Believe those who are seeking the truth. Doubt those who find it. Andre Gide

Saturday, January 19, 2013

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] ∑tβtU(yt)

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'(yt) = RtΠ-1t+1β U'(yt+1)

where Rt denotes the gross nominal interest rate from period t to t+1 and Πt+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] Rt = (1/β)Πt+1gt+1 [Euler Equation]

where gt+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] rt = (1/β)gt+1

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

[5] rt = RtΠ-1t+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 gt+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] Rt = R* + α( Πt - Π* ) + δln(yt/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 Rt and that its choice is governed in part by current (realized) inflation Πt, among other things. The implication is that if the Fed chooses Rt, 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 Rt to Πt+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 yt = y* and that growth is zero (gt = 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] Πt+1 = R*β + αβ( Πt - 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(ut), where ut is the unemployment rate:

[8] Wt+1 ≥ γ(ut)Wt  where γ(0) > β

with γ(ut) a decreasing function of ut, 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.


  1. At the risk of commenting before I really work through the model--I think that modern macro has been pretty sloppy about nominal interest rates. We used to talk about money supply and its impact on nominal interest rates. Now we talk about nominal interest rates and mostly assume that money supply does what ever it does to achieve the nominal interest rate target. That's fine from an abstract perspective--if the Fed controls the interest rate, it also determines the money supply by way of bank profit maximization and money demand.

    The problem, though, is that this is not very specific. There are two ways to raise the nominal interest rate--either through monetary expansion or through monetary contraction. The former works when the liquidity effect is greater than the inflation effect, and the latter works when the inflation effect is greater than the liquidity effect.

    So what Schmitt-Grohe and Uribe are actually saying is that the Fed should engage in so much quantitative easing that the interest rate actually rises. This is also what Milton Friedman argued when he said that low nominal interest rates were a sign that monetary policy was too tight rather than too loose.

    1. Matthew,

      I'm not sure if that is what the authors are actually saying. I read them as saying "increase the policy rate." In the present context, this would mean raising the IOR rate. I do not think it is "sloppy" thinking to imagine that such a policy change would have an impact on market interest rates independently of what the Fed does with its balance sheet. (Well, maybe I have in mind an appropriate support from the fiscal authority...)

    2. I also think Matthew is misinterpreting it (though I haven't fully worked thought it either). I think the authors are saying that the Fed should wake up one day and say, "ZOMG! We've already done so much QE (or whatever magic it is that we do) that the inflation rate (and/or real output) will move above our target if we don't tighten now, so we had better tighten!" whereupon private agents will say, "ZOMG! The Fed's magic has raised expected output and inflation to the point where they will be above target if the Fed doesn't tighten now! This is much stronger than we thought! We had better start consuming more stuff now before it gets more expensive, especially since our income is likely to be higher than we thought." It's all about self-fulfilling prophecies. And in this model, I gather, the Fed makes a prophecy by setting the target interest rate, so as long as it keeps the rate low, it's telling people the economy will remain weak. It's remotely plausible, but only remotely: I expect it depends on people having some confident knowledge of the Fed's reaction function, whereas in practice, the Fed's actions give new information about its reaction function as well as about its expectations.

    3. Interest on reserves changes the analysis a little bit in the sense that the Fed could potentially sterilize balance sheet expansion by paying higher interest on reserves. But, I don't see how it changes things that much. Interest rate policy still has two effects: raising interest on reserves causes banks to move funds from investments into reserves, which is the liquidity effect, but it also increases the total amount that banks have to invest by creating new money to pay the interest, which is the inflation effect.

      You mentioned that the authors' taylor rule makes an assumption about causality--this assumption basically says that the inflation effect is always larger than the liquidity effect. I see no reason to suppose that it is always this way. A lot of old research showed that timing matters--in the short run the liquidity effect exceeds the inflation effect, but in the long run the inflation effect dominates. This points to an interest rate hike being a very bad way to achieve short-run stimulus.

    4. A lot of old research showed that timing matters--in the short run the liquidity effect exceeds the inflation effect, but in the long run the inflation effect dominates. This points to an interest rate hike being a very bad way to achieve short-run stimulus.

      The authors argue (p. 21) that the conventional wisdom you appeal to here is likely to be overturned in a liquidity trap situation. I'll have to think about it some more.

    5. There is something that still doesn't make sense to me. Maybe I'm reading the paper wrong. The equation (3) above actually isn't that controversial in the literature--it differs from what you would typically see only in the fact that there's nothing stochastic about the parameters of the production function. So the causal claim about next period inflation isn't a big deal per se. However, there is a problem with the causal claim about price levels--if I'm not mistaken in a typical model a change in the interest rate should affect the current price level (or consumption, in the case of sticky prices), not the future price level. Thus, increasing the interest rate would decrease current prices, which, because wages are downward rigid, increases the real wage rate, which in turn cause movement up the labor demand curve, leading to less consumption. Even with the assumptions here about wage rigidity and the taylor rule, interest rate hikes should still be contractionary in the short run. I don't see how a liquidity trap changes this.

      That is, unless there is a way to raise the nominal rate by increasing future prices instead of decreasing current prices. In a model with money, this would be accomplished by "credibly" committing to a larger money supply next period. But then, there is no money in this model. I suppose that government debt could serve as money in this economy (I'm having a haunting flashback to the recent blog debate between Paul Krugman and Steve Randy Waldman), so I suppose issuing government bonds would be both inflationary and increase the interest rate.

      Also, there's something else I'm having trouble understanding. This goes to Nick Rowe's comment below. The labor market, by the author's own statement, is not in equilibrium when we are in the "bad" steady state--we are always on the labor demand schedule, but not the labor supply. By Walras' Law, at least one other market should also not be in equilibrium. But if the consumption Euler equation holds in all periods, which market is it?

    6. Matthew:

      Good questions. If I remember correctly, the authors do not even mention the price level. I'll have to go back and think this through.

      That is, unless there is a way to raise the nominal rate by increasing future prices instead of decreasing current prices.

      I suspect that this is what must be happening. And it would be easy to accomplish without money because expectations are determined by the Euler equation!

      With respect to your last question, note that the Euler equation is not something that determines the *level* of consumption--it determines the rate of change of consumption. The real wage is the relative price of contemporaneous labor vis-a-vis contemporaneous output. So, to use language loosely, if the labor market is not clearing, the goods market is not either. This has nothing to do with the Euler equation, I don't think.

  2. I'm not sure I understand how the authors' argument is fundamentally different than simply an application of the expectations-augmented Phillips curve. It seems quite similar in many respects to an armchair economist like me.

    Somewhat of an aside: Is anyone aware of any research on the effect of wage subsidies in a liquidty trap environment?

    1. Well, for one thing, I do not see a Phillips curve in the model above.

    2. Maybe you're not looking hard enough ;).

  3. David: sorry for not getting to this earlier.

    Your Euler equation does not tells us what {P,R} will *in fact* be. It tells us what {P,R} *would need to be* to keep actual and expected output at the natural rate.

    If prices are sticky, and if the economy is at a point below your Euler equation, that means the actual real interest rate is below the natural real interest rate. The standard story then says that that means output will be above the natural rate, so that actual and expected inflation will be rising. Because there is a Phillips Curve in the standard story too. Which means that point A is locally stable, and point B is locally unstable.

    1. Nick,

      Your Euler equation does not tells us what {P,R} will *in fact* be. It tells us what {P,R} *would need to be* to keep actual and expected output at the natural rate.

      I do not think you are correct here. Note that in their model, the Euler condition holds even in a liquidity trap, where output and expected output remain below potential forever.

      If prices are sticky...

      ...then we would have a different model. ;)

      I'm not so familiar with literature. Are you saying that by adding a standard Phillips curve (whilst maintaining rational expectations), the stability properties of A and B are reversed? Dang! Point me to a paper...

  4. David,

    I doesn't seem to me your interpration of the model is complete (I haven't read the paper).

    I believe your missing an important equation, the government's budget constraint. As I used to understand this class of models, the equations you've written down here determine the entire time path of inflation and the government's budget constraint, as well as nominal value of outstanding government bonds, then maps that path into a price level today.

    This implies that the way in which inflation expectations rise when the Fed raises the policy rate in the manner you describe (by setting a=0) is by the price level today falling.

    The logic basically is that the lower value of the nominal debt (higher nominal rate) must satisfy the government's budget constraint with equality and this requires a lower price level today.

    Nick: no, this economy is never at a point below the Euler equation. It always satisfies the equation. If you think it doesn't you need to explain to David why the poeple living in his model suddenly decided to stop acting according to their own preferences.

    1. Adam,

      They do have a government budget constraint (p. 8) which essentially says that a lump-sum tax is used to finance the carrying cost of the debt. The nominal debt does not appear to play any role in their analysis and, if I'm not mistaken, the price-level appears to be indeterminate.

      The logic basically is that the lower value of the nominal debt (higher nominal rate) must satisfy the government's budget constraint with equality and this requires a lower price level today.

      No, a higher lump-sum tax can be used to finance the higher carrying cost of the debt. No?

  5. I'm also struggling to see how we get a level drop in the real wage, that seems to require a discrete jump in the price level. You didn't even introduce a price level.

    Maybe I should just read the paper...

    1. Adam,

      Take a look at their diagrams on p. 22. Evidently, the inflation rate does not jump to the target rate; instead, it marches upward toward target. Unemployment only falls to zero gradually. So, evidently, there is no level drop in the real wage rate on impact -- it declines slowly (relative to trend) to its market-clearing level.

      Ya, read the paper and then explain it to me!