Why gold and bitcoin make lousy money

A desirable property of a monetary instrument is that it holds its value over short periods of time. Most assets do not have this property: their purchasing power fluctuates greatly at very high frequency. Imagine having gone to work for gold a few weeks ago, only to see the purchasing power of your wages drop by 10% in one day. Imagine having purchased something using Bitcoin, only to watch the purchasing power of your spent Bitcoin rise by 100% the next day. It would be frustrating. 

 

Is it important for a monetary instrument to hold its value over long periods of time? I used to think so. But now I'm not so sure. While I do not necessarily like the idea of inflation eating away at the value of fiat money, I don't think that a low and stable inflation rate is such a big deal. Money is not meant to be a long-term store of value, after all. Once you receive your wages, you are free to purchase gold, bitcoin, or any other asset you wish. (Inflation does hurt those on fixed nominal payments, but the remedy for that is simply to index those payments to inflation. No big deal.)

 

I find it interesting to compare the huge price movements in gold and Bitcoin recently, especially since the physical properties of the two objects are so different. That is, gold is a solid metal, while Bitcoin is just an abstract accounting unit (like fiat money). 

But despite these physical differences, the two objects do share two important characteristics:

[1] They are (or are perceived to be) in relatively fixed supply; and
[2] The demand for these objects can fluctuate violently.

The implication of [1] and [2] is that the purchasing power (or price) of these objects can fluctuate violently and at high frequency. Given [2], the property [1], which is the property that gold standard advocates like to emphasize, results in price-level instability. In principle, these wild fluctuations in purchasing power can be mitigated by having an "elastic" money supply, managed by some (private or public) monetary institution. This latter belief is what underlies the establishment of a central bank managing a fiat money system (though there are other ways to achieve the same result). 

 

The following graph depicts the rate of return on US money over the past century (the rate of return is actually the inverse of the inflation rate). The US was on and off the gold standard many times in its history. Early on in this sample, the gold standard was abandoned during times of war and re-instituted afterward. While inflation averaged around zero in the long-run, it was very volatile early in the sample. The U.S. last went off the gold standard in 1971. Later on in the sample, we see the great "peacetime inflation," followed by a period of low and stable inflation. 

 

Gold standard advocates are quick to point out the benefits of long-term price-level stability. The volatile nature of inflation early on in the sample is attributed to governments abandoning the gold standard. If only they would have kept the gold standard in place...

 

Of course, that is the whole point. A gold standard is not a guarantee of anything: it is a promise made "out of thin air" by a government to fix the value of its paper money to a specific quantity of gold. It is possible to create inflation under a gold standard simply by redefining the meaning of a "dollar." For example, in 1933, FDR redefined a dollar to be 1/35th of an ounce of gold (down from the previous 1/20th of an ounce). This simple act devalued the purchasing power of "gold backed money" by almost 60%. 

 

If the existence of a gold reserve does not prevent a government from reneging on its promises, then why bother with a gold standard at all? The key issue for any monetary system is credibility of the agencies responsible for managing the economy's money supply in a socially responsible manner. A popular design in many countries is a politically independent central bank, mandated to achieve some measure of price-level stability. And whatever faults one might ascribe to the U.S. Federal Reserve Bank, as the data above shows, since the early 1980s, the Fed has at least managed to keep inflation relatively low and relatively stable. 

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. 

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 IOER--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.

[Note: I have refined this idea in this post here: Excess Reserves and Inflation Risk (June 22, 2014) ]

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.

On the perils of Taylor rules

In the Seven Faces of "The Peril" (2010), St. Louis Fed president Jim Bullard speculated on the prospect of the U.S. falling into a Japanese-style deflationary outcome. His analysis was built on an insight of Benhabib, Schmitt-Grohe, and Uribe (2001) in The Perils of Taylor Rules.

These authors (BSU) showed that if monetary policy is conducted according to a Taylor rule, and if there is a zero lower bound (ZLB) on the nominal interest rate, then there are generally two steady-state equilibria. In one equilibrium--the "intended" outcome--the nominal interest rate and inflation rate are on target. In the other equilibrium--the "unintended" outcome--the nominal interest rate and inflation rate are below target--the economy is in a "liquidity trap."

As BSU stress, the multiplicity of outcomes occurs even in economies where prices are perfectly flexible. All that is required are three (non-controversial) ingredients: [1] a Fisher equation; [2] a Taylor rule; and [3] a ZLB.

Back in 2010, I didn't take this argument very seriously. In part it was because the so-called "unintended" outcome was more efficient than than the "intended" outcome (at least, in the version of the model with flexible prices). To put things another way, the Friedman rule turns out to be good policy in a wide class of models. But mostly, I figured that other factors were probably more important for explaining the events unfolding at that time.

Well, maybe I was a bit too hasty. Let me share with you my tinkering with a simple OLG model (similar to the one I developed here.) Unfortunately, what follows is a bit on the wonkish side. If you catch any errors, or otherwise have any comments to make, please let me know.

Basics 

People live for two periods; they are "young" and then "old." Everyone only values consumption when old. Their objective is simply to maximize (expected) future consumption.

The young are endowed with some output y. The are also each endowed with an investment technology such that k units of output invested today yields f(k) units of output tomorrow. Assume that f(k) is increasing and strictly concave; i.e., f'' < 0 < f'.

The autarkic (also competitive) outcome is one where the young save their entire endowment and consume f(y) when old. The competitive equilibrium (gross) real rate of interest is equal to the marginal product of capital, r = f'(k). [Note that time-preference plays no role in determining the real rate of interest here.]

An economy with real debt

Assume that there is a government that issues one-period real debt b. Let r denote the gross real rate of interest paid on this debt. I assume that the government finances the carrying cost of its debt via a lump sum t tax applied to old agents. In a steady state,

[1] t = (r-1)b.

By construction, the savings decision is trivial: the young save all their income y. The interesting decision entails a portfolio allocation choice problem between capital and bonds, y = k + b. Conditional on a choice of b (hence, k), future consumption is given by:

[2] c = f(y-b) + rb - t 

Assuming an interior solution, (expected) rate of return equality implies:

[3] r = f'(y-b)

Technically, [3] determines bond demand. In equilibrium, the supply of bonds (determined by policy) must equal the demand for bonds. Hence, by choosing b in this model, the government can choose the prevailing real rate of interest. Lump-sum taxes are simply adjusted by way of [1] to finance the carrying cost of the debt. Equilibrium consumption is then given by [2]; i.e., c = f(y-b) + b.

The real GDP in this economy is given by Y = y + f(y-b). Notice that this model delivers a standard IS curve. That is, by increasing b, the government increases r, capital is crowded out, and output falls. Likewise, lowering the real interest rate stimulates (investment) demand, leading to an increase in output.

In what follows, I assume that a socially desirable outcome is associated with some 0 < b* < y. The "natural" rate of interest is defined as r* = f'(y-b*), and potential GDP is defined by Y* = y + f(y-b*). [Note that r* may be either greater or less than one. Some of you may argue that r* should equal 1 here. That's fine. The qualitative results below do not hinge on this issue.]

An economy with nominal debt

Let P denote the price of output denominated in some abstract unit of account. Let B = Pb the nominal debt. Let P+ denote "next period's" price level. Then a young person faces the following sequence of budget constraints:

Py = Pk + B 
P+C+ = P+f(k) + RB - T+

where R denotes the gross nominal interest rate, and T is the nominal lump-sum tax. Define Π+ = P+/P, the expected gross rate of inflation. Then using b =B/P and t = T/P, rewrite the budget constraints above as

y = k + b
c+ = f(k) + (R/ Π+)b - t+ 

Desired real bond holdings must now satisfy the condition

[4]  f'(y - b) = R/ Π+

with the demand for nominal bond holdings given by B = Pb. If I define r = R/ Π+ as the expected real rate of interest, then we see that [4] is equivalent to [3].

The government budget constraint is given by T+ = RB - B+. In real terms,

[5] t+ = (R/ Π+)b - b+ 

so in a steady state with b = b+, we have t = (r-1)b, which is equivalent to [1]. Consumption is given by [2].

The model to this point is riddled with indeterminacy, even restricting attention to steady states. What determines the nominal interest rate, the inflation rate, the price level, etc.? Note that this indeterminacy is not present in the model with real debt. In that world, I assumed that b was a policy instrument. This (along with the lump-sum tax instrument) pins down an equilibrium. In the world I am describing now, the government does not pick b. It need not even pick B if, in particular, it is willing to let demand determine quantity at a given rate of interest. How to proceed? As usual, in small steps.

A Monetarist regime

We can think of B as interest-bearing money. The nominal interest rate on money is commonly assumed to be zero, so R = 1. But there is nothing that requires this to be the case; we are free to pick any interest rate supportable by taxes here. The key assumption is that R is determined and that it is constant over time.

The monetarist views B (and the time path for B) as determined by policy. With the demand for real money balances determined by [4], market clearing requires B = Pb for all time. Since B is determined by policy, and b is determined by agents, the price level is determined by P = B/b. The inflation rate must therefore be determined by

[6]  Π+ = (B+/B)(b/b+)

Let B+ = μB. Now combine [6] with [4] to derive:

[7]  b+ = (μ/R) f'(y - b)b

which is a first-order difference equation in real money balances. The model has two steady states. In one, b = 0; in the other, b > 0 satisfies f'(y - b) = (R/μ). It seems darn easy to construct the optimal policy here. Just set  (R/μ) = r* and we're done.

Well, not so fast. As it turns out, even for the case of a fixed stock of money B, there generally exists a continuum of nonstationary equilibria indexed by an initial condition 0 < b0 < y, with the time path for b asymptotically approaching zero; see Figure 1 in Woodford (1984). Of course, since P = B/b with B fixed, this implies that the price level approaches infinity (in fact, these are hyperinflation dynamics). Isn't it interesting to note that Friedman's k percent rule is dynamically unstable here?

The undesirable hyperinflation dynamic here appears to an artifact of (among other things) the assumed passivity of policy (the nominal interest rate and money growth rate are held fixed forever). But evidently, there exists a simple "activist" policy rule that uniquely implements the desired outcome:

[8]  ln(R) = ln(R*) + α[ ln(Π+) - ln(Π*) ]

where Π* = μ (arbitrary), R* = r*Π*, and α = 1.  The policy rule [8] is a Taylor rule. The rule dictates that the policy rate be increased one-for-one with expected inflation. Such a policy keeps the expected real rate of interest pinned to its natural rate. As such, the economy is always at potential (this would not necessarily be the case if I was to introduce other shocks, of course). Note that the price level is now determined, P = B/b* with P+ = Π*P.

Would the ZLB restriction R ≥ 1 limit the ability of policy here? I do not think so. First, the problem in this model is a the possibility of a self-fulfilling hyperinflation -- deflationary equilibria do not exist. As such, policy only needs to threaten to raise, not lower, the nominal interest rate. Second, I believe that the optimal monetary policy may alternatively be expressed as a money growth rate that varies in proportion to the expected growth rate in real money demand. That is, targeting the inflation rate is feasible here (and contrary to Eagle (2006), an inflation target policy seems consistent with price level determinacy here).

A Wicksellian regime

Following the approach taken in the New Keynesian literature, we might instead assume that the quantity of nominal debt B (money) is entirely demand-determined. The only policy instrument is R (and, of course, the lump-sum tax). I assume that policy follows the Taylor rule [8] with α = 1.

As far as I can tell, all of the math developed in the previous section continues to hold. But giving up the quantity variable B as a policy instrument must have some implication. Indeed, it does. What we seem to lose is any fundamental economic force determining the price level and inflation rate. That is, the level of debt and its growth rate simply accommodate themselves to the prevailing price level and inflation rate, respectively. According to [8], exogenous movements in the expected rate of inflation (inflation shocks) are met one-for-one with movements in the nominal interest rate, leaving the real rate of interest pegged to its natural rate.

Note something interesting here: the inflation target Π* is completely irrelevant. Inflation in this model can be whatever it "wants" to be. If the community expects an inflation rate Π+ < Π*, the inflation rate Π+ becomes a self-fulfilling expectation (and is hence a "rational expectation"). In this case, the monetary authority simply sets its policy rate R < R*. A situation like this can last indefinitely in this model.

Of course, everything works just fine here as long as the ZLB is not a constraint. Suppose, instead, that the policy rate is constrained by the ZLB, so that [8] becomes:

[9]  ln(R) = max{ 0, ln(R*) + α[ ln(Π+) - ln(Π*) ] }

Imagine that the economy is initially operating at potential with Π+ = Π* (without loss). Then, out of the blue, individuals suddenly believe that the inflation rate is going to be permanently lower Π+ = Π' < Π*. Moreover, suppose that this inflation shock is sufficiently large to make the ZLB bind. What happens?

What happens is that output drops permanently below potential (the economy continues to grow, however, at the rate implied by technological progress and population growth, both of which are normalized to zero here). Why does this happen?

It happens (here) because the real rate of interest rises above its natural rate, r' = 1/Π' > r*. The real interest rate is too high. The effect is to depress (investment) demand, r' = f'(k') implies k' < k*. The real GDP falls below potential, Y' = y + f(k') < Y*.

Because there is no nominal anchor for inflation in this economy, all sorts of bad things can happen at the lower bound. Contrary to the Friedman rule prescription, deflation is bad (generally, any inflation rate sufficiently low to make the ZLB bind). Not that the monetary authority could actually implement the Friedman rule if it wanted to. In this economy, the monetary authority has absolutely no control over the inflation rate!

A nominal anchor

An obvious way to provide a nominal anchor (in the model) is to adopt the monetarist approach and control the supply of the monetary aggregate. But perhaps this is something that is difficult to do in reality. What then?

Everything seems to hinge here on how individuals form inflation expectations. The theory here provides no guidance as to how these expectations should be formed. One can assert that individuals are likely to use the inflation target Π* as a nominal anchor. But this is just a bald-faced assertion. That is, if individuals do use Π* as a nominal anchor, then it will become a nominal anchor. The monetary authority, however, has no way enforcing the target Π* (unless it adopts a monetarist approach). 

Well then, let me assume a particular inflation expectation formation rule:

[10]  ln(Π+)  = (1 - ρ)ln(Π*) + ρln(Π) + δ[ ln(Y+) - ln(Y*) ] + ε

where 0 ≤ ρ ≤ 1, δ≥ 0, and where ε represents an inflation shock (say, i.i.d. and zero mean). Students may recognize [10] as a type of Phillips curve. 

Actually, now that I stare at [10], I see that the δ > 0 opens up another source of indeterminacy. It may be possible, for example, that if people suddenly expect a recession Y+ < Y*, that the downward revision in inflation forecasts implied by [10] could make the ZLB bind, generating a self-fulfilling prophecy.

Anyway, let's just set δ = 0 here. In the Wicksellian approach above, I adopted a special case of [10]; i.e., ρ = 1 and δ = 0;. But now, for 0 ≤ ρ < 1, any given inflation shock is mean-reverting (to the inflation target). The "lift off" date -- the date at which the monetary authority begins to raise its policy rate according to [9] depends on how quickly inflation expectations rise. The speed of adjustment here is governed by the parameter ρ--a lower ρ implies faster adjustment.

Is there anything the monetary authority can do here to "talk up inflation" (i.e., lower ρ)? We really can't say without a theory of expectation formation. But it seems to me that "promising to keep R = 1 for an extended period of time" may have the effect of increasing ρ, extending the period of adjustment. That is, by postponing the "lift off" date, agents may rationally expect inflation to remain below target for a longer period of time.

Concluding thoughts

Let me be clear that I do not think the 2008 drop in output below its previous trend was caused by a negative inflation shock. A negative inflation shock possibly played a role, but there had to be more to the story than this. In the analysis above, a negative inflation shock represents a movement along a stable IS curve; the real interest rate goes up, and output goes down. To make sense of recent events, we also have to consider shocks that shift the IS curve "leftward." (I describe just such a shock here.)

Nevertheless, I think it is interesting to explore what potential effects future downward revisions to inflation expectations may have on the economy at the ZLB. In the Wicksellian regime I study above, there appears to be no nominal anchor apart from what agents believe it to be. And if agents come to believe in a persistent deflation, it may come to pass, and the economy may be stuck below potential for a very long time. Convincing agents that the nominal interest rate is likely to remain at zero for a long time may be counterproductive, depending on how individuals interpret such policy announcements.

I want to stress, however, that while getting inflation and inflation expectations back to target (and firmly anchored to target) may be a solution to one problem, it is unlikely to be a solution to every problem currently facing the U.S. economy. To put it another way, suppose that the current real interest rate of -1% is too high relative to the current "natural" rate of -x%. Somehow driving the real return on bonds to -x% may then help things a bit, but it does nothing to address the more pressing question of why the "natural" rate is so low to begin with.