Interest Rate Rules, Endogenous Cycles, and Chaotic Dynamics in Open Economies
  • 1 https://isni.org/isni/0000000404811396, International Monetary Fund

Contributor Notes

We present an extensive analysis of the consequences for global equilibrium determinacy in flexible-price open economies of implementing active interest rate rules, i.e., monetary rules where the nominal interest rate responds more than proportionally to inflation. We show that conditions under which these rules generate aggregate instability by inducing liquidity traps, endogenous cycles, and chaotic dynamics depend on specific characteristics of open economies. In particular, rules that respond to expected future inflation are more prone to induce endogenous cyclical and chaotic dynamics the more open the economy to trade.

Abstract

We present an extensive analysis of the consequences for global equilibrium determinacy in flexible-price open economies of implementing active interest rate rules, i.e., monetary rules where the nominal interest rate responds more than proportionally to inflation. We show that conditions under which these rules generate aggregate instability by inducing liquidity traps, endogenous cycles, and chaotic dynamics depend on specific characteristics of open economies. In particular, rules that respond to expected future inflation are more prone to induce endogenous cyclical and chaotic dynamics the more open the economy to trade.

I. Introduction

In a recent paper, Bullard (2010) argues that the extremely low interest rates observed in the U.S. and in major developed economies, after the financial crisis, are consistent with the theoretical findings by Benhabib et al. (2001b). The latter show how, because of the zero lower bound (ZLB) on the nominal interest rate, simple Taylor rules can induce a liquidity trap—a situation where the nominal interest rate drifts away from its target towards an unintended low steady state—even if the rule satisfies the celebrated Taylor principle by responding more than proportionally to inflation (an active rule).

The liquidity trap in Benhabib et al. (2001b) is an equilibrium that is entirely driven by people’s self-fulfilling expectations. It is in fact the natural outcome from a non-linear analysis of the prototype micro-founded dynamic model that has become the workhorse for macroeconomic policy discussions. However, as Bullard (2010) points out, this outcome has surprisingly received scarce attention in the policy debate. Central banks often organize their discussions by looking at the impulse responses of linearized versions of the truly non-linear models. By doing so, they tend to rule out a priori, as a viable equilibrium, all the non-linear trajectories that move away from their targets, including liquidity traps.

Benhabib et al. (2001b, 2002a) also show that a global rather than a local analysis unveils the existence of other Taylor-rule-induced rational expectations equilibria, such as endogenous limit cycles and chaotic dynamics.1 This questions policy recommendations based on linearized models, since Taylor rules that ensure a unique local equilibrium may actually generate aggregate instability by inducing global cyclical fluctuations. The works by Benhabib et al., however, are based on closed economy models raising the question of whether similar results hold in the context of open economies. To this date, this still remains a valid question since most of the studies in open economies have been restricted exclusively to local analyses. De Fiore and Liu (2005), Llosa and Tuesta (2008), and Zanna (2003), for instance, pursue local analysis of Taylor rules in the context of a small open economy setup; while Benigno and Benigno (2008), Bullard and Singh (2008) and Leith and Wren-Lewis (2009), among others, present similar analyses in open economy models with more than one country.

The purpose of our paper is to assess how the existence of these self-fulfilling cyclical and chaotic fluctuations is affected by opening the economy to international trade in goods. We pursue a global non-linear equilibrium analysis of a traditional flexible-price small-open-economy model with traded and non-traded goods, where monetary policy takes the form of an active interest rate rule responding to expected future CPI inflation, i.e., forward-looking rules. By doing this, we bridge the gap between the closed economy literature on global analysis of Taylor rules and the open economy literature on local analysis of these rules.

Our main result is that forward-looking Taylor rules are more prone to induce endogenous cycles and chaos the more open the economy to trade. In contrast to the money-in-the-production-function (MIPF) set-up of Benhabib et al. (2002a), we use a money-in-the-utility-function (MIUF) set-up, whereby consumption and real money balances are non-separable in utility. As a result, these self-fulfilling complex fluctuations, whose existence depends on openness, can occur around either the target interest rate or the unintended low steady state, depending on whether consumption and money are Edgeworth substitutes or complements, respectively. Interestingly, the complements case allows for liquidity traps that converge non-monotonically to a limit cycle around the unintended steady state.

The global equilibrium dynamics in our model are driven by the interaction of an open-economy version of the Fisher equation and the non-linear Taylor rule. The modified Fisher equation is obtained from the combination of the uncovered interest parity condition and the definition of the consumer-price-index (CPI) inflation. In this modified equation, the dynamics of the ex-post real interest rate has to be consistent with future changes in the real exchange rate. Key to the existence of cycles and chaos is the elasticity of the real exchange rate to the policy interest rate. This elasticity is affected by the complementarity/substitutability between consumption and money and, more importantly for our purposes, by the degree of trade openness.

Relative to the works on global analysis in closed economies, we find that trade openness significantly enlarges the risk aversion parameter range under which forward-looking rules induce cycles and chaos. As in Benhabib et al.(2002a), the presence of a ZLB on nominal interest rates is a necessary condition for the existence of cyclical fluctuations. However, while these complex dynamics coexist in a MIUF closed economy model for a very restricted range of values of the risk aversion parameter, the same dynamics may occur for a much wider range of this parameter, as long as the economy is sufficiently open. By enlarging this range, openness plays a key qualitative role in determining the global determinacy properties of Taylor rules.

From a policy-making point of view, our results suggest that to avoid destabilizing endogenous cycles and chaos in open economies, the design of interest rate rules should take into account not only the interest response coefficient to inflation, but also specific structural characteristics, such as the degrees of openness. This complements the results of the aforementioned open economy literature on local analysis with one important caveat: as we show below, and in line with Benhabib et al. (2001b, 2002a), policy prescriptions that are derived from local analyses to ensure macroeconomic stability in an open economy can still lead to cycles and chaos (global indeterminacy), where the extent of disagreement between the local and global analyses depends on the degree of openness.

The remainder of this paper is organized as follows. In Section II, we present a flexible-price model with its main assumptions. We define the open economy equilibrium and derive some basic steady state results. In Section III, we pursue local and global equilibrium analyses for an interest rate rule that responds to expected future CPI inflation, focusing on the role played by the degree of openness. Section IV discusses the robustness of our main results under different extensions, including imperfect exchange rate pass-through, incomplete markets, alternative timings for the policy rule and money in utility, and alternative preferences and technologies. Finally, Section V concludes.

II. A Flexible-Price Model

A. The Household-Firm Unit

Consider a small open economy (SOE) populated by a large number of identical and infinitely lived household-firm units. Each unit derives utility from consumption (ct), real money balances (mtd), and not working (1htThtN) according to

E0Σt=0βt{[(ct)γ(mtd)1γ]1σ11σ+ψ(1htThtN)}(1)

and

ct = (ctT)α(ctN)(1α)(2)

where β, γ ∈ (0,1), and ψ, σ > 0 but σ ≠ 1; E0 is the expectations operator conditional on the set of information available at time 0; ctT and ctN denote the consumption of traded and non-traded goods; mtd = Mtdpt are real money balances defined in terms of the CPI pt htT and htN stand for labor supplied to the production of traded and non-traded goods, respectively, and α ∈ (0,1) is texchange rate pass-through. This andhe share of traded goods in the consumption aggregator (2). We interpret this share as a measure of the degree of trade openness of the economy. As α goes to zero, domestic agents do not value internationally traded goods and the economy is basically closed. Whereas if α goes to one, non-traded goods are negligible in consumption and the economy becomes completely open.

The utility specification in (1) is general enough to show analytically how Taylor-rule-induced cyclical dynamics depend on the degree of openness α. By defining U = (ctγmt1γ)1σ11σ note that the cross derivative of utility with respect to consumption and money Ucm satisfies sign {Ucm} = sign {1-σ}. So we can distinguish between the case of Edgeworth substitutes for which Ucm < 0, when σ >, and the case of complements for which Ucm >, when σ < 1.2

The representative unit produces traded and non-traded goods by employing labor and fixed stocks of capital, according to the technologies

ytT = zt(kT)1θT(htT)θTandytN = zt(kN)1θN(htN)θN(3)

where θT, θN ∈ (0,1) are the labor shares, and zt is an aggregate productivity shock following a stationary AR(1) stochastic process. We assume that zt is the sole source of fundamental uncertainty in the economy.

The law of one price holds for traded goods and their foreign price, PtTw, is normalized to one. Hence PtT = εt, where PtT is their domestic currency price and εt is the nominal exchange rate. In this regard, the model features complete exchange rate pass-through. This and equation (2) imply the following expression for the CPI:

pt(εt)α(PtN)1ααα(1α)1α,(4)

which can be used to derive the gross CPI-inflation rate

πtptpt1=εtα(πtN)(1α)(5)

As can be seen, the CPI inflation is just a weighted average of the (gross) nominal devaluation rate or traded goods inflation, εtεtεt1, and the (gross) non-traded goods inflation, πtNPtNPt1N, with weights related to the degree of openness, α. The real exchange rate, in turn, is defined as the ratio of the price of traded goods and the price of non-traded goods

etεtPtN(6)

and evolves according to

etet1=εtΠtN(7)

The household-firm units have access to a complete set of internationally traded claims, as in Galí and Monacelli (2005).3 In each period t ≥ 0, the agents can purchase two types of financial assets: fiat money Mtd and nominal state contingent claims, Dt+1. The latter pay one unit of foreign currency for aspecific realization of the fundamental shocks in t+1. Under complete markets the representative agent’s flow constraint can be written as

Mtd +EtQt,t+1Dt+1Wt+εtytT+PtNytNεtTtεtctTPtNctN(8)

where EtQt,t+1Dt+1 denotes the cost of all contingent claims bought at the beginning of period t and Qt,t+1 refers to the period-t price of a claim to one unit of currency delivered in a particular state of period t+1, divided by the probability of occurrence of that state and conditional on information available in period t. Constraint (8) says that the total end-of-period nominal value of the financial assets can be worth no more than the value of the financial wealth brought into the period, Wt =Mt1d + Dt,, plus non-financial income net of the value of taxes, εtTt, and the value of consumption spending. Since the expected period-t price of a claim is equal to the inverse of the risk-free gross nominal interest rate, that is EtQt,t+1 = 1Rt we can write constraint (8) as the following period-by-period constraint:

EtQt,t+1Wt+1Wt+εtytT + PtNytNεtTtRt1RtMtdεtctTPtNctN(9)

The representative unit is also subject to a Non-Ponzi game condition

limjEtqt+jWt+j0(10)

at all dates and under all contingencies, where qt+j = Qt t+1 Qt+1,t+2…..Qt+j-1,t+j.

The problem of the representative household-firm unit reduces then to choose the sequences {ct,TctN,htT,htN,Mtd,Wt+1}t=0 in order to maximize (1) subject to (2), (3), (9), and (10), given W0 and the time paths of Rt, εt, PtN, Qt,t+1, and Tt. Since the utility function specified in (1) implies that the preferences of the agent display non-satiation, both constraints (9) and (10) hold with equality. The first order conditions correspond to these constraints and

αγ(ctT)αγ(1σ)1(ctN)(1α)γ(1σ)(mtd)(1γ)(1σ)=λt(11)
αctN(1α)ctT = et(12)
λtetθN(htN)(θN1) = ψ = λtθT(htT)(θT1)(13)
mtd = (1γγ)(α1α)α(RtRt1)ctNetα(14)
λtεtQt,t+1 = βλt+1εt+1(15)

where λtεt is the Lagrange multiplier of the budget constraint.

The interpretation of conditions (11)-(15) is well-known. Equation (11) is the intertemporal envelope condition that makes the marginal utility of consumption of traded goods equal to the marginal utility of wealth measured in terms of traded goods (λt). Condition (12) implies that the marginal rate of substitution between traded and non-traded goods must be equal to the real exchange rate, while condition (13) equalizes the value of the marginal products of labor in both sectors. Equation (14) represents the demand for real money balances, and condition (15) describes a standard pricing equation for one-step-ahead nominal contingent claims for each period t and for each possible state of nature.

B. The Government

The government issues money, Mts, and a one period risk-free domestic bond, Bts, which pays a gross risk-free nominal interest rate Rt. It cannot issue or hold state contingent claims. It chooses the path of taxes to satisfy its intertemporal budget constraint in conjunction with a transversality condition at all times. As a result, we can ignore its budget constraint in the analysis to follow.

Monetary policy, on the other hand, is described as a forward-looking interest rate rule, which sets the nominal interest rate, Rt, as a continuous and increasing function of the deviation of the expected future CPI inflation rate, Etπt+1, from a target, π*. Our interest in this rule is motivated by the empirical evidence of Clarida et al. (1998), as well as Cogley and Sargent (2005), supporting the view that central banks are forward-looking in their monetary policy setting. For analytical purposes, we use the same functional form as that in Benhabib et al. (2002a) and Eusepi (2007):

Rt = ρ(Etπt+1)1+(R*1)(Etπt+1Π*)AR*1,(16)

where R* = π*β>1 corresponds to the interest rate target, and assume that Rt always satisfies the zero bound, i.e., Rt > 1. We focus on rules that satisfy the Taylor principle, also known as active rules, whose elasticity to inflation at the target steady state, ρ(π*)π*ρ(π*) = AR*, is strictly bigger than 1.

Assumption 0: The rule is active: ξAR*>1.

From now on we will refer to ξ as the degree of activism towards inflation.

C. International Capital Markets

We assume free international capital mobility. Then the no-arbitrage condition Qt,t+1w = Qt,t+1εt+1εt holds, where Qt,t+1w refers to the period-t foreign currency price of a claim to one unit of foreign currency delivered in a particular state of period t+1 divided by the probability of occurrence of that state and conditional on information available in period t. Furthermore, λtwPtTwQt,t+1w = λtw+1Pt+1Twβw and holds for the representative agent in the rest of the world (ROW), where λtw βw denote, respectively, the marginal utility of nominal wealth and the subjective discount rate of the ROW. Combining this with condition (15) and assumptions PtTw = 1,βw = βand PtT=εt yields λt+1λt = λt+1wλtw, which holds at all dates and under all contingencies. This condition implies that the domestic marginal utility of wealth is proportional to its foreign counterpart: λt = Λλtw, where Λ is a constant that determines the wealth difference between the SOE and the ROW. As in Schmitt-Grohé and Uribe (2003), we assume that λtw–and hence λt–is constant overtime

λt = λ = Λλw.(17)

Equations (15) and (17), together with EtQt,t+1 = 1Rt, imply the following expression:

Rt = 1βEt1εt+1,(18)

which is reminiscent of an uncovered interest parity (UIP) condition.

D. Equilibrium

We will focus on perfect foresight equilibria, where agents in the economy forecast correctly all the anticipated variables. Given our focus on endogenous fluctuations, we set zt = 1 for t ≥ 0 Therefore, for any variable xt, we have that Etxt+j= xt+j for j ≥ 0 implying that we can drop the expectation operator in the previous equations. For instance, condition (18) becomes

Rt = εt+1β,(19)

which corresponds to the UIP condition with 1β representing the foreign international interest rate.4

To provide a definition of the equilibrium dynamics, we derive a reduced non-linear form of the model. By combining conditions (11)-(14) with (17), and the market clearing conditions for money and the non-traded good–i.e., Mtd= Mts=Mt and ytN = (kN)1θNctN—we can express the real exchange rate et as a function of the nominal interest rate Rt

ete(Rt) = k(RtRt1)v,(20)

where k and v depend on structural parameters. In particular v (σ1)(1γ)(1θN)σ[θN+α(1θN)] + (1α)(1θN). For the discussion below, it is useful to note that5

e(Rt)0 for σ1.(21)

We can now combine equations (5), (7), and (19), to express the future real depreciation εt+1et in terms of the ex-post real interest rate Rtπt+1:

(et+1et)1α=βRtπt+1(22)

Equation (22) is an open economy version of the Fisher equation involving a term related to the real exchange rate depreciation, (et+1et)1α. It can be used with the rule (16), the definition R* = π*β, and real exchange rate equation (20) to derive the following first order difference equation, which represents the reduced non-linear form of the model and will be used to pursue both the local and global equilibrium analyses:

(Rt+1Rt+11)x = RtR*(R*1Rt1)R*1A(RtRt1)x,(23)

where

X(σ1)(1α)(1γ)(1θN)σ[θN + α(1θN)]+(1α)(1θN)=<>0 for σ=<>1.(24)

Using (23), we can provide the following definition of a perfect foresight equilibrium in our model.

Definition 1 Given the target R* and the initial condition R0, a perfect foresight equilibrium (PFE), under a forward-looking interest rate rule, is a deterministic process {Rt}t=0, with Rt> 1 for any satisfying equation (23).

Although this definition is stated exclusively in terms of the nominal interest rate (Rt), multiple PFE solutions to (23) imply real indeterminacy of all the endogenous variables, as a result of the non-separability between money and consumption in preferences.6 In fact, for any given Rt equations (5), (7), (11)-(14), (17), (19), and the market clearing conditions for money and the non-traded good, can be used to obtain all the remaining real endogenous variables.

For the equilibrium analysis, we first identify the steady state(s) of the economy. Setting Rt = Rt+1 = Rss in (23), we obtain:

(R*1)R*1ARss = R*(Rss1)R*1A(25)

Clearly the target R* of the central bank is a solution to (25) and therefore a feasible steady state. But since the rule is active at R* (Assumption 0), then a second unintended steady state RL ε (1,R*) exists. At this state, the elasticity of the rule to inflation is given by ARL<1. The following proposition formalizes the existence of this passive steady state. From now on, we are going to refer to R* as the active or target steady state and to RL as the passive or unintended steady state.

Proposition 1 If AR*>1 (an active rule) and Rss > 1 (the zero lower bound) then, besides the solution Rss = R*, there exists a value RL ∈ (1,R*) solving (25).

Proof. See the Appendix. ■

As in Benhabib et al. (2002a), the existence of two steady states will play a crucial role in the derivation of our results. Note that it depends exclusively on policy parameters. However, as we will show below, this does not preclude the possibility that the presence of global dynamics out of steady state may be driven by other structural parameters, including the degree of openness.

III. Local and Global Dynamics

In the main analysis, we will keep constant the structural parameters β γ and θN, as well as the policy parameters A and R*. This will allow us to compare the global dynamics of economies that implement the same monetary rule but differ in the degree of openness α and the risk aversion parameter σ. We will use the following two assumptions.

Assumption 1: 1γ>θN1θN12(R*1)(1R*A) and Assumption 2: R*1>A(RL1).

Assumption 1 is expressed in terms of the share of real money balances in utility 1–γ. It requires money to be suffciently important in providing transaction services and will be used in the proofs of local determinacy and existence of period-2 cycles when σ > 1 (Edgeworth substitutability). As in the closed economy models of Benhabib et al. (2002a) and Eusepi (2007), money is “essential” for the existence of cycles.7 In fact, a similar assumption would be needed in any closed or open economy money-in-the-utility-function (MIUF) model with endogenous labor choice to derive analytical results. Given the values of the share of labor θN and the share of money 1–γ that are commonly used in the literature, Assumption 1 holds for a very wide range of parametrizations of the interest rate target R* and the degree of policy activism ξAR*.8

Assumption 2 requires a suffciently positive spread between the active and the passive steady state and will be used in the proof of endogenous cycles around the unintended low steady state, when σ < 1 (Edgeworth complementarity). Models with money in the production function (MIPF) do not require this assumption, since cycles appear only around the active steady state. But a similar assumption would be needed in the context of a MIUF closed economy model, to prove analytically the existence of cycles around the low steady state.9 Moreover, our numerical simulations show that this assumption generally holds for any rule which is active at the target, i.e. AR*>1.10

A. Local Determinacy

The local determinacy analysis focuses on PFE which remain bounded within a small neighborhood of the target steady state. By log-linearizing equation (23) around R*, we obtain the following linear difference equation:

R^t+1 = (1+R*A1XR*1)R^t.(26)

Since Rt is a non-predetermined variable, local determinacy requires the previous equation to generate explosive dynamics.

Proposition 2 Suppose the government follows an active forward-looking rule like (16). Then, there exist threshold values σd> 1 and αd ∈ (0,1), for the risk aversion parameter σ and the degree of openness α, such that the active steady state is a locally determinate equilibrium for any α > 0 when σ ∈ (0,σd, but only for α > αd when σ > σd.

Proof. See the Appendix. ■

This proposition shows the importance of α and σ in the local equilibrium characterization. In a nutshell, active forward-looking rules guarantee local determinacy in the following cases: when, regardless of the degree of openness a, the risk aversion parameter σ is sufficiently low; and when, for higher values of σ, the economy is sufficiently open.11 Note that Proposition 2 also implies that, when consumption and money are Edgeworth substitutes (σ > 1), an active rule might induce local sunspot-driven fluctuations—i.e., multiple local equilibria—if the economy is sufficiently open (α < αd). In contrast, for any degree of trade openness, an active rule will always ensure local determinacy if consumption and money are complements (σ < 1). However, as we show next, local determinacy does not rule out global indeterminacy, which may be affected by the degree of openness.

B. Cycles and Chaos

B.1. The Main Results from a Global Non-linear Analysis

For the global analysis, we rewrite equation (23) as Rt+1 = f (Rt), where

f(Rt)11J(Rt)1x(27)

and

J(Rt)R*(R*1)R*1A(Rt1)X+R*1ARt1+x(28)

We study the global PFE that satisfies Rt+1 = f (Rt) for a given initial condition R0 > 1 and subject to the zero lower bound (ZLB) fn (Ro) > 1 for any n ≥ 1. The economy displays a globally indeterminate PFE, if there exists a continuum of initial conditions R0 for which (27) satisfies the ZLB at any point in time. We are interested in global indeterminacy taking the form of cyclical and chaotic dynamics conforming to the following definitions.

Definition 2 Period-n cycle. A value “R” is a point of a period-n cycle if it is a fixed point of the n-th iterate of the mapping f(.), i.e., R = fn (R), but not a fixed point of an iterate of any lower order. If “R” is such, we call the sequence {R, f (R), f2 (R),…, fn–1 (R)} a period-n cycle.

Definition 3 Topological chaos. The mapping f(.) is topologically chaotic if there exists a set “S” of uncountable many initial points, belonging to its domain, such that no orbit that starts in “S” will converge to one another or to any existing period orbit.

The global analysis requires the characterization of the function f in (27) over its entire domain. It is straightforward to prove that its analytical properties mainly depend on the composite parameter X defined in (24). More specifically, f can be single-peaked (respectively, single-troughed) for X positive (respectively, negative). This implies the possibility of a negative derivative of f at either one of the two steady states, which is a necessary condition for the existence of endogenous cycles in continuously differentiable maps.12 More precisely, by (24) and the equivalence sign {Ucm} = sign {1 - σ}, f is negatively sloped at the active steady state if σ> 1 (Edgeworth substitutability) and at the passive steady state if σ < 1 (Edgeworth complementarity).

The following Proposition contains our main result by stating the exact analytical conditions, with respect to the degree of trade openness a, under which endogenous period-2 cycles occur in our economy.

Proposition 3 Suppose the government follows an active forward-looking rule like (16). Then there exist threshold values σf ∈ (0,1) and σd> 1 for the risk aversion parameter σ, together with thresholds of αf ∈ (0,1) and αdε (0,1) for trade opennessα, such that

  • 1. period-2 equilibrium cycles occur around the passive steady state, for α αf when σ ∈ (0,σf] and for any α ≥ 0 when σ ∈ (σf, 1).

  • 2. period-2 equilibrium cycles occur around the active steady state, for α > αd when σ > σd and for any α ≥ 0 when σ ∈ 1, σd)

Proof. See the Appendix. ■

To simplify the statement of Proposition 3, we have abused of notation, to some extent, since the thresholds αf and αd can be in fact seen as functions αf (σ) and αd(σ) of the risk aversion parameter σ. Table 1 and Figure 1, which summarize and provide a graphical representation of Proposition 3, make these functions explicit. The table and figure also present the results of Proposition 2, allowing for a comparison of the results of both the local and the global determinacy analyses.

Using Table 1 and Figure 1, it is easy to grasp the following key messages from the global analysis.

Table 1:

Local versus Global Analysis

article image
Note: σf ∈ (0,1) and σd > 1. “x” stands for no cycles of any periodicity around the steady state.R* and RL stand for the target and the unintended low steady states, respectively.
Figure 1:
Figure 1:

Local and global equilibrium analyses for active forward-looking rules, as the degree of openness α and the risk aversion coefficient σ vary. R* and RL stand for the target and the unintended low steady state, respectively.

Citation: IMF Working Papers 2012, 121; 10.5089/9781475503500.001.A001

First, for σ falling outside the range (σf, σd) endogenous period-2 cycles exist as long as the economy is sufficiently open to trade. That is, openness is a necessary condition for endogenous cycles. In this sense, a rule that responds actively to expected future inflation is more prone to induce endogenous cyclical dynamics the more open the economy.

Second, depending on whether consumption and real money balances are Edgeworth complements (σ < 1) or substitutes (σ > 1), equilibrium cycles can appear either around the passive or the active steady state, respectively. In this regard, our MIUF approach is more general than the MIPF set-up of Benhabib et al. (2002a). Were the MIPF adapted to a small open economy, it would only display cycles around the active steady state.13 To the best of our knowledge, our paper is the first one to derive analytical conditions for the existence of cycles around the passive steady state, which depends on openness. These dynamics are rather interesting as one could think of them as cyclical liquidity traps, that is, a situation where the nominal interest rate is trapped in a small neighborhood around the low steady state.

Third, a comparison of the global and local analyses reveals some crucial differences with important policy implications. As displayed in Figure 1, global indeterminacy in the form of equilibrium cycles occurs in regions where, by local analysis, one concludes in favor of a unique equilibrium (white area).

Even more strikingly, for σ > σd, the condition for local equilibrium determinacy is identical to the sufficient condition for the existence of period-2 cycles. Therefore, while for σ > σd the local analysis concludes that higher openness (α) makes active Taylor rules more prone to stabilize the economy around the target, the global analysis instead identifies openness as a source of cycles around it (global indeterminacy). This contrast is due to the fact that, by log-linearizing around the target steady state, local analysis implicitly assumes that any path starting arbitrarily close to it and diverging cannot be part of an equilibrium, since it will eventually explode violating the transversality condition. This is not the case in the global analysis, where there may exist limit cycles towards which equilibrium paths–which start arbitrarily close to the target steady state–may converge to.14

Given the complicated functional form of the map Rt+1 = f (Rt) in (27), we cannot derive simple analytical conditions for the existence of higher order cycles and chaotic dynamics. We then resort to numerical simulations for a calibrated version of our economy. This analysis also allows us to assess the quantitative relevance of the analytical results of Proposition 3.

Table 2:

Parametrization

article image

Table 2 summarizes the benchmark parametrization, where Canada is mostly used as the representative economy, and the time unit is a quarter. From Mendoza (1995) we borrow the labor income share for the non-traded sector θN = 0.56. In line with empirical evidence by Holman (1998), we set the share of money in utility 1–γ equal to 0.03.15 The steady-state inflation and nominal interest rate are the averages of the CPI inflation and the central bank discount rate, during the last two decades. This yields π* = 1.031 14 and R* = 1.072 14, implying β = 0.99. For illustrative purposes we use an elasticity AR* of 2.25, which satisfies the Taylor principle and reflects a moderate degree of policy activism towards inflation, as suggested by Boivin (2006) and Cogley and Sargent (2005).

The simulations presented in Figure 2 show that cycles of various periodicities and chaos exist as long as the economy is sufficiently open. We vary openness α over the continuum (0,1), while setting σ equal to either 0.8 or 2.0, which correspond to consumption and money being Edgeworth complements or substitutes, respectively. The solid lines represent the stable period-n limit cycles of the mapping Rt+1 = f (R t in (27). The PFE dynamics settle on a steady state (flat solid lines), as long as α is sufficiently small. This attractive steady state is either the unintended steady state RL ≈ 1.01 14, under complementarity (σ = 0.8), or the target steady state R* =1.0723 14, under substitutability (σ = 2.0). Once α passes a certain threshold–about 0.15 for σ = 0.8 and 0.37 for σ = 2.0–a stable period-2 cycle appears, as displayed by the first split into two branches. As we continue increasing α the branches split again and a cascade of further period doubling bifurcations occurs, yielding cycles of period 4, 8, 16, and so on. Finally for sufficiently high values of a, the rule produces cycles of odd periodicity as well as aperiodic chaotic dynamics, where the attractor of the mapping changes from a finite to an infinite set of points.

Figure 2:
Figure 2:

Orbit-bifurcation diagrams for the degree of openness

They show the set of limit points for the nominal interest rate Rt, as a function of the degree of openness α, under Edgeworth complementarity (σ = 0.8) and sustitutability (σ = 2).

Orbit-Bifurcation Diagrams

Citation: IMF Working Papers 2012, 121; 10.5089/9781475503500.001.A001

Figure 3 confirms the existence of chaos for sufficiently open economies by looking at the Lyapunov exponent. A positive exponent measures the degree of exponential divergence between two series generated by the same map and starting from nearby, but not identical, initial conditions. This characterizes chaotic systems. Consistently with Figure 2, there is evidence of chaos for α > 0.38, when σ = 0.8, and for α > 0.5, when σ = 2.0.16

Figure 3:
Figure 3:

Lyapunov exponents for forward-looking rules, as a function of the degree of openness (α) and under Edgeworth complementarity (σ = 0.8) and sustitutability (σ = 2).

Lyapunov Exponents

Citation: IMF Working Papers 2012, 121; 10.5089/9781475503500.001.A001

Openness significantly enlarges the risk aversion parameter range under which forward-looking rules induce cycles. To see this, we look at our results from another angle. While Proposition 3 defines the bifurcation thresholds with respect to trade openness, it is possible to show that, for given α, equilibrium cycles require σ to belong to an interval (σF, σD). Formally σF and σD are functions of α and correspond to the inverse of the functions αf (σ) and αd (σ) depicted in Figure 1. These functions αF (α) and σD (α) are, respectively, strictly decreasing and strictly increasing in α. As a result, the interval (σF, σD) under which cycles can occur is strictly increasing in openness α, implying that it is much wider for an open economy (α > 0) than for a fully closed economy (α = 0). In the latter case (σF, σD) = (σf, σd) where σf and σd are depicted in Figure 1. We formalize all this analysis in Corollary 1.

Corollary 1 Assume σ ≠ 1 and a ∈ [0,1). For given α, period-2 cycles occur only for σ belonging to the interval (σF, σD), where σF and σD are, respectively, strictly decreasing and strictly increasing functions in α. Therefore, as α increases–i.e., openness increases–period-2 cycles occur for a much wider range of σ. The cycles are around the active steady state for σ > 1 and around the passive steady state for σ < 1.

Proof. The proof follows from the proof to Proposition 3 and is available upon request. ■

Table 3 highlights the quantitative importance of Corollary 1. Using the calibration of Table 2, it allows us to compare quantitatively the case of a completely closed economy, like in Benhabib et al. (2002a) and Eusepi (2007), with the case of an open economy. For a fully closed economy, i.e., α = 0, period-2 cycles exist for a very restricted range of values of the risk aversion, namely, (σF, σD = (0.82,1.47).17. As openness increases, this range also increases. When the economy is very open, e.g., α = 0.65, this range becomes (σF, σD) = (0.62,10.87), which is substantially much wider.

Table 3.

Openness, Risk Aversion, and Cycles

article image

Summarizing, the results in this subsection suggest that openness is an important factor for global indeterminacy under standard Taylor rules.

B.2. An Intuition of the Results

To build an intuition of our results, we use the rule (16) and equation (20) to rewrite the equilibrium condition (22) as:

[e(ρ(πt+2))]1α = [e(ρ(πt+1))]1αβρ(πt+1)πt+1,(29)

which has to be satisfied by any sequence of inflation rates in order to be a PFE. In addition, we focus on the case of σ < σf < 1. In this case we know, from Proposition 3 and Figure 1, that cyclical and chaotic dynamics exist around the passive steady state if the economy is sufficiently open. This is due to a flip bifurcation threshold αf, below which the passive steady state is dynamically stable and above which stability is lost. The switch from stability to instability depends on the elasticity of the real exchange rate to the interest rate factor RtRt1,,, which by equation (20), is given by v(σ1)(1γ)(1θN)σ[θN+α(1θN)] + (1α)(1θN). For σ < σf < 1, the elasticity v is always negative and decreasing in absolute”value as α approaches 1. That is, the more open the economy, the smaller the impact of the nominal interest rate on the real exchange rate.

Suppose now the economy is at the passive steady state πL ≡ βRL where the rule is passive, i.e., it responds less than proportionally to inflation with ARL<1. Assume that suddenly the private sector expects a higher inflation in the future. Since the rule is passive, the real interest rate ρ(Πt+1)Πt+1 declines. If openness α is high, the elasticity v is small and the increase in ρ(πt+1 has a small positive effect on et and even less on et1α18. As a result, the right-hand side of (29) decreases. But because of the small elasticity, to have a lower left-hand side it is necessary to have a substantial drop in πt+2 i.e., |Πt+2|>|Πt+1|.. Hence, when α is high, the passive steady state is dynamically unstable and the equilibrium series of inflation rates starts diverging from it.

The convergence to a cycle can be explained as follows. Suppose πt+1 = πh > πL, that is, agents expect an inflation level above the passive steady state πL. Since the rule is passive we have that ρ(Πh)Πh<ρ(ΠL)ΠL; and because of the high openness–hence, a low v–the term [e(ρ(Π))]1α is only slightly affected by changes in inflation, implying that the right-hand side of (29) is then lower than what it is at the steady state. But then, by (29), next period inflation must be below the steady state, i.e., πt+2 = πl < πL. Following the same argument, in the next period it must be that ρ(Πl)Πl>ρ(ΠL)ΠL, pushing the right-hand side of (29) above the steady state, thus requiring next period inflation to be at πh > πL, and so on.

Similar arguments can be invoked to construct a cyclical PFE around the active steady state for the case of σ > σd > 1.

B.3. Policy Activism and the Degree of Openness

To underscore the quantitative importance of our results, we investigate their implications in terms of policy activism. We use basins of attractions to unveil the interaction between the degree of openness a and the rule’s elasticity to inflation ξ. The left-hand side of Figure 4 shows these basins when σ = 0.8. The light grey area corresponds to trajectories converging to the passive steady state. Moving rightward, the darker grey area corresponds to period-2 cycles and the black area to period-4 cycles. The dotted area represents period-3 as well as higher order cycles and chaos. In the white area, the economy diverges. The areas of the right-hand side of the figure, when σ = 2, have the same interpretation with two caveats: the left-hand light grey area, associated with low a’s, corresponds to trajectories converging to the active steady state; while the right-hand light grey area represents trajectories violating the restriction Rt > 1 for any t.

Figure 4:
Figure 4:

Basis of attraction for forward-looking rules for different combinations of the degree of openness α and the elasticity of the forward-looking rule to inflation £, under Edgeworth complementarity (σ = 0.8) and substitutability (σ = 2). See the main text for the interpretation of the colored areas.

Basins of Attraction Diagrams

Citation: IMF Working Papers 2012, 121; 10.5089/9781475503500.001.A001

Two results stand out from Figure 4. When the economy is closed (α = 0) and either σ = 0.8 or σ = 2.0, cycles and chaos are ruled out for any degree of policy activism (£). As the economy opens up to trade, these complex dynamics can be induced by quantitatively realistic degrees of policy activism. For instance, when σ = 0.8 and α ≈ 0.3, period-2 cycles are possible for degrees of activism ξ that are between the famous Taylor coefficient of 1.5 and 3.0. Of course, judging the feasibility of these degrees of activism also depends on feasible values for the degree of openness and other structural parameters. In this regard, consider Table 4 where, given some calibrated values for these structural characteristics of several inflation targeting countries, we calculate the thresholds for the elasticity ξ above which, according to our model, complex global dynamics occur. Clearly these thresholds are to some extent possible in practice, in particular for Norway and Sweden. Interestingly, Bask (2002) and Gogas and Serletis (2000) suggest that the nominal and real exchange rates in the Scandinavian countries may have followed chaotic dynamics in the 1980s and 1990s.

Table 4.

Activism, Cycles, and Chaos

article image
Sources: Ogaki et al. (1996) for σ; Petursson (2004) for π; Bentolilla and Saint Paul (2003) for θN; and Campa and Goldberg (2006) for α.

The thresholds for ξ are for a forward-looking rule and assume 1–γ = 0.03 for all countries.

IV. Extensions and Robustness

This Section discusses the robustness of our results to alternative variations of our framework, including incomplete exchange rate pass-through, incomplete markets, different timings for the policy rule and for money in utility, and different preferences and technologies.19 Details of some of these extensions can be found in the Appendix.

A. Incomplete Exchange Rate Pass-Through

We have assumed perfect exchange rate pass-through into import prices. This, however, is at odds with empirical evidence. Campa and Goldberg (2006), for instance, suggest that the exchange rate pass-through into import prices varies across countries and has changed over time. We then introduce imperfect pass-through by assuming distribution costs as in Burnstein, Neves, and Rebelo (2003). More specifically, we assume that to consume one unit of the traded good, it is required η units of the non-traded good as a result of some distribution services.20 Then the consumer price of the traded good is

PtT =P˜tT +ηPtN,(30)

where P˜tT is the producer price in domestic currency. To simplify the analysis we assume that the law of one price holds for traded goods at the production level and normalize the foreign price of the traded good to one (P˜tTw = 1). Thus P˜tT=εtP˜tTw=εt.

The presence of distribution services leads to imperfect exchange rate pass-through into import prices. To see this we combine (30), P˜tT=εt, and εt=εt/PtN to express the gross inflation of import prices πtT as

πtT=(et1et1+η)εt+(ηet1+η)πtN,

where εtεtεt1 and πtNPtNPt1N correspond to the gross nominal depreciation and non-traded inflation, respectively. Clearly when η = 0—the case already studied—then there is perfect pass-through of the nominal depreciation rate into the inflation of import prices, since πtTεt = 1. But if η > 0 then we obtain imperfect pass-through, given that πtTεt =(εt1εt1+η)(0,1). As the parameter of distribution costs η increases, the degree of pass-through decreases.

Increasing η—decreasing the exchange rate pass-through—has a non-trivial impact on the global dynamics. Using the parametrization of Table 2 and setting α = 0.4, Figure 5 displays the related orbit-bifurcation diagrams for η. When η = 0—perfect pass-through–chaos and period-2 cycles occur when σ = 0.8 and σ = 2.0, respectively. But these complex dynamics disappear once η passes a certain threshold, as the economy eventually settles on either the passive or the active steady state. Such thresholds for η are about 0.84 for σ = 0.8 and 0.07 for σ = 2. In this regard, openness plays a more fundamental role for cyclical and chaotic dynamics, the lower η (the higher the exchange rate pass-through).

Figure 5:
Figure 5:

Orbit-bifurcation diagrams for the degree of exchange rate pass-through

Orbit-Bifurcation Diagrams for The Degree of Exchange Rate Pass-through

Citation: IMF Working Papers 2012, 121; 10.5089/9781475503500.001.A001

They show the set of limit points for the nominal interest rate Rt as a function of the distribution costs parameter η, under Edgeworth complementarity (σ = 0.8) and substitutability (σ = 2). A lower η means a higher exchange rate pass-through into import prices.

Proposition 4 Given openness, forward-looking rules are more prone to induce cyclical and chaotic dynamics the higher the degree of exchange rate pass-through.

B. Incomplete Markets

The assumption about complete markets was made in order to allow a meaningful local determinacy analysis. However, it is not essential for the existence of cyclical and chaotic dynamics. To see this, suppose that the agent has access to only one international bond btw, paying a constant risk-free rate Rw, and a domestic bond Bt, paying the time-varying risk-free interest rate Rt. By the first-order conditions and the standard assumption βRw = 1, we would still obtain λt=λt+1 together with the UIP condition εt=βRt. By the same steps highlighted in Section D, it is immediate to show that the global PFE dynamics are entirely described by the same non-linear difference equation (23). Hence, under incomplete international markets, the results on the existence of cycles/chaos would be exactly identical to those of the sub-Section B.1. In addition to that, one could also show that cycles and chaos may occur in the accumulation of foreign bonds, btw, and therefore in the current account, cat=btwbt1w.

C. Alternative Timings for the Policy Rule

Our focus on forward-looking interest rate rules has been mostly motivated by their tractability and the strong empirical support in the data. Alternative timings include contemporaneous rules of which Lubik and Schorfheide (2007) provide evidence, as well as backward-looking rules.

Contemporaneous Rules

Suppose the central bank sets the nominal interest rate in response to current CPI inflation according to Rt=1+(R*1)(πtπ*)AR*1 with R*=π*β and ξ=AR*>1 (active rule). We find that the degree of openness affects the existence of complex dynamics, but only for the case of σ > 1 (Edgeworth substitutability). However, in contrast to the previous results, openness also determines whether the cycles are centered around the passive or active steady state.21 More specifically, cycles and chaos occur around the active (respectively, passive) steady state for low (respectively, intermediate) degrees of openness.22

Backward-Looking Rules

For a policy rule that responds to past CPI inflation, we find that regardless the degree of openness, the PFE dynamics always converge to either the active or the passive steady state, depending on whether consumption and money are Edgeworth substitutes or complements, respectively. Thus the only long-run equilibrium is a liquidity trap. The non-existence of cycles is in line with the findings by Eusepi (2007).23

D. Cash-in-Advance Timing of Money in Utility

Our set-up follows the “Cash-When-I’m-Done” (CWID) timing for real money balances, traditionally adopted in the literature. Carlstrom and Fuerst (2001) have promoted the alternative “Cash-In-Advance” (CIA) timing, where the real money balances entering the agent’s utility are those left after leaving the bond market, but before entering the goods market. As the next proposition shows, openness still plays a key role for the presence of period-2 cycles under this CIA timing.

Proposition 5 Suppose that the government follows the active forward-looking rule (16), and let σ > 1 (consumption and money are Edgeworth substitutes). Then, under the CIA timing of money in utility, there exists thresholds σcia > 1 and αcia ε (0,1), such that period-2 equilibrium cycles occur around the active steady state for any α ε (0,1) if σ ε (1, σcia) and for α > αcia if σ ≥ σcia.

Proof. See the Appendix.■

Figure 6:
Figure 6:

Orbit-bifurcation diagrams with CES preferences

Orbit-Bifurcation Diagrams with CES Preferences

Citation: IMF Working Papers 2012, 121; 10.5089/9781475503500.001.A001

They show the set of limit points for the nominal interest rate Rt, as a function of the degree of openness (α), for Edgeworth complementarity (σ = 2 and μ =–1.1) and substitutability (σ = 2 and μ = –0.9).

E. Constant-Elasticity-of-Substitution (CES) Preferences

We replace the Cobb-Douglas specification in (1) with the CES specification [γctμ+(1γ)(mtd)π]1μ, where μ < 1 and (1 –μ)–1 is the intratemporal elasticity of substitution between ct and mtd. Note that for μ = 0, this elasticity is equal to 1, and the model collapses to the Cobb-Douglas case. Moreover, we now have that sign{Ucm}=sign{1σ1μ}. Then consumption and money are Edgeworth complements for μ < 1–σ and substitutes for μ > 1–σ. Under these CES preferences, the model is less tractable than under the Cobb-Douglas specification, preventing us from deriving simple analytical conditions for the existence of cycles as we did in Proposition 3.24 Nevertheless, numerical simulations are sufficient to confirm that our results are robust to this change in specification. Figure 6 presents the orbit bifurcation diagrams with respect to the degree of openness α. It considers the cases of Edgeworth substitutability (σ = 2 and μ =–0.9) and complementarity (σ = 2 and μ = –1.1).

Three results clearly emerge from Figure 6. First, despite the CES preferences, more open economies are more prone to endogenous cycles/chaos under active forward-looking rules. Second, whether the cycles are around the passive or the active steady state still depends on whether consumption and money are Edgeworth complements or substitutes. And third, under a CES specification, cycles and chaos seem to start appearing at much lower degrees of openness relative to the Cobb-Douglas case.

F. Constant-Return-to-Scale (CRS) Technologies

The production functions in (3) imply decreasing returns to scale with respect to labor. Our analysis and the main messages would not change under CRS technologies. However, to introduce these technologies, it is not sufficient to set θN = θ T = 1 in (3). The reason is that to ensure an interior solution in the model, it is necessary to assume some convexity on the disutility of labor. We then modify (1) by assuming that the disutility from working takes the form ψ[(htT)1+φ1+φ+(htN)1+φ1+φ], where φ is the inverse of the Frisch elasticity of labor.25 From this, and the CRS technologies ytT=zthtT, the global PFE dynamics would still be defined by the mapping Rt+1 = f (Rt) in (27), but with χ(1α)φ(σ1)(1γ)σ+φ+(σ1)αφ. As a result, the model is isomorphic to that extensively studied in the sub-Section B.1. Hence, it is possible to derive analytical conditions for the existence of period-2 cycles similar to those of Proposition 3, which depend on openness, as well as to show numerically the occurrence of higher order cycles and chaos.26

V. Conclusions

In this paper, we fully characterize the set of global perfect foresight equilibria of a standard two-good small-open-economy model, where monetary policy takes the form of an active forward-looking Taylor-type interest rate rule. We show that a higher degree of trade openness—measured by the share of traded goods in aggregate consumption—makes the rule more prone to induce endogenous complex dynamics, such as equilibrium cycles of various periodicities and chaos. Interestingly, the results on chaos should not be perceived as theoretical curiosities. They are to some extent consistent with the evidence reported by Bask (2002) and Goga and Serletis (2000) on chaotic dynamics for the nominal and real exchange rates of some inflation targeting countries.27

To the best of our knowledge, our paper is the first to bridge the gap between the closed economy literature on global analysis of Taylor rules and the open economy literature on local analysis of these rules. And by doing this, we unveil some crucial differences. On one hand, relative to the global analyses on closed economy models, we show that trade openness considerable widens the range of values of the risk aversion parameter under which cycles and chaos can arise. In this regard, our results are in the same spirit as those from Weder (2001) and Meng and Velasco (2003) of the RBC literature on production externalities: openness makes it easier to obtain expectations-driven fluctuations. On the other hand, relative to the local analysis on open economy models, we show that rules that imply local determinacy can still lead to global indeterminacy by inducing cyclical and chaotic dynamics. This is certainly in line with Benhabib et al (2001b, 2002a), but the key difference here is that the extent of disagreement between the local and global analyses depends on the degree of openness.

Our model can be extended in different directions. Because of tractability, our analytical results are restricted to the case of a flexible-price and perfectly-competitive economy. Using a continuous time framework, in Airaudo and Zanna (2012), we extend the current model to allow for monopolistic competition and nominal price rigidities in one sector (e.g., the non-traded goods sector). We prove analytically that cycles appear through a Hopf bifurcation with respect to the weight on the inflation of the flexible-price sector (e.g., the traded sector), which in the small open economy set-up is related to the degree of trade openness. Furthermore, in this paper we have assumed the government follows a fixed monetary policy rule. Many authors, including Benhabib et al. (2002b), have advocated for an explicit commitment to switch to a different monetary rule (exchange-rate-based or money-growth-based rules) to escape from or rule out liquidity traps. It would be interesting to assess whether cyclical and chaotic dynamics still occur in such regime-switching environments. We leave this extension for future research.

Interest Rate Rules, Endogenous Cycles, and Chaotic Dynamics in Open Economies
Author: Luis-Felipe Zanna and Mr. Marco Airaudo
  • View in gallery

    Local and global equilibrium analyses for active forward-looking rules, as the degree of openness α and the risk aversion coefficient σ vary. R* and RL stand for the target and the unintended low steady state, respectively.

  • View in gallery

    Orbit-bifurcation diagrams for the degree of openness

    They show the set of limit points for the nominal interest rate Rt, as a function of the degree of openness α, under Edgeworth complementarity (σ = 0.8) and sustitutability (σ = 2).

    Orbit-Bifurcation Diagrams

  • View in gallery

    Lyapunov exponents for forward-looking rules, as a function of the degree of openness (α) and under Edgeworth complementarity (σ = 0.8) and sustitutability (σ = 2).

    Lyapunov Exponents

  • View in gallery

    Basis of attraction for forward-looking rules for different combinations of the degree of openness α and the elasticity of the forward-looking rule to inflation £, under Edgeworth complementarity (σ = 0.8) and substitutability (σ = 2). See the main text for the interpretation of the colored areas.

    Basins of Attraction Diagrams

  • View in gallery

    Orbit-bifurcation diagrams for the degree of exchange rate pass-through

    Orbit-Bifurcation Diagrams for The Degree of Exchange Rate Pass-through

  • View in gallery

    Orbit-bifurcation diagrams with CES preferences

    Orbit-Bifurcation Diagrams with CES Preferences