Interest rate rules and equilibrium (in)determinacy in a small open economy: the role of internationally traded capital

Wen-ya Chang; Hsueh-fang Tsai; Juin-jen Chang; Hsieh-yu Lin

doi:10.1515/bejm-2016-0128

Article

Interest rate rules and equilibrium (in)determinacy in a small open economy: the role of internationally traded capital

Wen-ya Chang , Hsueh-fang Tsai , Juin-jen Chang and Hsieh-yu Lin

Published/Copyright: March 21, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal The B.E. Journal of Macroeconomics Volume 18 Issue 2

Abstract

This study develops a small-open-economy version of Benhabib, J., S. Schmitt-Grohé, and M. Uribe. 2001. “Monetary Policy and Multiple Equilibria.” American Economic Review 91: 167–186. We systematically explore the role of international capital mobility and the portfolio balance channel in terms of macroeconomic (in)stability when the government follows a commonly-adopted interest-rate feedback rule. In a one-traded-good model, the steady-state equilibrium, in general, is locally determinate; international capital mobility stabilizes the economy against business cycle fluctuations under a simple interest-rate feedback rule. In a two-good (traded and non-traded goods) model, the relationship between equilibrium (in)determinacy and the aggressiveness of interest rate rules is not monotonic, and crucially depends on households’ portfolio preferences. These results suggest that a unified interest rate rule can end up with very different consequences of macroeconomic (in)stability in an open economy from those in a closed economy.

Keywords: indeterminacy; interest rate rules; international capital mobility; portfolio preference

JEL Classification: F41; E52; E31

Acknowledgements

We thank two anonymous referees as well as Been-Lon Chen, Jang-Ting Guo, Ching-Chong Lai, and Ping Wang for their helpful suggestions and insightful comments on an earlier version of this paper, whose inputs have led to a much improved paper. The usual disclaimer applies. Financial support from the National Science Council is gratefully acknowledged.

Appendix

Appendix A

The proof of Propositions 1 and 2

The exact derivatives of (13) are given by:

cλ=1Δ{λt(1−ψ′)(umm+v11−ψ(πt)ucm)+λtψ′[v11−v12+ucm(Rtf+πt−ψ(πt))]},ca=1Δ(−λtucm)(v11−ψ′v12),cbf=1Δ(−λtucm)(v12−v11+ψ′v21−ψ′v22−λtψ′R′f),mλ=1Δ{λt(1−ψ′)(uccψ(πt)−ucm)−λtψ′ucc[Rtf+πt−ψ(πt)]},ma=1Δλtucc(v11−ψ′v12),mbf=1Δλtucc(v12−v11+ψ′v21−ψ′v22−λtψ′Rf′),πλ=−1Δ{(v11−v12)(uccψ(πt)−ucm)+(Rtf+πt−ψ(πt))[ucc(umm+v11)−ucm2]},πa=1Δ(v11−v12)(uccumm−ucm2),πbf=1Δ{(v21−v22−v11+v12−λtRf′)[ucc(umm+v11)−ucm2]+ucc(v11−v12)2},

where Δ=λt(1−ψ′)[ucc(umm+v11)−ucm2]+λtψ′(v11−v12)ucc. Apparently, this dynamic system should be complicated. Thus, we impose Assumption 2 in order to make our model analytically tractable.

Under Assumption 1 and Assumption 2, we can obtain the following Jacobian matrix of this dynamic system from (14)–(16):

(50)[λ˙tb˙tfa˙t]=[λ^(1−ψ′)πλ+ρ+π^−R^λ^(1−ψ′)πbf0−cλ−cbf+R^f(1−η)0ΦλΦbfΦa][λt−λ^btf−b^fat−a^],

where Φλ=−cλ+πλ[(ψ′−1)(a^−b^f)−ψ′m^]−R^mλ,Φbf=−cbf+R^f(1−η)+πbf[(ψ′−1)(a^−b^f)−ψ′m^]−R^mbf+π^−R^, and Φa=(R^−π^).

Let μ₁, μ₂, and μ₃ be the eigenvalues of the dynamic system. From (50), it is easy to obtain:

(51)μ1+μ2=v2λ^+R^f(1−η)−cbf,

(52)μ1μ2=v2λ^[R^f(1−η)−cbf]+cλλ^(1−ψ′)πbf=ξ−v22+λ^R′fuccumm−ucm2(umm−ψ(π^)ucm)⏟+,

(53)μ3=R^−π^>0,

where ξ≡v2λ^R^f(1−η)−v1λ^cbf. As noted in the context, the steady-state equilibrium is locally determinate when there are two roots with positive real parts and one root with a negative real part. The equilibrium is locally indeterminate when the dynamic system has more than two roots with negative real parts. It is a source when there are three roots with positive real parts.

By focusing on Proposition 1, we first assume that v₁ = v₂ = 0, which rules out the portfolio balance channel. Under such a situation, ξ = 0 is true and hence μ₁μ₂ < 0. Thus, there are two roots with positive real parts and one root with a negative real part in the dynamic system. The equilibrium is then characterized by local determinacy.

We now turn to the analysis in which households exhibit a portfolio preference between domestic and foreign bonds. To focus on more meaningful cases, we abstract the source (three roots with positive real parts) from the analysis. Accordingly, it is easy to infer that if μ₁ + μ₂ > 0, we can rule out the possibility of indeterminacy, given that μ₃ > 0. In what follows, we will prove that if μ₁ + μ₂ < 0, equilibrium determinacy can also be true. It is clear from (51) that R^f(1−η)<cbf must hold under the case with μ₁ + μ₂ < 0. There are two scenarios: v₁ > v₂ and v₂ > v₁. In the scenario where v₁ > v₂, the condition R^f(1−η)<cbf guarantees that ξ < 0 is true. From (52), we can refer to μ₁μ₂ < 0, given that ξ < 0. Accordingly, we have μ₁ + μ₂ < 0 and μ₁μ₂ < 0, implying that there is one negative eigenvalue and two positive eigenvalues in the dynamic system. Therefore, the equilibrium is locally determinate. In the scenario where v₂ > ν₁, we have R^−π^>R^f, since the condition ρ=v1λ^+ψ(π^)−π^=v2λ^+R^f is true in the steady state. Thus, we can further infer that under the condition ρ2>R^−π^ the following inequality holds:

v1λ^>R^−π^>R^f>R^f(1−η).

Since v2λ^+R^f(1−η)<cbf is true in the case where μ₁ + μ₂ < 0, we can further obtain:

ξ=v2λ^R^f(1−η)−v1λ^cbf<v2λ^R^f(1−η)−v1λ^[v2λ^+R^f(1−η)]<0,

implying that μ₁μ₂ < 0. Therefore, the dynamic system has two roots with positive real parts and one root with a negative real part. The equilibrium is also locally determinate in the scenario where v₂ > v₁.

Appendix B

The characterization of the two-good model with flexible prices

First, we define:

ct=h(cT,t,cH,t)=cT,tαcH,t1−α,and h1=αctcT,t>0, h2=(1−α)ctcH,t>0, h12=α(1−α)ctcT,tcH,t>0, h11=−α(1−α)ctcT,t2<0, and h22=−α(1−α)ctcH,t2<0.

Thus, by substituting (23) into (20)–(22), we can obtain the instantaneous relationships of the traded good consumption, real money balances, and inflation as follows:

cT,t=cT(et),mt=m(λt,et),πt=π(λt,et),

where

cT,e=h22h11h2−h21h1, mλ=1h2ucm, me=−(ucch1h2+u2h21)cT,eh2ucm, πe=cT,eλtψ′h2ucm[h1h2(ucm2−uccumm)−ummuch21], and πλ=1λtπ′(ummh2ucm−ψ).

Given that v(bt,etbtf)=bt+A(etbtf)1−ε1−ε, (25)–(27) can be rewritten as:

λ˙t=λt[ρ+π(λt,et)−ψ(π(λt,et))]−1,b˙tf=[R¯f+Rf(btf)]btf−cT(et),e˙t=−et{A(e^b^f)−ε−1λt+[R¯f+Rf(btf)]+π(λt,et)−ψ(π(λt,et))},

which implies that the dynamic system can be reduced to a 3×3 one in terms of λ_t, btf, and e_t. Therefore, the Jacobian matrix of this dynamic system is given by:

[λ˙tb˙tfe˙t]=[λ^(1−ψ′)πλ0λ^(1−ψ′)πe0R^f(1−η)−cT,ee^[A(e^b^f)−ε−1λ^2−(1−ψ′)πλ]−e^[−e^Aε(e^b^f)−ε−1λ^+R′f]−e^[−b^fAε(e^b^f)−ε−1λ^+(1−ψ′)πe]][λt−λ^btf−b^fet−e^]

Appendix C

The characterization of the two-good model with price stickiness

Given that v(bt,etbtf)=bt+A(etbtf)1−ε1−ε, we can use (33)–(35) and (40) to obtain:

cT,t=cT(λt,et,πt),cH,t=cH(λt,et,πt),mt=m(λt,et,πt), and mp,t=mp(λt,et,πt),where cT,λ=1∇(etumm−ψ(πt)ucmh1)(h12−eth22),cT,e=−h2∇[h1h2(uccumm−ucm2)+uceth22umm],cT,π=−λtψ′∇ucmh1(h12−eth22),cH,λ=1∇(h11−eth21)(ψ(πt)ucmh1−etumm),cH,e=h2∇[h12(uccumm−ucm2)+uceth21umm],cH,π=λtψ′∇(h11−eth21)ucmh1,mλ=1∇(ψ(πt)ucch1−etucm)h2(2h1h2h12−h22h11−h12h22),me=−1∇λtumch1(h12−eth22),mπ=λtψ′∇uccet(2h1h2h12−h22h11−h12h22),mp,λ=cH,λyH′,mp,e=cH,eyH′,mp,π=cH,πyH′,and∇=et(uccumm−ucm2)(2h1h2h12−h22h11−h12h22).

By substituting (45) into (46)–(49), we thus have a 4×4 dynamic system in terms of λ_t, e_t, π_t, and btf and, accordingly, the corresponding Jacobian matrix is given by:

[λ˙te˙tπ˙tb˙tf]=[1λ^0λ^(1−ψ′)0e^[1−A(e^b^f)−ελ^2)e^[−εA(e^b^f)−ε−1]b^fλ^−e^(1−ψ′)e^[−εA(e^b^f)−ε−1λ^e^−R′f]−Ωλ−Ωeρ−Ωπ0−cT,λ−cT,e−cT,πR^f(1−η)][λt−λ^et−e^πt−π^btf−b^f]

where Ωλ=[d′(1)y^H/ς][1/(λ^y^H′)+(y^H″/y^H′2)mp,λ(1+λ^ψ(π^))],Ωe=[d′(1)y^H/ς](y^H″/y^H′2)mp,e(1+λ^ψ(π^)), and Ωπ=[d′(1)y^H/ς][−λ^ψ′/y^H′+(y^H″/y^H′2)mp,π(1+λ^ψ(π^))].

References

Airaudo, M., and L. Zanna. 2012. “Interest Rate Rules, Endogenous Cycles and Chaotic Dynamics in Open Economies.” Journal of Economic Dynamics and Control 36: 1566–1584.10.1016/j.jedc.2012.06.003Search in Google Scholar

Alpanda, S., and S. Kabaca. 2015. International Spillovers of Large-Scale Asset Purchases. Bank of Canada.Search in Google Scholar

Bardhan, P. K. 1967. “Optimum Foreign Borrowing.” In Essays on the Theory of Optimal Growth, edited by K. Shell, 117–128. Cambridge, MA: MIT Press.Search in Google Scholar

Benes, J., A. Berg, R. A. Portillo, and D. Vavra. 2015. “Modeling Sterilized Interventions and Balance Sheet Effects of Monetary Policy in a New-Keynesian Framework.” Open Economies Review 26: 81–108.10.1007/s11079-014-9320-1Search in Google Scholar

Benhabib, J., S. Schmitt-Grohé, and M. Uribe. 2001. “Monetary Policy and Multiple Equilibria.” American Economic Review 91: 167–186.10.1257/aer.91.1.167Search in Google Scholar

Benhabib, J., S. Schmidt-Grohé, and M. Uribe. 2003. “Backward-Looking Interest Rate Rules, Interest-Rate Smoothing, and Macroeconomic Instability.” Journal of Money, Credit, and Banking 35: 1379–1412.10.1353/mcb.2004.0020Search in Google Scholar

Bernanke, B., and M. Woodford. 1997. “Inflation Forecasts and Monetary Policy.” Journal of Money, Credit, and Banking 29: 653–684.10.2307/2953656Search in Google Scholar

Blanchard, O., F. Giavazzi, and F. Sa. 2005. The US Current Account and the Dollar. No. w11137. National Bureau of Economic Research.10.3386/w11137Search in Google Scholar

Calvo, G. A. 1980. Financial Opening, Crawling Peg and the Real Exchange Rate. Unpublished Manuscript, Centro de Estudios Macroeconomicos de Argentina, Buenos Aires.Search in Google Scholar

Calvo, G. A. 1983. “Staggered Prices in a Utility-Maximizing Framework.” Journal of Monetary Economics 12: 383–398.10.1016/0304-3932(83)90060-0Search in Google Scholar

Canzoneri, M. B., and B. T. Diba. 2005. “Interest Rate Rules and Price Determinacy: The Role of Transactions Services of Bonds.” Journal of Monetary Economics 52: 329–343.10.1016/j.jmoneco.2004.05.009Search in Google Scholar

Carlstrom, C. T., and T. S. Fuerst. 2001. “Timing and Real Indeterminacy in Monetary Models.” Journal of Monetary Economics 47: 285–298.10.1016/S0304-3932(01)00048-4Search in Google Scholar

Carlstrom, C. T., and T. S. Fuerst. 2002a. “Taylor Rules in a Model that Satisfies the Natural-Rate Hypothesis.” American Economic Review 92: 79–84.10.1257/000282802320189041Search in Google Scholar

Carlstrom, C. T., and T. S. Fuerst. 2002b. “Optimal Monetary Policy in a Small Open Economy: A General-Equilibrium Analysis.” In Monetary Policy: Rules and Transmission Mechanisms, edited by N. Loayza and K. Schmidt-Hebbel, 275–298. Santiago, Chile: Central Bank of Chile.Search in Google Scholar

Chen, S. 2000. “Endogenous Real Exchange Rate Fluctuations in an Optimizing Open Economy Model.” Journal of International Money and Finance 19: 185–205.10.1016/S0261-5606(00)00003-6Search in Google Scholar

Chen, Z. 1992. “Long-Run Equilibria in a Dynamic Heckscher-Ohlin Model.” Canadian Journal of Economics 23: 923–943.10.2307/135772Search in Google Scholar

Clarida, R., J. Gali, and M. Gertler. 2000. “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory.” Quarterly Journal of Economics 115: 147–180.10.1162/003355300554692Search in Google Scholar

De Fiore, F., and Z. Liu. 2005. “Does Trade Openness Matter for Aggregate Instability?” Journal of Economic Dynamics and Control 29: 1165–1192.10.1016/j.jedc.2004.06.001Search in Google Scholar

Dib, Ali, C. Mendicino, and Y. Zhang. 2008. “Price Level Targeting in a Small Open Economy with Financial Frictions: Welfare Analysis.” No. 2008, 40. Bank of Canada Working Paper.Search in Google Scholar

Dornbusch, R. 1983. “Real Interest Rates, Home Goods and Optimal External Borrowings.” Journal of Political Economy 91: 141–153.10.1086/261132Search in Google Scholar

Dupor, B. 2001. “Investment and Interest Rate Policy.” Journal of Economic Theory 98: 85–113.10.1006/jeth.2000.2765Search in Google Scholar

Eicher, T. S., S. F. Schubert, and S. J. Turnovsky. 2008. “Dynamic Effects of Terms of Trade Shocks: The Impact on Debt and Growth.” Journal of International Money and Finance 27: 876–896.10.1016/j.jimonfin.2007.04.015Search in Google Scholar

Fischer, S. 1979. “Capital Accumulation on the Transition Path in a Monetary Optimizing Model.” Econometrica 47: 1433–1439.10.2307/1914010Search in Google Scholar

Fisher, W. H. 1995. “An Optimizing Analysis of the Effects of World Interest Disturbances on the Open Economy Term Structure of Interest Rates.” Journal of International Money and Finance 14: 105–126.10.1016/0261-5606(94)00017-USearch in Google Scholar

Greiner, A., and B. Fincke. 2009. Public Debt and Economic Growth. Berlin: Springer.10.1007/978-3-642-01745-2Search in Google Scholar

Krishnamurthy, A., and A. Vissing-Jorgensen. 2012. “The Aggregate Demand for Treasury Debt.” Journal of Political Economy 120: 233–267.10.1086/666526Search in Google Scholar

Kumhof, M. 2010. “On the Theory of Sterilized Foreign Exchange Intervention.” Journal of Economic Dynamics and Control 34: 1403–1420.10.1016/j.jedc.2010.04.005Search in Google Scholar

Leeper, E. M. 1991. “Equilibria under ‘Active’ and ‘Passive’ Monetary and Fiscal Policies.” Journal of Monetary Economics 27: 129–147.10.1016/0304-3932(91)90007-BSearch in Google Scholar

Linnemann, L., and A. Schabert. 2002. Monetary Policy, Exchange Rates and Real Indeterminacy. University of Cologne, Manuscript.Search in Google Scholar

Linnemann, L., and A. Schabert. 2006. “Monetary Policy and the Taylor Principle in Open Economies.” International Finance 9: 343–367.10.1111/j.1468-2362.2006.00189.xSearch in Google Scholar

Ludvigson, S. C., and A. Michaelides. 2001. “Does Buffer-Stock Saving Explain the Smoothness and Excess Sensitivity of Consumption?” American Economic Review 91: 631–647.10.1257/aer.91.3.631Search in Google Scholar

McKnight, S. 2007. “Real Indeterminacy and the Timing of Money in Open Economies.” Working Papers No. 046, University of Reading.Search in Google Scholar

Obstfeld, M. 1982. “Aggregate Spending and the Terms of Trade: Is There a Laursen-Metzler Effect?” Quarterly Journal of Economics 97: 251–270.10.2307/1880757Search in Google Scholar

Osang, T., and S. J. Turnovsky. 2000. “Differential Tariffs, Growth, and Welfare in a Small Open Economy.” Journal of Development Economics 62: 315–342.10.1016/S0304-3878(00)00087-0Search in Google Scholar

Pitchford, J. D. 1989. “Optimum Borrowing and the Current Account When There are Fluctuations in Income.” Journal of International Economics 26: 345–358.10.1016/0022-1996(89)90008-1Search in Google Scholar

Reis, R. 2007. “The Analytics of Monetary Non-Neutrality in the Sidrauski Model.” Economics Letters 94: 129–135.10.1016/j.econlet.2006.08.017Search in Google Scholar

Rotemberg, J. J. 1982. “Sticky Prices in the United States.” Journal of Political Economy 90: 1187–1211.10.1086/261117Search in Google Scholar

Sargent, T. 1987. Dynamic Macroeconomic Theory. Cambridge, MA: Harvard University Press.Search in Google Scholar

Schmitt-Grohé, S., and M. Uribe. 2000. “Price Level Determinacy and Monetary Policy under a Balanced-Budget Requirement.” Journal of Monetary Economics 45: 211–246.10.1016/S0304-3932(99)00046-XSearch in Google Scholar

Srinivasan, N., V. Mahambare, and M. Ramachandran. 2006. “UK Monetary Policy Forecasting under Inflation Targeting: Is Behaviour Consistent with Symmetric Preferences?” Oxford Economic Papers 58: 706–721.10.1093/oep/gpl009Search in Google Scholar

Taylor, J. B. 1993. “Discretion versus Policy Rules in Practice.” Carnegie-Rochester Conference Series on Public Policy 39: 195–214.10.1016/0167-2231(93)90009-LSearch in Google Scholar

Taylor, J. B. 1999. Monetary Policy Rules. Chicago: University of Chicago Press.10.7208/chicago/9780226791265.001.0001Search in Google Scholar

Turnovsky, S. J. 1997. International Macroeconomic Dynamics. Cambridge, MA: MIT Press.Search in Google Scholar

Woodford, M. 1995. “Price-Level Determinacy without Control of a Monetary Aggregate.” Carnegie-Rochester Conference Series on Public Policy 43: 1–46.10.1016/0167-2231(95)90033-0Search in Google Scholar

Woodford, M. 2003. Interest and Prices. Princeton: Princeton University Press.Search in Google Scholar

Zanna, L. F. 2003. “Interest Rate Rules and Multiple Equilibria in the Small Open Economy.” International Finance Discussion Papers No. 785, Board of Governors of the Federal Reserve System.10.17016/IFDP.2003.785Search in Google Scholar

Zhang, Y. 2013. “Unemployment Fluctuations in a Small Open-Economy Model with Segmented Labour Markets: The Case of Canada.” No. 2013–40. Bank of Canada Working Paper.Search in Google Scholar

Published Online: 2018-03-21

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/bejm-2016-0128

Keywords for this article

indeterminacy; interest rate rules; international capital mobility; portfolio preference