### 'Some Unpleasant Properties of Log-Linearized Solutions when the Nominal Interest Rate is Zero'

[This one is wonkish. It's (I think) one of the more important papers from the St. Louis Fed conference.]

One thing that doesn't get enough attention in DSGE models, at least in my opinion, is the constraints, implicit assumptions, etc. imposed when the theoretical model is log-linearized. This paper by Tony Braun and Yuichiro Waki helps to fill that void by comparing a theoretical true economy to its log-linearized counterpart, and showing that the results of the two models can be quite different when the economy is at the zero bound. For example, multipliers that are greater than two in the log-linearized version are smaller -- usually near one -- in the true model (thus, fiscal policy remains effective, but may need to be more aggressive than the log-linear model would imply). Other results change as well, and there are sign changes in some cases, leading the authors to conclude that "we believe that the safest way to proceed is to entirely avoid the common practice of log-linearizing the model around a stable price level when analyzing liquidity traps."

Here's part of the introduction and the conclusion to the paper:

Some Unpleasant Properties of Log-Linearized Solutions when the Nominal Interest Rate is Zero, by Tony Braun and Yuichiro Waki:AbstractDoes fiscal policy have qualitatively different effects on the economy in a liquidity trap? We analyze a nonlinear stochastic New Keynesian model and compare the true and log-linearized equilibria. Using the log-linearized equilibrium conditions the answer to the above question is yes. However, for the true nonlinear model the answer is no. For a broad range of empirically relevant parameterizations labor falls in response to a tax cut in the log-linearized economy but rises in the true economy. While the government purchase multiplier is above two in the log-linearized economy it is about one in the true economy.

1 IntroductionThe recent experiences of Japan, the United States, and Europe with zero/near-zero nominal interest rates have raised new questions about the conduct of monetary and fiscal policy in a liquidity trap. A large and growing body of new research has emerged that provides answers using New Keynesian (NK) frameworks that explicitly model the zero bound on the nominal interest rate. One conclusion that has emerged is that fiscal policy has different effects on the economy when the nominal interest rate is zero. Eggertsson (2011) finds that hours worked fall in response to a labor tax cut when the nominal interest rate is zero, a property that is referred to as the “paradox of toil,” and Christiano, Eichenbaum, and Rebelo (2011), Woodford (2011) and Erceg and Lindé (2010) find that the size of the government purchase multiplier is substantially larger than one when the nominal interest rate is zero.

These and other results ( see e.g. Del Negro, Eggertsson, Ferrero, and Kiyotaki (2010), Bodenstein, Erceg, and Guerrieri (2009), Eggertsson and Krugman (2010)) have been derived in setups that respect the nonlinearity in the Taylor rule but loglinearize the remaining equilibrium conditions about a steady state with a stable price level. Log-linearized NK models require large shocks to generate a binding zero lower bound for the nominal interest rate and the shocks must be even larger if these models are to reproduce the measured declines in output and inflation that occurred during the Great Depression or the Great Recession of 2007-2009.[1] Log-linearizations are local solutions that only work within a given radius of the point where the approximation is taken. Outside of this radius these solutions break down (See e.g. Den Haan and Rendahl (2009)). The objective of this paper is to document that such a breakdown can occur when analyzing the zero bound.

We study the properties of a nonlinear stochastic NK model when the nominal interest rate is constrained at its zero lower bound. Our tractable framework allows us to provide a partial analytic characterization of equilibrium and to numerically compute all equilibria when the zero interest state is persistent. There are no approximations needed when computing equilibria and our numerical solutions are accurate up to the precision of the computer. A comparison with the log-linearized equilibrium identifies a severe breakdown of the log-linearized approximate solution. This breakdown occurs when using parameterizations of the model that reproduce the U.S. Great Depression and the U.S. Great Recession.

Conditions for existence and uniqueness of equilibrium based on the log-linearized equilibrium conditions are incorrect and offer little or no guidance for existence and uniqueness of equilibrium in the true economy. The characterization of equilibrium is also incorrect.

These three unpleasant properties of the log-linearized solution have the implication that relying on it to make inferences about the properties of fiscal policy in a liquidity trap can be highly misleading. Empirically relevant parameterization/shock combinations that yield the paradox of toil in the log-linearized economy produce orthodox responses of hours worked in the true economy. The same parameterization/shock combinations that yield large government purchases multipliers in excess of two in the log-linearized economy, produce government purchase multipliers as low as 1.09 in the nonlinear economy. Indeed, we find that the most plausible parameterizations of the nonlinear model have the property that there is no paradox of toil and that the government purchase multiplier is close to one.

We make these points using a stochastic NK model that is similar to specifications considered in Eggertsson (2011) and Woodford (2011). The Taylor rule respects the zero lower bound of the nominal interest rate, and a preference discount factor shock that follows a two state Markov chain produces a state where the interest rate is zero. We assume Rotemberg (1996) price adjustment costs, instead of Calvo price setting. When log-linearized, this assumption is innocuous - the equilibrium conditions for our model are identical to those in Eggertsson (2011) and Woodford (2011), with a suitable choice of the price adjustment cost parameter. Moreover, the nonlinear economy doesn’t have any endogenous state variables, and the equilibrium conditions for hours and inflation can be reduced to two nonlinear equations in these two variables when the zero bound is binding.[2]

These two nonlinear equations are easy to solve and are the nonlinear analogues of what Eggertsson (2011) and Eggertsson and Krugman (2010) refer to as “aggregate demand” (AD) and “aggregate supply” (AS) schedules. This makes it possible for us to identify and relate the sources of the approximation errors associated with using log-linearizations to the shapes and slopes of these curves, and to also provide graphical intuition for the qualitative differences between the log-linear and nonlinear economies.

Our analysis proceeds in the following way. We first provide a complete characterization of the set of time invariant Markov zero bound equilibria in the log-linearized economy. Then we go on to characterize equilibrium of the nonlinear economy. Finally, we compare the two economies and document the nature and source of the breakdowns associated with using log-linearized equilibrium conditions. An important distinction between the nonlinear and log-linearized economy relates to the resource cost of price adjustment. This cost changes endogenously as inflation changes in the nonlinear model and modeling this cost has significant consequences for the model’s properties in the zero bound state. In the nonlinear model a labor tax cut can increase hours worked and decrease inflation when the interest rate is zero. No equilibrium of the log-linearized model has this property. We show that this and other differences in the properties of the two models is precisely due to the fact that the resource cost of price adjustment is absent from the resource constraint of the log-linearized model.[3] ...

...

5 Concluding remarksIn this paper we have documented that it can be very misleading to rely on the log-linearized economy to make inferences about existence of an equilibrium, uniqueness of equilibrium or to characterize the local dynamics of equilibrium. We have illustrated that these problems arise in empirically relevant parameterizations of the model that have been chosen to match observations from the Great Depression and Great Recession.

We have also documented the response of the economy to fiscal shocks in calibrated versions of our nonlinear model. We found that the paradox of toil is not a robust property of the nonlinear model and that it is quantitatively small even when it occurs. Similarly, the evidence presented here suggests that the government purchase GDP multiplier is not much above one in our nonlinear economy.

Although we encountered situations where the log-linearized solution worked reasonably well and the model exhibited the paradox of toil and a government purchase multiplier above one, the magnitude of these effects was quantitatively small. This result was also very tenuous. There is no simple characterization of when the log-linearization works well. Breakdowns can occur in regions of the parameter space that are very close to ones where the log-linear solution works. In fact, it is hard to draw any conclusions about when one can safely rely on log-linearized solutions in this setting without also solving the nonlinear model. For these reasons we believe that the safest way to proceed is to entirely avoid the common practice of log-linearizing the model around a stable price level when analyzing liquidity traps.

This raises a question. How should one proceed with solution and estimation of medium or large scale NK models with multiple shocks and endogenous state variables when considering episodes with zero nominal interest rates? One way forward is proposed in work by Adjemian and Juillard (2010) and Braun and Körber (2011). These papers solve NK models using extended path algorithms.

We conclude by briefly discussing some extensions of our analysis. In this paper we assumed that the discount factor shock followed a time-homogeneous two state Markov chain with no shock being the absorbing state. In our current work we relax this final assumption and consider general Markov switching stochastic equilibria in which there are repeated swings between episodes with a positive interest rate and zero interest rates. We are also interested in understanding the properties of optimal monetary policy in the nonlinear model. Eggertsson and Woodford (2003), Jung, Teranishi, and Watanabe (2005), Adam and Billi (2006), Nakov (2008), and Werning (2011) consider optimal monetary policy problems subject to a non-negativity constraint on the nominal interest rate, using implementability conditions derived from log-linearized equilibrium conditions. The results documented here suggest that the properties of an optimal monetary policy could be different if one uses the nonlinear implementability conditions instead.

[1] Eggertsson (2011) requires a 5.47% annualized shock to the preference discount factor in order to account for the large output and inflation declines that occurred in the Great Depression. Coenen, Orphanides, and Wieland (2004) estimate a NK model to U.S. data from 1980-1999 and find that only very large shocks produce a binding zero nominal interest rate.

[2] Under Calvo price setting, in the nonlinear economy a particular moment of the price distribution is an endogenous state variable and it is no longer possible to compute an exact solution to the equilibrium.

[3] This distinction between the log-linearized and nonlinear resource constraint is not specific to our model of adjustment costs but also arises under Calvo price adjustment (see e.g. Braun and Waki (2010)).

Posted by Mark Thoma on Friday, October 12, 2012 at 06:13 PM in Academic Papers, Economics, Macroeconomics, Methodology |
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