposed afterward). Please refer to: Some Unpleasant Properties of Log-Linearized Solutions When the Nominal Rate is Zero.

 

Germany: The Price of Stability?

My colleague, Fernando Martin, has an interesting chart that plots the real per capita GDP of five industrialized countries since 1991:

 

The data above are expressed as percentage deviations from the U.S. level in 1991, with the initial position calculated using PPP converted GDP per capita from the Penn World Tables. Thus, in 1991, the U.K. is estimated to have had a real per capita income that was 30% lower than the U.S. Germany's real per capita income was only 10% lower than the U.S. in 1991, and so on.

Each of these countries experienced a similar decline in output during the most recent recession. But only Germany has the recession been "temporary." That is, only in Germany has real per capita income returned to its pre-recession trend. The other four countries exhibit persistent "output gaps"--their real per capita income remains below their respective pre-recession trends. Ah, Germany. All hail that Teutonic economic juggernaut.

But just hold on a second. While it is true that Germany appears to have weathered the economic storm better than others, it seems to have done so at considerable cost.

From 1991-2007, German real per capita income grew at a paltry 1.3% per annum. Compare this to the U.S. (2.1%), Canada (2.2%), France (1.6%), and the U.K. (2.9%). These are huge differences in long-run growth rates. In particular, while German income was only 10% below that of the U.S. in 1991, it is presently about 18% below U.S. income. That's called falling behind (albeit, at a steady pace).

And yes, the U.K. presently looks ugly. But it looks considerably less ugly when we take into account the growth record. In 1991, real per capita income in the U.K. was 20% below that of Germany. Just prior to the recession, the U.K. had just about closed that gap. Of course, things have not looked good for the U.K. snce then.

Fernando and I are also led to speculate on the role played by monetary policy for shaping the economic recoveries of these nations. France and Germany, as members of the EMU, operated under the same monetary policy--and yet, their recovery dynamics look very different. Also, since inflation was pretty low over this sample period, real and nominal GDP look practically the same. So for the NGDP targeters out there: it appears that German NGDP is back on target, but not French NGDP. Care to comment?

Moreover, out of this set of countries, only the U.K. has experienced a significant rise in inflation (and hence, NGDP). It's not exactly clear from this data how this "looser" monetary policy has contributed to a more rapid recovery dynamic (although, as usual, we have to be careful because many other things are happening, especially on the fiscal front).
 

Fed Balance Sheet Risks

The Balance Sheet

As everyone knows, the Fed's balance sheet has more than tripled since the financial crisis. Here is a look at the liability side of the Fed's balance sheet:

 

What's interesting here is that prior to the crisis, almost all the Fed's liabilities were in the form of (zero-interest) cash -- that is, currency in circulation (the blue area). No one knows for sure, but probably over half of this cash is circulating outside the U.S.

At the time of the financial crisis, the balance sheet doubled over a very short period of time, and has continued to grow since then. What's interesting here is that subsequent to the crisis, most of the growing liabilities are in the form of interest-bearing reserves (held by depository institutions with accounts at the Fed).

Now, three trillion dollars sounds like a heck of a lot of liabilities. There is no danger of bankruptcy, however. That's because Fed liabilities are not debt; or, if they are (as in the case of reserves), they are made redeemable in cash, which the Fed can print at any time. Fed liabilities are more like equity shares, than debt. That is, there is a risk of dilution (inflation), but no risk of bankruptcy.

Next question: what did the Fed do with all this money it "printed" up (out of thin air, I might add)? Many people are likely to conjure up an image of "Helicopter Ben."

Alas, that's not quite how it works (even if it does work this way in some other countries, like the recent experience in Zimbabwe). You see, while the Fed is "pumping money into the economy," it is simultaneously sucking some other group of assets out of the economy.

To put things in a slightly different way, the Fed is acting much like, um...well, much like a bank. That is, the Fed finances acquisitions of less liquid assets with more liquid liabilities. The Fed's liquid liabilities look a lot like super short duration Treasuries. What do the assets look like? Take a look here:

 

Most of the assets consist of interest-bearing securities, primarily U.S. government debt and agency MBS. In 2012, this portfolio yielded over 3%. In 2012, the Fed remitted $89B to the Treasury (historically, remittances have been around $25B per year). Not a bad return, for a year of "helicopter drops."

In the meantime, inflation appears to be fairly well centered around the Fed's 2% inflation target (recent data is coming in below target):

 

TIPS based measures of inflation expectations appear to be fairly well centered around 2% as well (recent data appears to be rising a bit away from target):

While inflation and inflation expectations appear to be muted for the time-being, a number of economists and Fed officials still worry about the various risks associated with the Fed's large and growing balance sheet. The most obvious worry is the risk of inflation. The extent to which one worries about inflation depends a lot on one's theory of inflation, which I will explain below.

(New) Keynesian View

In the extreme version of this view (Woodford's cashless economy), Fed liabilities serve only as a unit of account; and the private sector manufactures the "money" it needs. The Fed determines (influences) the nominal interest rate, which influences the aggregate demand (AD) for goods and services. Inflation is determined in part by the pricing decisions of firms. When AD is strong, prices rise more rapidly; and conversely when AD is weak. Inflation is also determined in part by the Fed's policy function (Taylor rule), which stipulates a long-term inflation target (serving as the nominal anchor) together with the promise to alter the interest rate (hence AD) in response to undesirable movements in inflation away from target.

Conspicuously absent from the theory of inflation above is any role played by the money supply. The Fed's balance sheet plays no role in determining inflation according to this view. It follows as a corollary that the size of the Fed's balance sheet poses no economic risk.

This world view likely explains the statements made by the more dovish members of the FOMC. Inflation is low because aggregate demand is weak. We need to keep interest rates low. Additional QE by the Fed is mostly innocuous--except possibly for political reasons.  What do I mean by this? Let me explain.

Federal Reserve Board economist Seth Carpenter (and his coauthors) have recently distributed an interesting working paper that offers a methodology for making projections about the way the Fed's balance sheet is to evolve over time, see: The Federal Reserve's Balance Sheet: A Primer and Projections. In some of their projections, the Fed actually incurs an operating loss and remittances to the U.S. Treasury cease for a while. The economic consequences of this are innocuous if you're coming at things from a strict New Keynesian perspective. But the optics can be made to look bad, something that politicians hostile to the Fed are likely to exploit. Let me consider a simple example.

The Fed currently pays 1/4% on excess reserves, which are presently close to $2T. In one year's time, and under currently policy, reserves will be closer to $3T. The annual interest expense associated with these reserve liabilities is presented in the following table, for various interest rates (IOR):

 

Now, perhaps this is unlikely, but it is certainly not outside the realm of possibility: Imagine that inflation and inflation expectations begin to rise sharply at the end of 2013. The Fed's policy response is to jack up it's policy rate (IOR) sharply, say, to 3%.  If reserves remain close to $3T, that's about $90B in interest payments to banks and hence, $90B less in remittances to the Treasury.

 

From an economic (a consolidated Fed and Treasury balance sheet) perspective, interest-bearing reserves look a lot like interest-bearing Treasuries. So whether the Fed or the Treasury services this debt makes little difference to the American taxpayer. Naturally, this is not the way things will be portrayed in the political arena.

(New) Monetarist View

According to this view, there are financial market imperfections (limited commitment, asymmetric information, etc.) that allow Fed and Treasury liabilities to be valued for their liquidity/collateral properties. Inflation, in the long-run at least, is determined by the supply and demand for currency (a special type of Fed liability).

In normal times, currency is dominated in rate of return, so their is a well-defined demand for the stuff. As well, in normal times, reserves are dominated in rate of return, so Fed liabilities are mostly in the form of currency (see first diagram above, prior to 2008). A well-defined demand for currency plus Fed control over the supply of currency means that the Fed can control inflation.

In abnormal times, however, reserves and Treasuries earn (roughly) the same rate of return. In this case, the Fed only controls the total supply of its liabilities--the composition of these liabilities between currency and reserves is determined by banks. Reserves are like a demand deposit liability--convertible into currency on demand. The Fed can influence bank redemption policies by manipulating IOR--if it has this tool available. Note that the Fed has only had this tool available since 2008. (And in light of the political risks outlined above, one could easily imagine Congress taking this tool away.)

If inflation and inflation expectations begin to rise, so should the nominal demand for currency (even if the real demand remains more or less fixed). One might expect a flood of currency into the economy as banks exercise their redemption option on reserves. The flood of currency could potentially validate the higher inflation expectations -- a self-fulfilling prophesy.

In many of our models, we assume passive support from the Treasury to support whatever needs to be done to keep inflation in check. But how realistic is this? What if that support does not materialize? And moreover, suppose that the Fed is no longer permitted to use IOR as a policy tool? What if inflation and inflation expectations start to rise? What then?

In this case, the only way the Fed can "suck out" excess liquidity is via asset sales. In a sense, the value of the Fed's assets represents the extent to which the Fed can credibly commit to withdraw cash from circulation (the Fed has no ability to tax). But if inflation expectations rise, so will longer term interest rates. The Fed's assets will decline in value. And with the decline in value, the ability to purchase cash.

What sort of capital losses are we talking here? Obviously, it depends. The average maturity of the Fed's asset portfolio is around 10 years (up significantly from historical norms, thanks to Operation Twist, etc.). The following formula provides a rough approximation of exposure to interest rate risk:

1 ppt increase in the interest rate = (average duration)% decline in bond price

So, to take a bad (but not worse case) scenario, suppose interest rates rise by 5 ppt (e.g., China decides to unload its holdings of U.S. Treasuries?). We are talking about a 50% (somewhat less) loss on the Fed's $3T portfolio. The remaining $1.5T in asset value would not be enough to suck out the current $2T in reserves. There would be $0.5T in reserves remaining--representing $0.5T in potential new currency (a 50% increase over the current supply of $1T).

Conclusions

No one knows for sure which of the two theories of inflation above is the better approximation for our current reality. Central bankers are charged with the task of evaluating the risk of their policies under different theoretical scenarios. If the monetarist view is correct, then continued expansion of the Fed's balance sheet exposes the economy to ever higher inflation risk. Of course, this is not to say that the risk is not worth taking. Policymakers just need to be aware of the risk and make provisions for it.

It is interesting to note, however, that independent of one's theory of inflation, the large and growing balance sheet may expose the Fed to a certain type of political risk. If tightening needs to happen in the future, the Fed will have to raise interest rates (IOR) and/or sell off its assets. IOR may be made to look like Fed Reserve (instead of Treasury) transfers to the banking sector, at taxpayer expense. Capital losses on asset sales would similarly reduce remittances to the Treasury. It's not going to look very pretty.
 

Is it time for the Fed to raise its policy rate?

What would happen if the Fed was to adopt a different policy rule--one that entailed an immediate increase in its policy rate? The effect of such a policy change in the OLG model I studied here would be to raise the real interest rate and contract the level of output.

But I recently came across a paper by Stephanie Schmitt-Grohe and Martin Uribe who ask the same question using a more conventional model: The Making of a Great Contraction with a Liquidity Trap and a Jobless Recovery. Here is the abstract:

The great contraction of 2008 pushed the U.S. economy into a protracted liquidity trap (i.e., a long period with zero nominal interest rates and inflationary expectations below target). In addition, the recovery was jobless (i.e., output growth recovered but unemployment lingered). This paper presents a model that captures these three facts. The key elements of the model are downward nominal wage rigidity, a Taylor-type interest-rate feedback rule, the zero bound on nominal rates, and a confidence shock. Lack-of-confidence shocks play a central role in generating jobless recoveries, for fundamental shocks, such as disturbances to the natural rate, are shown to generate recessions featuring recoveries with job growth. The paper considers a monetary policy that can lift the economy out of the slump. Specifically, it shows that raising the nominal interest rate to its intended target for an extended period of time, rather than exacerbating the recession as conventional wisdom would have it, can boost inflationary expectations and thereby foster employment.

I highlighted the policy conclusion in blue because I find it interesting and because it is likely to be controversial. In what follows, I discuss their model and conclusions. I hope that some readers more familiar with this literature than myself might want to comment.

The basic framework should be familiar to most macroeconomists. There is a representative agent with preferences defined over sequences of consumption:

[1] ∑_tβ^tU(y_t)

where 0 < β < 1. Each agent has a unit of time which they are willing to supply as labor at any wage (since they derive no disutility from labor here). Firms hire labor at a competitive wage, and remit any profit to shareholders (the representative agent). There is a single asset--a risk free nominal government bond. The government makes no purchases and has a lump-sum tax that it uses to finance the interest cost of its debt.

There is an Euler equation describing the time path for consumption (equals GDP):

[2] U'(y_t) = R_tΠ^-1_t+1β U'(y_t+1)

where R_t denotes the gross nominal interest rate from period t to t+1 and Π_t+1 denotes the gross rate of inflation (actual and expected) from period t to t+1. With logarithmic preferences, we can rewrite [2] as:

[3] R_t = (1/β)Π_t+1g_t+1 [Euler Equation]

where g_t+1 denotes the (gross) rate of growth of real GDP.

Note that if we had instead assumed that the government issued real debt (or nominal debt perfectly indexed to the price level), then condition [3] would instead become

[4] r_t = (1/β)g_t+1

where  r_t denotes the gross real rate of interest. If we compare [3] to [4], we see that the following must be true:

[5] r_t = R_tΠ^-1_t+1 [Fisher Equation]

One way to interpret the Fisher equation is that it represents a no-arbitrage-condition that must hold between real and nominal debt with identical risk characteristics (their real rates of return must be the same, if both instruments are to be willingly held in the wealth portfolios of individuals).

Now, for any given expected growth rate g_t+1, condition [3] asserts a very tight link between the nominal interest rate and expected inflation. In and of itself, however, condition [3] does not make any statement about the direction of causality: it is perfectly consistent with the idea of inflation expectations causing the nominal interest rate, or the nominal interest rate causing inflation expectations. [Keep in mind that the authors are working strictly within the rational expectations paradigm.]

The authors implicitly take a stand on the direction of causality by specifying a Taylor rule of the form:

[6] R_t = R* + α( Π_t - Π* ) + δln(y_t/y*) [Taylor Rule]

with (R*/Π*) = (g/β). I'm not deep into this literature, but as far as I can tell, this is a perfectly standard Taylor rule. Embedded in this rule is the assumption that the Fed chooses R_t and that its choice is governed in part by current (realized) inflation Π_t, among other things. The implication is that if the Fed chooses R_t, then expected inflation must be determined by the Euler equation [3]. That is to say, the direction of causality here is assumed to run from R_t to Π_t+1. (And I think this will be the source of controversy, along the lines of the discussion that surrounded Narayana Kocherlakota's speech; see Nick Rowe.)

Suppose that the economy is always at full employment, so y_t = y* and that growth is zero (g_t = 1). Later on, I will follow the authors and add a nominal wage rigidity, but for now what I have to say is independent of full employment equilibrium.

Now, let's combine the Euler equation [3] with the Taylor rule [6]. Eliminating the interest rate, we can derive a first-order difference equation in the inflation rate:

[7] Π_t+1 = R*β + αβ( Π_t - Π* )

Now, I'm not sure about you, but where I grew up we'd say that the stability condition here (for monotone dynamics) is 0 < αβ < 1. If this is the case, then inflation converges monotonically to the inflation target Π* from any given initial inflation rate.

The only "problem" here is that the initial inflation rate is not determined. Consequently, there is a continuum of inflation paths consistent with equilibrium. And, since there are multiple equilibria, I guess this opens the door for extraneous inflation rate shocks (what the authors call "confidence shocks")? Is there some way for policy to eliminate the potential instability that induced by this indeterminacy?

The "solution" to the indeterminacy problem seems weird (to me). The basic idea goes like this. Suppose that the policy maker sets α > 1/β (so that αβ > 1). Evidently, this is known as the "Taylor Principle," which is the idea that the Fed should respond aggressively to inflation (i.e., increase the interest rate by more than one-for-one with any rise in the inflation rate).

But wait a second -- doesn't αβ > 1 imply that the steady state equilibrium is unstable? Yes. But suppose we restrict attention to inflation trajectories that remain within a bounded neighborhood of the steady state? Alright, let's suppose. Well then, if αβ > 1, then the only equilibrium trajectory that remains within that neighborhood is the steady state itself. In other words, the steady state is the locally unique equilibrium. In contrast, if αβ < 1, there are multiple equilibrium inflation trajectories within a given neighborhood (that converge to the steady state).

What justifies restricting attention to equilibria that are locally unique in the sense defined above? Beats me. From a global perspective, if  α > 1/β, then it seems that a hyperinflation dynamic is possible if the initial inflation rate starts above the steady state Π*. (Evidently, Eduardo Loyo uses just such a case to interpret the Brazilian hyperinflation; see Tight Money Policy on the Loose: A Fiscalist Hyperinflation.) And as Benhabib, Schmitt-Grohe, and Uribe 2001 demonstrate, if there is a zero-lower-bound (ZLB) on the interest rate, then there exists a second steady state -- a liquidity trap equilibrium. Moreover, for any initial inflation rate below Π*, inflation converges to the liquidity trap equilibrium (zero nominal interest rate and deflation).

The following diagram summarizes these ideas. The diagram is plotted in (R,Π) space (sorry, but P = Π in the diagram -- could not find the Greek letters in Coreldraw). The Euler equation is from [3]; i.e., R = (1/b)P. The Taylor Rule is from [6] with a ZLB (R ≥ 1) imposed, and with α > 1/β ( a > 1/b). Point A denotes the "intended" steady state and point B denotes the "unintended" steady state, where the net nominal interest rate is zero, and inflation is below target.

Start with an inflation rate close to but below target. Then trace your pencil up to the Taylor Rule line--this is the policy rate associated with the initial inflation rate. Now ask: at this policy rate, what does the Euler Equation imply about the expected (one period ahead) inflation rate? Move your pencil from left to right until it hits the Euler Equation line, then project down on the P-axis. The next period inflation rate is lower than the initial inflation rate. The economy heads toward the liquidity trap equilibrium, point B.

Sticky Wages 

But so what if the economy gets stuck at point B? In the model economy, the liquidity trap equilibrium is consistent with full employment, with nominal variables, including the nominal wage rate, declining over time in this equilibrium at the rate of time preference (deflation).

Here is where Schmitt-Grohe and Uribe add a relatively weak form of nominal wage rigidity. In particular, assume that nominal wages are downwardly rigid in the sense that they may decline, but at a rate no faster than γ(u_t), where u_t is the unemployment rate:

[8] W_t+1 ≥ γ(u_t)W_t  where γ(0) > β

with γ(u_t) a decreasing function of u_t, so that nominal wages are permitted to fall more rapidly as the unemployment rate rises.

The fact that γ(0) > β is critical here. It implies that nominal wages cannot fall as fast as the rate of deflation implied by the Friedman rule when the economy is at full employment. Consequently, the liquidity trap equilibrium at point B in the figure above is inconsistent with full employment.

Assuming g =1 (zero real growth), the inflation rate at point B is Π = β < 1 (deflation). In the steady state, it must be the case that nominal wages are falling as fast as prices. The condition which ensures that this is the case is:

[9] γ(u') = β  where 0 < u' < 1

That is, the unemployment rate must rise in order to permit the nominal wage to decline fast enough to keep the real wage growing at its long-run "natural" rate of g ( = 1 here). While real wage growth is consistent with a balanced growth path, the level of the real wage is too high (which is what causes firms to demand less than the full employment amount of labor). While real GDP is growing along its balanced growth path, it remains forever below "potential."

Policy Implications

Many recent papers model the great recession shock as some event that lowers the natural rate of interest (an exogenous increase in β would do the trick). In contrast, the authors here appeal to evidence which suggests that a "confidence shock" -- an exogenous decline in inflation, interacting with the Taylor Rule above, leading to a liquidity trap -- may have been the culprit.

Assume that this is the case. Then what can be done?

Well, one thing you would not want to do is adopt language suggesting that the Fed is prepared to keep R =1 for an extended period of time. Nor would you want to adopt the Evans Rule (which essentially accomplishes the same thing). These are policies that (in the context of this model) lead agents to expect deflation (or inflation below target) off into the indefinite future. These policies, if anything, reinforce the liquidity trap outcome.

What about balance sheet policies? Nope -- this is a "cashless economy." The Fed's balance sheet plays no role in this model.

The type of policy that works here entails changing the parameters of the Taylor rule. In particular, instead of α > 1/β, the Fed should set α < 1/β. The effect of this change would be to make the Taylor Rule line in the figure above "flatter" than the Euler Equation line. The equilibrium inflation rate would then eventually rise back to the target rate. (Of course, this opens the door to multiplicity, but what the heck -- at least we get back to full employment eventually.) In other words, the Fed should move to increase its policy rate -- so that inflation expectations (and inflation) will follow.

In fact, the authors suggest an extreme form of this policy change: set α = 0 in [6]. The effect of this policy change in the model is for individuals to revise their forecast of inflation immediately to Π*. While nominal wages begin to rise at the rate of inflation, there is a level drop in the real wage as the unemployed are absorbed into the workforce. The economy moves from B to A in the figure above.

This policy conclusion has to hinge critically on the manner in which agents are assumed to form expectations. Falling into the liquidity trap appears possible even under an adaptive learning rule; see, e.g., Benhabib, Evans and Honkapohja 2012. But getting out would seem to entail more than a simple adjustment to the Taylor Rule.