IDEAS home Printed from
MyIDEAS: Login to save this paper or follow this series

Linear-Quadratic Approximation of Optimal Policy Problems

  • Benigno, Pierpaolo
  • Woodford, Michael

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 forward-looking 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.

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.

File URL:
Download Restriction: CEPR Discussion Papers are free to download for our researchers, subscribers and members. If you fall into one of these categories but have trouble downloading our papers, please contact us at

As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

Paper provided by C.E.P.R. Discussion Papers in its series CEPR Discussion Papers with number 5964.

in new window

Date of creation: Dec 2006
Date of revision:
Handle: RePEc:cpr:ceprdp:5964
Contact details of provider: Postal: Centre for Economic Policy Research, 77 Bastwick Street, London EC1V 3PZ.
Phone: 44 - 20 - 7183 8801
Fax: 44 - 20 - 7183 8820

Order Information: Email:

References listed on IDEAS
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:

as in new window
  1. Pierpaolo Benigno & Michael Woodford, 2004. "Optimal monetary and fiscal policy: a linear-quadratic approach," International Finance Discussion Papers 806, Board of Governors of the Federal Reserve System (U.S.).
  2. Jinill Kim & Sunghyun Henry Kim, 1999. "Spurious Welfare Reversals in International Business Cycle Models," Virginia Economics Online Papers 319, University of Virginia, Department of Economics.
  3. Robert King & Alexander L. Wolman, 1999. "What Should the Monetary Authority Do When Prices Are Sticky?," NBER Chapters, in: Monetary Policy Rules, pages 349-404 National Bureau of Economic Research, Inc.
  4. Marc P. Giannoni & Michael Woodford, 2003. "Optimal Interest-Rate Rules: I. General Theory," NBER Working Papers 9419, National Bureau of Economic Research, Inc.
  5. Andrea Ferrero, 2005. "Fiscal and Monetary Rules for a Currency Union," Macroeconomics 0508020, EconWPA.
  6. Stephanie Schmitt-Grohe & Martin Uribe, 2002. "Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function," NBER Technical Working Papers 0282, National Bureau of Economic Research, Inc.
  7. Pierpaolo Benigno & Michael Woodford, 2004. "Optimal Stabilization Policy When Wages and Prices are Sticky: The Case of a Distorted Steady State," NBER Working Papers 10839, National Bureau of Economic Research, Inc.
  8. Andrew Blake, 2001. "A Timeless Perspective on Optimality in Forward-Looking Rational Expectations Models," NIESR Discussion Papers 188, National Institute of Economic and Social Research.
  9. Christopher J. Erceg & Dale W. Henderson & Andrew T. Levin, 1999. "Optimal monetary policy with staggered wage and price contracts," International Finance Discussion Papers 640, Board of Governors of the Federal Reserve System (U.S.).
  10. Aubhik Khan & Robert King & Alexander L. Wolman, 2002. "Optimal monetary policy," Working Papers 02-19, Federal Reserve Bank of Philadelphia.
  11. Pierpaolo Benigno & Michael Woodford, 2005. "Inflation Stabilization And Welfare: The Case Of A Distorted Steady State," Journal of the European Economic Association, MIT Press, vol. 3(6), pages 1185-1236, December.
  12. Richard Clarida & Jordi Gali & Mark Gertler, 1999. "The Science of Monetary Policy: A New Keynesian Perspective," NBER Working Papers 7147, National Bureau of Economic Research, Inc.
  13. Benigno, Gianluca & Benigno, Pierpaolo, 2006. "Designing targeting rules for international monetary policy cooperation," Journal of Monetary Economics, Elsevier, vol. 53(3), pages 473-506, April.
  14. Schmitt-Grohe, Stephanie & Uribe, Martin, 2004. "Optimal fiscal and monetary policy under sticky prices," Journal of Economic Theory, Elsevier, vol. 114(2), pages 198-230, February.
  15. Söderlind, Paul, 1998. "Solution and Estimation of RE Macromodels with Optimal Policy," SSE/EFI Working Paper Series in Economics and Finance 256, Stockholm School of Economics.
  16. Jinill Kim & Sunghyun Henry Kim, 2007. "Two pitfalls of linearization methods," Finance and Economics Discussion Series 2007-64, Board of Governors of the Federal Reserve System (U.S.).
  17. Schmitt-Grohé, Stephanie & Uribe, Martín, 2004. "Optimal Simple and Implementable Monetary and Fiscal Rules," CEPR Discussion Papers 4334, C.E.P.R. Discussion Papers.
  18. Lars Peter Hansen & Thomas J. Sargent, 1990. "Recursive Linear Models of Dynamic Economies," NBER Working Papers 3479, National Bureau of Economic Research, Inc.
  19. Bianca De Paoli, 2004. "Monetary policy and welfare in a small open economy," LSE Research Online Documents on Economics 19950, London School of Economics and Political Science, LSE Library.
  20. Christian Jensen & Bennett C. McCallum, 2002. "The Non-Optimality of Proposed Monetary Policy Rules Under Timeless-Perspective Commitment," NBER Working Papers 8882, National Bureau of Economic Research, Inc.
  21. Christopher A. Sims & Jinill Kim & Sunghyun Kim, 2003. "Calculating and Using Second Order Accurate Solution of Discrete Time Dynamic Equilibrium Models," Computing in Economics and Finance 2003 162, Society for Computational Economics.
  22. Kendrick, David A., 2005. "Stochastic control for economic models: past, present and the paths ahead," Journal of Economic Dynamics and Control, Elsevier, vol. 29(1-2), pages 3-30, January.
  23. Magill, Michael J. P., 1977. "A local analysis of N-sector capital accumulation under uncertainty," Journal of Economic Theory, Elsevier, vol. 15(1), pages 211-219, June.
  24. Jinill Kim & Sunghyun Kim & Ernst Schaumburg & Christopher A. Sims, 2003. "Calculating and Using Second Order Accurate Solutions of Discrete Time," Levine's Bibliography 666156000000000284, UCLA Department of Economics.
  25. LeRoy, Stephen F & Waud, Roger N, 1977. "Applications of the Kalman Filter in Short-Run Monetary Control," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 18(1), pages 195-207, February.
  26. Michael Woodford, 2001. "Inflation Stabilization and Welfare," NBER Working Papers 8071, National Bureau of Economic Research, Inc.
  27. Alan Sutherland, 2002. "A Simple Second-Order Solution Method for Dynamic General Equilibrium Models," Discussion Paper Series, Department of Economics 200211, Department of Economics, University of St. Andrews.
  28. Paul Levine & Joseph Pearlman & Richard Pierse, 2006. "Linear-Quadratic Approximation, Efficiency and Target-Implementability," Computing in Economics and Finance 2006 441, Society for Computational Economics.
  29. Backus, David & Driffill, John, 1986. "The Consistency of Optimal Policy in Stochastic Rational Expectations Models," CEPR Discussion Papers 124, C.E.P.R. Discussion Papers.
  30. Barro, Robert J, 1979. "On the Determination of the Public Debt," Journal of Political Economy, University of Chicago Press, vol. 87(5), pages 940-71, October.
  31. Benigno, Pierpaolo & Woodford, Michael, 2006. "Optimal taxation in an RBC model: A linear-quadratic approach," Journal of Economic Dynamics and Control, Elsevier, vol. 30(9-10), pages 1445-1489.
  32. John H. Kalchbrenner & Peter A. Tinsley, 1976. "On the use of optimal control in the design of monetary policy," Special Studies Papers 76, Board of Governors of the Federal Reserve System (U.S.).
  33. Hans M. Amman & David A. Kendrick, 1996. "The DUALI/DUALPC Software for Optimal Control Models: Introduction," CARE Working Papers 9602, The University of Texas at Austin, Center for Applied Research in Economics.
  34. Kydland, Finn E & Prescott, Edward C, 1982. "Time to Build and Aggregate Fluctuations," Econometrica, Econometric Society, vol. 50(6), pages 1345-70, November.
Full references (including those not matched with items on IDEAS)

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

When requesting a correction, please mention this item's handle: RePEc:cpr:ceprdp:5964. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: ()

The email address of this maintainer does not seem to be valid anymore. Please ask to update the entry or send us the correct address

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

If references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

This information is provided to you by IDEAS at the Research Division of the Federal Reserve Bank of St. Louis using RePEc data.