Note that Mid-Term Exams are graded and available in Room 6, 37 Hillhouse. Maximum score was 50, median 28. Scores 40 and above are good, below 30 problematic, and below 20 very worrisome. There should be a plotted cdf for the grades posted on the door to room 5 at 37 HH.

Answers to Final Exam  Final Exam

Answer to One Last Exercise (MS Word Version)

One last Exercise. (MS Word Version.) .

Dynamic Programming Exercise Answers. (MS Word Version) (The MS Word file is zipped. If you need the MS Word version and have difficulty with zipped files, let me know by email.) Answer is now complete. (4/22, 5PM)

Two-thirds term exam answers (MS Word Version)

Dynamic Programming Notes (These are in two pieces, dynprog and dynprog2. The link takes you to the course directory, where you pick out the files for yourself.)

Corrected Permanent Income Exercise (4/9/97 2PM) The corrections replace the exponential discounting terms with discrete ones and fix the equation of evolution for Y in problem 2 so that Y is lagged on the right-hand side. (.pdf version)

Change in Sticky Price Exercise: The 4th problem has no solution as stated. It should be modified to drop the L=1 assumption and add a term proportional to log (1-L) to the individual's single-period utility function. This makes labor supply elastic.

Answers to Sticky Price Exercise Complete answer, 3/30 1:45PM MS Word Version

Note that some errors on the answer to 4 in the 3/27 version have been fixed in this version.

First Exercise  due Wednesday, 1/29/97 (MS Word Version)

Second Exercise due Wednesday, 2/13/97 (MS Word Version)

Notes on Price Level Determination and Answers to Second Exercise(MS Word Version)

Comments and hints on the first exercise.
a) As noted in class, because the initial B is zero, you can't loglinearize the third problem's model. Linearize in levels instead.

b) In problems 1 and 3, there is not a unique steady state. If you set derivatives to zero, you end up with a singular equation system. There are multiple steady states, in fact infinitely many, indexed by the initial level of wealth (W in problem 1, B in problem 3). Linearization is justified in such a framework only by the assumption that the path is only slightly perturbed from some steady state. Thus it is justifiable to fix W and B at their initial values, then solve the singular system defining a steady state for the other variables. This gives you a complete set of steady state values to use in forming the linearization.

c) In problem 2, it is not made explicit that you should use the same value of gamma as was specified in problem 1.

NTBHO=Notes to be handed out.

1. Methods
• Readings: NTBHO
• *Stokey, Nancy and R.E. Lucas, Recursive Methods in Economic Dynamics, Harvard University Press, 1990, Chapter 9.
• Sargent, Thomas J. Dynamic Macroeconomic Theory, Harvard University Press 1987, Chapter 1.
• Gregory Chow, Analysis and Control of Dynamic Economic Systems, Wiley 1975, section on this topic.
• (Chow has a new book that may replace the reference above.)
• Hansen and Sargent, "Linear Rational Expectations Models for Interrelated Variables" in Rational Expectations and Econometric Practice, Robert E. Lucas and Thomas J. Sargent, editors, University of Minnesota Press 1981 (and also available as a journal article.)
• Blanchard, Olivier and Charles M. Kahn, "The Solution of Linear Difference Models under Rational Expectations," Econometrica 48, 1305-1313; NTBHO;
• Solving Linear Rational Expectations Models 
2. Dynamic Competitive Equilibrium. Consumption and Investment with Time-Varying Interest Rates.
• Readings: NTBHO
• (Much of the reading in the section "Dynamic Stochastic General Equilibrium" below is also relevant, but it is all based on discrete time, stochastic models, whereas our initial approach in class has used continuous time, non-stochastic models.)
3. Price Determination
• Readings: NTBHO
• Aiyagari, Rao and Mark Gertler, "The Backing of Government Bonds and Monetarism", Journal of Monetary Economics, v. 16, 1985.
• Blanchard and Fischer, Chapter 5.
• Leeper, E.M. [1991]. "Equilibria under `Active' and `Passive' Monetary and Fiscal Policies," Journal of Monetary Economics 27, 129-147.
• Lucas, R.E. and Nancy Stokey, "Money and Interest in a Cash-in-Advance Economy," Econometrica May 1987.
• Sargent, T.J. and N. Wallace, "Unpleasant Monetarist Arithmetic," Minneapolis Federal Reserve Bank Quarterly Review, Fall 1981.
• Sims, C.A., "A Simple Model for the Study of the Determination of the Price Level and the Interaction of Monetary and Fiscal Policy", Economic Theory, 1994.
• Woodford, Michael, "Monetary Policy and Price Level Indeterminacy in a Cash-in-Advance Economy," Economic Theory, 1994.
• Woodford, Michael, "Control of the Public Debt: A Requirement for Price Stability?" Unpublished, 1996.
• Bergin, Paul, "Fiscal Restrictions in a Currency Union: Further Lessons on the Interactions of Monetary and Fiscal Policy", unpublished, 1995.
4. Equilibrium with Non-Neutralities
• Blanchard and Fischer, Chapters 8 and 9.
1. Barriers to Information Flow: Natural Rate Models
• Lucas, Robert E., Jr. "Some International Evidence on Output--Inflation Tradeoffs," in his book Studies in Business Cycle Theory, MIT Press 1981, or American Economic Review 63 (June 1973), 326-334.
• Lucas, Robert E., Jr. "Expectations and the Neutrality of Money," Journal of Economic Theory 4 (April 1972) 103-124.
2. Wages and Price Stickiness
1. Overlapping Contracts
• Calvo
• Taylor
2. Menu Costs with Monopolistic Competition
• Blanchard and Fischer, Chapter 8.
• Blanchard-Kiyotaki.
• Caplin, Andrew S. and Daniel F. Spulber, "Menu Costs and the Neutra-lity of Money," Quarterly Journal of Economics 102 (November 1987), 703-725. Reprinted in Mankiw and Romer, New Keynesian Economics, volume 1.
• Rotemberg (82 RES? 83 AER? 87NBER?)
3. Search
• Pissarides, C. [1985], "Short Run Equilibrium Dynamics of Unemployment, Vacancies and Real Wages," AER 75, 676-690.
4. Efficiency Wages
5. Credit Channel Models
5. Life Cycle Consumption
• Readings: Hall, R. E., "Stochastic Implications of the Life Cycle-Permanent Income Hypothesis: Theory and Evidence," Journal of Political Economy, December 1978.
6. Dynamic Stochastic General Equilibrium
1. Complete markets vs. bond-only equilibrium.  Co-ordinating a firm with a representative agent; two representative agents. What is lost in linearization.
• Readings: Campbell, John Y., "Inspecting the Mechanism: An Analytical Approach to the Stochastic Growth Model", NBER discussion paper No. 4188, 1992.
• Kydland, F. and E. Prescott, "Time to Build and Aggregate Fluctuations," Econometrica, November 1982.