## _wp12121 ## Source details **Canonical URL:** [_wp12121](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12121.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12121.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12121.pdf.json) --- ### Background and motivation - Benhabib et al. (2001b) show that, because of the zero lower bound (ZLB) on the nominal interest rate, simple Taylor rules can induce a liquidity trap driven by self-fulfilling expectations. - Benhabib et al. (2001b, 2002a) document that a global (non-local) analysis uncovers additional Taylor-rule-induced rational expectations equilibria, including endogenous limit cycles and chaotic dynamics. - Bullard (2010) interprets extremely low post-crisis interest rates in major developed economies as consistent with the Benhabib et al. (2001b) theoretical findings. ### Purpose and approach of the paper - Assess how self-fulfilling cyclical and chaotic fluctuations are affected by opening the economy to international trade in goods. - Conduct a global non-linear equilibrium analysis of a flexible-price small-open-economy model with traded and non-traded goods. - Monetary policy is a forward-looking active interest rate rule responding to expected future CPI inflation (Benhabib et al. (2002a) / Eusepi (2007) functional form). - Bridge closed-economy global Taylor-rule literature and open-economy local Taylor-rule literature. ### Main results and findings - Forward-looking Taylor rules are more prone to induce endogenous cycles and chaos the more open the economy is to trade (higher α). - Using a money-in-the-utility-function (MIUF) set-up: - Self-fulfilling complex fluctuations can occur around the target interest rate if consumption and money are Edgeworth substitutes (σ>1). - Self-fulfilling complex fluctuations can occur around the unintended low steady state if consumption and money are Edgeworth complements (σ<1). - In the complements case, liquidity traps can converge non-monotonically to a limit cycle around the unintended steady state. - Trade openness significantly enlarges the risk aversion parameter range under which forward-looking rules induce cycles and chaos, relative to closed-economy analyses. - Presence of a ZLB on nominal interest rates is a necessary condition for cyclical fluctuations. - MIUF closed-economy models generate complex dynamics only for a very restricted range of σ, but similar dynamics may occur for a much wider range of σ if the economy is sufficiently open. ### Mechanisms and key drivers - Global dynamics driven by interaction of: - an open-economy Fisher-type relation (from UIP and CPI definition), and - the non-linear forward-looking Taylor rule. - The modified Fisher equation requires ex-post real interest rate dynamics to be consistent with future real exchange rate changes. - Key to cycles and chaos is the elasticity of the real exchange rate to the policy interest rate; this elasticity depends on: - complementarity/substitutability between consumption and money (sign of U_{cμ} determined by σ), and - degree of trade openness α. ### Policy implications and recommendations - Interest rate rule design should account for structural characteristics such as degree of openness, not only the interest response coefficient to inflation. - Policy prescriptions from local analyses that ensure stability can still lead to cycles and chaos (global indeterminacy); the divergence between local and global results depends on openness α. - Imperfect exchange rate pass-through (distribution costs) can mitigate the propensity for cycles/chaos, implying trade and distribution structure matter for monetary policy risk. ### II. Model setup and key identities (selected exact forms and parameters) - Preferences (exact): E_0 ∑_{τ=0}^{∞} β^τ { [(h(c_τ)^θ − μ^d_τ) ]^{1−σ} −1 /(1−σ) + ψ(1−η^T_τ − η^N_τ) }. - Consumption aggregator: c_τ = (c^T_τ)^α (c^N_τ)^{(1−α)}. - Parameters: β, θ ∈ (0,1); ψ, σ > 0 but σ ≠ 1; α ∈ (0,1). - Real money balances: μ^d_τ = M^d_τ / π_τ. - Cross-derivative sign: sign{U_{cμ}} = sign{1−σ} ⇒ U_{cμ}<0 when σ>1 (Edgeworth substitutes), U_{cμ}>0 when σ<1 (Edgeworth complements). - Production: y^T_τ = z_τ (k^T)^{1−θ_T} (η^T_τ)^{θ_T}; y^N_τ = z_τ (k^N)^{1−θ_N} (η^N_τ)^{θ_N}. - CPI and inflation (as presented): π_τ ≡ (E_τ)^α (Π^N_τ)^{1−α} ^[ α/(1−α) ? ] (equation as in source); Π_τ ≡ π_τ / π_{τ−1} = 𝚤_τ^α (Π^N_τ)^{(1−α)} with traded inflation 𝚤_τ ≡ E_τ/E_{τ−1}. - Real exchange rate: e_τ ≡ E_τ / Π^N_τ; e_τ/e_{τ−1} = 𝚤_τ / Π^N_τ. - Uncovered interest parity / UIP-like relation under perfect foresight: R_τ = 𝚤_{τ+1} / β (equation (19)). ### II.B Monetary policy rule (exact form preserved) - Forward-looking rule: R_τ = ρ(E_τ Π_{τ+1}) ≡ 1 + (R^* −1) [ (E_τ Π_{τ+1} / Π^* )^{A_{R^*−1}} ] (equation (16)). - Target: R^* = Π^* / β > 1; zero lower bound satisfied R_τ > 1. - Active rule (Taylor principle): Assumption 0: ς ≡ A_{R^*} > 1 (denoted ς as degree of activism). ### II.D Reduced-form equilibrium and steady states (exact forms) - Real exchange rate as function of nominal interest: e_τ ≡ e(R_τ) = μ (R_τ / (R_τ −1))^ν (equation (20)), with ν ≡ (σ−1)(1−θ)(1−α_N) / [ σ(α_N + α(1−α_N)) + (1−α)(1−α_N) ]. - Sign relation: e'(R_τ) ≷ 0 for σ ≷ 1 (equation (21)). - Fisher-type open-economy relation: (e_{τ+1}/e_τ)^{1−α} = β R_τ / Π_{τ+1} (equation (22)). - Reduced first-order difference equation in R (equation (23) as presented). - Composite parameter Θ (exact): Θ ≡ (σ−1)(1−α)(1−θ_N) / [ σ(θ_N + α(1−θ_N)) ] + (1−α)(1−θ_N) with sign Θ ≷ 0 for σ ≷ 1 (equation (24)). - Definition 1: Perfect foresight equilibrium (PFE) — deterministic R_τ > 1 satisfying (23). - Existence result: under active rule (A_{R^*} > 1) and ZLB R_ss > 1, there exists a second unintended steady state R_Λ ∈ (1, R^*) (Proposition 1). ### III.A Local determinacy (exact linearization and proposition) - Log-linearization around R^* (equation (26)): - ĤR_{τ+1} = [ 1 + (R^*/A −1) Θ / (R^* −1) ] ĤR_τ. - Proposition 2 (exact implications): - There exist threshold values σ_δ > 1 and α_δ ∈ (0,1) such that: - the active steady state is a locally determinate equilibrium for any α ≥ 0 when σ ∈ (0, σ_δ), - but only for α > α_δ when σ ≥ σ_δ. - Interpretation: - Active forward-looking rules guarantee local determinacy when σ<1 for any openness. - When σ>1, sufficient openness (α>α_δ) is needed for local determinacy. - For σ>1 and α<α_δ, active rule can induce local sunspot-driven fluctuations. ### III.B Cycles and Chaos — global non-linear analysis (structure and sign results) - Reduced map: R_{τ+1} = φ(R_τ) with φ as in equations (27) and (28). - φ single-peaked for Θ>0 and single-troughed for Θ<0. - Negative derivative of φ at a steady state is necessary for endogenous cycles. - Sign results: - φ negatively sloped at active steady state if σ>1 (Edgeworth substitutability). - φ negatively sloped at passive steady state if σ<1 (Edgeworth complementarity). - Definitions: - Period-n cycle: fixed point of φ^n not of any lower order. - Topological chaos: φ topologically chaotic if there exists an uncountable set S of initial points with orbits that neither converge to one another nor to any periodic orbit. - Proposition 3 (summary): existence of flip bifurcation thresholds α_φ and α_δ (functions of σ) determine when passive or active steady state loses stability and period-2 cycles emerge. ### Numerical calibration, bifurcations, and country thresholds (benchmarks and exact values) - Benchmark parametrization (Table 2, time unit = quarter): - α_N =0.56 - β=0.99 - Π^* =1.031 1 4 (as presented in source) - R^* =1.072 1 4 - A_{R^*} =2.25 - 1−θ =0.03 - Orbit-bifurcation findings (Figure 2): - For σ=0.8 (Edgeworth complementarity) a stable period-2 cycle appears at about α≈0.15. - For σ=2.0 (Edgeworth substitutability) a stable period-2 cycle appears at about α≈0.37. - As α increases, classical period-doubling cascade to higher periods and eventual chaos. - Lyapunov exponents: - Chaos evidence for α>0.38 when σ=0.8. - Chaos evidence for α>0.5 when σ=2.0. - Openness widens σ-range for cycles (Table 3 exact intervals): - α=0: (0.82,1.47) - α=0.15: (0.80,1.60) - α=0.35: (0.75,1.95) - α=0.65: (0.62,10.87) - Basins of attraction and policy activism (Table 4 country thresholds, 1−θ=0.03 assumed): - Norway (σ=1.58, α=0.34, Π^*=0.025, α_N=0.57): 2-period cycles if ς>1.75; 1/Chaos if ς>2.26 - Sweden (σ=1.58, α=0.26, Π^*=0.02, α_N=0.65): 2-period cycles if ς>1.7; 1/Chaos if ς>2.17 - United Kingdom (σ=1.59, α=0.34, Π^*=0.02, α_N=0.45): 2-period cycles if ς>2.61; 1/Chaos if ς>5.47 - Australia (σ=1.58, α=0.31, Π^*=0.025, α_N=0.48): 2-period cycles if ς>2.26; 1/Chaos if ς>3.80 - New Zealand (σ=1.58, α=0.38, Π^*=0.015, α_N=0.56): 2-period cycles if ς>1.94; 1/Chaos if ς>2.75 ### Intuition and mechanism behind cycles (selected exact relations) - Representative equilibrium condition (equation (29) as presented) yields alternating high/low inflation realizations when passive rule instability combines with muted exchange-rate response under high openness. - Elasticity of real exchange rate to interest rate factor ϵ (exact expression as presented): - ϵ ≡ (σ−1)(1−θ)(1−α_N) / [ σ(α_N + α(1−α_N)) + (1−α)(1−α_N) ] (preserved form as in text). - For σ<1 with σ<σ_φ<1, ϵ is negative and its absolute value decreases as α→1, so higher openness can reduce exchange-rate response and facilitate instability and cycles. ### Extensions and robustness (exact findings) - Incomplete exchange rate pass-through (distribution cost ϖ, equation (30)): - Imperfect pass-through if ϖ>0; pass-through declines as ϖ increases. - Simulations (α=0.4): complex dynamics disappear when ϖ passes thresholds: about 0.84 for σ=0.8 and 0.07 for σ=2.0. - 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. - Incomplete markets: - With only one international bond and one domestic bond, same nonlinear difference equation (23) obtains; cycles/chaos results identical to complete-markets case. - Alternative policy timings: - Contemporaneous rules: openness affects cycles only when σ>1; cycles can be centered around active or passive steady state depending on α. - Backward-looking rules: PFE dynamics converge to either active or passive steady state (no cycles); liquidity trap is the only long-run equilibrium type. - Cash-in-Advance (CIA) timing: - Mapping differs only by coefficient Ψ (exact form preserved); Ψ>0 for σ∈(0;1) and Ψ<0 for σ>1. - Proposition 5: For σ>1 under CIA there exist thresholds σ_CIA>1 and α_CIA∈(0,1) such that period-2 cycles occur around active steady state for any α∈(0,1) if σ∈(1,σ_CIA) and for α>α_CIA if σ≥σ_CIA. - CES preferences and CRS technologies: - Main messages robust: greater openness increases propensity for cycles/chaos; center of cycles determined by complementarity/substitutability; cycles can appear at lower openness under CES. - Nominal price rigidities (quadratic adjustment costs): - Active steady state dynamically unstable for any α and ς>1, but numerical experiments show no cycles of any periodicity; liquidity traps are the only global indeterminacy found under sticky prices. ### Main conclusions (concise) - The global PFE set in a standard two-good SOE with an active forward-looking Taylor rule includes endogenous cycles of various periodicities and chaos for sufficiently high trade openness α. - Openness widens the σ-range under which cycles/chaos arise and can reverse stabilizing implications from local analyses: local determinacy may coexist with global indeterminacy. - Results robust to incomplete markets, alternative timings, CES preferences, and certain technology specifications; exchange rate pass-through and distribution costs materially affect occurrence of complex dynamics. *Source: _wp12121 - Section IV discusses the robustness of our main results under different extensions, including imperfect (PDF chapter/section provided).* ### References ............................................................................................................. ### _wp12121 - References ### Background and motivation - Bullard (2010) argues that 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). - Benhabib et al. (2001b) show that, 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 entirely driven by people’s self-fulfilling expectations; it is the natural outcome from a non-linear analysis of the prototype micro-founded dynamic model commonly used in macroeconomic policy discussions. - Benhabib et al. (2001b, 2002a) also show that a global rather than a local analysis unveils other Taylor-rule-induced rational expectations equilibria, such as endogenous limit cycles and chaotic dynamics. ### Purpose and approach of the paper - Assess how self-fulfilling cyclical and chaotic fluctuations are affected by opening the economy to international trade in goods. - Pursue a global non-linear equilibrium analysis of a traditional flexible-price small-open-economy model with traded and non-traded goods. - Monetary policy is modeled as an active forward-looking interest rate rule responding to expected future CPI inflation. - Bridge the gap between closed-economy literature on global analysis of Taylor rules and open-economy literature on local analysis of these rules. ### Main results and findings - Forward-looking Taylor rules are more prone to induce endogenous cycles and chaos the more open the economy is to trade. - Using a money-in-the-utility-function (MIUF) set-up (consumption and real money balances non-separable in utility), self-fulfilling complex fluctuations can occur around either: - the target interest rate if consumption and money are Edgeworth substitutes, or - the unintended low steady state if consumption and money are Edgeworth complements. - In the complements case, liquidity traps can converge non-monotonically to a limit cycle around the unintended steady state. - Trade openness significantly enlarges the risk aversion parameter range under which forward-looking rules induce cycles and chaos, relative to closed-economy global analyses. - 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. - While MIUF closed economy models generate complex dynamics only for a very restricted range of the risk aversion parameter, the same dynamics may occur for a much wider range of this parameter if the economy is sufficiently open. ### Mechanisms and key drivers - Global equilibrium dynamics are driven by the interaction of: - an open-economy version of the Fisher equation (obtained from the uncovered interest parity condition and the definition of CPI inflation), and - the non-linear Taylor rule. - The modified Fisher equation requires the ex-post real interest rate dynamics 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 - the degree of trade openness. ### Policy implications and recommendations - To avoid destabilizing endogenous cycles and chaos in open economies, the design of interest rate rules should account not only for the interest response coefficient to inflation but also for structural characteristics such as the degree of openness. - Policy prescriptions derived from local analyses that ensure macroeconomic stability in an open economy can still lead to cycles and chaos (global indeterminacy); the extent of disagreement between local and global analyses depends on the degree of openness. ### Paper organization (as stated) - Section II: presentation of the flexible-price model with main assumptions; definition of the open economy equilibrium; derivation of basic steady state results. - Section III: local and global equilibrium analyses for an interest rate rule responding to expected future CPI inflation, focusing on the role of the degree of openness. *Source: _wp12121 - References* ### Section IV discusses the robustness of our main results under different extensions, including imperfect ### _wp12121 - Section IV discusses the robustness of our main results under different extensions, including imperfect ### II. A Flexible-Price Model — setup and key identities - Economy: small open economy (SOE) with identical infinitely lived household-firm units. - Preferences (exact form): - Lifetime utility: E_0 ∑_{τ=0}^{∞} β^τ { [(h(c_τ)^θ − μ^d_τ) ]^{1−σ} −1 /(1−σ) + ψ(1−η^T_τ − η^N_τ) } (equation (1) as given). - Consumption aggregator: c_τ = (c^T_τ)^α (c^N_τ)^{(1−α)} (equation (2)). - Parameters: β, θ ∈ (0,1); ψ, σ > 0 but σ ≠ 1; α ∈ (0,1). - Real money balances: μ^d_τ = M^d_τ / π_τ where π_τ is CPI. - Interpretation: α measures degree of trade openness; α→0 closed, α→1 fully open. - Cross-derivative sign: sign{U_{cμ}} = sign{1−σ} so U_{cμ}<0 when σ>1 (Edgeworth substitutes), U_{cμ}>0 when σ<1 (complements). (Footnote: σ=1 separable, no equilibrium cycles.) - Production: - Traded and non-traded goods: y^T_τ = z_τ (k^T)^{1−θ_T} (η^T_τ)^{θ_T}; y^N_τ = z_τ (k^N)^{1−θ_N} (η^N_τ)^{θ_N} (equation (3)). - Labor shares: θ_T, θ_N ∈ (0,1); z_τ is aggregate productivity (stationary AR(1) process) and sole fundamental shock. - Prices and CPI: - Law of one price for traded goods; foreign traded price Π^T_w_τ normalized to 1, so Π^T_τ = E_τ (nominal exchange rate). - CPI (exact): π_τ ≡ (E_τ)^α (Π^N_τ)^{1−α} ^[ α/(1−α) ? ] (equation (4) as given in source; preserve as presented). - Gross CPI inflation: Π_τ ≡ π_τ / π_{τ−1} = 𝚤_τ^α (Π^N_τ)^{(1−α)} (equation (5) with traded inflation 𝚤_τ ≡ E_τ/E_{τ−1} and non-traded Π^N_τ). - Real exchange rate: e_τ ≡ E_τ / Π^N_τ (equation (6)); evolves as e_τ/e_{τ−1} = 𝚤_τ / Π^N_τ (equation (7)). - Model feature: complete exchange rate pass-through (Π^T_w_τ normalized to one). - Financial markets and household budget: - Assets: fiat money M^d_τ and nominal state-contingent claims Δ_{τ+1}. - Complete markets: flow constraint (equation (8)), period-by-period constraint (equation (9)), Non-Ponzi condition (equation (10)). - With expectations and risk-free gross nominal rate R_τ, E_τ Q_{τ,τ+1} = 1/R_τ. - Under free international capital mobility and assumptions Π^T_w_τ=1, β_ω=β, representative domestic marginal utility proportional to foreign: λ_τ = Λ λ^ω_τ and λ_τ constant over time (equation (17)). - Uncovered interest parity / UIP-like relation under perfect foresight: R_τ = ι_{τ+1} / β (equation (19)), where 1/β represents foreign international interest rate. - First-order conditions (exact forms preserved): - Intertemporal envelope: equation (11). - MRS traded vs non-traded: equation (12). - Labor marginal products equality: equation (13). - Money demand: equation (14). - Pricing of nominal contingent claims: equation (15). ### II.B Government and monetary policy - Government issues money M^s_τ and one-period domestic bond B^s_τ paying gross nominal rate R_τ; cannot issue/hold state-contingent claims; budget constraint ignored because taxes chosen to satisfy intertemporal budget. - Monetary policy: forward-looking interest rate rule (functional form from Benhabib et al. (2002a) / Eusepi (2007)): - R_τ = ρ(E_τ Π_{τ+1}) ≡ 1 + (R^* −1) [ (E_τ Π_{τ+1} / Π^* )^{A_{R^*−1}} ] (equation (16) as in source). - Target R^* = Π^* / β > 1; zero lower bound satisfied R_τ > 1. - Taylor principle (active rule): Assumption 0: rule active: ς ≡ A_{R^*} > 1. From now on ς referred to as degree of activism towards inflation. ### II.C International capital markets (key condition) - No-arbitrage linking domestic and foreign claim prices leads to λ_τ = Λ λ^ω_τ and constant λ_τ (equation (17)). - Combined with pricing eqn (15) yields R_τ = 1 / [ β E_τ (1/𝚤_{τ+1}) ] and under perfect foresight simplifies to equation (19): R_τ = 𝚤_{τ+1} / β. ### II.D Equilibrium — reduced form and steady states - Real exchange rate as function of nominal interest: e_τ ≡ e(R_τ) = μ (R_τ / (R_τ −1))^ν (equation (20)), where μ and ν depend on structural parameters; ν ≡ (σ−1)(1−θ)(1−α_N) / [ σ(α_N + α(1−α_N)) + (1−α)(1−α_N) ] (as given). - Sign relation: e'(R_τ) ≷ 0 for σ ≷ 1 (equation (21). - Fisher-type open-economy relation linking real exchange rate depreciation and ex-post real interest: - (e_{τ+1}/e_τ)^{1−α} = β R_τ / Π_{τ+1} (equation (22)). - Reduced first-order difference equation (core dynamic equation), representing the model in R: - [ (R_{τ+1} / (R_{τ+1} −1)) ]^Θ = (R_τ / R^*) ( (R^* −1) / (R_τ −1) )^{R^*−1} A ( (R_τ / (R_τ −1)) )^Θ (equation (23) as presented; dependent on composite parameter Θ defined below). - Composite parameter Θ (exact form preserved): - Θ ≡ (σ−1)(1−α)(1−θ_N) / [ σ(θ_N + α(1−θ_N)) ] + (1−α)(1−θ_N) for σ≷1 implying Θ ≷ 0 (equation (24) as given). - Definition 1: Perfect foresight equilibrium (PFE) — deterministic process {R_τ} with R_τ > 1 for all τ satisfying equation (23) given R^* and R_0. - Existence of two steady states: - Target (active) steady state R^* is a solution of steady-state condition (25). - Under active rule (A_{R^*} > 1) and zero lower bound R_ss > 1, there exists a second unintended (passive) steady state R_Λ ∈ (1, R^*) (Proposition 1). ### III. Local and Global Dynamics — assumptions and results - Parameter held constant for comparisons: β, θ, θ_N, and policy parameters A and R^*; vary openness α and risk aversion σ. - Additional assumptions introduced for analytical results: - Assumption 1: 1−θ > θ_N (1−θ_N)^(1/2) (R^* −1) [ 1 − (R^*/A) ] (exact inequality as in source); requires money sufficiently important (1−θ) large enough; used for local determinacy and period-2 cycles when σ>1. - Assumption 2: R^* −1 > A (R_Λ −1); requires positive spread between active and passive steady state; used for cycles around unintended low steady state when σ<1. - Discussion: Assumption 1 holds for typical calibration (example given: 1−θ = 0.03, θ_N = 0.56 leads RHS < 0.015 for wide parameter ranges); Assumption 2 generally holds for any active rule A_{R^*} > 1 (analytic approximation argument ln A < A−1). - A. Local determinacy - Log-linearization of (23) around R^* yields: - ĤR_{τ+1} = [ 1 + (R^*/A −1) Θ / (R^* −1) ] ĤR_τ (equation (26) exact form). - Local determinacy requires the linearized system to generate explosive dynamics because R_τ is non-predetermined. - Proposition 2 (exact statement): - Suppose the government follows an active forward-looking rule like (16). Then, there exist threshold values σ_δ > 1 and α_δ ∈ (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, σ_δ), but - only for α > α_δ when σ ≥ σ_δ. - Interpretation: - Active forward-looking rules guarantee local determinacy when consumption and money are complements (σ<1) for any openness; when σ>1, sufficient openness (α>α_δ) is needed. - When σ>1 (Edgeworth substitutes) and economy sufficiently open (α<α_δ), active rule can induce local sunspot-driven fluctuations (multiple local equilibria). - B. Cycles and Chaos — global non-linear analysis - Rewrite reduced map as R_{τ+1} = φ(R_τ) with φ defined by equations (27) and (28) (exact composite forms preserved). - Analytical properties of φ depend primarily on Θ (equation (24)). - φ can be single-peaked for Θ>0 or single-troughed for Θ<0; negative derivative of φ at a steady state is necessary for endogenous cycles in continuously differentiable maps. - Sign result: - φ is negatively sloped at active steady state if σ>1 (Edgeworth substitutability). - φ is negatively sloped at passive steady state if σ<1 (Edgeworth complementarity). - Definitions used for global dynamics (preserved): - Definition 2: Period-n cycle — fixed point of nth iterate φ^n but not of any lower order. - Definition 3: Topological chaos — φ topologically chaotic if there exists an uncountable set S of initial points in domain such that orbits starting in S neither converge to one another nor to any existing periodic orbit. - Proposition 3 (partial statement from source; exact thresholds noted as existing): - Suppose the government follows an active forward-looking rule like (16). Then there exist threshold values σ_φ ∈ (0,1) and σ_δ > 1 for σ, together with thresholds α_φ ∈ (0,1) and α_δ ∈ (0,1) for trade openness α, such that ... (Proposition 3 continues beyond provided text). - Key implications stated prior to Proposition 3: - Existence of endogenous period-2 cycles depends on degree of openness α and risk aversion σ via Θ. - Global indeterminacy (cycles and potentially topological chaos) can arise even when local determinacy holds, and the location (around active or passive steady state) depends on σ (σ>1 cycles around active steady state; σ<1 cycles around passive steady state). *Italic: Source — _wp12121 - Section IV discusses the robustness of our main results under different extensions, including imperfect (PDF chapter/section provided).* ### 1. period-2 equilibrium cycles occur around thepassive steady state,for ### _wp12121 - 1. period-2 equilibrium cycles occur around thepassive steady state,for ### Main finding - 1. period-2 equilibrium cycles occur around thepassive steady state,for  ### Parameter conditions - when∈ ¡ 0  ¤ - and for any≥0when∈ ¡   1 ¢  *Source: _wp12121 - 1. period-2 equilibrium cycles occur around thepassive steady state,for (PDF chapter/section).* ### 2. period-2 equilibrium cycles occur around theactive steady state,for ### _wp12121 - 2. period-2 equilibrium cycles occur around theactive steady state,for ### Global versus Local Equilibrium Analysis - Table 1 and Figure 1 contrast global analysis (existence of cycles around   and  ∗ ) with local analysis (local determinacy around  ∗ ). - Key global messages: - For  outside the range ¡  ;  ¢, endogenous period-2 cycles exist provided the economy is sufficiently open to trade. Openness () is a necessary condition for endogenous cycles. - If consumption and real money balances are Edgeworth complements (1) cycles can appear around the passive steady state; if Edgeworth substitutes (1) cycles can appear around the active steady state. - Global indeterminacy (cycles) can occur in parameter regions where local analysis indicates unique equilibrium (white area in Figure 1). For   the local condition for determinacy is identical to the sufficient condition for period-2 cycles — implying a direct contrast between local and global conclusions about openness (). ### Analytical Conditions, Propositions, and Corollary - Proposition 3 (summary implications): - Existence of flip bifurcation thresholds   and   (functions of ) determine when the passive or active steady state loses stability and period-2 cycles emerge. - Thresholds   (for passive steady state loss) and   (for active steady state loss) can be expressed as functions   () and   (). - Corollary 1: - Assume 6=1 and ∈[0,1). For given , period-2 cycles occur only for  in the interval ¡  ;  ¢, where   and   are strictly decreasing and strictly increasing functions in , respectively. - Therefore as  increases, the interval ¡  ;  ¢ widens and period-2 cycles occur for a much wider range of . - Cycles are around the active steady state for 1 and around the passive steady state for 1. ### Numerical Calibration and Simulations (benchmark and results) - Benchmark parametrization (Table 2, time unit = quarter): -   =056 - =099 -  ∗ =1031 1 4 -  ∗ =1072 1 4 -   ∗ =225 - 1−=003 - Orbit-bifurcation findings (Figure 2, varying  over (0,1), two  cases): - For =08 (Edgeworth complementarity) a stable period-2 cycle appears at about ≈015. - For =20 (Edgeworth substitutability) a stable period-2 cycle appears at about ≈037. - As  increases, classical period-doubling cascade: period-4, period-8, period-16, … and eventually odd periodicities and aperiodic chaotic dynamics. - Lyapunov exponents (Figure 3): - Evidence of chaos for 038 when =08. - Evidence of chaos for 05 when =20. - Openness widens risk-aversion range for cycles (Table 3): - =0: ¡  ;  ¢=(082147) - =015: ¡  ;  ¢=(080160) - =035: ¡  ;  ¢=(075195) - =065: ¡  ;  ¢=(0621087) - Basins of attraction and policy activism (Figure 4 and Table 4): - When =0 (closed economy) cycles and chaos are ruled out for any degree of policy activism . - As  increases, complex dynamics can be induced for quantitatively realistic . - Example thresholds (Table 4, forward-looking rule, 1−=003 assumed): - Norway (=158, =034,  ∗ =0025,   =057): 2-period cycles if 175; 1/Chaos if 226 - Sweden (=158, =026,  ∗ =002,   =065): 2-period cycles if 17; 1/Chaos if 217 - United Kingdom (=159, =034,  ∗ =002,   =045): 2-period cycles if 261; 1/Chaos if 547 - Australia (=158, =031,  ∗ =0025,   =048): 2-period cycles if 226; 1/Chaos if 380 - New Zealand (=158, =038,  ∗ =0015,   =056): 2-period cycles if 194; 1/Chaos if 275 ### Intuition and Mechanism Behind Cycles - Representative equilibrium condition (equation (29)): - [(( +2 ))]1−=[(( +1 ))]1−  ( +1 )  +1 - For   1, a flip bifurcation threshold   exists: below it the passive steady state is stable; above it stability is lost. - Role of elasticity  of real exchange rate to interest rate factor: - ≡(−1)(1−)(1−  ) [  +(1−  )]+(1−)(1−  ) - For   1,  is always negative and its absolute value decreases as →1. Thus higher openness reduces the impact of nominal interest rate changes on the real exchange rate, facilitating instability of the passive steady state and the emergence of cycles. - Mechanism for cyclical PFE: - A shock that raises expected inflation under a passive rule reduces the real interest rate; with high openness the exchange rate response is muted, requiring a larger compensating swing in future inflation—leading to alternating high/low inflation realizations (period-2 cycle). ### Extensions and Robustness (main findings) - Incomplete exchange rate pass-through (distribution cost , equation (30)): - Imperfect pass-through obtained when 0; pass-through declines as  increases. - Simulations (Figure 5, set =04): - Complex dynamics (chaos and period-2 cycles) disappear once  passes thresholds: about 084 for =08 and 007 for =20. - 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. - Incomplete markets: - With only one international bond and one domestic bond, UIP condition   =  and   = +1 still imply the same nonlinear difference equation (23); cycles/chaos results are identical to complete-market case. Cycles may also appear in foreign bond accumulation and current account dynamics. - Alternative policy rule timings: - Contemporaneous rules (  responds to  ): openness affects cycles only when 1, and openness determines whether cycles are centered around active or passive steady state (low vs intermediate openness). - Backward-looking rules (respond to past inflation): PFE dynamics always converge to either active or passive steady state (no cycles); liquidity trap as only long-run equilibrium. - Cash-in-Advance (CIA) timing for money in utility: - Proposition 5: For 1 under CIA there exist thresholds   >1 and   ∈(0,1) such that period-2 cycles occur around the active steady state for any ∈(0,1) if ∈¡1,  ¢ and for   if ≥  . - CES preferences (parameter ): - Replace Cobb-Douglas with CES: intratemporal elasticity = (1−)−1; Edgeworth complementarity if 1−, substitutes if 1−. - Numerical results (Figure 6): main messages robust — more openness increases propensity for cycles/chaos; cycles' center determined by complementarity/substitutability; cycles can appear at lower openness under CES. - CRS technologies and labor disutility modification: - Under CRS production (with appropriate labor disutility specification), global PFE dynamics still described by mapping  +1 =(  ); analytical conditions and numerical occurrence of cycles/chaos remain. ### Policy Implications - Trade openness () is a key determinant of global indeterminacy: higher openness enlarges parameter regions (in  and ) where active forward-looking Taylor-type rules induce endogenous period-2 cycles, higher-order cycles, and chaos. - Local determinacy conclusions can be misleading: policies that satisfy local determinacy conditions may still produce global indeterminacy (cycles/chaos) when openness is sufficiently high. - Policy activism (higher elasticity  of the interest-rate rule to expected inflation) can trigger global complex dynamics in open economies for realistic parameter values — country-specific calibrations (Table 4) indicate thresholds within plausible ranges for some inflation-targeting countries. - Imperfect exchange rate pass-through (higher distribution costs ) can mitigate the propensity for cycles/chaos; thus structural features of trade and distribution affect monetary policy risks. ### Main Conclusions (concise) - The set of global perfect-foresight equilibria in a standard two-good small-open-economy with an active forward-looking Taylor rule includes endogenous cycles of various periodicities and chaos for sufficiently high trade openness. - Openness widens the risk-aversion parameter range under which cycles/chaos arise and can reverse the stabilizing implication of openness found in local analyses: local determinacy may coexist with global indeterminacy. - Results are robust to incomplete markets, alternative timing assumptions for policy and money in utility, CES preferences, and certain technology specifications; exchange rate pass-through and distribution costs materially affect the occurrence of complex dynamics. *Source: _wp12121 - 2. period-2 equilibrium cycles occur around theactive steady state,for (IMF working paper PDF content unit provided).* ### 1. Assume that1Recall that, by the definition (24), this implies0Consider the two ### _wp12121 - 1. Assume that1Recall that, by the definition (24), this implies0Consider the two ### 1. Case split: 1 (0) - Two subcases: (a)  ≤ −  ∗ −1  and (b) −  ∗ −1     0. - (a) If  ≤ −  ∗ −1 : - The mapping  is monotonically increasing for any    1, with  0 ¡   ¢  0 and  0 ( ∗ )  1. - This rules out cycles of any periodicity; the only PFE dynamics are paths that monotonically converge to the passive steady state. - (b) If −  ∗ −1     0: - Properties of  on ¡     ¢: - a) : ¡     ¢ → (1; +∞), where   and   are solutions to (  ) = 1 with   ∈ ¡1;   ¢ and     ∗. - b) (·) is continuously differentiable on ¡     ¢. - c)  has a unique minimum at  . - d) lim   → + (  ) = lim   → − (  ) = +∞. - e)  0 ( ∗ )  1 always while  0 ¡   ¢ T 0. - If  0 ¡   ¢  0 (hence  attains minimum at   ∈ ¡    ∗ ¢) define auxiliary function () ≡  −  2 (). A period-2 cycle is a solution to () = 0. - Three alternatives depending on relation of (  ) to ̃ ≡  −1 ( ∗ ): i. If (  ) ≥ ̃: - The set [̃;  ∗] is invariant;  is continuously differentiable on [̃;  ∗]. - (  ) = ( ∗ ) = 0;  0 (  ) = 1 − [ 0 (  )] 2 for   =   ,  ∗. - By Assumption 0 and algebra,  0 ( ∗ ) =   ∗  1 ⇒  0 ( ∗ )  0. - The Intermediate Value Theorem implies existence of   ∈ ¡    ∗ ¢ with (  ) = 0 if  0 ¡   ¢ = 1 − [ 0 ¡   ¢ ] 2  0 ⇔  0 ¡   ¢  −1. - Differentiation yields condition (31):  0 ¡   ¢ S −1 iff  T Υ  , where Υ  ≡ 1 2 £   ¡ 1 −  ∗ −1  ¢ −1 ¤ and −  ∗ −1   Υ   0. - Hence: if  ∈ (Υ  ; 0):  0 ¡   ¢  −1 ⇒ period-2 cycles occur around   (equation (32)). ii. If (  ) ∈ (  ; ̃): -  ν : [̃;  ∗] → [  ;  ] for ν = 1, 2 ⇒  is continuously differentiable on [̃;  ∗]. - Same condition (32) yields period-2 cycles around passive steady state. iii. If (  ) ≤  : - There exist closed intervals I1 ≡ [ A ;  B ] with () <   for  ∈ I1, and I2 ⊂ (̃;  ) and I3 ⊂ ( B ;  ∗) with : I2 → I1 and : I3 → I1, so  2 () <   for  ∈ I2 ∪ I3. -  is continuously differentiable only over [̃;  ∗] \ ∪3 i=1 Ii. - Considering interval (̃;  C ), with  C =  −1 ( B ), we have (̃) < 0 and lim → C () > 0. By Intermediate Value Theorem there exists   ∈ (̃;  C) with (  ) = 0. Thus period-2 cycles always exist in this case. - Unified statement: when   1,  ∈ (Υ  ; 0) is a sufficient condition for existence of period-2 cycles around the passive steady state. - Using definition of  in (24) this is equivalent to: -   (1 − )(1 −   )(1 − ) + Υ  (  + 1 −   ) (1 − )(1 −   )(1 −  + Υ  ) ≡   - Properties: -    1 always. -   ≥ 0 iff  ≤ (1 −   ) ¡ 1 −  + Υ  ¢ (1 −   )(1 − ) − Υ    ≡   with    1. - Conclusion: - Period-2 cycles around passive steady state occur for     when  ∈ ¡0;   ¤. - For any  ≥ 0 when  ∈ ¡  ; 1 ¢. ### 2. Case split:   1 (  0) - Under definition (24)  > 0. The mapping  has properties on (1; +∞): - a) :(1; +∞) → (1; +∞), continuously differentiable. - b)  has global maximum at   ≡ (1 + ) ¡ 1 −  ∗ −1  ¢ −1 where     . - c) lim   →1+ (  ) = lim   →∞ (  ) = 0. - d)  0 ¡   ¢  1 while  0 ( ∗ ) T 0. - Single-peaked on (1; +∞). From d) there cannot be cycles around the passive steady state. - Suppose  0 ( ∗ )  0 (otherwise no cycles around active steady state). Then maximum occurs at   ∈ ¡    ∗ ¢. - Define () ≡  −  2 (), continuously differentiable on (1; +∞). Period-2 cycles solve () = 0. - (  ) = ( ∗ ) = 0 and  0 ¡   ¢ = 1 − [ 0 ¡   ¢ ] 2  0. - Intermediate Value Theorem ⇒ existence of   ∈ ¡    ∗ ¢ where (  ) = 0 if  0 ( ∗ )  −1. - Differentiation yields condition (33):  0 ( ∗ ) S −1 iff  S Υ  , where Υ  ≡ 1 2 ( ∗ −1) ¡ 1 −  ∗  ¢  0 (since   ∗  1 and  ∗  1). - Hence: if  ∈ (0; Υ  ):  0 ( ∗ )  −1 ⇒ period-2 cycles occur around  ∗. - Using definition of  in (24) this condition is equivalent to: -   (1 − )(1 − )(1 −   ) + Υ  (  + 1 −   ) (1 − )(1 −   )(1 −  + Υ  ) ≡   - Properties: -    1 always. -   ≥ 0 iff  ≥ (1 −   ) ¡ 1 −  + Υ  ¢ (1 − )(1 −   ) − Υ    ≡   with    1 by Assumption 1. - Conclusion: - Period-2 cycles around the active steady state occur for     when    . - For any  ≥ 0 when  ∈ ¡1;   ¢. ### CIA timing for money in utility (Appendix A.4) - Modified global PFE dynamics under forward-looking interest rate rule (16) (equation (34)): - The mapping differs from CWID counterpart (23) only by coefficient Ψ ≡ 1 −   (1 − ) (1 − )(1 − )(1 −   ) . - Ψ > 0 for  ∈ (0; 1) and Ψ < 0 for  > 1. - For  > 1, Ψ < 0 and  > 0: - K(·) strictly increasing on (1; +∞) and globally invertible; hence (  ) = K −1 ((  )) well-defined. - If  <  1 −  ∗ −1  × (1 − Ψ) then (·) is single-peaked with critical point   = 1 + Ψ (1 −  ∗ −1  )  1, and  behaves like a logistic map on (1; +∞). - By analogous arguments to Proposition 3, a sufficient condition for period-2 cycles around the active steady state is  0 ( ∗ )  −1. - This is equivalent to    1 −  ∗ −1  ×  ∗ −1 2 ∗ . - There exist thresholds    1 and   ∈ (0; 1) such that: - Period-2 cycles exist for any  ∈ (0; 1) if    . - For    , period-2 cycles exist for    . ### B. Contemporaneous Rules (summary and numerical evidence) - Contemporaneous rule:   = 1 + ( ∗ − 1) ¡    ∗ ¢   ∗ −1 with  ∗ =  ∗  and    1. - Equilibrium dynamics described by implicit system (35); explicit forward/backward maps not available. - Numerical simulations (using parametrization of Table 2,  = 2.5) produce Figure 8: - For  ∈ {0.01; 0.38; 0.90}, plot  +1 = (  ),  2 (·) and  3 (·). - Results: -  ∈ {0.01; 0.38} (almost closed and moderately open economies): second and third iterates have fixed points different from steady states  ∗ and   ⇒ cycles of periods 2 and 3 exist. - By Sarkovskii’s Theorem and Li and Yorke’s Theorem:  features cycles of any order and aperiodic cyclical dynamics (topological chaos). - For  = 0.01 cycles and chaos appear around the active steady state; for  = 0.38 they occur around the passive steady state. - For very open economies ( = 0.90) no cycles or chaotic dynamics appear;  is monotone and standard liquidity traps are the only global equilibrium multiplicity. - Additional numeric thresholds (footnote): - Period-2 cycles around active steady state appear when  ∈ (0.001; 0.22). - Period-3 cycles around active steady state occur when  ∈ (0.001; 0.16). - Period-2 cycles around passive steady state for  ∈ (0.25; 0.43). - Period-3 cycles (and chaos) exist for  ∈ (0.33; 0.38). - For   0.39 only liquidity traps exist. - Proposition 6 (summary): 1. If consumption and money are Edgeworth complements ( ∈ (0; 1)), no equilibrium cycles of any periodicity for any  ∈ (0; 1). 2. If consumption and money are Edgeworth substitutes (  1): - (a) No equilibrium cycles if economy sufficiently open. - (b) Cyclical and chaotic dynamics occur around passive steady state for intermediate degrees of openness. - (c) Cyclical and chaotic dynamics around active steady state occur if economy sufficiently closed. ### C. Backward-Looking Rules (summary) - Backward-looking rule:   ≡ 1 + ( ∗ − 1) ¡  −1  ∗ ¢   ∗ −1 with  ≡   ∗  1. - System forms two first-order difference equations (from combination of (20) and (22)). - Numerical simulations for  ∈ {0.8; 1.5; 2.0; 2.5} and varying  show no cycles or chaos; interest rate converges to active or passive steady state. ### D. Nominal Price Rigidities (sticky prices) - Introduce quadratic price adjustment costs; representative agent maximizes expected utility with adjustment cost term: - −  2 Ã      −1 −  ∗ ! 2 - Under contemporaneous interest rate rule and symmetry equilibrium conditions given (see text). - Two steady states persist. Numerical experiments varying  ∈ [2.8; 44] (Dib (2003) range), elasticity   1 and combinations of  and : - Active steady state is dynamically unstable for any  and   1. - Across wide experiments, no cycles of any periodicity detected around either steady state. - Under sticky prices, liquidity traps are the only type of global indeterminacy found. ### E. MIUF versus MIPF isomorphism and Proposition 7 - MIPF small open economy (fixed traded endowment , non-traded technology    =    1− , utility   =  1−  1− with composite consumption) yields: - Modified Fisher equation as (22): ( +1 /   ) 1− =     +1. - Global dynamics described by non-linear difference equation with same form as (23) but with  ≡ (1 − )     + (1 − )(1 − )  0. - Proposition 7 (MIPF set-up): - With an active forward-looking rule like (16), there exist threshold values    0 and   ∈ (0; 1) such that period-2 equilibrium cycles occur around the active steady state: - for any  ≥ 0 when  ∈ (0;   ), - and for     when  ≥  . - Conclusion: A SOE-MIPF model can be isomorphic to the MIUF (with Edgeworth substitutability) in terms of existence of period-2 cycles under active interest rate rules. *Italic: Content unit drawn from provided IMF PDF chapter/section.* ### References ### References ### Monetary policy rules, determinacy, and liquidity traps - Benhabib, J., S. 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