This appendix presents the proofs of the Propositions in the paper and further elaborates and shows the results of some extensions of the model, including contemporaneous rules, backward-looking rules, nominal prince rigidities and money in the production function set-ups.
Alstadheim, R. and D. Henderson, 2006, “Price-Level Determinacy, Lower Bounds on the Nominal Interest Rate, and Liquidity Traps,” The B.E. Journal of Macroeconomics, Vol. 6, pp. 1437–1437.
Bask, M., 2002, “A Positive Lyapunov Exponent in Swedish Exchange Rates?,” Chaos, Solitons and Fractals, Vol. 14, pp. 1295–1304.
Benhabib, J., S. Schmitt-Grohé, and M. Uribe, 2001a, “Monetary Policy Rules and Multiple Equilibria,” American Economic Review, Vol. 91, pp. 167–184.
Benhabib, J., S. Schmitt-Grohé, and M. Uribe, 2001b, “The Perils of the Taylor Rules,” Journal of Economic Theory, Vol. 96, pp. 40–69.
Benhabib, J., S. Schmitt-Grohé, and M. Uribe, 2002a, “Chaotic Interest Rate Rules” American Economic Review, Vol. 92, pp. 72–78.
Benhabib, J., S. Schmitt-Grohé, and M. Uribe, 2002b, “Avoiding Liquidity Traps,” Journal of Political Economy, Vol. 110, pp. 535–563.
Benigno, G. and P. Benigno, 2008, “Exchange Rate Determination under Interest Rate Rules,” Journal of International Money and Finance, Vol. 27, pp. 971–993.
Bentolilla, S. and G. Saint-Paul, 2003, “Explaining Movements in the Labor Share,” The B.E. Journal of Macroeconomics, Vol. 3, pp. 1103–1103.
Boivin, J., 2006, “Has U.S. Monetary Policy Changed? Evidence from Drifting Coefficients and Real-Time Data,” Journal of Money, Credit and Banking, Vol. 38, pp. 1149–1174.
Bullard, J. and A. Singh, 2008, “Worldwide macroeconomic stability and monetary policy rules,” Journal of Monetary Economics, Vol. 55, pp. S34–S47.
Bullard, J., 2010, “Seven Faces of ‘The Peril’,” Federal Reserve Bank of St. Louis Review, Vol. 92(5), pp. 339–52 (St. Louis: Federal Reserve Bank).
Burnstein, A., J. Neves, and S. Rebelo, 2003, “Distribution Costs and Real Exchange Rate Dynamics,” Journal of Monetary Economics, Vol. 50, pp. 1189–1214.
Campa, J. M., and L. Goldberg, 2006, “Pass Through of Exchange Rates to Consumption Prices: What has Changed and Why?,” NBER Working Papers No. 12547, (Cambridge, Massachusetts: National Bureau of Economic Research).
Carlstrom, C., and T. Fuerst, 2001, “Timing and Real Indeterminacy in Monetary Models,” Journal of Monetary Economics, Vol. 47, pp. 285–298.
Carlstrom, C.T., T.S. Fuerst, and F. Ghironi, 2006, “Does It Matter (For Equilibrium Determinacy) What Price Index the Central Bank Targets?,” Journal of Economic Theory, Vol. 128, pp. 214–231.
Clarida, J., J. Galí and M. Gertler, 1998, “Monetary Policy Rules in Practice: Some International Evidence,” European Economic Review, Vol. 42, pp. 1033–1067.
Cogley, T. and T. Sargent, 2005, “Drifts and Volatilities: Monetary Policy and Outcomes in the Post WWII US,” Review of Economic Dynamics, Vol. 8, pp. 262–302.
De Fiore, F. and L. Zheng, 2005, “Does Trade Openness Matters for Aggregate Instability?” Journal of Economic Dynamics and Control, Vol. 29, pp. 1165–1192.
Dib, A., 2003, “An Estimated Canadian DSGE Model with Nominal and Real Rigidities,” Canadian Journal of Economics, Vol. 36(4), pp. 949–972.
Diks, C., C. Hommes, V. Panchenko, and R. Van der Weide, 2008, “E&F Chaos: A User Friendly Software Package for Non-Linear Economic Dynamics,” Computational Economics, Vol. 32, pp. 221–244.
Eusepi, S., 2005, “Comparing Forecast-Based and Backward-Looking Taylor Rules: a Global Analysis,” Federal Reserve Bank of New York Staff Report, No. 198 (New York City, Federal Reserve Bank).
Eusepi, S., 2007, “Learnability and Monetary Policy: A Global Perspective,” Journal of Monetary Economics, Vol. 54, pp. 1115–1131.
Evans, G., E. Guse, and S. Honkapohja, 2008, “Liquidity Traps, Learning and Stagnation,” European Economic Review, Vol. 52, pp. 1438–1463.
Galf, J., and T. Monacelli, 2005, “Monetary Policy and Exchange Rate Volatility in a Small Open Economy,” Review of Economic Studies, Vol. 72, pp. 707–734.
Gogas, P., and A. Serletis, 2000, “Purchasing Power Parity, Non-linearity and Chaos,” Applied Financial Economics, Vol. 10, pp. 615–622.
Holman, J., 1998, “GMM Estimation of a Money-in-the-Utility-Function Model: The Implications of Functional Forms,” Journal of Money, Credit and Banking, Vol. 30, pp. 679–698.
Llosa, G., and V. Tuesta, 2008, “Determinacy and Learnability of Monetary Policy Rules in Small Open Economies,” Journal of Money, Credit and Banking, Vol. 40, pp. 1033–1063.
Lubik, T. and F. Schorfheide, 2007, “Do Central Banks Target Exchange Rates? A Structural Investigation,” Journal of Monetary Economics, Vol. 54, pp. 1069–1087.
Mendoza, E., 1995, “The Terms of Trade, The Real Exchange rate and Economic Fluctuations,” International Economic Review, Vol. 36, pp. 101–137.
Meng, Q., and A. Velasco, 2003, “Indeterminacy in a Small Open Economy with Endogenous Labor Supply,” Economic Theory, Vol. 22, pp. 661–670.
Monacelli, T., 2005, “Monetary Policy in a Low Pass-Through Environment,” Journal of Money Credit and Banking, Vol. 37, pp. 1047–1066.
Ogaki, M., J.D. Ostry, and C.M. Reinhart, 1996, “Saving Behavior in Low and Middle-Income Developing Countries: A Comparison,” IMF Staff Papers, Vol. 43, pp. 38–71.
Petursson, T., 2004, “The Effects of Inflation Targeting on Macroeconomic Performance” Central Bank of Iceland Working Papers No. 23(June).
Schmitt-Grohé, S. and M, Uribe 2003, “Closing Small Open Economy Models,” Journal of International Economics, Vol. 61, pp. 137–139.
Zanna, L. F., 2003, “Interest Rate Rules and Multiple Equilibria in the Small Open Economy,” International Finance Discussion Papers No. 7895.
The authors would like to thank the following people for comments and suggestions: Gian-Italo Bischi, David Bowman, Chris Erceg, Jon Faust, Laura Gardini, Dale Henderson, two anonymous referees, the associate editor, and the seminar participants at the Federal Reserve Board, LUISS, the University of Urbino, the Society for NonLinear Dynamics and Econometrics and the Latin American Meeting of the Econometric Society. Part of this work was completed while Airaudo was Junior Fellow and then Visiting Fellow at Collegio Carlo Alberto. He is grateful to the Collegio for all the hospitality and research support. The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reflecting those of the International Monetary Fund.
The case of σ = 1 corresponds to the case of separability between consumption and money in the utility function. It can be easily shown that no equilibrium cycles occur in this case.
From the technical point of view, complete markets serve the purpose of ruling out the unit root problem of the small open economy (see Schmitt-Grohé and Uribe, 2003). This allows us to derive meaningful local determinacy of equilibrium results and compare them to the ones derived under our global equilibrium analysis.
This holds by the previous analysis since
To see why, consider the case of σ < 1 (Edgeworth complements) and, without loss of generality, assume that the economy has a fixed endowment of non-traded goods, such that
implying that the marginal utility from traded-goods consumption must be constant. By the definition of ct in (2) and the fact that
By indeterminacy we refer to a situation where one or more real variables are not pinned down by the model. We use the following terms interchangeably: (i) “indeterminacy” and “multiple equlibria”, and (ii) “determinacy” and a “unique equilibrium.”
Benhabib et al. (2002a) use a flexible-price money-in-the-production-function (MIPF) set-up and rely on numerical simulations to show the existence of cycles. They do not have to state explicitly this type of assumptions, since they do not derive analytical results for cycles. But clearly money is “essential”, given that it is used to some degree in productive activities. Similar considerations apply to the part of the paper by Eusepi (2007) that studies global determinacy under a MIPF set-up. Eusepi (2005) also looks at an endowment MIUF set-up. And although he relies on numerical simulations, it is possible to see that in his endowment set-up, Assumption 1 reduces to the inequality 1–γ > 0, which holds for any positive share of money in utility. In fact for θN in (3), our economy would be equivalent to one with a non-traded good endowment and the same inequality 1–γ 0 would hold.
Based on empirical evidence, in Table 2 below, we assign a value of 0.03 to 1–γ and of 0.56 to θN. A simple numerical evaluation shows that the right-hand side of the inequality of Assumption 1 is always smaller than 0.015 for any combination of R* ∈ [1.0025, 1.025] and ξ ∈ [1.01, 10]. This corresponds to an annual interest rate target between 1% and 10% and spans the universe of active rules from weakly to extremely active.
As mentioned before, Benhabib et al. (2002a) and Eusepi (2007) focus on cycles in MIPF models. Eusepi (2005) studies an endowment MIUF model of a closed economy and resorts to numerical simulations to investigate the existence of cycles around the passive steady state.
In fact, the assumption holds for an approximation. Let RHS(R) and LHS(R) be, respectively, the right- and the left-hand sides of (25). Then Assumption 2 written as
A quick inspection of Proposition 2 also suggests that if the economy is very open, an active rule always leads to local equilibrium determinacy.
Benhabib et al. (2001a) show that, in closed economies, a MIUF model with Edgeworth substitutability (Ucm < 0) and a MIPF model are isomorphic for what concerns the local stability properties of Taylor rules. The literature has not proved analytically this isomorphism in the context of global analysis of open economies. In the Appendix we establish this global isomorphism.
We say “may converge” rather than “will converge” because, due to the complicated functional form of Rt+1 = f (Rt) in (27), it is not possible to obtain meaningful analytical conditions for the stability of cycles. However, as the numerical analysis will show, our economy features stable limit cycles of various periodicity.
This is roughly the average of the GMM estimates using U.S. data. Because of the similar degree of financial development, we take this as a reasonable value for Canada.
Figure 3 is constructed using the E&F Chaos software package by Diks, Hommes, Panchenko, and Van der Weide developed at the Center for Non-Linear Dynamics in Economics and Finance (CeNDEF), University of Amsterdam. This software quickly iterates the non-linear equilibrium mapping f and computes the set of limit points. Diks et al. (2008) provide a detailed description of the E&F Chaos software and its functionalities.
By numerical analysis, we have also found that, when α = 0, the range (σF, σD), i.e., (σf, σd) shrinks–and hence openness plays a bigger role to some extent–if we assume a) a lower interest rate target; b) a lower degree of activism towards inflation; c) a larger role for real money balances in preferences; or d) a lower labor share in production. For instance, other things equal, setting
Recall the results in (21): eʹ (Rt) > 0 if σ < 1.
An additional extension is the inclusion of nominal price rigidities (see the Appendix). Similar to Eusepi (2005), we find that under sticky prices, liquidity traps that converge to the “unintended” low steady state are the only type of global PFE dynamics.
This is certainly not the only approach to model imperfect exchange rate pass-through. For other approaches see Monacelli (2005), among others.
Recall that under a forward-looking rule whether cycles/chaos occur around the passive or active steady state depend entirely on σ, while α determines whether they exist or not.
Keeping all the remaining structure unchanged, extensive algebra shows that the PFE dynamics of our economy are described by a non linear difference equation
This functional form is quite common in multi-sector DSGE models. See, for instance, Horvath (2000) and Carlstrom et al. (2006).
Note that under CRS, one would have to make an assumption similar to Assumption 1 of the main text. For the functional form used in this section this would be
As hinted above, in our model, a chaotic nominal interest rate implies chaotic dynamics for other variables including the real exchange rate.
If Assumption 1 did not hold, then αd ≤ 0 and the equilibrium would be locally determinate for any degree of trade openness. However, Assumption 1 holds for any realistic calibration of the model.
Although, the two set-ups are non-nested, one could think of the CWID model as one where Ψ = 1.
It is also possible to find the exact numerical values of the α thresholds triggering a qualitative switch in dynamics. We find that period-2 cycles appear around the active steady state when α ∈ (0.001, 0.22); and period-3 cycles occur around the active steady state when α ∈ (0.001, 0.16). Pushing α up, period-2 cycles appear around the passive steady state for α ε, (0.25, 0.43), whereas period-3 cycles (and therefore chaos) exist for α ∈ (0.33, 0.38). Finally for α > 0.39 only liquidity traps exist.
The derivation of these conditions is available from the authors upon request.