Stephanie Schmitt-Grohé and Martín Uribe are Professors of Economics at Duke University. Their main line of interest lies in monetary macroeconomics, in particular issues of optimal stabilisation policy. Schmitt-Grohé's
entry. Uribe's RePEc/IDEAS entry.
Much of our recent research has been devoted to developing and applying
tools for the evaluation of macroeconomic stabilization policy. This
choice of topic was motivated by an important development in
business-cycle theory. By the late 1990s, a frictionless model of the
macroeconomy was viewed by many as no longer providing a satisfactory
account of aggregate fluctuations. As a response, the new Keynesian
paradigm emerged as an alternative framework for understanding business
cycles. A key difference between the neoclassical and the new Keynesian
paradigms is that in the latter, the presence of various nominal and real
distortions provide a meaningful role for stabilization policy, opening
the door once again, after decades of dormancy, for policy evaluation.
Developing Tools For Policy Evaluation
An obstacle we encountered early on in executing the research agenda described here was the lack of appropriate tools to evaluate stabilization policies in the context of distorted economies. An important part of our effort was therefore devoted to developing such tools.
Most models used in modern macroeconomics are too complex to allow for exact solutions. For this reason, researchers have appealed to numerical approximation techniques. One popular and widely used approximation technique is a first-order perturbation method delivering a linear approximation to the policy function. One reason for the popularity of first-order perturbation techniques is that they do not suffer from the `curse of dimensionality.' That is, problems with a large number of state variables can be handled without much computational demands. Because models that are successful in accounting for many aspects of observed business cycles are bound to be large (e.g., Smets and Wouters, 2004; and Christiano, Eichenbaum, and Evans, 2003), this advantage of perturbation techniques is of particular importance for policy evaluation. However, an important limitation of first-order approximation techniques is that the solution displays the certainty equivalence property. In particular, the first-order approximation to the unconditional means of endogenous variables coincides with their non-stochastic steady state values. This limitation restricts the range of questions that can be addressed in a meaningful way using first-order perturbation techniques.
One such question that is of particular relevance for our research agenda is welfare evaluation in stochastic environments featuring distortions or market failures. For example, Kim and Kim (2003) show that in a simple two-agent economy, a welfare comparison based on an evaluation of the utility function using a linear approximation to the policy function may yield the spurious result that welfare is higher under autarky than under full risk sharing. The problem here is that some second- and higher-order terms of the equilibrium welfare function are omitted while others are included. Consequently, the resulting criterion is inaccurate to order two or higher. The same problem arises under the common practice in macroeconomics of evaluating a second-order approximation to the objective function using a first-order approximation to the decision rules. For in this case, too, some second-order terms of the equilibrium welfare function are ignored while others are not. See Woodford (2003, chapter 6) for a discussion of conditions under which it is correct up to second order to approximate the level of welfare using first-order approximations to the policy function. In general, a correct second-order approximation of the equilibrium welfare function requires a second-order approximation to the policy function.
This is what we set out to accomplish in Schmitt-Grohé and Uribe (2004a). Building on previous work by Collard and Juillard, Sims, and Judd among others, we derive a second-order approximation to the solution of a general class of discrete-time rational expectations models. Specifically, our technique is applicable to nonlinear models whose equilibrium conditions can be written as: Et f(yt+1,yt,xt+1,xt)=0, where the vector xt is predetermined and the vector yt is nonpredetermined.
The main theoretical contribution of Schmitt-Grohé and Uribe (2004a) is to show that for any model belonging to this general class, the coefficients on the terms linear and quadratic in the state vector in a second-order expansion of the decision rule are independent of the volatility of the exogenous shocks. In other words, these coefficients must be the same in the stochastic and the deterministic versions of the model. Thus, up to second order, the presence of uncertainty affects only the constant term of the decision rules. But the fact that only the constant term is affected by the presence of uncertainty is by no means inconsequential. For it implies that up to second order the unconditional mean of endogenous variables can in general be significantly different from their non-stochastic steady state values. Thus, second-order approximation methods can in principle capture important effects of uncertainty on average rate of return differentials across assets with different risk characteristics and on the average level of consumer welfare.
An additional advantage of higher-order perturbation methods is that like their first-order counterparts, they do not suffer from the curse of dimensionality. This is because given the first-order approximation to the policy function, finding the coefficients of a second-order approximation simply entails solving a system of linear equations.
The main practical contribution of Schmitt-Grohé and Uribe (2004a) is the development of a set of MATLAB programs that compute the coefficients of the second-order approximation to the solution to the general class of models described above. This computer code is publicly available at the authors' websites. Our computer code coexists with others that have been developed recently by Chris Sims and Fabrice Collard and Michel Juillard to accomplish the same task. We believe that the availability of this set of independently developed codes, which have been shown to deliver identical results for a number of example economies, helps build confidence across potential users of higher-order perturbation techniques.
Optimal Operational Monetary Policy for the U.S. Economy
After the completion of the second-order approximation toolkit, we felt that we were suitably equipped to undertake a systematic and rigorous evaluation of stabilization policy. A contemporaneous development that highly facilitated our work was the emergence of estimated medium-scale dynamic general equilibrium models of the U.S. economy with the ability to explain the behavior of a relatively large number of macroeconomic variables at business-cycle frequency (e.g., Christiano, Eichenbaum, and Evans, 2003; and Smets and Wouters, 2004).
A central characteristic of the studies on optimal monetary policy that existed at the time we initiated our research on policy evaluation, was that they were conducted in the context of highly stylized environments. An important drawback of that approach is that highly simplified models are unlikely to provide a satisfactory account of cyclical movements for but a few macroeconomic variables of interest. For this reason, the usefulness of this strategy to produce policy advise for the real world is necessarily limited.
In a recent working paper (Schmitt-Grohé and Uribe, 2004b), we depart from the literature extant in that we conduct policy evaluation within the context of a rich theoretical framework capable of explaining observed business cycle fluctuations for a wide range of nominal and real variables.
Following the lead of Kimball (1995), the model emphasizes the importance of combining nominal and real rigidities in explaining the propagation of macroeconomic shocks. Specifically, the model features four nominal frictions, sticky prices, sticky wages, money in the utility function, and a cash-in-advance constraint on the wage bill of firms, and four sources of real rigidities,
investment adjustment costs, variable capacity utilization, habit formation, and imperfect competition in product and factor markets. Aggregate fluctuations are assumed to be driven by supply shocks, which take the form of stochastic variations in total factor productivity, and demand shocks stemming from exogenous innovations to the level of government purchases.
Altig et al. (2003) and Christiano, Eichenbaum, and Evans (2003) argue that the model economy for which we seek to design optimal operational monetary policy
can indeed explain the observed responses of inflation, real wages, nominal interest rates, money growth, output, investment, consumption, labor productivity, and real profits to productivity and monetary shocks in the postwar United States. In this respect, Schmitt-Grohé and Uribe (2004b) aspires to be a step ahead in the research program of generating monetary policy evaluation that is of relevance for the actual practice of central banking.
In our quest for the optimal
monetary policy scheme we restrict attention to what we call operational interest rate rules. By an operational interest-rate rule we mean an interest-rate rule that satisfies three requirements. First, it prescribes that the nominal interest rate is set as a function of a few readily observable macroeconomic variables. In the tradition of Taylor (1993), we focus on rules whereby the nominal interest rate depends on measures of inflation, aggregate activity, and possibly its own lag. Second, the operational rule must induce an equilibrium satisfying the zero lower bound on nominal interest rates. And third, operational rules must render the rational expectations equilibrium unique. This last restriction closes the door to expectations driven aggregate fluctuations.
The object that monetary policy aims to maximize in our study is the expectation of lifetime utility of the representative household conditional on a particular initial state of the economy. Our focus on a conditional welfare measure represents a fundamental departure from most existing normative evaluations of monetary policy, which rank policies based upon unconditional expectations of utility. Exceptions are Kollmann (2003) and Schmitt-Grohé and Uribe (2004c). As Kim et al. (2003) point out, unconditional welfare measures ignore the welfare effects of transitioning from a particular initial state to the stochastic steady state induced by the policy under consideration. Indeed, we document that under plausible initial conditions, conditional welfare measures can result in different rankings of policies than the more commonly used unconditional measure. This finding highlights the fact that transitional dynamics matter for policy evaluation.
In our welfare evaluations, we depart from the widespread practice in the neo-Keynesian literature on optimal monetary policy of limiting attention to models in which the nonstochastic steady state is undistorted. Most often, this approach involves assuming the existence of a battery of subsidies to production and employment aimed at eliminating the long-run distortions originating from monopolistic competition in factor and product markets. The efficiency of the deterministic steady-state allocation is assumed for purely computational reasons. For it allows the use of first-order approximation techniques to evaluate welfare accurately up to second order, a simplification that was pioneered by Rotemberg and Woodford (1999). This practice has two potential shortcomings. First, the instruments necessary to bring about an undistorted steady state (e.g., labor and output subsidies financed by lump-sum taxation) are empirically uncompelling. Second, it is ex ante not clear whether a policy that is optimal for an economy with an efficient steady state will also be so for an economy where the instruments necessary to engineer the nondistorted steady state are unavailable. For these reasons, we refrain from making the efficient-steady-state assumption and instead work with a model whose steady state is distorted.
Departing from a model whose steady state is Pareto efficient has a number of important ramifications. One is that to obtain a second-order accurate measure of welfare it no longer suffices to approximate the equilibrium of the model up to first order. Instead, we obtain a second-order accurate approximation to welfare by solving the equilibrium of the model up to second order. Specifically, we use the methodology and computer code developed in Schmitt-Grohé and Uribe (2004a).
Our numerical work suggests that in the model economy we study, the optimal operational interest-rate rule takes the form of a real-interest-rate targeting rule. For it features an inflation coefficient close to unity, a mute response to output, no interest-rate smoothing, and is forward looking. The optimal rule satisfies the Taylor principle because the inflation coefficient is greater than unity albeit very close to 1.
Optimal operational monetary policy calls for significant inflation volatility. This result stands in contrast to those obtained in the related literature.
The main element of the model driving the desirability of inflation volatility is indexation of nominal factor and product prices to 1-period lagged inflation. Under the alternative assumption of indexation to long-run inflation, the conventional result of the optimality of inflation stability reemerges.
There remain many challenging unanswered questions in this research program. One is to investigate the sensitivity of the parameters of the optimal operational policy rule to changes in the sources of uncertainty driving business cycles. This question is of importance in light of the ongoing quest in business-cycle research to identify the salient sources of aggregate fluctuations. One alternative would be to incorporate the rich set of shocks identified in econometric estimations of the model considered here (e.g., Smets and Wouters, 2004).
The class of operational rules discussed here is clearly not exhaustive. It would be of interest to investigate whether the inclusion of macroeconomic indicators other than those considered here would improve the policymaker's ability to stabilize the economy.
In particular, the related literature has emphasized the use of measures of the output gap that are different from that used by us. Additionally, it has been argued that in models with nominal wage and price rigidities the optimal policy should target an average of wage and price inflation as opposed to only price inflation, which is the case we analyze.
The optimal policy problem we analyze takes the central bank's inflation target as exogenously given. A natural extension is to endogenize this variable. However, in our theoretical framework, the optimal inflation target is the one associated with the Friedman rule. This is because the assumption of full indexation to past inflation implies the absence of inefficient price and wage dispersion in the long run.
Thus the only remaining nominal frictions are the demand for money by households and firms. These frictions call for driving the opportunity cost of holding money to zero in the long run. In other words, the zero bound on nominal interest rate binds in the non-stochastic steady state. The perturbation technique we employ is ill suited to handle this case. Therefore, analyzing the case of an endogenous inflation target entails either changing the model so that the Friedman rule is no longer optimal in the long-run or adopting alternative numerical techniques for computing welfare accurately up to second-order or higher.
One of our findings is that the initial state of the economy plays a role in determining the parameters defining the optimal interest-rate rule. This finding suggests that the optimal operational rule identified here is time inconsistent. In Schmitt-Grohé and Uribe (2004b), we assume that the government is able to commit to the policy announcements made at time 0. It would be of interest to characterize optimal operational rules in an environment without commitment.
Finally, we limit attention to the special case of passive fiscal policy, taking the form of a balanced-budget rule with lump-sum taxation.
It is well known that the set of operational monetary rules depends on the stance of fiscal policy. For instance, the determinacy properties of the rational expectations equilibrium associated with a particular monetary rule can change as fiscal policy is altered.
Therefore, it would be of interest to introduce operational fiscal rules as an additional policy instrument.
Altig, David, Lawrence J. Christiano, Martin Eichenbaum, and Jesper Lindé (2003): Technology Shocks and Aggregate Fluctuations
manuscript, Northwestern University.
Christiano, Lawrence J., Martin Eichenbaum, and Charles Evans (2003): Nominal
Rigidities and the Dynamic Effects of a Shock to Monetary Policy. Northwestern University.
Kim, Jinill, and Sunghyun Henry Kim (2003):
Spurious Welfare Reversals in International
Business Cycle Models,
Journal of International Economics
vol. 60, pages 471-500.
Kim, Jinill, Sunghyun Henry Kim, Ernst Schaumburg, and Christopher Sims (2003):
Calculating and Using Second Order Accurate Solutions of Discrete Time Dynamic Equilibrium Models, Finance and Economics Discussion Series 2003-61, Board of Governors of the Federal Reserve System.
Kimball, Miles S. (1995):
The Quantitative Analytics of the Basic Neomonetarist Model,
Journal of Money, Credit and Banking, vol. 27, pages
Kollmann, Robert (2003):
Welfare Maximizing Fiscal and Monetary Policy Rules
mimeo, University of Bonn.
Rotemberg, Julio J., and Michael Woodford (1999):
Interest Rate Rules in an Estimated Sticky Price Model, in:
John B. Taylor, ed.,
Monetary policy rules
NBER Conference Report series. Chicago and London: University of Chicago Press, pages 57-119.
Schmitt-Grohé, Stephanie and Martín Uribe (2004a):
Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function, Journal of Economic Dynamics and Control, vol. 28, pages 755-775.
Schmitt-Grohé, Stephanie, and Martín Uribe (2004b):
Optimal Operational Monetary Policy in the Christiano-Eichenbaum-Evans Model of the U.S. Business Cycle, NBER working paper 10724.
Schmitt-Grohé, Stephanie and Martín Uribe (2004c):
Optimal Simple And Implementable Monetary and Fiscal Rules,
NBER working paper 10253.
Smets, Frank and Raf Wouters (2004):
Comparing shocks and frictions in US and Euro area business cycles: a Bayesian DSGE approach,
Working paper 61, Nationale Bank van Belgie.
Taylor, John B. (1993):
Discretion versus Policy Rules in Practice
Carnegie Rochester Conference Series on Public Policy, vol. 39,
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