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

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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 und Hsieh-yu Lin

Veröffentlicht/Copyright: 21. März 2018

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Aus der Zeitschrift The B.E. Journal of Macroeconomics Band 18 Heft 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^))].

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Published Online: 2018-03-21

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