## Monday, November 5, 2012

### Managing a Liqudity Trap: Monetary and Fiscal Policy

Ivan Werning's paper on liquidity traps is getting attention in the Federal Reserve System. For example, Narayana Kocherlakota cited the paper in a recent speech. Ivan presented a version of the paper at the 2011 St. Louis Fed Policy conference, which was where I saw it.

What's the paper about, and why would the FOMC be interested in it? The basic model Werning uses is a well-worked-over linearized New Keynesian sticky price model, much like what can be found, for example, in Clarida, Gali, and Gertler's survey paper. The key differences here are that Werning works in continuous time with no aggregate uncertainty, which is going to lend tractability to the problem. Further, he's going to solve an optimal policy problem. Perhaps surprisingly, New Keynesians do not often do that. The typical approach is to assume a Taylor rule for monetary policy, and go from there.

Here's the basic model(changing notation a bit):

(1) dx/dt = a[i(t)-r(t)-p(t)]
(2) dp/dt = bp(t)-kx(t)
(3) i(t) >= 0

x is the output gap, i is the nominal interest rate, r is the natural real rate of interest, and p is the inflation rate. a, b, and k are positive parameters. Equation (1) is an inverted Euler equation, equation (2) is a Phillips curve, and (3) imposes the zero lower bound on the nominal interest rate. Given an exogenous path r(t), the central bank determines i(t), and this determines a solution p(t) and x(t). Werning assumes a typical quadratic loss function, where bliss is taken to be a zero output gap and zero inflation.

What is an optimal monetary policy in this environment? If r(t)>=0 for all t, then the answer is easy. Bliss is attainable for all t. An optimal monetary policy is i(t)=r(t), which implies x(t)=p(t)=0.

The interesting question is what happens if r(t)<0 for some t. A simple example is r(t)=r1 for t<=T and r(t)=r2 for t>T, where r1<0 and r2>0. This will imply that bliss is not feasible for all t, and the zero lower bound (3) must bind at some dates. This is intended to look like the type of monetary policy problem that the Fed is currently faced with, as we'll see.

It will make a big difference whether or not the central bank is able to commit to a policy. With no commitment, we know that the central bank chooses i(t)=r(t) from date T on, with x(t)=p(t)=0 for t>T. Then we can work back to find the dynamic path up to time T. Before time T, we are in a liquidity trap, with i(t)=0, and things can be very bad. The output gap and the inflation rate are increasing, but until the time of liftoff there is a negative output gap and deflation. The problem gets worse the larger is T, i.e. the longer the liquidity trap scenario lasts.

Basically, the problem is that the real rate of interest is too high, and there is nothing the central bank can do about it, being constrained by the zero lower bound on the nominal interest rate. Further, the problem is compounded because of deflation during the liquidity trap period.

But, the central bank can do better if it can commit to a policy different from i(t)=r(t) after date T. After date T, the optimal policy with commitment is for the central bank to generate inflation above zero and an output boom (i.e. a positive output gap) after date T, which feeds back to the liquidity trap period, increasing inflation and output before period T.

You can see why people looking for a rationale for the FOMC's recent policy decisions like this paper. If you buy Werning's results, you might write something like the following in an FOMC statement:
To support continued progress toward maximum employment and price stability, the Committee expects that a highly accommodative stance of monetary policy will remain appropriate for a considerable time after the economic recovery strengthens.
The Fed is under a lot of pressure to "do something" about the weak labor market, and that pressure has been fed (no wordplay intended) by the Fed's confident statements about the efficacy of its policies. Apparently they are looking for serious research that will support what they are doing.

But how seriously do we want to take Werning's paper? Is this a basis for sound monetary policy in the circumstances we find ourselves in? I don't think so.

What's the shock that is driving the policy? As in much recent Keynesian analysis, the reason for being in a prolonged state with a binding zero lower bound on the nominal interest interest rate is that the "natural real rate of interest" is low (negative, in the model). In the underlying model, the natural interest rate is the real rate of interest under flexible prices. What would cause the real rate to be low? The underlying model is a standard neoclassical growth model, in which the natural rate of interest depends (with a constant relative risk aversion utility function) on the discount factor (from preferences) and consumption growth. The natural real rate goes down if the discount factor rises (people care more about the future) or if efficient consumption growth falls. We're supposed to think that the financial crisis was about everyone simultaneously taking a greater interest in the future? At the optimum, this would make them want to save more. Or maybe the financial crisis shock - whatever it was - should have resulted in lower than average consumption growth coming out of the recession? These things are inconsistent with the Keynesian ideas about intervention that I have been hearing.

In terms of the logic of the model, and what we know about the recent recession, the negative natural real rate shock does not make any sense. More fundamentally, this is a model which does not incorporate any financial factors. There is no credit, no banks, and in fact no money. Monetary policy is about setting a market nominal interest rate, and that's it. The model has nothing to say about quantitative easing, why it may or may not work, and how that policy fits into the forward guidance policy that Werning's paper is about. It seems particularly bizarre to be attempting to address monetary policy in the wake of the financial crisis in a model that can exhibit nothing that looks like a financial crisis.

Approximations. There are at least a couple of papers in which it is argued that nonlinearities are important at the zero lower bound, one by Villaverde and coauthors, the other by Braun and coauthors. Sometimes accounting properly for the nonlinearities doesn't just change the results quantitatively, but gives you qualitatively different results. It's a big deal. Werning uses the standard approach of linearizing around zero inflation, and then using a quadratic loss function to approximate welfare loss. I think the latter is suspect as well. As evidence that approximation may be leading Werning astray, look at his results on the effects of allowing for more price flexibility. Monetary policy in this model is all about correcting the price distortions that arise from price stickiness. Thus, one would expect that any monetary policy problems become less foreboding as prices get less sticky. That's not true in Werning's linearized model. In fact, in the economy without commitment, the output gap goes to minus infinity as price stickiness goes away in the limit. Something wrong there, I'm afraid.

Basic New Keynesian problems. I've come to think of the standard New Keynesian framework as a model of fiscal policy. The basic sticky price (or sticky wage) inefficiency comes from relative price distortions. Particularly given the zero lower bound on the nominal interest rate, monetary policy is the wrong vehicle for addressing the problem. Indeed, in Werning's model we can always get an efficient allocation with appropriately-set consumption taxes (see Correia et al., for example). I don't think the New Keynesians have captured what monetary policy is about.

Commitment. The original idea about monetary policy and commitment came from an example in Kydland and Prescott's paper. In that model, in the absence of commitment the central bank is always tempted to use inflation to increase the output gap. The result is a bad equilibrium with high inflation. That model was used to justify commitment to policy rules that would lower long-run inflation with no cost in terms of output. The Werning paper, and related work, is being used to turn that argument on its head. Now, we are supposed to think that committing to high inflation in the future, when the central bank would otherwise choose low inflation, will be a good thing. Whether Kydland and Prescott's monetary policy model was any good (it has its problems), the idea certainly played out well in the policy realm, beginning with the Volcker disinflation.

The forward guidance idea in FOMC policy, backed up by Werning's work (and Woodford's), may prove to be harmless. But maybe not. Some FOMC members, particularly Evans and Kocherlakota, seem bent on writing down explicit numerical criteria for future policy tightening. I hope they run up against resistance.

1. Slightly off-topic: I'm one who hasn't had to sit through a macro lecture since circa 1988 (thank God). Lately I've been trying to get to grips with current theory, more for mental exercise than for any special project. Two books by Benassy seem quite good to me: his Macro Theory (graduate textbook) and Money, Interest and Policy. Any thoughts on his approach?

1. I have not read Benassy's books. I don't use a textbook in 1st year graduate macro, but I know people who do. I looked up the Benassy book, and it appears to cover the standard material. Sargent and Ljungvist's book is used a lot, as is David Romer's. If you want a rundown of basic New Keynesian ideas, there is Interest and Prices by Woodford.

2. Thanks. I've read most of Gali's book on the NK model which is as much as I want on that. Actually it was your remark that NK types don't usually look to solve for an optimal policy which made me think of Benassy.

3. Stephen, I sympathise with your concern regarding the discontinuity of linearized new keynesian dynamics as price flexibility increases (though Braun et al highlight that this depends on which of the multiple equilibria you select). But preference shocks can be justified to an extent as a reduced form for increases in uninsured idiosyncratic income risk in the Bewley/Aiygari model. Google papers by Algan,Ragot and coauthors or papers by Tao Zha and coauthors for formal demonstrations in simplified heterogenous agent economies. In fact in an economy with fixed capital (as in the 3 eq NK model used by Werning) the preference shock is often isomorphic to an increase in the external finance premium or a reduction in the maximum borrowing limit, again in an economy with heterogenous agents. Of course, Chari et al who formalised this link between shocks to the representative agent/firm and richer microfoundations economies (business cycle accounting) explicitly rejected any use of their framework to conduct welfare analysis.

4. "But preference shocks can be justified to an extent as a reduced form for increases in uninsured idiosyncratic income risk in the Bewley/Aiygari model."

I'm not sure what that means. I have a representative agent model with preference shocks. That is somehow supposed to be "like" having fluctuations in (I guess) the dispersion in idiosyncratic income risk in a heterogeneous agent model. Even if you could convince me that was true, why does it make me feel any better? If what I'm interested in is idiosyncratic income risk, I guess I should want to work with the Bewley model. You think the financial crisis is about an increase in uninsurable risk? Maybe. That's getting closer to what I like to think about.

"...the preference shock is often isomorphic to an increase in the external finance premium or a reduction in the maximum borrowing limit, again in an economy with heterogenous agents."

A reduction in the borrowing limit makes the risk-free rate go down. It seems that works in the wrong direction. Seems the idea is that policy should fix whatever is making borrowing constraints bite more, which would then increase the risk-free rate.

1. Yes, but the point is that in some circumstances it may be faster to solve and simulate the dynamics of variables such as aggregate consumption or gdp, or a benchmark risk free interest raate by using the approximate equivalence of the aggregate dynamics of the Bewley model and the representative agent model with the appropriate preference or tax shocks (e.g for some financing frictions acting like consumption, labour, capital taxes).Then, for welfare evaluation , you can always simulate a panel of heterogenous households under the aggregate capital, prices etc... process you found using a representative agent+the appropriate wedges/shocks. Why do this? Why would you simulate many households trading a vast array of Arrow Debreu securities to analyse aggregates if you know the dynamics are equivalent to a representative household? Also the intuition in terms of the representative agent can be simpler than the one in terms of many different households trading many different kinds of contracts.
As for the 2nd part, yes in the flexible price equilibrium, a low risk free rate is a symptom of credit problems. Welfare improving government action would typically involve trying to relax credit constraints in a way that ends up raising credit demand and real interest rates. But in the sticky price model, the risk free rate is stuck at a too high level in the 1st place relative to the flex price equilibrium, making things even worse. So central banks seeing this, think monetary policy can step in and lower interest rates. This assumes central banks control the relevant interest rates, which I'd probably agree with you is far from obvious. Eugene Fama has a recent paper reviewing the last issue.

2. It seems that I have to know what the solution is in a more elaborate model, or I don't. How am I supposed to know what I am supposed to be approximating? We are stuck with the models that are tractable, either in the sense that we can compute solutions efficiently, or solve using pencil and paper. If I'm using the simpler model, because it's tractable, it's hard to see how I don't lose something relative to the more complicated model I might like to use, but can't because it is intractable.

"But in the sticky price model, the risk free rate is stuck at a too high level in the 1st place relative to the flex price equilibrium, making things even worse."

Exactly. So the sticky price model is not capturing the nature of the problem.

5. Math and models aside, I think the CB commitment to inflation trending through inflationary expectations is a failure because of the past history of most adults in perceiving the Fed as the primary inflation fighter. The current model of "QE3" does not push any increased inflationary funding into the federal government to fund new programs, fiscal expansion, or indeed anything beyond indirectly propping up the housing market.

The Fed promising to be "irresponsible" is not credible when most of the Fed seems to committed to marginal, easily reversible interventions designed to accommodate the banking system's distress. The crisis style interventions do not provide inflationary impetus to anything beyond the bond and equity markets.

Further, the Fed still seems to be thinking in terms of the past static demands for funds, and missing the drain of capital involved in funding our deficits, with that capital not returning in any meaningful way through increased export demands that would stimulate employment. In short, our Fed looks a lot like the Swiss National Bank, trying to deal with large capital flows, while trying to prevent further damage to the real economy.

In short, employment and wages have to rise quite a bit before all of this monetary "stimulus" could have any chance of providing higher inflation.

I would note the Roosevelt currency devaluations had a much more salutary effect on labor and employment, along with massive federal fiscal stimulus.

If the Fed wanted to start transmitting inflation, they might as well start monetizing gold again at say \$3k per ounce, and stimulate through that useless Keynesian hole digging activity.

Very interesting to watch the arguments, while sitting here watching a stagnant, slowly improving economy debating the impact of the next shock.

In short, the Fed can provide plenty of inflation in asset prices in the monetary side of the system, but in a liquidity trap, utterly fails to do more than support the system in maintaining asset prices.

6. "What's the shock that is driving the policy? As in much recent Keynesian analysis, the reason for being in a prolonged state with a binding zero lower bound on the nominal interest interest rate is that the "natural real rate of interest" is low (negative, in the model)."

The shock is that the economy has gone from real aggregate supply constrained to real aggregate demand constrained, CONTRARY to the most common definition of economics, unlimited wants/needs and limited resources.

1. That's another model altogether.

7. Which model? All the ones I have seen assume whatever potential output is that is what real GDP should be.

1. "...whatever potential output is that is what real GDP should be."

That, I think, would be how most people would define potential output. Go on...

8. Potential output = real Aggregate Supply (real AS)

There is also real Aggregate Demand (real AD).

All the models I see assume that if real AD is below real AS that is an aggregate demand shock that needs to be corrected.

What if real AS can get above the level of real AD without it being an aggregate demand shock violating the most common definition of economics?

1. If only you would realize how meaningless your statements were, you might start learning something from Steve.

2. "All the models I see..."

It looks like the only models you see are the ones in some undergraduate textbooks. I don't find those very helpful in understanding what is going on in the world currently.

9. In our paper, we also highlight the role of expectations. When the exit from the ZLB is endogenous and the economy is buffeted by aggregate shocks, the results are quite different. To start with, multipliers are smaller because households put a positive probability to escaping the ZLB in one quarter. Equally interesting is the evolution of the expectations of duration of the ZLB. The longer households stay at the ZLB, the longer they expect the economy will be at the liquidity trap.

10. Anonymous at 6:52 a.m. , the models I use weren't meaningless when we sold all of our mutual funds in early fall 2007. They said there was a housing bubble and a lower/middle class debt bubble while greenspan, bernanke, and others said there was not.

Stephen Williamson, what economic model(s) does not assume the most common definition of economics is correct?

Plus and sorry, the reply button is not working for me for some reason.

11. Totally agree about the fact that many (most) of the papers on monetary policy at the ZLB simply assume that the constraint is the result of a negative shock to the natural real rate (the Eggertsson-Krugman paper is the one that springs to mind that doesn't do this). However I still think it is a reasonable assumption to invoke to keep things tractable, given that we are looking at monetary policy when zero nominal rates are too tight to spur aggregate demand. But also agree that more needs to be done to ascertain the sources of the binding constraint and whether flagging aggregate demand is really the cause.

For instance as you rightly argue, financial frictions are all too often overlooked as a potential source for a binding ZLB. But they are considered in models that purely look at the effectiveness of QE and the LSAPs, such as in Curdia & Woodford, and Gertler & Karadi.

However, I do think that 'reversed' time inconsistency problem does make sense. Once you take into account the inefficiencies resulting from price dispersion and inertia in the NK models, the welfare loss functions that are derived using second order approximations to the utility function naturally result in an optimal policy where policy makers promise future inflation during the binding ZLB to minimise negative output and inflation target gaps, but when the ZLB is no longer binding re-optimisation by a discretionary central banker would result in the policy maker reneging on the earlier promise of higher inflation and positive output gaps made earlier. This is something that has been around since the Krugman '97 paper, and fleshed out in the DSGE models by Eggertsson and Woodford on several occasions (which I'm sure you're aware of but some of your readers may not be).

Great review of a vast amount of literature though. And thanks for telling everyone about Ivan's paper. I wasn't aware of it but I am now.

12. Werning is very overrated.