Linear-quadratic approximation of optimal policy problems

We consider a general class of nonlinear optimal policy problems involving forward-looking constraints (such as the Euler equations that are typically present as structural equations in DSGE models), and show that it is possible, under regularity conditions that are straightforward to check, to derive a problem with linear constraints and a quadratic objective that approximates the exact problem. The LQ approximate problem is computationally simple to solve, even in the case of moderately large state spaces and flexibly parameterized disturbance processes, and its solution represents a local linear approximation to the optimal policy for the exact model in the case that stochastic disturbances are small enough. We derive the second-order conditions that must be satisfied in order for the LQ problem to have a solution, and show that these are stronger, in general, than those required for LQ problems without forwardlooking constraints. We also show how the same linear approximations to the model structural equations and quadratic approximation to the exact welfare measure can be used to correctly rank alternative simple policy rules, again in the case of small enough shocks. ∗An earlier version of this paper was presented as a Plenary Lecture at the 10th Annual Conference on Computing in Economics and Finance, Amsterdam, July 2004. We would like to thank Filippo Altissimo, Vasco Curdia, Wouter Den Haan, Ken Judd, Jinill Kim, Andy Levin, Paul Levine, Diego Rodriguez Palenzuela, Joseph Pearlman, and Harald Uhlig for comments, and the National Science Foundation for research support through a grant to the NBER. Linear-quadratic (LQ) optimal-control problems have been the subject of an extensive literature. General characterizations of their solutions and useful numerical algorithms to compute them are now available, allowing models with fairly large state spaces, complicated dynamic linkages, and a range of alternative informational assumptions to be handled. And the extension of the classic results of the engineering control literature to the case of forward-looking systems of the kind that naturally arise in economic policy problems when one allows for rational expectations on the part of the private sector has proven to be fairly straightforward. An important question, however, is whether optimal policy problems of economic interest should take this convenient form. It is easy enough to apply LQ methodology if one specifies an ad hoc quadratic loss function on the basis of informal consideration of the kinds of instability in the economy that one would like to reduce, and posits linear structural relations that capture certain features of economic time series without requiring these relations to have explicit choice-theoretic foundations, as in early applications to problems of monetary policy. But it is highly unlikely that the analysis of optimal policy in a DSGE model will involve either an exactly quadratic utility function or exactly linear constraints. We shall nonetheless argue that LQ problems can usefully be employed as approximations to exact optimal policy problems in a fairly broad range of cases. Since an LQ problem necessarily leads to an optimal decision rule that is linear, the most that one could hope to obtain with any generality would be for the solution to the LQ problem to represent a local linear approximation to the actual optimal policy — that is, a first-order Taylor approximation to the true, nonlinear optimal policy rule. In this paper we present conditions under which this will be the case, and show how to derive an LQ approximate problem corresponding to any member of a general class of optimal policy problems. The conditions under which the solution to an LQ approximate problem will yield a correct local linear approximation to optimal policy are in fact more restrictive Important references include Bertsekas (1976), Chow (1975), Hansen and Sargent (2004), Kendrick (1981), Kwakernaak and Sivan (1972), and Sargent (1987). See Kendrick (2005) for an overview of the use of LQ methods in economics. For numerical algorithms see, among others, Amman (1996), Anderson et al. (1996), Amman and Kendrick (1999), Diaz-Gimenez (1999), Gerali and Lippi (2005), Hansen and Sargent (2004), and Söderlind (1999). See, e.g., Backus and Driffill (1986) for a useful review. Notable examples include Kalchbrenner and Tinsley (1975) and Leroy and Waud (1977).