## wpiea2022112-print-pdf

## Source details

**Canonical URL:** [wpiea2022112-print-pdf](https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022112-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2022/english/wpiea2022112-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2022/english/wpiea2022112-print-pdf.pdf.json)

---

### Introduction and motivation
- Expectations crucial for central bank effectiveness as natural interest rates decline and forward guidance used to address the ELB.
- New Keynesian models generate a “forward guidance puzzle” (FGP); behavioral alternatives (finite planning horizons, cognitive discounting, partial myopia) have been proposed to address it.
- This paper embeds cognitive discounting into an incomplete-asset-market New Open Economy Macroeconomics (NOEM) framework, extending Gabaix (2020) to an open economy.

### Main contributions and headline findings
- Behavioral extensions materially alter equilibrium conditions and exacerbate unit root/stationarity issues typical of incomplete-market small open economy models.
- For degrees of myopia in the authors’ estimated range, the behavioral model:
  - Largely resolves several UIP-related puzzles: forward premium puzzle (Fama, 1984); predictability reversal puzzle (Bacchetta and van Wincoop, 2010); Engel puzzle (Engel, 2016); and forward guidance exchange rate puzzle (Galí, 2020).
  - Explains why UIP holds exactly in agents’ subjective expectations but appears to fail under a rational-expectations testing framework: realized exchange rate outcomes are biased indicators of non-rational ex ante expectations.
  - Weakens efficacy of announcements about future interest paths and of “low for longer” (LFL) policies, helping resolve the FGP in an open economy.
  - Attenuates the effects of persistent monetary shocks or interest-rate-smoothing because agents underestimate persistence of interest-rate changes.
  - Under complete markets and optimal monetary policy, behavioral discounting generates a unit root in the nominal exchange rate: temporary domestic cost-push shocks produce permanent appreciations (if positive) or depreciations (if negative).
  - Greater cognitive discounting implies more persistent net foreign assets (NFA) and real exchange rates and larger international monetary spillovers because cognitive discounting attenuates expenditure switching.

### Theoretical setup and myopia specification
- Two-country NOEM with Home share ζ∈(0,1) and Foreign share 1−ζ; focus on Home.
- Households: infinitely-lived with utility
  - U^h_t = Ê_t Σ_{T=t}^∞ β^{T−t} [ (C^h_T)^{1−σ}/(1−σ) − (N^h_T)^{1+φ}/(1+φ) ], parameters: 0<β<1; σ>0; φ>0.
- Consumption aggregator:
  - C^h_t = [ (1−α)^{1/η} (C^h_{H,t})^{(η−1)/η} + α^{1/η} (C^h_{F,t})^{(η−1)/η} ]^{η/(η−1)}, α∈(0,1); η>0.
- Financial assets: one-period Home bonds B^h_t and foreign-currency bonds B^{*,h}_t; real exchange rate Q_t ≡ ε_t P^*_t / P_t; risk premium Φ_t = Φ(B^*_t).
- Firms: Dixit–Stiglitz final goods, producer currency pricing, Calvo friction θ∈(0,1); production Y = z_t N_t.
- Myopia (cognitive discounting) via perceived law of motion:
  - X_{t+1} − X = m G_X(X_t − X, ε_{t+1}), with 0 ≤ m ≤ 1.
  - m = 1 → rational expectations; lower m → faster perceived reversion to steady state.
- Behavioral k-period ahead expectations approximation:
  - Ê_t{X_{t+k} − X} = m^k E_t{X_{t+k} − X}.

### Key linearized equilibrium relations (behavioral)
- Behavioral IS curve (Equation (13)):
  - ˆC_t = m E_t ˆC_{t+1} − (1/σ)(ˆi_t − m E_t ˆπ_{t+1}) + (1−m) (1−β)/(1+σ μ φ) ˆB^*_t.
- Behavioral UIP (Equation (14)):
  - ˆi_t − m E_t{ˆπ_{t+1}} = ˆi^*_t − m E_t{ˆπ^*_{t+1} − ˆQ_{t+1}} − φ ˆB^*_t, where φ = Φ′(0).
- Phillips curve (Equation (15)):
  - ˆπ_{H,t} = m β E_t{ˆπ_{H,t+1}} + κ ˆMC_t, κ ≡ (1−θ)(1−βθ)/θ.
- Real marginal cost (Equation (16)):
  - ˆMC_t = σ ˆC_t + φ ˆY_t + α/(1−α) ˆQ_t − (1+φ) ˆz_t.
- CPI inflation decomposition (Equation (17)):
  - ˆπ_t = ˆπ_{H,t} + α/(1−α) (ˆQ_t − ˆQ_{t−1}).
- Net foreign asset law of motion (Equation (19)):
  - ˆB^*_t = β^{−1} (ˆB^*_{t−1} + ˆY_t − ˆC_t).
- Linearized policy rule (Equation (20)):
  - ˆi_t = ρ ˆi_{t−1} + (1−ρ)(φ_π ˆπ_t + φ_y ˆY_t) + ν_t.

### Calibration, estimation, and empirical fit
- Calibration (Table 1):
  - α 0.4 Openness
  - β 0.99 Discount Factor
  - η 1 Elasticity of Substitution Between Home and Foreign Goods
  - φ 3 Inverse Frisch Elasticity of Labor Supply
  - σ 1 Inverse Intertemporal Elasticity of Substitution
  - μ 1.2 Gross Product Markup
  - θ 0.85 Calvo Probability of No Price Adjustment
  - ρ 0.9 Interest Rate Smoothing Parameters
  - φ_π 1.5 Inflation Feedback Taylor Rule
  - φ_y 0.125 Output Feedback Taylor Rule
  - φ 0.01 Intermediation Costs Debt Sensitivity
- Model frequency: quarterly; β = 0.99 implies steady state nominal interest rate of 4 per cent per annum (zero average inflation).
- Values explored for m ∈ [0.5, 1]; literature references: Gabaix (2020) m = 0.85; Fuhrer and Rudebusch (2004) m = 0.65; Gust et al. (2021) m ≈ 0.5; Ilabaca et al. (2020) m_households = 0.71 and m_firms = 0.41.
- Bayesian estimation (Canada as Home, US as Foreign), baseline sample 1972–2007: posterior mean of m ≈ 0.53.
- Estimation posterior highlights (Table E.1 excerpts):
  - m Cognitive Discounting: Posterior Mean 0.53 [0.44, 0.62]
  - ρ^H Interest Rate Smoothing: Posterior Mean 0.84 [0.80, 0.87]
  - φ_π^H Taylor Rule, Inflation: Posterior Mean 1.63 [1.23, 2.01]
  - σ Intertemporal Elasticity: Posterior Mean 3.11 [2.27, 3.91]
  - θ Calvo Probability: Posterior Mean 0.95 [0.93, 0.97]
- Shock parameter posteriors (Table E.2 excerpts):
  - ρ_g AR H Preference: Posterior Mean 0.97 [0.96, 0.98]
  - ρ_ρ AR Risk Premium: Posterior Mean 0.98 [0.97, 0.99]
  - σ_ν SD H Monetary Policy: Posterior Mean 0.32 [0.28, 0.35]

### Stationarity, determinacy, and market completeness
- Incomplete markets models exhibit stationarity issues; debt-elastic risk premium φ>0 commonly introduced for stationarity.
- Behavioral discounting (m<1) exacerbates stationarity problems:
  - IS equation (13) can become explosive when m<1 absent sufficient stationarizing mechanism.
  - Lower m requires larger φ for stationarity; openness (higher α) mitigates stationarity concerns.
- Myopia reduces indeterminacy regions: weaker φ_π responses may suffice for uniqueness when m<1 (example: baseline calibration m = 0.85 determinacy if φ_π > 0.555; reducing α by half lowers threshold to 0.545).
- Complete markets case (Appendix C): last term in IS curve (13) vanishes, yielding
  - ˆC_t = m E_t ˆC_{t+1} − 1/σ [ ˆi_t − m E_t ˆπ_{t+1} ].
  - Combined relations give
    - ˆi_t − m E_t {ˆπ_{t+1}} = ˆi^*_t − m E_t {ˆπ^*_{t+1}} − ˆQ_{t+1} − ˆQ_t.
  - Stationarizing risk premium φ ˆB^*_t absent; because φ small, behavioral discounting can also address UIP anomalies under complete markets.
- Empirically, Fama regression coefficients are similar in complete- and incomplete-market variants for given parameter combinations.

### Exchange rate dynamics and resolution of UIP-related puzzles
- Behavioral UIP reformulation yields (Equation (25)–(26)):
  - ˆi_t − ˆι^*_t = m E_t{∆ˆε_{t+1}} − (1−m) ˆQ_t, with ˆι^*_t ≡ ˆi^*_t − φ ˆB^*_t.
  - Rearranged: E_t{∆ˆε_{t+1}} = (1/m)(ˆi_t − ˆι^*_t) + (1/m − 1) ˆQ_t.
- Population regression slope a_1 (Equation (28)):
  - E a_1 = 1/m + (1/m − 1) Corr(ˆQ_t, ˆi_t − ˆι^*_t) × Std(ˆQ_t)/Std(ˆi_t − ˆι^*_t).
- Quantitative Fama regression coefficients (Table 2):
  - For m=0.50: ρ=0.95 → −0.07; ρ=0.90 → 0.17; ρ=0.75 → 0.51; ρ=0.50 → 0.75.
  - For m=0.75: ρ=0.95 → −0.04; ρ=0.90 → 0.36; ρ=0.75 → 0.69; ρ=0.50 → 0.86.
  - For m=0.90: ρ=0.95 → 0.37; ρ=0.90 → 0.67; ρ=0.75 → 0.86; ρ=0.50 → 0.94.
  - For m=1.00: all ρ → 1.00.
  - Estimated model: 0.44 (for m = 0.53, ρ = 0.84).
- Mechanism: m < 1 increases the (1/m) term above 1 (worsening standard Fama expectation) but the (1/m − 1) ˆQ_t term, when ˆQ_t negatively correlated with interest differentials, offsets and can produce empirically consistent low or negative slopes.
- Predictability sign reversal and Engel condition:
  - Behavioral discounting generates sign reversals in Engel regressions b_{s,1} as horizon s increases; rational expectations (m=1) fails to produce such reversals.
  - Estimated model (m ≈ 0.53, moderately high ρ) satisfies sign reversal and Engel’s excess volatility condition.
  - Intuition: domestic easing → persistent real depreciation → initial NFA accumulation then reversal → excess returns flip sign over horizons; high interest-rate inertia (high ρ) amplifies persistence and can violate Engel condition if excessive.

### Monetary policy transmission, forward guidance, and LFL
- Forward-iterated UIP for real exchange rate (Equation (34)):
  - ˆQ_t = −E_t ∑_{T=t}^∞ m^{T−t} [ˆr_T − ˆr^*_T + φ ˆB^*_T].
  - Cognitive discounting dampens the effect of future real interest rate changes on current exchange rate, mitigating the exchange rate forward guidance puzzle.
- Consumption iteration (Equation (35)):
  - ˆC_t = −(1/σ) E_t ∑_{T=t}^∞ m^{T−t} ˆr_T + (1−m)/(1−β) 1/(1 + σ μ φ) E_t ∑_{T=t}^∞ m^{T−t} ˆB^*_T.
  - Absent discounting and with forward guidance horizon H, an anticipated 1pp decrease in real rate ten periods ahead implies consumption is (1/σ) percent above steady state for exactly eleven periods.
- IS equation and iterated form (Equations (36)–(37)) show openness and trade elasticity influence direct effects of real rate changes on output; criterion for stronger direct effect versus closed economy:
  - η < (1−α)/(2−α) σ^{−1} ≤ (1/2) σ^{−1}.
- Inflation representations (Equations (38)–(39)):
  - For m = 1: inflation sensitivity to FG horizon grows approximately linearly.
  - For lower m: relationship can be non-monotonic or decreasing; open-economy import-price component diminishes FGP for inflation.
- Empirical IRF findings:
  - Conventional 25bp (100 bp annualized) negative monetary shock: discounting dampens effects on inflation more than on exchange rate; output response in open economy less sensitive to discounting; discounting increases persistence of NFA and delays sign reversal in excess returns.
  - Low-for-longer experiments: with m = 1 LFL very effective; for moderate discounting (e.g., m = 0.9) LFL can be less efficient than comparable conventional stimulus; potency of LFL more affected by discounting in closed economy.

### International spillovers and cross-country heterogeneity
- Iterated foreign-home output relation (Equation (40)):
  - ˆY_t = α ˆY^*_t (Demand Channel) + η α(2−α)/(1−α) ˆQ_t (Expenditure Switching) − (1−α)/σ E_t ∑_{T=t}^∞ m^{T−t} ˆr_T + (1−m)(1−α)/(1−β) 1/(1 + σ μ φ) E_t ∑_{T=t}^∞ m^{T−t} ˆB^*_T.
- Conditional spillover (Home keeps real rate constant) approximation (Equation (41)):
  - ˆY_t ≈ −[α/σ + η α(2−α)/(1−α)] E_t ∑_{T=t}^∞ m^{T−t} ˆr^*_T (omitting small NFA terms).
- Implications:
  - Unless η very low, coefficient on foreign real interest path is positive → foreign monetary easing tends to be expansionary for Home if Home does not tighten.
  - Greater myopia reduces impact of future foreign real rate changes on Home output; international transmission weaker with lower m.
  - If foreign agents less myopic (m^* > m), demand channel strengthens and spillovers can be less negative or even positive.

### Optimal monetary policy with behavioral agents
- Approximate welfare loss (Equation (43)):
  - U_t ≈ −(1−α)/2 E_t Σ_{T=t}^∞ β^{T−t} [ μ/(κ(μ−1)) ˆπ_{H,T}^2 + (1 + φ) ˆx_T^2 ] + t.i.p.
- Phillips curve in output-gap terms (Equation (44)):
  - ˆπ_{H,t} = m β E_t {ˆπ_{H,t+1}} + κ (1 + φ) ˆx_t + ξ_t.
- Timeless-standards optimal targeting rule (Equation (45)):
  - ˆπ_{H,t} + (μ−1)/μ (ˆx_t − m ˆx_{t−1}) = 0.
- Optimal producer price level (Equation (46)):
  - ˆP_{H,t} = (μ−1)/μ [ ˆx_t − (1−m) ∑_{T=0}^{t−1} ˆx_T ].
- Key implications:
  - When m < 1, price-level targeting is not optimal in the long run: positive cost-push shocks produce permanent increases in price levels.
  - With commitment and m = 1, optimal policy induces deeper/recurrent recessions to promise future deflation; as m falls, motive weakens and optimal policy resembles discretionary outcomes more closely.
  - For m = 0, optimal responses equal discretionary outcome; output falls only in shock period while price level increases on impact and remains flat thereafter.
  - Open-economy: optimal tightening appreciates exchange rate on impact and then depreciates; with myopia exchange rate can be permanently weaker. Nominal exchange rate can follow a random walk under optimal policy, reducing desirability of exchange rate as nominal anchor.
  - Behavioral agents make NFA less responsive to shocks; for m = 0, NFA does not respond to temporary cost-push shocks.

### Mechanisms summary and trade-offs
- Distinction between subjective expectations (where UIP holds) and realized outcomes (biased under non-rational agents) reconciles UIP anomalies with survey evidence.
- Myopia dampens forward-looking channels: forward guidance, LFL, and international spillovers are less powerful when agents discount future events cognitively.
- Market incompleteness matters: NFA dynamics under incomplete markets are central to generating predictability sign reversal; in complete markets some channels are muted but many attenuation effects from behavioral discounting remain.
- Trade-offs: behavioral modeling complicates equilibrium (stationarity/determinacy issues) and alters optimal policy prescriptions (weakens price-level targeting and exchange-rate anchoring), but yields better empirical fit to UIP-related puzzles.

_Italic: Source — wpiea2022112-print-pdf_

### 1.    Introduction1

### 1. Introduction

### Background and motivation
- Expectations play a central role in determining the effectiveness of central bank actions, especially as natural interest rates decline and policymakers increasingly rely on forward guidance to mitigate the effective lower bound (ELB) on interest rates.
- The New Keynesian (NK) framework produced mechanisms that made forward guidance appear counterfactually powerful, known as the “forward guidance puzzle” (FGP).
- Replacing full rationality with behavioral alternatives (finite planning horizons, cognitive discounting, partial myopia) has been a leading approach to addressing the FGP (examples: Woodford (2019); Gabaix (2020); Gust et al. (2021)).

### Contribution and key findings
- The paper develops an open economy extension of Gabaix’s (2020) behavioral model by embedding cognitive discounting into a standard incomplete asset market new open economy macroeconomics (NOEM) framework.
- Behavioral extensions significantly modify several key equilibrium conditions and exacerbate the well-known unit root problem in small open economy models arising from asset market incompleteness.
- For degrees of myopia in the authors’ estimated range, the behavioral model:
  - Largely resolves several uncovered interest parity (UIP)-related puzzles: the forward premium puzzle (Fama, 1984), the predictability reversal puzzle (Bacchetta and van Wincoop, 2010), the Engel puzzle (Engel, 2016), and the forward guidance exchange rate puzzle (Galí, 2020).
  - Explains why UIP holds exactly in terms of agents’ subjective expectations but appears to fail when tested under a rational expectations assumption: because agents are not rational, ex post exchange rate realizations are a biased read of their ex ante expectations.
- Behavioral discounting weakens the efficacy of future interest path announcements and of “low for longer” policies, helping to resolve the FGP in an open economy context.
- Persistent monetary shocks or interest-rate-smoothing practices have attenuated effects in economies with cognitive discounting, as agents underestimate the persistence of interest rate changes.
- Differences between closed- and open-economy behavioral models:
  - Domestic price sensitivity to forward guidance increases approximately linearly with the horizon under full rationality, but exchange rate sensitivity does not follow the same linear pattern.
  - The FGP is less dramatic in an open economy to begin with, so myopia produces relatively less dampening of future interest rate changes, especially when exchange rate pass-through to import prices is high (as in the dominant currency paradigm of Gopinath et al., 2020).
- Under complete markets and optimal monetary policy, behavioral discounting generates a unit root in the nominal exchange rate: temporary domestic cost-push shocks produce permanent appreciations (if positive) or depreciations (if negative), reducing the desirability of the exchange rate as a nominal anchor.
- More severe cognitive discounting implies:
  - Greater persistence of net foreign assets and the real exchange rate in response to monetary shocks.
  - Larger international monetary policy spillovers: monetary easing in one economy is more likely to be expansionary for its trading partners if those partners’ agents are behavioral, because cognitive discounting attenuates expenditure switching.

### Mechanisms and theoretical placement
- The paper works within the New Open Economy Macroeconomics (NOEM) paradigm, using an incomplete asset market setup.
- Cognitive discounting alters linearized IS and Phillips curve relationships, in line with earlier implementations of behavioral discounting (Gabaix, 2020; Woodford, 2019) and related literature (Angeletos and Lian, 2018).
- The model’s success in reconciling UIP-related puzzles stems from the distinction between subjective expectations (where UIP holds) and rational-expectations-based tests (which would reject UIP because realized exchange rates are biased indicators of agents’ expectations).

### Relation to prior literature
- The paper is related to structural models with bounded rationality and learning, including:
  - Brock and Hommes (1997); Evans and Honkapohja (2001); Bullard and Mitra (2002); Preston (2005); Branch and McGough (2009); De Grauwe (2011).
  - Recent approaches: Bordalo et al. (2018) on diagnostic expectations; Bianchi et al. (2021) on introducing diagnostic expectations into DSGE models.
  - Open-economy bounded-rationality/learning work: Llosa and Tuesta (2008); Zanna (2009); Du et al. (2021).
- The paper contrasts behavioral solutions with recent “rational” solutions to UIP puzzles (e.g., delayed portfolio adjustment in Bacchetta and van Wincoop (2021); financial frictions in Valchev (2020) and Itskhoki and Mukhin (2021)), and is consistent with empirical evidence in Kalemli-Ozcan and Varela (2021) regarding the interpretation of exchange rate expectations.

### Paper structure (overview)
- Section 2: Theoretical setup with two countries and boundedly rational agents.
- Section 3: Linearized small open economy version; how behavioral discounting alters key equilibrium relationships and parameter choices for numerical experiments.
- Section 4: Effects of agents’ myopia on stationarity and equilibrium determinacy.
- Section 5: How behavioral discounting resolves selected open economy puzzles.
- Subsequent sections (listed in the table of contents) develop dynamics, transmission mechanisms, international spillovers, optimal policy, and empirical/Bayesian results.

*wpiea2022112-print-pdf - 1. Introduction*

### Section 6, we analytically characterize the implications of myopia for the transmission of “surprise”

### Section 6, we analytically characterize the implications of myopia for the transmission of “surprise”

### Theoretical setup and model structure
- Two-country NOEM model with Home and Foreign economies; world population normalized to unity with ζ∈(0,1) the Home share and 1−ζ the Foreign share. Focus on Home due to isomorphism.
- Agents: continuum of infinitely-lived households and monopolistically competitive firms.
- Trade and finance link the economies via goods trade and cross-border borrowing; separate monetary authorities.
- Key household utility (Equation (1)):
  - U^h_t = Ê_t Σ_{T=t}^∞ β^{T−t} [ (C^h_T)^{1−σ}/(1−σ) − (N^h_T)^{1+φ}/(1+φ) ]
  - Parameters: 0<β<1; σ>0; φ>0; Ê_t denotes subjective expectation operator.
- Consumption aggregator (Equation (2)):
  - C^h_t = [ (1−α)^{1/η} (C^h_{H,t})^{(η−1)/η} + α^{1/η} (C^h_{F,t})^{(η−1)/η} ]^{η/(η−1)}
  - α∈(0,1) openness parameter; η>0 trade elasticity.
- Financial assets:
  - One-period Home bonds B^h_t and internationally traded Foreign-currency bonds B^{*,h}_t (paying i_t and i^*_t).
  - Real budget constraint (Equation (3)) with real exchange rate Q_t ≡ ε_t P^*_t / P_t and risk premium Φ_t = Φ(B^*_t).
- Firms:
  - Final goods assembled with Dixit-Stiglitz aggregator (Equation (4)); intermediate firms set prices in domestic currency (producer currency pricing) and face Calvo friction (probability of no reoptimization θ∈(0,1)). Production: Y^f_{H,t}+Y^{*,f}_{H,t} = z_t N^f_t (Equation (5)).
  - Firms discount profits using Λ_{t,T} ≡ β^{T−t} u_1(C_T,N_T) consistent with household SDF (Equation (6)).

### Myopia specification
- Agents form subjective expectations Ê_t that shrink rational expectations toward steady state following Gabaix (2020).
- Perceived law of motion for any variable X_t (Equation (7)):
  - X_{t+1} − X = m G_X(X_t − X, ε_{t+1}), with 0 ≤ m ≤ 1 cognitive discounting parameter.
  - m = 1 → rational expectations; lower m → faster perceived reversion to steady state.
- Agents correctly perceive constraints of their optimization problems; misperception applies only to aggregate variables they take as given.

### Monetary authority and equilibrium
- Standard Taylor-like rule (Equation (8)):
  - i_t = ρ i_{t−1} + (1−ρ) [ i + φ_π (Π_t − Π) + φ_y log(Y_t / Y) ] + ν_t, with 0 ≤ ρ < 1.
- Equilibrium aggregation: identical household choices imply C^h_t = C_t, N^h_t = N_t, B^{*,h}_t = B^*_t, B^h_t = B_t = 0. Market clearing conditions (Equations (9)–(10)) link labor and goods markets.

### Linearized small open economy model
- Behavioral k-period ahead expectations approximation (Equation (11)):
  - Ê_t{X_{t+k} − X} = m^k E_t{X_{t+k} − X}
- Choice to form projections about levels (not rates) of variables (discussion around Equation (12)).
- Transformations and hat notation: ˆi_t ≡ log(1+i_t) − log(β^{−1}), ˆπ_t ≡ log(Π_t), ˆB^*_t ≡ (B^*_t Q_t − B^*)/Y, and ˆX_t ≡ (X_t − X)/X (percent deviations).

Key linearized equilibrium relationships (all as in source):
- Behavioral IS curve (Equation (13)):
  - ˆC_t = m E_t ˆC_{t+1} − (1/σ)(ˆi_t − m E_t ˆπ_{t+1}) + (1−m) (1−β)/(1+σ μ φ) ˆB^*_t
  - Note: extra term due to open economy net foreign asset position when m<1.
- Behavioral UIP (Equation (14)):
  - ˆi_t − m E_t{ˆπ_{t+1}} = ˆi^*_t − m E_t{ˆπ^*_ {t+1} − ˆQ_{t+1}} − φ ˆB^*_t, where φ = Φ′(0).
- Phillips curve for domestic prices (Equation (15)):
  - ˆπ_{H,t} = m β E_t{ˆπ_{H,t+1}} + κ ˆMC_t, κ ≡ (1−θ)(1−βθ)/θ.
- Real marginal cost (Equation (16)):
  - ˆMC_t = σ ˆC_t + φ ˆY_t + α/(1−α) ˆQ_t − (1+φ) ˆz_t.
- CPI inflation decomposition (Equation (17)):
  - ˆπ_t = ˆπ_{H,t} + α/(1−α) (ˆQ_t − ˆQ_{t−1}).
- Goods market clearing (Equation (18)):
  - ˆY_t = (1−α) ˆC_t + α ˆY^*_t + η α(2−α)/(1−α) ˆQ_t.
- Net foreign asset law of motion (Equation (19)):
  - ˆB^*_t = β^{−1} (ˆB^*_{t−1} + ˆY_t − ˆC_t).
- Linearized policy rule (Equation (20)):
  - ˆi_t = ρ ˆi_{t−1} + (1−ρ)(φ_π ˆπ_t + φ_y ˆY_t) + ν_t.
- Foreign (closed-economy) counterparts (Equations (21)–(23)):
  - ˆY^*_t = m E_t ˆY^*_{t+1} − (1/σ)(ˆi^*_t − m E_t ˆπ^*_{t+1})
  - ˆπ^*_t = m β E_t{ˆπ^*_{t+1}} + κ(σ+φ)(ˆY^*_t − ˆz^*_t)
  - ˆi^*_t = ρ ˆi^*_{t−1} + (1−ρ)(φ_π ˆπ^*_t + φ_y ˆY^*_t) + ν^*_t

### Calibration and estimation (parameter values preserved exactly)
- Calibration (Table 1 values):
  - α 0.4 Openness
  - β 0.99 Discount Factor
  - η 1 Elasticity of Substitution Between Home and Foreign Goods
  - φ 3 Inverse Frisch Elasticity of Labor Supply
  - σ 1 Inverse Intertemporal Elasticity of Substitution
  - μ 1.2 Gross Product Markup
  - θ 0.85 Calvo Probability of No Price Adjustment
  - ρ 0.9 Interest Rate Smoothing Parameters
  - φ_π 1.5 Inflation Feedback Taylor Rule
  - φ_y 0.125 Output Feedback Taylor Rule
  - φ 0.01 Intermediation Costs Debt Sensitivity
- Model frequency: quarterly. β = 0.99 implies steady state nominal interest rate of 4 per cent per annum (zero average inflation).
- Values explored for cognitive discounting m ∈ [0.5, 1]. Reference values from literature: Gabaix (2020) m = 0.85; empirical estimates consistent with m = 0.65 (Fuhrer and Rudebusch, 2004); Gust et al. (2021) close to m = 0.5; Ilabaca et al. (2020) estimate m_households = 0.71 and m_firms = 0.41.
- Bayesian estimation: Home and Foreign represented by Canada and the US; baseline quarterly sample 1972–2007 with robustness subsamples. Posterior mean of m ≈ 0.53 for baseline sample, indicating high empirical relevance of behavioral discounting.

### Stationarity and determinacy implications
- Incomplete markets NOEMs exhibit stationarity issues; debt-elastic risk premium (φ>0) commonly introduced for stationarity.
- Behavioral discounting (m<1) exacerbates stationarity problems:
  - IS equation (Equation (13)) implies possible explosiveness when m<1 absent sufficient stationarizing mechanism.
  - Myopia reduces sensitivity to future risk premia; therefore a larger φ is needed for stationarity as m decreases.
  - Openness (higher α) mitigates stationarity problems since exchange rate movements more strongly affect consumption.
- Myopia shrinks indeterminacy regions:
  - Behavioral discounting reduces the range of policy feedback φ_π required to guarantee uniqueness (weaker responses to inflation may suffice).
  - Example: baseline calibration implies for m = 0.85 determinacy if φ_π > 0.555; reducing α by half lowers threshold to 0.545.
- Non-trivial interaction:
  - For moderate discounting, stability may be achieved either via sufficient deviation from Taylor principle or via a positive risk premium φ.
  - Chosen operational solution: set φ = 0.01 to ensure stationarity across considered m while keeping short-term dynamics minimally affected.

### Exchange rate dynamics and empirical puzzles
- Behavioral UIP reexpressed (Equation (24), repeated):
  - ˆi_t − m E_t{ˆπ_{t+1}} = ˆι^*_t − m E_t{ˆπ^*_{t+1} − ˆQ_{t+1}} − ˆQ_t, with ˆι^*_t ≡ ˆi^*_t − φ ˆB^*_t the premium-adjusted foreign nominal rate.
- Forward premium (Fama puzzle):
  - Behavioral UIP rewritten (Equation (25)):
    - ˆi_t − ˆι^*_t = m E_t{∆ˆε_{t+1}} − (1−m) ˆQ_t
  - Rearranged for Fama regression analogy (Equation (26)):
    - E_t{∆ˆε_{t+1}} = (1/m)(ˆi_t − ˆι^*_t) + (1/m − 1) ˆQ_t
  - Population regression expression for slope a_1 (Equation (28)):
    - E a_1 = 1/m + (1/m − 1) Corr(ˆQ_t, ˆi_t − ˆι^*_t) × Std(ˆQ_t)/Std(ˆi_t − ˆι^*_t)
  - Insight:
    - m < 1 pushes the first term above 1 (worsening the Fama puzzle) but the second term is negative when real exchange rate and interest differential are negatively correlated, potentially offsetting and improving fit.
  - Quantitative results (Table 2 Fama Regression Coefficients):
    - For combinations of m and ρ:
      - m=0.50: ρ=0.95 → −0.07; ρ=0.90 → 0.17; ρ=0.75 → 0.51; ρ=0.50 → 0.75
      - m=0.75: ρ=0.95 → −0.04; ρ=0.90 → 0.36; ρ=0.75 → 0.69; ρ=0.50 → 0.86
      - m=0.90: ρ=0.95 → 0.37; ρ=0.90 → 0.67; ρ=0.75 → 0.86; ρ=0.50 → 0.94
      - m=1.00: all ρ → 1.00
      - Estimated model: 0.44 (for m = 0.53, ρ = 0.84)
    - Interpretation: large discounting (low m) combined with high interest rate smoothing (high ρ) can generate negative or substantially lower Fama coefficients, improving empirical fit.
- Predictability sign reversal and Engel condition:
  - Engel-style regressions (Equation (29)):
    - r^x_{t+1} ≡ ˆi_t − ˆi^*_t − ∆ˆε_{t+1} = b_{s,0} + b_{s,1} (ˆi_{t−s} − ˆi^*_{t−s}) + ε_t
  - Empirical facts: b_{s,1} flips from positive to negative for some s; Engel condition Σ_{s=0}^∞ b_{s,1} < 0 equivalent to Cov(E_t Σ_{s=0}^∞ r^x_{t+s+1}, ˆi_t − ˆi^*_t) < 0.
  - Model implications (Figure 2 summary):
    - Behavioral discounting (m<1) generates sign reversals in b_{s,1} as s increases; rational expectations (m=1) fails to produce sign reversal.
    - Cumulative sums Σ_{s=0}^T b_{s,1} can be negative (satisfying Engel condition) when discounting is sufficient and interest rate smoothing is not excessive.
    - Estimated model (posterior m ≈ 0.53 and moderately high ρ) satisfies both sign reversal and Engel’s excess volatility condition.
  - Intuition (Equation (30)):
    - E_t r^x_{t+1} = (m−1) E_t{ˆQ_{t+1} + ˆπ_{t+1} − ˆπ^*_{t+1}}
    - Domestic policy easing → persistent real depreciation → initial accumulation of net foreign assets then later appreciation → excess returns follow mirror-image path producing sign reversal.
    - Interest rate inertia (high ρ) amplifies persistence and can violate Engel condition if too large.

### Role of market incompleteness
- Model adopts incomplete markets (standard NOEM). Under complete markets, perfect international risk sharing obtains (Equation (31)):
  - σ (ˆC_t − ˆC^*_t) = ˆQ_t
- Perfect risk sharing relation is unaffected by myopia if subjective probabilities are same across agents.
- Further implications and comparisons between complete and incomplete markets are discussed subsequently (text truncated in provided content).

*Italic: Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022112-print-pdf.pdf*

### Appendix C, the last term in the IS curve (13) vanishes when markets are complete, leading to

### wpiea2022112-print-pdf - Appendix C, the last term in the IS curve (13) vanishes when markets are complete, leading to

### Modified IS curve under complete markets
- Equation (32) (as stated):
  - ˆC_t = m E_t ˆC_{t+1} − 1/σ [ ˆi_t − m E_t ˆπ_{t+1} ] .
- Context:
  - This expression follows from the last term in the IS curve (13) vanishing when markets are complete.
  - The modified IS curve is combined with its Foreign analog and the risk sharing condition (31) in subsequent derivations.

### UIP relationship and key algebraic result
- Combining the modified IS curve, its Foreign analog, and the risk sharing condition (31) yields equation (33):
  - ˆi_t − m E_t {ˆπ_{t+1}} = ˆi^*_t − m E_t {ˆπ^*_{t+1}} − ˆQ_{t+1} − ˆQ_t .
- Comparison with baseline:
  - This is like formula (14) in the baseline model except for the stationarizing risk premium component φ ˆB^*_t no longer being present.

### Implications emphasized in the text
- The stationarizing risk premium component φ ˆB^*_t is absent under complete markets.
- Because φ is assumed small, the authors state that behavioral discounting can help address UIP-related anomalies also when markets are complete.

*Source: Appendix C, wpiea2022112-print-pdf*

### Appendix C, this version of our model implies very similar Fama regression coefficients to those

### wpiea2022112-print-pdf - Appendix C, this version of our model implies very similar Fama regression coefficients to those

### Market incompleteness and the predictability sign reversal
- When markets are incomplete, net foreign asset accumulation driven by the initially-weak exchange rate must eventually be reversed, allowing the trade balance to deteriorate; as indicated by formula (30), this flips the excess return sign.
- In the complete markets case, net foreign assets no longer play a role in equilibrium exchange rate dynamics, which eliminates the flip in the exchange rate and excess returns.
- Figure 3 findings (qualitative):
  - Left panel: impulse response of the real exchange rate to a 25bp (100 bp annualized) negative monetary policy shock, expressed in percent deviation from the steady state; calibrated with m = 0.85.
  - Right panel: corresponding Engel regression coefficients, i.e., b_{s,1} from Equation (29), as a function of s.
- Implication: market incompleteness is key to resolving the predictability sign reversal puzzle via the role of NFA dynamics.

### Monetary policy transmission — inspecting the mechanism
- Setup:
  - Define Home and Foreign real interest rates as ˆr_t ≡ ˆi_t − m E_t ˆπ_{t+1} and ˆr^*_t ≡ ˆi^*_t − m E_t ˆπ^*_{t+1}.
  - Approach follows Gabaix (2020) with slight differences in definitions (see footnote 12).

- Real exchange rate (RER)
  - Forward-iterated UIP condition (Equation (34)):
    - ˆQ_t = −E_t ∑_{T=t}^∞ m^{T−t} [ˆr_T − ˆr^*_T + φ ˆB^*_T]
  - Cognitive discounting dampens effects of future real interest rate changes on the current exchange rate, addressing the exchange rate forward guidance puzzle of Galí (2020).
  - Figure 4 observations:
    - Left panel: normalized initial RER response as a function of forward guidance horizon with UIP premium φ = 0.01.
    - Right panel: same with φ = 0.
    - Even absent discounting (m = 1), RER response can decline with FG horizon due to endogenous NFA response feeding back into the UIP premium; when φ = 0 the purple line (m = 1) remains flat.

- Consumption and NFA link
  - Iterating the consumption Euler (Equation (35)):
    - ˆC_t = −(1/σ) E_t ∑_{T=t}^∞ m^{T−t} ˆr_T + (1−m)/(1−β) 1/(1 + σ μ φ) E_t ∑_{T=t}^∞ m^{T−t} ˆB^*_T
  - Absent discounting and with a single forward guidance horizon H, an anticipated 1pp decrease in the real interest rate ten periods ahead implies consumption is (1/σ) percent above steady state for exactly eleven periods.
  - Longer FG horizon → longer period of trade surpluses → larger NFA accumulation.

- Output (IS curve)
  - IS equation (36):
    - ˆY_t = m E_t ˆY_{t+1} − [ (1−α)/σ + η α(2−α)/(1−α) ] ˆr_t − [η α(2−α)/(1−α) φ ˆB^*_t] − (1−m)(1−α)/(1−β) 1/(1 + σ μ φ) ˆB^*_t
  - Iterated form (37):
    - ˆY_t = −[ (1−α)/σ + η α(2−α)/(1−α) ] E_t ∑_{T=t}^∞ m^{T−t} ˆr_T − [η α(2−α)/(1−α) φ − (1−m)(1−α)/(1−β) 1/(1 + σ μ φ)] E_t ∑_{T=t}^∞ m^{T−t} ˆB^*_T
  - Criterion: relative to a closed economy, same change in the real interest path has stronger direct effect on output unless η is very low; formal criterion:
    - η < (1−α)/(2−α) σ^{−1} ≤ (1/2) σ^{−1}
  - Discounting (lower m) dampens effects of future real interest rate changes on output, mitigating the forward guidance puzzle (FGP).

- Inflation
  - Domestic inflation (Equation (15) repeated):
    - ˆπ_{H,t} = β m E_t {ˆπ_{H,t+1}} + κ [ σ ˆC_t + φ ˆY_t + α/(1−α) ˆQ_t − (1 + φ) ˆz_t ]
  - Approximate relation (38):
    - ˆπ_t ≈ −κ a / [β(1−m)] E_t ∑_{T=t}^∞ [m^{T−t+1} − (β m)^{T−t+1}] ˆr_T − α/(1−α) E_t ∑_{T=t}^∞ m^{T−t} ˆr_T − α/(1−α) ˆQ_{t−1}, with a ≡ φ[(1−α)/σ + η α(2−α)/(1−α)] + 1/(1−α)
  - Limiting case β → 1 (Equation (39)):
    - ˆπ_t ≈ −κ a E_t ∑_{T=t}^∞ (T−t+1) m^{T−t} ˆr_T − α/(1−α) E_t ∑_{T=t}^∞ m^{T−t} ˆr_T − α/(1−α) ˆQ_{t−1}
  - Implications:
    - For m = 1: relationship between inflation effect and FG horizon is linear.
    - For small discounting: non-monotonic (increasing then decreasing).
    - For strong discounting: decreasing.
    - Open economy: presence of import price component reduces the FGP for inflation; forward guidance puzzle less pronounced in open economy, and mitigating effect of discounting on FGP is relatively smaller compared to closed economy.

### Dynamic effects of monetary policy
- Conventional monetary shocks (Figure 6)
  - Impulse responses to 25bp (100 bp annualized) negative monetary policy shock.
  - Discounting dampens effects of current monetary policy shocks because interest rate smoothing creates a lower future policy path that behavioral agents "cognitively discount".
  - Discounting matters more for domestic inflation than for the exchange rate, hence also for CPI. Output response in the open economy is less sensitive to discounting.
  - Discounting makes NFA accumulation more persistent; more inertial NFA → more persistent RER appreciation delay → postpones sign reversal in excess returns.

- Low for Longer (LFL) policies (Figure 7)
  - Experiment: central bank desires 100bp annualized deviation but, at ELB, cuts by 10bp and keeps it for 10 quarters; impulse responses shown.
  - Findings:
    - With no discounting (m = 1), LFL is very effective in both closed and open economies (reflecting FGP).
    - For moderate discounting (e.g., m = 0.9), LFL can become less efficient than the corresponding conventional stimulus.
    - Potency of LFL is relatively more affected by cognitive discounting in the closed economy.

### International monetary policy spillovers
- Iterated foreign-home output relation (Equation (40)):
  - ˆY_t = α ˆY^*_t (Demand Channel) + η α(2−α)/(1−α) ˆQ_t (Expenditure Switching Channel) − (1−α)/σ E_t ∑_{T=t}^∞ m^{T−t} ˆr_T (Endogenous Home Policy Response) + (1−m)(1−α)/(1−β) 1/(1 + σ μ φ) E_t ∑_{T=t}^∞ m^{T−t} ˆB^*_T (Myopia “Damper”)
- Conditional analysis (Home keeps real interest rate constant)
  - Approximated spillover (Equation (41)):
    - ˆY_t ≈ −[α/σ + η α(2−α)/(1−α)] E_t ∑_{T=t}^∞ m^{T−t} ˆr^*_T (omitting small NFA terms)
  - Unless η is very low (same criterion as Section 6.2), coefficient on foreign real interest path is positive → output spillovers from foreign monetary easing are negative when Home keeps its rate constant.
- Effect of discounting:
  - Greater agent myopia reduces the impact of future foreign real rate changes on Home output (FG less powerful for spillovers); international transmission weaker with lower m.
  - If Foreign agents are less myopic than Home (m^* > m), the demand channel gains relative importance and spillovers can become less negative or even positive.
- Figure 8 illustrates:
  - First row: m^* = m case responses to 25bp (100 bp annualized) negative foreign monetary shock.
  - Second row: m^* = 1 case (fully rational foreign agents), showing that less-myopic foreign agents can produce less negative or positive spillovers for Home when Home keeps its real rate constant.

### Optimal monetary policy
- Setup and welfare loss (quadratic approximation, Equation (43)):
  - U_t ≈ −(1−α)/2 E_t ∑_{T=t}^∞ β^{T−t} [ μ/(κ(μ−1)) ˆπ_{H,T}^2 + (1 + φ) ˆx_T^2 ] + t.i.p.
  - Output gap: ˆx_t ≡ log(Y_t / Ȳ_t)
- Phillips curve in output gap terms (44):
  - ˆπ_{H,t} = m β E_t {ˆπ_{H,t+1}} + κ (1 + φ) ˆx_t + ξ_t (cost-push shock)
- Optimal commitment rule (timeless perspective):
  - Targeting rule (45):
    - ˆπ_{H,t} + (μ−1)/μ (ˆx_t − m ˆx_{t−1}) = 0
  - Optimal producer price level (46):
    - ˆP_{H,t} = (μ−1)/μ [ ˆx_t − (1−m) ∑_{T=0}^{t−1} ˆx_T ]
- Implications:
  - Unless m = 1, price level targeting is no longer optimal even in the long run: positive cost-push shocks push the price level permanently up.
  - Figure 9 (impulse to 0.25% i.i.d. cost-push shock):
    - With forward-looking agents and commitment (m = 1), optimal policy induces a longer lasting recession to promise future deflation.
    - With myopic agents (lower m), that motive weakens; optimal policy tightens more on impact but yields higher permanent producer prices and makes policy closer to discretionary outcome.
    - Extreme m = 0: optimal responses equal discretionary outcome—output falls only in the shock period while price level increases on impact and then stays flat.
  - Open economy implication:
    - Optimal tightening appreciates exchange rate on impact and then depreciates; with myopia exchange rate becomes permanently weaker (consistent with permanent producer price increase).
    - Nominal exchange rate follows a random walk under optimal policy even when relative prices are stationary → using exchange rate as nominal anchor becomes less appealing with myopia.
  - Optimal policy with behavioral agents makes NFA less responsive to shocks; for m = 0, NFA does not respond to temporary cost-push shocks.

### Conclusions — key takeaways
- Introducing behavioral agents (cognitive discounting/myopia) into an open economy New Keynesian model:
  - Helps resolve UIP-related puzzles in a manner consistent with survey-based evidence.
  - Decreases efficacy of policies relying on announcements of future actions (e.g., “low for longer”), mitigating the forward guidance puzzle.
  - Reduces the relative strength of the exchange rate channel, improving explanations for international output comovement.
  - Alters optimal monetary policy: weakens the case for price-level targeting and using exchange rate as a nominal anchor; makes optimal policy responses closer to discretionary outcomes as myopia increases.
- Trade-offs:
  - Incorporating behavioral aspects adds complexity, but yields better empirical fit and more reasonable implications for both closed and open economy phenomena, particularly for anomalies cognitive discounting helps eliminate.

*Italic: Source — wpiea2022112-print-pdf - Appendix C, this version of our model implies very similar Fama regression coefficients to those*

### References

### Appendices (Key Derivations and Additional Results)

### Appendix A — Key Derivations
- Linearized household budget constraint and optimality:
  - Linearized budget constraint (A.1) and Euler equations (A.2), (A.3) for Home and Foreign bond holdings.
  - Intratemporal labor supply condition (A.4): ˆWt = σ ˆCht + φ ˆNht.
  - Combining Euler equations yields UIP condition (A.6), identified as equation (14) in the main text.
- Individual consumption function:
  - Iteration of the budget constraint and Euler equation lead to the individual consumption function (A.11) that incorporates labor supply choice and aggregate expectations.
- Aggregate IS curve:
  - Application of behavioral discounting to (A.11) and market clearing B t = 0 yields recursive expressions (A.12)–(A.14) and the aggregate IS curve (A.15), which is equation (13) in the main text:
    - ˆCt = mEt ˆCt+1 − (1/σ)(ˆit − mEt ˆπt+1) + (1−m)(1−β)/(1 + σ/(μφ)) ˆB∗t.
- Phillips curve derivation:
  - From firm pricing first-order condition (A.19), linearization gives (A.20) and, after applying behavioral discounting, recursive form (A.22).
  - Using the price index relation (A.23) yields Phillips curve (A.24), corresponding to equation (15) in the main text:
    - ˆπH,t = mβ Et{ˆπH,t+1} + (1−βθ)(1−θ)/θ ˆMCt.
- Marginal cost:
  - Aggregate price index and production relations lead to marginal cost expression (A.32), equation (16) in the main text:
    - ˆMCt = σ ˆCt + φ ˆYt + (α/(1−α)) ˆQt − (1 + φ) ˆzt.
- Goods market clearing:
  - Starting from aggregate output definition (A.29) and consumption composition, linearization yields (A.37), equation (18) in the main text:
    - ˆYt = (1−α) ˆCt + α ˆY∗t + η α(2−α)/(1−α) ˆQt.

### Appendix B — Additional Derivations and Closed-Form Representations
- Derivation of Equation (36):
  - Eliminating consumption and exploiting UIP yields (B.3), equation (36) in the main text:
    - ˆYt = mEt ˆYt+1 − [ (1−α)/σ + η α(2−α)/(1−α) ] ˆrt − [ η α(2−α)/(1−α) φ − (1−m)(1−α)/(1−β)(1 + σ/(μφ)) ] ˆB∗t.
- Long-horizon inflation representations (Equations (38) and (39)):
  - Combination of (15) and (16) and forward iteration give (B.4)–(B.6), leading to CPI inflation expression (B.7), equation (38).
  - In the limit β → 1 (so M → m), one obtains (B.8) and (B.9), equation (39) in the text.
  - The relative weight of the penultimate component is shown to be increasing in openness α (final formula in B.2).
- Iterated IS results (Equations (40) and (41)):
  - Forward iteration of the output IS curve gives (B.11), equation (40):
    - ˆYt = α ˆY∗t + η α(2−α)/(1−α) ˆQt − (1−α)/σ Et ∑_{T=t}∞ m^{T−t} ˆrT + (1−m)(1−α) Et ∑_{T=t}∞ m^{T−t} ˆB∗T.
  - Under assumed conditions and dropping net foreign asset terms, (B.12) yields equation (41):
    - ˆYt ≈ [ −α/σ + η α(2−α)/(1−α) ] Et ∑_{T=t}∞ m^{T−t} ˆr∗T.

### Appendix C — Complete Markets Case
- Complete markets IS curve:
  - With complete markets, Arrow–Debreu asset structure leads to linearized asset pricing relation (C.2).
  - The individual consumption function analogous to (A.11) is (C.3).
  - Applying behavioral discounting to aggregate Arrow–Debreu payoffs yields aggregate consumption Euler equation (C.4), which is equation (32) in the main text:
    - ˆCt = mEt ˆCt+1 − (1/σ)(ˆit − mEt ˆπt+1).
  - Key difference from incomplete markets: the country’s net foreign assets position does not appear in the complete-markets IS (compare (C.4) vs (13)/(A.15)).
- Empirical and IRF implications:
  - Table C.1 reports Fama regression coefficients for the complete markets model across parameter combinations (m=0.50, m=0.75, m=0.90, m=1.00) and ρ values:
    - For ρ= 0.95: −0.12, −0.17, 0.25, 1.00.
    - For ρ= 0.90: 0.13, 0.28, 0.60, 1.00.
    - For ρ= 0.75: 0.49, 0.66, 0.83, 1.00.
    - For ρ= 0.50: 0.74, 0.84, 0.93, 1.00.
  - Figure 10 impulse responses (note) — monetary policy shock:
    - The shock is a 25bp (100 bp annualized) negative monetary policy shock.
    - Behavioral discounting generates a very similar degree of attenuation in short-term responses of output, inflation, and the real exchange rate under complete markets as in the baseline incomplete markets model.
    - In the IRF figure, lines correspond to m values: purple m=1, blue m=0.5, red m=0.75, yellow m=0.9.

*wpiea2022112-print-pdf - References*

### 0.75 and 0.9 respectively.

### Appendix D. Optimal Monetary Policy; Appendix E. Bayesian Estimation Results

### D.1. Deriving a Welfare Loss Function
- Social planner maximizes
  - Ut = Et Σ_{T=t}^{∞} β^{T−t} [ (C_T)^{1−σ}/(1−σ) − (N_T)^{1+φ}/(1+φ) ].
- Second order Taylor expansion of period utility ut around steady state:
  - ut ≈ u + u_C C [ Ĉ_t + (1−σ)/2 Ĉ_t^2 ] + u_N N [ Ñ_t + (1+φ)/2 Ñ_t^2 ].
- Optimal steady state allocation for a small open economy implies −u_N / u_C = (1−α) C/N.
- Under complete markets and assuming σ = η = 1 and C*_t = Y*_t:
  - Ĉ_t = (1−α) Ŷ_t + α Ŷ*_t.
- Rewriting welfare deviations:
  - ut − u / (u_C C) ≈ (1−α) [ Ŷ_t − Ñ_t − (1+φ)/2 Ñ_t^2 ] + t.i.p.
- Using aggregate production function and noting price dispersion ∆_t is second order:
  - ut − u / (u_C C) ≈ −(1−α)/2 [ 2 ∆̂_t + (1+φ)(Ŷ_t − ẑ_t)^2 ] + t.i.p.
- Flexible price output proportional to productivity via real marginal cost:
  - MĈ_t = (1+φ)(Ŷ_t − ẑ_t).
  - Define output gap x̂_t ≡ log(Y_t / Ȳ_t).
- Using Lemmas from Gali and Monacelli (2005) / Woodford (2003):
  - Σ_{T=t}^{∞} β^{T−t} ∆̂_T = μ / [2(μ−1)] Σ_{T=t}^{∞} β^{T−t} Var P̂^f_{H,T} = μ / [κ(μ−1)] Σ_{T=t}^{∞} β^{T−t} π̂^2_{H,T}.
- Final welfare approximation:
  - U_t ≈ −(1−α)/2 E_t Σ_{T=t}^{∞} β^{T−t} [ μ / (κ(μ−1)) π̂^2_{H,T} + (1+φ) x̂_T^2 ] + t.i.p.
  - This corresponds to Equation (43) in the main text.

### D.2. Optimal Stabilization
- Minimize welfare loss function (43) subject to constraint (44).
- Lagrangian:
  - E_t Σ_{T=t}^{∞} β^{T−t} [ μ/(κ(μ−1)) π̂^2_{H,T} + (1+φ) x̂_T^2 + δ_T(π̂_{H,T} − mβ π̂_{H,T+1} − κ(1+φ) x̂_T − ξ_T) ].
- First-order conditions under commitment:
  - 2 x̂_t − κ δ_t = 0.
  - 2 μ/(κ(μ−1)) π̂_{H,t} + δ_t − m δ_{t−1} = 0.
- Combining yields:
  - π̂_{H,t} + (μ−1)/μ ( x̂_t − m x̂_{t−1} ) = 0.
  - This is the formulation reported in the main text (Equation D.15).

### E.1. Data
- Domestic block: Canada. Foreign block: US (closed economy).
- Quarterly data on output, inflation, interest rates, and bilateral exchange rate from FRED.
- Transformations and measurement equations:
  - (1) Per-Capita Real Output Growth (y_obs_t):
    - y_obs_t = 100 [ ln(GDP_t / Pop_t) − ln(GDP_{t−1} / Pop_{t−1}) ] using NGDPRSAXD-CCAQ and population 17-10-0009-01.
  - (2) Inflation (π_obs_t):
    - π_obs_t = 100 ln(Def_t / Def_{t−1}) using CANGDPDEFQISMEI.
  - (3) Interest Rate (i_obs_t):
    - i_obs_t = R_t / 4 using IR3TIB01CAM156N.
  - (4) Foreign Per-Capita Real Output Growth (y_obs,∗_t):
    - y_obs,∗_t = 100 [ ln(GDP*_t / Pop*_t) − ln(GDP*_{t−1} / Pop*_{t−1}) ] using GDPC1 and CNP16OV.
  - (5) Foreign Inflation (π_obs,∗_t):
    - π_obs,∗_t = 100 ln(Def*_t / Def*_{t−1}) using GDPDEF.
  - (6) Foreign Interest Rate (r_obs,∗_t):
    - r_obs,∗_t = R*_t / 4 using FEDFUNDS.
  - (7) Exchange Rate (e_obs_t):
    - ∆e_obs_t = 100 ln(RER_t / RER_{t−1}) using CCUSMA02CAQ618N.
- Measurement equations linking data to model variables (intercepts y, π, r, y*, π*, r* capture trends instead of demeaning):
  - y_obs,∗_t = ˆY*_t − ˆY*_{t−1} + y*.
  - π_obs,∗_t = ˆπ*_t + π*.
  - i_obs,∗_t = ˆi*_t + π* + r*.
  - y_obs_t = ˆY_t − ˆY_{t−1} + y.
  - π_obs_t = ˆπ_{H,t} + π.
  - i_obs_t = ˆi_t + π + r.
  - e_obs_t = Q_t − Q_{t−1} + ˆπ_t + π − ˆπ*_t − π*.
- Baseline sample: 1972:Q1–2007:Q4. Robustness samples: 1982:Q2–2007:Q4 and 1972:Q1–2019:Q4.
  - In 1972:Q1–2019:Q4 sample, replace interest rate with Wu and Xia (2016) shadow rate for periods with ZLB and asset purchases.

### E.2. Shocks
- Seven stochastic shocks:
  - Monetary policy shocks: ν_t and ν*_t (enter Taylor rule; assumed to be white noise).
  - Firm cost (cost-push) shocks: ξ_t and ξ*_t (enter Phillips curve).
  - Household intertemporal preference shocks: g_t and g*_t (shifters in subjective discount factor β, appear in modified IS curve).
  - International risk premium shock: ρ_t (modifies UIP condition).
- Modified IS curve with preference shocks:
  - Ĉ_t = m E_t Ĉ_{t+1} − (1/σ)( ˆi_t − m E_t ˆπ_{t+1} ) + (1−m)/(1−β) 1/(1+σ μ φ) ˆB^*_t + g_t − m E_t g_{t+1}.
- UIP condition with risk premium shock:
  - ˆi_t − m E_t{ˆπ_{t+1}} = ˆi*_t − m E_t{ˆπ*_{t+1} − ˆQ_{t+1}} − ˆQ_t − φ ˆB^*_t + ρ_t.
- All shocks are independent AR(1) processes except monetary policy shocks (white noise).

### E.3. Priors and Estimation Method
- Fix values for weakly identified parameters: openness α, discount factor β, elasticity η, intermediation-cost sensitivity φ set to baseline calibration in Table 1 (not reproduced here).
- Remaining parameters estimated jointly via Bayesian methods.
- Cognitive discounting parameter m ~ Beta with mean 0.85 and SD 0.05 (mean chosen as in Gabaix, 2020).
- MCMC details:
  - Metropolis-Hastings jump size scaled for target acceptance ~25 percent.
  - 4 parallel chains × 250,000 draws each.
  - Burn-in phase: 100,000 draws.
  - Convergence verified via trace plots and potential scale reduction factors.
  - Estimation conducted with Dynare 4.6.4.

### E.4. Estimation Results (baseline sample 1972:Q2–2007:Q4)
- Table E.1: Structural parameters (prior distribution, prior mean, prior SD, posterior mean, posterior [5, 95] interval)
  - m Cognitive Discounting: Prior B, Mean 0.85, SD 0.05 → Posterior Mean 0.53 [0.44, 0.62]
  - φ_π^H Taylor Rule, Inflation: Prior N, Mean 1.5, SD 0.3 → Posterior Mean 1.63 [1.23, 2.01]
  - φ_π^*^F Taylor Rule, Inflation: Prior N, Mean 1.5, SD 0.3 → Posterior Mean 1.39 [1.07, 1.72]
  - φ_y^H Taylor Rule, Output: Prior N, Mean 0.125, SD 0.13 → Posterior Mean 0.13 [0.04, 0.23]
  - φ_y^*^F Taylor Rule, Output: Prior N, Mean 0.125, SD 0.13 → Posterior Mean 0.09 [0.006, 0.18]
  - ρ^H Interest Rate Smoothing: Prior B, Mean 0.90, SD 0.05 → Posterior Mean 0.84 [0.80, 0.87]
  - ρ^*_F Interest Rate Smoothing: Prior B, Mean 0.90, SD 0.05 → Posterior Mean 0.80 [0.74, 0.86]
  - θ Calvo Probability: Prior B, Mean 0.875, SD 0.10 → Posterior Mean 0.95 [0.93, 0.97]
  - φ Inverse Frisch Elasticity: Prior G, Mean 3.0, SD 0.75 → Posterior Mean 2.74 [1.58, 3.83]
  - σ Intertemporal Elasticity of Substitution: Prior G, Mean 1.0, SD 0.40 → Posterior Mean 3.11 [2.27, 3.91]
  - r_SS^H Real Rate: Prior N, Mean 0.5, SD 0.25 → Posterior Mean 0.23 [−0.16, 0.62]
  - r_SS^*_F Real Rate: Prior N, Mean 0.5, SD 0.25 → Posterior Mean 0.68 [0.43, 0.92]
  - π_SS^H Inflation: Prior N, Mean 1.0, SD 0.25 → Posterior Mean 1.07 [0.91, 1.24]
  - π_SS^*_F Inflation: Prior N, Mean 1.0, SD 0.25 → Posterior Mean 0.79 [0.64, 0.95]
  - y_SS^H Output Growth: Prior N, Mean 0.5, SD 0.25 → Posterior Mean 0.44 [0.38, 0.49]
  - y_SS^*_F Output Growth: Prior N, Mean 0.5, SD 0.25 → Posterior Mean 0.44 [0.41, 0.46]
  - Note: B = Beta, G = Gamma, N = Normal, H = Home, F = Foreign, SS = steady state, SD = standard deviation.
- Table E.2: Shock parameters (prior distribution, prior mean, prior SD, posterior mean, posterior [5,95] interval)
  - ρ_g AR H Preference: Prior B mean 0.7 SD 0.1 → Posterior Mean 0.97 [0.96, 0.98]
  - ρ^*_g AR F Preference: Prior B mean 0.7 SD 0.1 → Posterior Mean 0.92 [0.89, 0.96]
  - ρ_ξ AR H Cost-Push: Prior B mean 0.7 SD 0.1 → Posterior Mean 0.46 [0.36, 0.55]
  - ρ^*_ξ AR F Cost-Push: Prior B mean 0.7 SD 0.1 → Posterior Mean 0.89 [0.85, 0.93]
  - ρ_ρ AR Risk Premium: Prior B mean 0.7 SD 0.1 → Posterior Mean 0.98 [0.97, 0.99]
  - σ_ν SD H Monetary Policy: Prior IG mean 0.25 SD Inf. → Posterior Mean 0.32 [0.28, 0.35]
  - σ_ν^* SD F Monetary Policy: Prior IG mean 0.25 SD Inf. → Posterior Mean 0.36 [0.32, 0.39]
  - σ_g SD H Preference: Prior IG mean 0.25 SD Inf. → Posterior Mean 4.46 [3.19, 5.65]
  - σ_g^* SD F Preference: Prior IG mean 0.25 SD Inf. → Posterior Mean 2.80 [2.07, 3.50]
  - σ_ξ SD H Cost-Push: Prior IG mean 0.25 SD Inf. → Posterior Mean 0.50 [0.43, 0.56]
  - σ_ξ^* SD F Cost-Push: Prior IG mean 0.25 SD Inf. → Posterior Mean 0.13 [0.11, 0.16]
  - σ_ρ SD Risk Premium: Prior IG mean 0.25 SD Inf. → Posterior Mean 0.37 [0.31, 0.43]
  - Note: B = Beta, IG = Inverted Gamma, AR indicates AR(1) coefficient, SD indicates standard deviation of innovation, H = Home, F = Foreign.

*Monetary Policy and Exchange Rate Dynamics in a Behavioral Open Economy Model — Working Paper No. WP/2022/112*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022112-print-pdf.pdf_