## wp17264

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---

### Dominant Currency Paradigm (DCP): framework and key mechanisms
- Firms set export prices in a dominant currency (most often the dollar) and change them infrequently.
- Firms face strategic complementarities in pricing so desired mark-ups vary over time and across destination markets (Γ > 0 captures elasticity of the markup).
- Production uses domestic and foreign intermediate inputs (intermediate share α; benchmark α = 2/3).
- DCP departs fundamentally from the canonical New Keynesian small open economy model a` la Galí and Monacelli (2005).

### Main quantitative and qualitative predictions (model results)
- Terms-of-trade:
  - Terms-of-trade is stable at both short and medium horizons and plays little to no role in expenditure switching.
- Exchange rate pass-through (ERPT):
  - Dominant currency exchange rate pass-through into export and import prices is high regardless of destination or origin.
  - Pass-through of non-dominant currencies is negligible when dominant-currency exchange rate movements are accounted for.
- Trade quantities and expenditure switching:
  - Depreciations have limited expansionary impact on exports to dominant-currency destinations; if dominant currency prices are unchanged there is no increase in exports.
  - For firms that change prices, higher imported input prices and pricing complementarities dampen incentives to reduce prices, leaving exports mostly unchanged.
  - Impact on exports to non-dominant destinations depends on destination currency movements relative to the dominant currency.
- Relative prices driving import demand:
  - Expenditure switching occurs through imports via fluctuations in the relative price of imported to domestic goods driven by the country’s exchange rate relative to the dominant currency.
  - When a country’s currency depreciates relative to the dominant currency, it reduces demand for imports from all countries, all else equal.
- Global trade effects:
  - A strengthening of the dominant currency relative to non-dominant ones can negatively impact global trade: decline in imports across the periphery without a commensurate increase in exports.

### Numerical impulse-response example (domestic monetary policy shock: negative 25 basis point interest-rate cut)
- Parameter highlights (quarterly):
  - β = 0.99; σ_c = 2.00; φ^−1 (Frisch) = 0.50; κ = 1.00; α = 2/3; σ = 2.00; ε = 1.00; δ_w = 0.85; δ_p = 0.75; ρ_m = 0.50; φ_M = 1.50; ρ_ε_i = 0.50.
  - Set Γ = 1 (ε = 1 given Γ = ε/(σ−1) and σ = 2); home bias shares {γ_H, γ_U, γ_R} = {3/5, 1/5, 1/5}; η = 1.
- Outcomes (DCP vs PCP vs LCP, impact values reported in text):
  - Exchange rates and inflation: E_U and E_R depreciate by around 0.8% raising inflationary pressures; inflation increase larger under DCP and PCP than under LCP.
  - Trade-weighted imports decline by 0.43% (DCP); trade-weighted exports increase by 0.1% (DCP).
  - World trade (sum of export and import quantities):
    - DCP: trade declines by 0.2%.
    - PCP: trade expands by 0.47%.
    - LCP: trade increases by 0.27%.
  - Output and inflation trade-off (impact):
    - DCP: inflation rises by 0.35% on impact and output by 0.67% ⇒ ratio 0.52.
    - PCP: ratio ≈ 0.35/1.2 = 0.3.
    - LCP: ratio 0.1.
  - Real interest rate impact:
    - LCP: −0.24%.
    - PCP: −0.03%.
    - DCP: −0.01%.

### Equilibrium analytics and ERPT (closed-form and special cases)
- Flexible prices (Proposition 1) — export price ERPT:
  - ERPT_x ≡ ∆p^H_i,t / ∆e_i,t = 1 − (1−α)/((1 + Γ)(1−αγ_H)). Special cases:
    - α = 0 and Γ = 0 ⇒ ERPT_x = 0 (pass-through into destination-currency prices is 100%).
    - γ_H = 1 and Γ = 0 ⇒ ERPT_x = 0.
    - γ_H < 1 or Γ > 0 ⇒ ERPT_x > 0 (imperfect pass-through).
- Flexible prices — import price ERPT:
  - ERPT_m ≡ ∆p^i_H,t / ∆e_i,t = 1/(1 + Γ) + Γ/(1 + Γ) (1−γ_H)/(1−αγ_H).
    - Γ = 0 ⇒ ERPT_m = 1 (100%).
    - Γ > 0 ⇒ ERPT_m < 1 (incomplete pass-through).
- Fully rigid prices (Proposition 2) — pricing-regime extremes (stickiness δ_p = 1):
  - PCP: export prices independent of bilateral e_i,t; import prices load on e_i,t.
  - LCP: export prices load on e_i,t; import prices independent of e_i,t.
  - DCP: both export and import prices load exclusively on the dominant currency e_U,t; terms-of-trade are constant.

### Price dynamics (general case, Calvo-style reset equation approximations)
- Reset-price inflation equations (log-linearized):
  - π^j_Hi,t = λ_p/(1 + Γ) [ (mc^j_H,t − p^j_Hi,t) + Γ (p^j_i,t − p^j_Hi,t) + μ ] + β E_t π^j_Hi,t+1.
  - π^j_iH,t = λ_p/(1 + Γ) [ (mc^j_i,t − p^j_iH,t) + Γ (p^j_H,t − p^j_iH,t) + μ ] + β E_t π^j_iH,t+1.
- Interpretations:
  - Γ > 0 dampens direct impact of marginal cost on export price inflation but raises sensitivity to relative price ratios (competition).

### Canonical representation and law-of-one-price departures (special parametrization)
- Under ε = α = φ = 0, σ_c = 1, complete markets (Proposition 3):
  - π_HH,t = λ_p γ [ ̃y_t − (1−γ) ̃s_t ] + βE_t π_HH,t+1.
  - ̃y_t = E_t ̃y_t+1 − (i_t − E_t π_HH,t+1 − r^n_t) + (1−γ)E_t(∆ ̃m_t+1).
  - ̃m_t = 1/γ (̃y_t − ̃s_t), where ̃m_t captures deviations from the law of one price against the dominant currency.
- Implication: terms-of-trade ̃s_t evolves independently of monetary policy under these parameter restrictions; “divine coincidence” breaks down under DCP.

### Welfare loss and optimal monetary policy (second-order approximation)
- Welfare loss function under DCP (Proposition 4, when ε = α = φ = 0, σ_c = 1, τ = 1/σ):
  - W_DCP ≈ E_0 ∑_{t=0}^{∞} β^t [ 1/2 ̃y_t^2 + γ/(σ 2 λ_p) π_HH,t^2 + γ(1−γ)/2 ̃m_t^2 ] + t.i.p.
  - Additional misalignment term γ(1−γ)/2 ̃m_t^2 arises from failure of the law of one price; absent under PCP.
- Discretionary optimal trade-off under DCP:
  - ̃y_t + (1−γ) ̃m_t = −σ π_HH,t.
  - Contrast PCP: ̃y_t = −σ π_HH,t.
- Under discretionary optimal policy (no cost-push shocks), optimal policy calls for:
  - π_HH,t = 0 (domestic producer price inflation targeting).
  - ̃y_t = (1−γ) ̃s_t (output gap fluctuates with terms-of-trade).

### Commitment vs. discretion
- Commitment solution (t = 0):
  - ̃y_t = (1−γ) ̃s_t − γσ (p_HH,t − p_HH,−1).
  - Under absence of cost-push shocks discretion and commitment coincide because discretion yields π_HH,t = 0 each period.

### Policy interpretation and example
- Under DCP monetary policy cannot control terms-of-trade under the stated assumptions; optimal policy therefore targets producer price inflation and the misalignment ̃m_t in addition to the output gap.
- Example: positive domestic productivity shock can generate a negative output gap under DCP because dollar exchange rate movements imply import-relative-price behavior that does not induce the desired terms-of-trade movement.

---

### Empirical evidence from Colombia (customs-level data)
- Data and scope:
  - International trade data from DIAN and DANE; accessed via Banco de la República.
  - Universe of Colombian importers and exporters; 10-digit product code (Nandina); FOB values in U.S. dollars; net kilograms; country of destination/origin.
  - Monthly data aggregated to annual or quarterly; sample 2000–2015.
  - Starting 2007, export data include invoicing currency for each transaction.
- Currency invoicing distribution (Table 2; percentages):
  - All destinations: US Dollar: 98.28%; Euro: 0.72%; Colombian Peso: 0.67%.
  - Manufactures — All destinations: US Dollar: 98.39%; Euro: 0.70%; Colombian Peso: 0.52%.
  - Exports to US: US Dollar: 99.71%; Manufactures — US Dollar: 99.93%.
  - European Union: US Dollar: 90.73%; Euro: 8.64%; Manufactures — US Dollar: 86.19%; Manufactures — Euro: 13.28%.
- Conclusion: Colombia relies extensively on dollar invoicing; representative of many economies with extensive dollar invoicing.

### Empirical ERPT estimates (preserving reported estimates)
- Export prices to dollar destinations:
  - Contemporaneous pass-through: 84%.
  - Cumulative pass-through after two years: 56%.
  - Table 3 ∆e COP/USD coefficients reported: 0.699***, 0.677***, 0.830***, 0.863***, 0.798***, 0.821***.
- Import prices from dollar origins:
  - Contemporaneous pass-through: around 100%.
  - Cumulative effect after two years: 81%.
  - Table 5 ∆e COP/USD coefficients reported: 0.976***, 0.975***, 1.003***, 1.034***, 0.969***, 0.970***.
- Export prices to non-dollar destinations:
  - Initial pass-through: around 86%.
  - Decreases to 47% after two years.
  - Table 4 examples for ∆e COP/USD: 0.660***, 0.652***, 0.626***, 0.681***, 0.671***.
- Imports from non-dollar origins:
  - Initial pass-through: around 87%.
  - Decreases to 49% after two years.
  - Table 6 ∆e COP/LCU and ∆e COP/USD examples: ∆e COP/LCU (col 1) = 0.742***; other ∆e COP/LCU reported: 0.301***, 0.289***, 0.461***, 0.257**, 0.282***, 0.289**; ∆e COP/USD examples: 0.540***, 0.484***, 0.547***, 0.628***, 0.624***.
- Bilateral versus dominant-currency effects:
  - Unconditional bilateral exchange rate pass-through into peso export prices to non-dollar destinations: 70% at the annual horizon.
  - Conditional on the peso-dollar exchange rate, the coefficient on the bilateral exchange rate drops to 9% while the coefficient on the peso-dollar exchange rate is 70%.
  - Conditional on the peso-dollar exchange rate, the bilateral exchange rate is quantitatively insignificant as an explanatory factor in bilateral transactions with non-dollar economies.

### Quantities responses (empirical findings, preserved coefficients)
- Export quantities to dollar destinations: pass-through mainly insignificantly different from zero in most specifications; selected coefficients (Table 7) ∆e COP/USD: -0.608*, -0.466, -0.421, -0.0447.
- Import quantities from dollar origins: pronounced decline (Table 8) ∆e COP/USD: -1.104***, -0.939**, -1.123***, -0.950*.
- Exports to non-dollar destinations (Table 9): ∆e COP/LCU (col 1) = -0.872***; ∆e COP/USD examples: -1.057***, -0.972**, -1.156***, -0.966**.
- Imports from non-dollar origins (Table 10): ∆e COP/LCU (col 1) = -0.569**; ∆e COP/USD examples: -0.881***, -0.942***, -0.908***, -0.983**.
- Empirical regularity: when quantities respond, the relevant exchange rate is the peso/dollar exchange rate rather than bilateral rates for both export and import quantities.

### Terms-of-trade and commodity currency nature
- Colombia is a commodity-currency economy; peso movements strongly negatively correlated with commodity prices.
- Correlation between Colombian peso and overall (log) terms-of-trade: 0.62; regression coefficient: 1.15; R^2: 0.38.
- Non-commodity terms-of-trade: regression coefficient: 0.33; R^2: 0.36 — non-commodity terms-of-trade is far more stable, consistent with DCP predictions.

---

### Model estimation, simulation fit, and role of key features
- Estimation approach: calibration plus minimum distance estimation minimizing m(~τ)' Ω^{-1} m(~τ) over nine parameters ~τ and eleven moments.
- Selected fitted parameter and moment values (preserved exactly):
  - Measured export invoicing shares: to U: θ^U_HU = 1.00; to R: θ^U_HR, θ^R_HR = 0.93, 0.07.
  - Shocks — commodity prices: σ_ζ, ρ_ζ = 0.09, 0.74.
  - Estimated home-bias/trade shares: γ_H = 0.88; γ_U = 0.06; γ_R = 0.06.
  - e_R process: η, ρ_ε^r, σ_ε^r = 0.74, 0.82, 0.016.
  - a process: σ_a, ρ_a, ρ_{a,ζ} = 0.13, 0.49, -0.26.
- Moment matching (selected data vs model moments, preserved exactly):
  - β^U_{0,UH}: Data 0.98, Model 0.97.
  - β^U_{0,RH}: Data 0.89, Model 0.80.
  - β^H_{0,RH}: Data 0.18, Model 0.13.
  - ˆη: Data 0.54, Model 0.54.
  - ˆσ_ε^r: Data 0.018, Model 0.017.
  - ˆρ_ε^r: Data 0.78, Model 0.78.
  - ˆρ_{a,ζ}: Data 0.84, Model 0.87.
  - ˆσ_a: Data 0.023, Model 0.026.
  - ˆρ_a: Data 0.64, Model 0.64.
  - β^U_{0,HR}: Data 0.86, Model 0.81.
  - β^U_{0,RH}: Data 0.87, Model 0.90.
- Simulated quantity responses (Table 14, preserved exactly):
  - Responses of (∆y_HU, ∆y_UH, ∆y_HR, ∆y_RH) to exchange rates:
    - ∆e_U effects: 0.26, -1.60, -1.33, -1.19.
    - ∆e_R effects: -0.18, 0.28, 1.43, -0.11.
  - Interpretation: simulated model generates a weak expansion in exports to U following a depreciation and a pronounced contraction in imports from both U and R; exports to R are negatively impacted by depreciations relative to the dollar.

### Role of strategic complementarities (Γ) and imported inputs (α)
- Sensitivity exercises:
  - Benchmark: Γ = 1, α = 2/3 produces export pass-through at four quarters to both dollar and non-dollar destinations ≈ 65% (model).
  - Shutting down strategic complementarities and imported input use (Γ = 0 and α = 0) reduces predicted pass-through to 30% at one year — benchmark with Γ = 1 and α = 2/3 matches data best.

### Model selection and empirical fit
- Data strongly rejects PCP and LCP in favor of DCP.
- Estimated model and DCP reproduce empirical patterns:
  - Pass-through starts high and declines over time for exports to both U and R and for imports from U.
  - Bilateral exchange rates appear large when included alone for non-dollar partners but their coefficients fall substantially when the dollar exchange rate is included — captured by estimated model/DCP; PCP and LCP do not match this fact.

---

### Policy implications and conclusions
- Monetary policy:
  - Under DCP the inflation-output trade-off in response to a monetary policy shock worsens relative to PCP: a monetary rate cut raises inflation by much more than it increases output compared to PCP.
  - Terms-of-trade is independent of monetary policy under common parameter restrictions (contrast PCP).
  - Breakdown of “divine coincidence”: cannot simultaneously attain zero inflation and zero output gap under DCP in cases where divine coincidence would obtain under PCP.
  - Optimal monetary policy targets deviations from the law of one price arising from dominant currency fluctuations, in addition to inflation and output gap.
  - In absence of cost-push shocks optimal policy calls for domestic producer price inflation targeting while the output gap fluctuates with the terms-of-trade.
- Trade and macro implications:
  - Following a depreciation of the domestic currency, imports from all origins will decrease.
  - Exports to dominant-currency destinations will not be responsive to currency movements; exports to other destinations depend on co-movement of the destination country’s exchange rate with the dominant currency.
  - DCP implies that weakening of emerging market currencies relative to the dominant currency can be associated with a decline in world trade relative to PCP or LCP.
- Empirical validation:
  - Colombia’s customs data and pass-through estimates strongly support DCP and highlight the roles of dollar invoicing, strategic complementarities (Γ > 0), and imported input use (α > 0).

*Source: IMF Working Paper — wp17264.*

### 1. Introduction ........................................................................................................

### 1. Introduction

### Motivation and pricing paradigms
- Nominal exchange rate fluctuations, together with price rigidities, affect relative prices and real variables such as the trade balance, consumption, and output.
- The relationship between nominal exchange rate fluctuations and other nominal and real variables depends critically on the currency in which prices are rigid.
- Two canonical paradigms in the literature:
  - Producer currency pricing (PCP): prices sticky in the currency of the producing country; law of one price holds; a nominal depreciation raises the price of imports relative to exports (the terms-of-trade), improving competitiveness.
  - Local currency pricing (LCP): prices sticky in the currency of the destination market; a nominal depreciation lowers the price of imports relative to exports (a decline in the terms-of-trade), worsening competitiveness.
- Empirical observations challenge both PCP and LCP:
  - Vast majority of trade invoiced in a small number of dominant currencies, with the U.S. dollar playing an outsized role.
  - Invoiced prices are found to be rigid for significant durations in their currency of invoicing.
  - Exporters face strategic complementarities in pricing leading to time-varying desired mark-ups.
  - Most exporting firms use imported inputs in production, reducing value added content of exports.

### Dominant Currency Paradigm (DCP): assumptions
- Firms set export prices in a dominant currency (most often the dollar) and change them infrequently.
- Firms face strategic complementarities in pricing so desired mark-ups vary over time and across destination markets.
- Production uses domestic and foreign intermediate inputs (roundabout production).
- The model departs fundamentally from the canonical NK small open economy model a` la Galí and Monacelli (2005).

### DCP: quantitative and qualitative predictions (main results)
- Terms-of-trade is stable at both short and medium horizons and plays little to no role in expenditure switching.
- Exchange rate pass-through (ERPT):
  - Dominant currency exchange rate pass-through into export and import prices is high regardless of destination or origin.
  - Pass-through of non-dominant currencies is negligible.
- Trade quantities and expenditure switching:
  - Depreciations have limited expansionary impact on exports to dominant-currency destinations; for firms whose dominant currency prices are unchanged there is no increase in exports.
  - For firms that change prices, the rise in marginal cost from higher imported input prices and pricing complementarities dampens incentives to reduce prices, leaving exports mostly unchanged.
  - Impact on exports to non-dominant destinations depends on destination currency movements relative to the dominant currency.
- Relative prices driving import demand:
  - Expenditure switching still occurs through imports via fluctuations in the relative price of imported to domestic goods.
  - These fluctuations are driven by the country’s exchange rate relative to the dominant currency, regardless of origin of imported goods.
  - When a country’s currency depreciates relative to the dominant currency, it reduces demand for imports from all countries, all else equal.
- Global trade effects of dominant currency movements:
  - A strengthening of the dominant currency relative to non-dominant ones can negatively impact global trade: decline in imports across the periphery without a commensurate increase in exports.
  - Under PCP, a depreciation would raise export competitiveness and increase exports; under LCP both import and export responses are muted and the impact on global trade is weak but positive.
- Monetary policy implications:
  - Under DCP, the inflation-output trade-off in response to a monetary policy shock (under an inflation targeting rule) worsens relative to PCP: a monetary rate cut raises inflation by much more than it increases output compared to PCP.
  - Terms-of-trade is independent of monetary policy under common parameter restrictions, in contrast to PCP.
  - Breakdown of “divine coincidence”: cannot simultaneously attain zero inflation and zero output gap under DCP in cases where divine coincidence would obtain under PCP.
  - Optimal monetary policy targets deviations from the law of one price arising from dominant currency fluctuations, in addition to inflation and output gap.
  - Optimal policy calls for domestic producer price inflation targeting while the output gap fluctuates with the terms-of-trade.

### Empirical evidence (Colombia)
- Colombia is representative of emerging markets in heavy reliance on dollar invoicing:
  - 98.3% of exports invoiced in dollars.
  - 98.4% of manufacturing exports invoiced in dollars.
- Pass-through estimates (elasticity relative to the peso-dollar exchange rate):
  - Exports to dollar destinations:
    - Contemporaneous pass-through: 84%.
    - Cumulative pass-through after two years: 56%.
  - Imports from dollar origins:
    - Contemporaneous pass-through: around 100%.
    - Cumulative effect after two years: 81%.
  - Exports to non-dollar destinations:
    - Initial pass-through: around 86%.
    - Decreases to 47% after two years.
  - Imports from non-dollar origins:
    - Initial pass-through: around 87%.
    - Decreases to 49% after two years.
- Bilateral versus dominant currency exchange rate effects:
  - Unconditionally, bilateral exchange rate pass-through into peso export prices to non-dollar destinations is 70% at the annual horizon.
  - Conditional on the peso-dollar exchange rate, the coefficient on the bilateral exchange rate drops to 9% while the coefficient on the peso-dollar exchange rate is 70%.
  - Conditional on the peso-dollar exchange rate, the bilateral exchange rate is quantitatively insignificant as an explanatory factor in bilateral transactions with non-dollar economies.
- Quantities response to peso-dollar depreciation:
  - For export quantities to dollar destinations, pass-through is mainly insignificantly different from zero.
  - There is a pronounced decline in quantities imported from both dollar and non-dollar countries.
  - Exports to non-dollar destinations also decline.
  - When quantities respond, the relevant exchange rate is the peso/dollar exchange rate rather than the bilateral exchange rate for both export and import quantities.
- Terms-of-trade evidence:
  - Colombia’s overall terms-of-trade is very volatile and strongly correlated with the exchange rate.
  - Stripping out commodity prices, the terms-of-trade is highly stable—consistent with DCP predictions.

### Model validation, extensions, and quantitative performance
- Section 5 simulates an economy subject to commodity price shocks, productivity shocks, and third country exchange rate shocks; the model nests DCP, PCP, and LCP and is used to evaluate which paradigm matches the data.
- Data strongly rejects PCP and LCP in favor of DCP.
- The data also favors a model with strategic complementarities in pricing and imported input use:
  - Under benchmark DCP specification, export pass-through at four quarters to both dollar and non-dollar destinations is 65%.
  - Shutting down strategic complementarities and imported input use reduces predicted pass-through to 30%.

### Contribution to literature
- Develops a quantitative new Keynesian small open economy model combining dynamic dominant currency pricing, variable mark-ups (strategic complementarities), and imported input use.
- Provides a counterpart for empirical pass-through regressions.
- Empirically evaluates DCP using novel tests on Colombia customs data.
- Derives target criteria for optimal monetary policy under dominant currency pricing.
- Situates contribution relative to prior dollar pricing literature and highlights differences (dynamic pricing, three-region world, variable mark-ups, imported inputs, incomplete international asset markets).

*Source: IMF Working Paper — 1. Introduction (wp17264).*

### 2.1  Households

### 2.1  Households

### Preferences and consumption aggregator
- Per-period utility (separable in consumption and labor):
  - U(C_t,N_t) = 1/(1−σ_c) C_t^(1−σ_c) − κ/(1+φ) N_t^(1+φ)
  - Parameters: σ_c > 0, φ > 0, κ scales disutility of labor.
- Consumption aggregator C defined implicitly by a Kimball (1995) homothetic demand aggregator:
  - ∑_i 1/|Ω_i| ∫_{ω∈Ω_i} γ_i Υ(|Ω_i| C_{iH}(ω) / (γ_i C)) dω = 1.
  - C_{iH}(ω) is consumption in country H of variety ω produced in country i, i ∈ {H, U, R}.
  - γ_i captures home bias in H with ∑_i γ_i = 1.
  - |Ω_i| is the measure of varieties produced in region i.
  - Υ satisfies Υ(1) = 1, Υ′(.) > 0, Υ′′(.) < 0.
- Demand structure implies strategic complementarities in pricing and variable mark-ups; captures Dornbusch (1987) and Krugman (1987) channel of variable mark-ups and pricing to market.

### Household problem and budget constraint
- Households maximize expected lifetime utility:
  - max_{C_t,W_t,B_t} E_0 ∑_{t=0}^∞ β^t U(C_t,N_t)
- Per-period budget constraint (expressed in home currency):
  - P_t C_t + E_{U,t} (1 + i_{U,t−1}) B_{U,t} + B_t = W_t(h) N_t(h) + Π_t + E_{U,t} B_{U,t+1} + ∑_{s′∈S} Q_t(s′) B_{t+1}(s′) + E_{U,t} ζ_t
  - Variables and definitions:
    - P_t: price index for domestic consumption aggregator C_t.
    - Π_t: domestic profits transferred to households (owners).
    - B_{U,t+1}: dollar debt holdings of the risk-free international bond; i_{U,t} nominal interest rate in dollars.
    - Q_t(s): period-t price of the security that pays one unit of home currency in period t+1 in state s ∈ S.
    - B_{t+1}(s): holdings of state-contingent securities (domestic H currency), zero net supply.
    - ζ_t: exogenous dollar income shock (captures commodity price movements for small commodity exporters).

### Demand system, elasticities, and price index
- Optimal demand system:
  - C_{iH,t}(ω) = γ_i ψ(D_t P_{iH,t}(ω) / P_t) C_t
  - ψ(.) ≡ Υ′^−1(.) > 0 so that ψ′(.) < 0.
  - D_t ≡ ∑_i ∫_{Ω_i} Υ′(|Ω_i| C_{iH,t}(ω) / (γ_i C_t)) C_{iH,t}(ω) / C_t dω.
  - P_{iH,t}(ω): home price of variety ω produced in country i and sold in H.
- Elasticity of demand:
  - σ_{iH,t}(ω) ≡ −∂ log C_{iH,t}(ω) / ∂ log Z_{iH,t}(ω), where Z_{iH,t}(ω) ≡ D_t P_{iH,t}(ω) / P_t.
  - Log optimal flexible price mark-up: μ_{iH,t}(ω) ≡ log(σ_{iH,t} / (σ_{iH,t} − 1)), time-varying.
  - Γ_{iH,t}(ω) ≡ ∂ μ_{iH,t} / ∂ log Z_{iH,t}(ω): elasticity of the markup.
- Price index relation:
  - P_t C_t = ∑_i ∫_{Ω_i} P_{iH,t}(ω) C_{iH,t}(ω) dω.

### Intertemporal Euler equations
- For Ubonds (dollar bonds):
  - C_t^(−σ_c) = β (1 + i_{U,t}) E_t [ C_{t+1}^(−σ_c) P_t / P_{t+1} E_{U,t+1} / E_{U,t} ]
- For H bonds (domestic H currency bonds):
  - C_t^(−σ_c) = β (1 + i_t) E_t [ C_{t+1}^(−σ_c) P_t / P_{t+1} ]
- Relation: (1 + i_t) = (∑_{s′∈S} Q_t(s′))^(−1) is the inverse of the price of a risk-free nominal H currency bond at time t delivering one unit of H currency in every state in period t+1.

### Wage setting and Calvo friction
- Calvo friction on wage setting in H currency:
  - In any period, households can adjust wage with probability 1 − δ_w; otherwise maintain previous-period nominal wage.
- Individual labor demand (downward sloping) for variety supplied by household h:
  - N_t(h) = (W_t(h) / W_t)^(−θ) N_t, with θ > 1 the constant elasticity of labor demand and W_t the aggregate wage rate.
- Optimality condition for wage setting (standard Calvo wage condition):
  - E_t ∑_{s=t}^∞ δ_w^{s−t} Θ_{t,s} N_s W_s^{θ(1+φ)} [ θ/(θ−1) κ P_s C_s^σ N_s^φ − \bar W_t(h)^{1+θφ} / W_s^{θφ} ] = 0
  - Θ_{t,s} ≡ β^{s−t} C_s^(−σ_c) / C_t^(−σ_c) P_t / P_s is the stochastic discount factor between periods t and s ≥ t.
  - \bar W_t(h) is the optimal reset wage in period t.
- Interpretation: \bar W_t(h) is preset as a constant markup over the expected weighted-average of future marginal rates of substitution between labor and consumption and aggregate wage rates during the duration of the wage (standard New Keynesian result).

*Source: wp17264 - 2.1  Households (IMF working paper PDF).*

### 2.4 Equilibrium and Some Analytics

### 2.4 Equilibrium and Some Analytics

### Definition of equilibrium
- Monopolistically competitive equilibrium of the small open economy H consists of:
  - Households maximizing utility over consumption, labor supply and portfolio choice, and firms maximizing profits over labor demand, intermediate inputs and prices in each market.
  - Market clearing: L_t = N_t, B^h_t = 0, Y^H_i,t = C^H_i,t + X^H_i,t.
  - Real exchange rates of R and U related according to eq.(17).
  - Exogenous shocks to domestic monetary policy, ε^M,t, the budget constraint, ζ_t, productivity a_t, and the real exchange rate ε^R,t that follow AR(1) processes.
- The model is solved by log-linearizing around a symmetric zero inflation steady state. Pricing equations approximated up to first order depend on the constant Γ_ij,t = Γ evaluated at the steady state.

### Exchange Rate Pass-through (ERPT)
- Variables: p = ln P, w = ln W, e = ln E. Foreign prices and quantities kept fixed at exogenous values.
- Proposition 1 (Flexible prices, δ_p = 0):
  - Export prices (p^H_i,t) and import prices (p^i_H,t) in H currency:
    - ∆p^H_i,t =
      1/(1 + Γ) [ (αγ_i/(1−αγ_H) + Γ) ∆e_i,t + (αγ_j/(1−αγ_H)) ∆e_j,t + (1−α)/(1−αγ_H) ∆w_t − 1/(1−αγ_H) ∆a_t ]  (Eq. 18)
    - ∆p^i_H,t =
      1/(1 + Γ) [ (1 + Γ γ_i/(1−αγ_H)) ∆e_i,t + Γ γ_j/(1−αγ_H) ∆e_j,t + Γ γ_H (1−α)/(1−αγ_H) ∆w_t − Γ γ_H/(1−αγ_H) ∆a_t ]  (Eq. 19)
    - j ≠ i, for i,j ∈ {U,R}.
  - Export price ERPT (under ∆w_t = 0, ∆a_t = 0, η = 1):
    - ERPT_x ≡ ∆p^H_i,t / ∆e_i,t = 1 − (1−α)/((1 + Γ)(1−αγ_H))  (Eq. 20)
    - Special cases:
      - α = 0 and Γ = 0 ⇒ ERPT_x = 0 (pass-through into destination currency prices is 100%).
      - γ_H = 1 and Γ = 0 ⇒ ERPT_x = 0 (marginal cost depends only on local wages/productivity).
      - γ_H < 1 or Γ > 0 ⇒ ERPT_x > 0 (imperfect pass-through).
  - Import price ERPT (same assumptions):
    - ERPT_m ≡ ∆p^i_H,t / ∆e_i,t = 1/(1 + Γ) + Γ/(1 + Γ) (1−γ_H)/(1−αγ_H)  (Eq. 21)
    - Special cases:
      - Γ = 0 ⇒ ERPT_m = 1 (100%): foreign firms set constant price in foreign currency.
      - Γ > 0 ⇒ ERPT_m < 1 (incomplete pass-through) because strategic complementarities dampen direct pass-through but imported-input channel can work in opposite direction.
- Proposition 2 (Fully rigid prices, δ_p = 1):
  - ∆p^H_i,t = θ^U_Hi ∆e_U,t + I_{i=R} · θ^R_Hi ∆e_R,t  (Eq. 22)
  - ∆p^i_H,t = θ^U_iH ∆e_U,t + I_{i=R} · θ^R_iH ∆e_R,t  (Eq. 23)
  - Pricing regime extremes:
    - PCP (producer currency pricing): θ_H^Hi = 1 and θ_i^iH = 1 for i ∈ {U,R}:
      - ∆p^H_i,t = 0·∆e_i,t + 0·∆e_{j≠i,t}; ∆p^i_H,t = 1·∆e_i,t + 0·∆e_{j≠i,t}; tot_iH,t = 1·∆e_i,t ∀i.
    - LCP (local currency pricing): θ_i^Hi = 1 and θ_H^iH = 1 for i ∈ {U,R}:
      - ∆p^H_i,t = 1·∆e_i,t + 0·∆e_{j≠i,t}; ∆p^i_H,t = 0·∆e_i,t + 0·∆e_{j≠i,t}; tot_iH,t = −1·∆e_i,t ∀i.
    - DCP (dominant currency pricing): θ^U_Hi = 1 and θ^U_iH = 1 for i ∈ {U,R}:
      - ∆p^H_i,t = 1·∆e_U,t + 0·∆e_{i≠U,t}; ∆p^i_H,t = 1·∆e_U,t + 0·∆e_{i≠U,t}; tot_iH,t = 0 ∀i.
  - Implications:
    - Under DCP and extreme stickiness, the only relevant exchange rate is the dollar exchange rate e_U,t and terms-of-trade are constant.
    - Under PCP and LCP one of export or import prices loads on the bilateral exchange rate e_i,t and terms-of-trade movements fully load on e_i,t (PCP: depreciation worsens tot; LCP: reverse).

### Price dynamics: the general case
- Destination/source and currency-specific inflation representations (log-linearized reset price equation):
  - π^j_Hi,t = λ_p/(1 + Γ) [ (mc^j_H,t − p^j_Hi,t) + Γ (p^j_i,t − p^j_Hi,t) + μ ] + β E_t π^j_Hi,t+1  (Eq. 24)
  - π^j_iH,t = λ_p/(1 + Γ) [ (mc^j_i,t − p^j_iH,t) + Γ (p^j_H,t − p^j_iH,t) + μ ] + β E_t π^j_iH,t+1  (Eq. 25)
  - Definitions:
    - λ_p = (1 − δ_p)(1 − β δ_p)/δ_p.
    - mc^j_i,t = ln(MC_t / E_j,t) is nominal marginal cost of firms in country i expressed in currency j.
    - p^j_i,t is log aggregate price level of country i in currency j.
    - μ is log steady-state desired gross markup; Γ is steady-state elasticity of that markup.
- Interpretations:
  - π^j_Hi,t depends on (a) destination/currency-specific markup (p^j_Hi,t − mc^j_H,t), (b) ratio of export prices to destination price index (p^j_Hi,t − p^j_i,t), and (c) expected future export price inflation.
  - Γ > 0 dampens direct impact of real marginal cost or markup on export price inflation but raises sensitivity to relative price ratios (firms pay more attention to competitors’ prices).
  - Marginal costs incorporate imported inputs; cost shocks in U and R directly affect H firms’ pricing.

### Impulse Response to a Monetary Policy Shock (numerical)
- Demand aggregator: Klenow and Willis (2016) functional form used:
  - Y_iH,t(ω) ≡ C_iH,t(ω) + X_iH,t(ω) = γ_i (1 + ε ln((σ−1)/σ) − ε ln Z_iH,t)^{σ/ε} (C_t + X_t)
  - Z ≡ P_iH(ω) / P
  - σ_iH,t = σ (1 + ε ln((σ−1)/σ) − ε ln Z_iH,t)
  - Γ_iH,t = ε (σ−1 − ε ln((σ−1)/σ) + ε ln Z_iH,t)
  - In symmetric steady state Z_iH,t = (σ−1)/σ, elasticity σ and markup elasticity Γ ≡ ε/(σ−1).
- Parameter values (Table 1; time period: quarter):
  - Discount factor β 0.99
  - Risk aversion σ_c 2.00
  - Frisch elasticity of N φ^−1 0.50
  - Disutility of labor κ 1.00
  - Intermediate share α 2/3
  - Elasticity σ 2.00
  - Super-elasticity ε 1.00
  - Wage stickiness δ_w 0.85
  - Price δ_p 0.75
  - Monetary inertia ρ_m 0.50
  - Inflation sensitivity φ_M 1.50
  - Shock persistence ρ_ε_i 0.50
  - Additional calibrations (in text):
    - Set Γ = 1 ⇒ ε = 1 given Γ = ε/(σ−1) and σ = 2.
    - Home bias shares {γ_H, γ_U, γ_R} = {3/5, 1/5, 1/5} (implies steady state spending on imported goods equal to forty percent).
    - Set η = 1 (both currencies depreciate identically to H monetary shock).
    - Wage stickiness δ_w = 0.85 (≈ year and a half average wage duration).
- Shock studied: negative 25 basis point exogenous cut in interest rates (domestic monetary policy shock).
- Main numerical outcomes (comparisons across DCP, PCP, LCP):
  - Exchange rates and inflation:
    - Domestic interest rates decline but less than one-to-one; exchange rates E_U and E_R depreciate by around 0.8%, raising inflationary pressures.
    - Inflation increase: DCP and PCP show much larger inflation responses than LCP.
  - Terms-of-trade:
    - Under PCP: terms-of-trade depreciates almost one-to-one with exchange rate.
    - Under LCP: terms-of-trade appreciates one-to-one.
    - Under DCP: terms-of-trade depreciates negligibly and remains stable (export and import prices stable in dominant currency).
  - Exports and imports (trade-weighted):
    - DCP: H-currency price of exports and imports rise with depreciation; trade-weighted imports decline by 0.43%; trade-weighted exports increase by 0.1%.
    - PCP: large increase in exports; imports behavior differs due to imported inputs and export expansion.
    - LCP: increase in imports driven by demand expansion.
  - World trade (sum of export and import quantities):
    - DCP: trade declines by 0.2% (imports fall without commensurate export increase).
    - PCP: trade expands by 0.47% (exports increase outweigh decrease in imports).
    - LCP: trade increases by 0.27% (driven mainly by imports).
  - Output:
    - Expansionary impact on output muted under DCP relative to PCP; lowest impact under LCP.
    - Under DCP: expenditure-switching from imports toward domestic output present but export expansion absent.
    - Impact magnitudes and inflation-output trade-off (impact values given in text):
      - DCP: inflation rises by 0.35% on impact and output by 0.67% ⇒ ratio 0.52.
      - PCP: ratio ≈ 0.35/1.2 = 0.3.
      - LCP: ratio 0.1 (lowest).
  - Consumption:
    - Consumption increases most under LCP compared to PCP and DCP.
    - Real interest rates decline most under LCP on impact: −0.24% (LCP) vs −0.03% (PCP) vs −0.01% (DCP).
  - Mark-up and pricing-to-market:
    - DCP: increase in mark-ups due to stable prices in dominant currency and rigidity of wages in home currency; increase is similar but more muted than LCP because higher import input prices raise marginal costs in DCP.
    - PCP: mark-ups decline as marginal costs increase while prices in home currency remain stable.
    - Pricing-to-market: large decline in relative price of goods sold at home in LCP and DCP; much more muted in PCP (arises through variable mark-up channel).
- Graphical results referenced (Figures 1 and 2) illustrate impulse responses across DCP, PCP, LCP for shock, interest rates, inflation, exchange rate, terms-of-trade, export/import prices, export/import quantities, trade, output, consumption, mark-up, and pricing-to-market.

### Empirical evidence (model testing)
- The model’s implications are tested using unique customs data from Colombia on exports and imports at the firm level. Empirical pass-through results for import and export prices and quantities are presented and later compared to the model’s predictions (details in subsequent sections).

*Source: wp17264 - 2.4 Equilibrium and Some Analytics (IMF Working Paper).*

### Section 5.

### Section 5

### Data sources
- International trade data are from the customs agency (DIAN) and the department of statistics (DANE), and include information on the universe of Colombian importers and exporters.
- Data accessed through the Banco de la República.
- Key data fields: trading firm’s tax identification number, the 10-digit product code (Nandina classification system, based on the Harmonized System), the FOB value (in U.S. dollars), volume (net kilograms) of exports (imports), and the country of destination (origin), among other details.
- Temporal frequency: data available on a monthly basis; analysis aggregates exports and imports at the annual or quarterly level.
- Sample period: available for the period between 2000 and 2015.
- Starting in 2007, export data include invoicing currency for each transaction.
- For imports, when the imported good was produced in one country but arrived from a third country (e.g., goods produced in China arriving to Colombia from either the United States or Panama), regressions keep only observations where the country of origin and purchase are the same to avoid introducing unnecessary noise.

### Currency invoicing distribution (Table 2)
- The vast majority of Colombian exports are priced in dollars; the U.S. dollar accounts for over 98% of all exports.
- Distribution calculated for number of invoices in each currency.
- Table 2 — Currency Distribution, by Destination (values are percentages):

  - Destination: US
    - US Dollar: 99.71%
    - Euro: 0.02%
    - Colombian Peso: 0.27%
    - Manufactures — US Dollar: 99.93%
    - Manufactures — Euro: 0.03%
    - Manufactures — Colombian Peso: 0.03%

  - Destination: Dollar economies
    - US Dollar: 99.73%
    - Euro: 0.03%
    - Colombian Peso: 0.23%
    - Manufactures — US Dollar: 99.91%
    - Manufactures — Euro: 0.04%
    - Manufactures — Colombian Peso: 0.03%

  - Destination: CAN
    - US Dollar: 99.75%
    - Euro: 0.07%
    - Colombian Peso: 0.18%
    - Manufactures — US Dollar: 99.90%
    - Manufactures — Euro: 0.07%
    - Manufactures — Colombian Peso: 0.03%

  - Destination: Latin America
    - US Dollar: 99.18%
    - Euro: 0.13%
    - Colombian Peso: 0.22%
    - Bolívar (Ven): 0.44%
    - Mexican Peso: 0.02%
    - Colón (CR): 0.01%
    - Manufactures — US Dollar: 99.34%
    - Manufactures — Euro: 0.13%
    - Manufactures — Colombian Peso: 0.03%
    - Manufactures — Bolívar (Ven): 0.45%
    - Manufactures — Mexican Peso: 0.02%
    - Manufactures — Colón (CR): 0.01%

  - Destination: European Union
    - US Dollar: 90.73%
    - Euro: 8.64%
    - Colombian Peso: 0.31%
    - Sterling Pound: 0.28%
    - Manufactures — US Dollar: 86.19%
    - Manufactures — Euro: 13.28%
    - Manufactures — Colombian Peso: 0.21%
    - Manufactures — Sterling Pound: 0.26%

  - Destination: Euro zone
    - US Dollar: 88.78%
    - Euro: 10.80%
    - Colombian Peso: 0.39%
    - Sterling Pound: 0.01%
    - Manufactures — US Dollar: 84.48%
    - Manufactures — Euro: 15.22%
    - Manufactures — Colombian Peso: 0.25%
    - Manufactures — Sterling Pound: 0.01%

  - Destination: All destinations
    - US Dollar: 98.28%
    - Euro: 0.72%
    - Colombian Peso: 0.67%
    - Venezuelan Bolívar: 0.27%
    - Sterling Pound: 0.02%
    - Manufactures — US Dollar: 98.39%
    - Manufactures — Euro: 0.70%
    - Manufactures — Colombian Peso: 0.52%
    - Manufactures — Venezuelan Bolívar: 0.33%
    - Manufactures — Sterling Pound: 0.01%

- Notes:
  - (1) Exports of coke, refined petroleum products, and nuclear fuel (ISIC 23), and basic metals (ISIC 27) excluded from “Manufactures”.
  - (2) Distribution calculated for number of invoices in each currency.
- Conclusion: The distribution is very similar when looking at the value of exports negotiated in each currency instead of the number of transactions; Colombian economy relies extensively on dollar invoicing and is representative of many economies with extensive dollar invoicing.

### Exchange rate, commodity currency nature, and terms-of-trade
- Exchange rate data obtained from the International Monetary Fund.
- The Colombian exchange rate (peso) is a commodity currency; fluctuations in the peso are strongly negatively correlated with fluctuations in commodity prices.
- Commodity prices can be considered as exogenous to the economy: while mining output makes up 58.4% of total exports for Colombia, it is small relative to world commodity markets. Example: Colombia’s oil production was 1.1% of world oil production in 2014.
- Figure 3: Relation between the Colombian peso (solid black line) and the overall (log) terms-of-trade (dashed blue line), defined as the log difference between import and export prices.
  - Correlation between the two series: 0.62.
  - Regression coefficient: 1.15.
  - R^2: 0.38.
- Non-commodity terms-of-trade (dots-and-dash red line) results:
  - Regression coefficient: 0.33.
  - R^2: 0.36.
  - Interpretation: non-commodity terms-of-trade is far more stable, consistent with the predictions of the model under DCP.

*Source: Authors’ calculations based on data from DIAN/DANE; exchange rate data from the International Monetary Fund.*

### 4.2 Results

### 4.2 Results

### Data, sample and measurement
- Focus: manufactured goods (excluding petrochemicals and basic metals); ISIC Rev. 3.1 classification. Robustness: differentiated products subsample using Rauch (1999).
- Observations unit: 10-digit product × country × year (or quarter).
- Prices: FOB value per net kilogram. Quantities: total net kilograms. Exchange rates: annual (or quarterly) average.

### Exchange rate pass-through (ERPT) — dynamic evidence
- Estimation: dynamic lag regression (∆x_t = α + sum_{s=0}^8 β_s ∆e_{t−s} + Z_t + ε_t) with contemporaneous effect and eight lags; controls include firm-industry-country fixed effects and quarter dummies; standard errors clustered at quarter-year for cumulative plots.
- Main stylized pattern: pass-through starts high (close to one) and declines over time across export and import prices and for dollar and non-dollar origins/destinations — consistent with Dominant Currency Pricing (DCP).
- Key point estimates (cumulative / contemporaneous descriptions reported in text):
  - Export prices to dollar destinations: contemporaneous estimate = 0.84; cumulative pass-through after two years = 0.56.
  - Import prices from dollar origins: contemporaneous pass-through ≈ 1; cumulative after two years = 0.81.
  - Export prices to non-dollar destinations: start ≈ 0.86 and decrease to 0.47 after two years.
- Bilateral vs dollar exchange rate: when the peso/dollar exchange rate is included, coefficients on bilateral exchange rates fall substantially — consistent with DCP.

### Selected regression coefficient highlights (preserved exactly as reported)
- Table 3: ERPT into Colombian Export Prices (Dollarized economies, U)
  - ∆e COP/USD coefficients across columns: 0.699***, 0.677***, 0.830***, 0.863***, 0.798***, 0.821***.
- Table 4: ERPT into Colombian Export Prices (Non-Dollarized economies, R)
  - ∆e COP/LCU (column 1) = 0.697***; when bilateral and peso/dollar included coefficients vary (examples preserved in table).
  - ∆e COP/USD (included in some columns) = 0.660***, 0.652***, 0.626***, 0.681***, 0.671*** (as reported).
- Table 5: ERPT into Colombian Import Prices (Dollarized, U)
  - ∆e COP/USD coefficients: 0.976***, 0.975***, 1.003***, 1.034***, 0.969***, 0.970***.
- Table 6: ERPT into Colombian Import Prices (Non-Dollarized, R)
  - ∆e COP/LCU (column 1) = 0.742***; other reported ∆e COP/LCU coefficients: 0.301***, 0.289***, 0.461***, 0.257**, 0.282***, 0.289**.
  - ∆e COP/USD examples: 0.540***, 0.484***, 0.547***, 0.628***, 0.624*** (as reported).

### Quantities: differential import and export responses (theoretical prediction and empirical findings)
- Theoretical first-order approximations:
  - Exports: ∆y_Hi / ∆e_U = −σ ( ∆p_Hi / ∆e_U − ∆e_i / ∆e_U ). (Eq. 28)
  - Imports: ∆y_iH / ∆e_U = −σ ( ∆p_iH / ∆e_U − ∆p / ∆e_U ) + ∆y_d / ∆e_U. (Eq. 29)
- Empirical regularities:
  - Exports to dollarized economies (U): pass-through to export quantities is insignificantly different from zero in most specifications; one specification shows exports decline.
  - Imports from U: pronounced decline in quantities across specifications.
  - For non-dollarized economies (R): imports from R decline significantly; the relevant exchange rate is the peso/dollar rate rather than the bilateral rate.
  - Exports to R can decline following a weakening of the peso relative to the dollar if the co-movement between R’s currency and the dollar is sufficiently weak.
- Selected quantity regression coefficients (Tables 7–10):
  - Table 7 (Export quantities to Dollarized U): ∆e COP/USD coefficients: -0.608*, -0.466, -0.421, -0.0447.
  - Table 8 (Import quantities from Dollarized U): ∆e COP/USD coefficients: -1.104***, -0.939**, -1.123***, -0.950*.
  - Table 9 (Export quantities to Non-Dollarized R): ∆e COP/LCU (col 1) = -0.872***; ∆e COP/USD examples: -1.057***, -0.972**, -1.156***, -0.966**.
  - Table 10 (Import quantities from Non-Dollarized R): ∆e COP/LCU (col 1) = -0.569**; ∆e COP/USD examples: -0.881***, -0.942***, -0.908***, -0.983**.

### Model estimation, parameter values and moment matching
- Estimation approach: combination of calibration and minimum distance estimation minimizing m(~τ)' Ω^{-1} m(~τ), with ~τ = {θ^U_UH, θ^U_RH, θ^R_RH, η, σ_ε^r, ρ_ε^r, σ_a, ρ_a, ρ_{a,ζ}} (nine parameters). Eleven moments used, including invoicing shares, pass-through coefficients, exchange rate co-movement, and moments for productivity and commodity shocks.
- Table 11: Parameter values (selected entries preserved exactly)
  - Measured — Export Invoicing Shares:
    - to U: θ^U_HU = 1.00
    - to R: θ^U_HR, θ^R_HR = 0.93, 0.07
  - Shocks — commodity prices: σ_ζ, ρ_ζ = 0.09, 0.74
  - Estimated — Home bias and trade shares:
    - γ_H = 0.88
    - γ_U = 0.06
    - γ_R = 0.06
  - Exports: to U D_U = -2.38; to R D_R = -0.87
  - Oil endowment ζ̄ = 0.27
  - Import invoicing shares:
    - from U: θ^U_UH = 1.00
    - from R: θ^U_RH, θ^R_RH = 0.93, 0.07
  - e_R process parameters: η, ρ_ε^r, σ_ε^r = 0.74, 0.82, 0.016
  - a process: σ_a, ρ_a, ρ_{a,ζ} = 0.13, 0.49, -0.26
  - Note: other parameter values as reported in the text.
- Moment matching (Table 12): selected data vs model moments (preserved exactly)
  - β^U_{0,UH}: Data 0.98, Model 0.97
  - β^U_{0,RH}: Data 0.89, Model 0.80
  - β^H_{0,RH}: Data 0.18, Model 0.13
  - ˆη: Data 0.54, Model 0.54
  - ˆσ_ε^r: Data 0.018, Model 0.017
  - ˆρ_ε^r: Data 0.78, Model 0.78
  - ˆρ_{a,ζ}: Data 0.84, Model 0.87
  - ˆσ_a: Data 0.023, Model 0.026
  - ˆρ_a: Data 0.64, Model 0.64
  - β^U_{0,HR}: Data 0.86, Model 0.81
  - β^U_{0,RH}: Data 0.87, Model 0.90

### Model simulations and pricing paradigms (DCP vs PCP vs LCP)
- Approach: simulate estimated model and contrast with DCP, Producer Currency Pricing (PCP) and Local Currency Pricing (LCP) paradigms (the latter two implemented by forcing invoicing shares to extreme paradigm values while keeping other parameters unchanged).
- Fit:
  - Estimated model and DCP replicate the pattern of pass-through declining over time for export prices to U and R and for import prices from U fairly closely.
  - Fit is weaker for import prices from R but still shows pass-through starting high and declining.
  - PCP implies low initial pass-through for export prices (prices sticky in exporting currency), which contradicts the data.
  - LCP implies low initial pass-through for import prices (prices sticky in destination currency), which contradicts the data.
- Relevance of bilateral exchange rates: estimated model and DCP reproduce the empirical fact that bilateral exchange rates appear large and significant when included alone for non-dollar partners, but their coefficient falls substantially when the dollar exchange rate is included. PCP and LCP do not match this fact.
- Quantity pass-through from simulated model (Table 14, preserved exactly):
  - Columns (1)-(4) (∆y_HU, ∆y_UH, ∆y_HR, ∆y_RH) responses to exchange rates:
    - ∆e_U effects: 0.26, -1.60, -1.33, -1.19
    - ∆e_R effects: -0.18, 0.28, 1.43, -0.11
  - Interpretation: simulated model generates a weak expansion in exports to U following a depreciation and a pronounced contraction in imports from both U and R; exports to R are negatively impacted by depreciations relative to the dollar in simulations — consistent with empirical findings.

### Role of key model features (α and Γ)
- Sensitivity: setting Γ = 0 and α = 0 (relative to benchmark Γ = 1 and α = 2/3) materially alters pass-through:
  - Export price pass-through into home prices declines by a half at the one-year horizon when Γ and α are both set equal to 0 (compared to the benchmark and data).
  - Import pass-through differences are smaller but benchmark specification (Γ = 1, α = 2/3) matches the data best.

### Preview of optimal monetary policy context
- The paper derives a linear quadratic representation of the optimal monetary policy problem for a small open economy with Dominant Currency Pricing, under special assumptions (C.E.S. demand, production using only labor, complete international asset markets).
- Comparison: analytic derivation and comparison to producer currency pricing results in Galí and Monacelli (2005) are provided; details are given in the appendix.

*Source: wp17264 - 4.2 Results (IMF Working Paper PDF: wp17264 - 4.2 Results)*

### 6.1  Canonical Representation

### 6.1  Canonical Representation

### Canonical representation of the small open economy under DCP
- System of three equations (deviations from the flexible price allocation, tilde notation) plus a monetary policy rule fully characterizes the domestic economy.
- Proposition 3 (when ε = α = φ = 0, σ_c = 1, and international asset markets are complete):
  - Evolution of inflation, output gap and law of one price departures:
    - π_HH,t = λ_p γ [ ̃y_t − (1−γ) ̃s_t ] + βE_t π_HH,t+1  (33)
    - ̃y_t = E_t ̃y_t+1 − (i_t − E_t π_HH,t+1 − r^n_t) + (1−γ)E_t(∆ ̃m_t+1)  (34)
    - ̃m_t = 1/γ (̃y_t − ̃s_t)  (35)
      - where ̃m_t = ̃e_U,t + ̃p_UHU,t − ̃p_HH,t = ̃e_U,t + ̃p_UHR,t − ̃p_HH,t captures deviations from the law of one price against the dominant currency.
      - r^n_t = log β + E_t ∆a_t+1 is the natural real rate.
      - γ measures home-bias.
      - λ_p = (1−δ_p)(1−βδ_p)/δ_p.
  - The terms-of-trade ̃s_t evolves independently of monetary policy under DCP (given the parameter restrictions).
- Interpretation:
  - Eq. (33) is the New Keynesian Phillips Curve (NKPC) in the DCP environment.
  - Eq. (34) is the dynamic IS curve in the DCP environment.
  - Eq. (35) characterizes deviations from the law of one price against the dominant currency.
  - Under DCP there is no “divine coincidence”: it is not possible to attain simultaneously zero inflation and a zero output gap because inflation depends on ̃s_t which evolves exogenously and independently from monetary policy under the parameter restrictions.
  - Contrast with PCP: under PCP (same parameter restrictions) the relation is π_HH,t = λ_p ̃y_t + βE_t π_HH,t+1, which allows the monetary authority to achieve zero inflation and zero output gap.

### Additional mechanism notes
- Exogeneity of import price in the dominant currency follows from the small open economy assumption.
- Exogeneity of export price in the dominant currency follows from complete markets and φ = 0: wage expressed in the dominant currency equals nominal consumption in U, so export prices in U’s currency are exogenous from policy.
- With both export and import prices exogenous, the terms-of-trade is exogenous.

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### 6.2  Welfare Loss Function

### Second-order approximation to planner’s welfare (DCP)
- Setup: tax τ = 1/σ on labor income, σ is elasticity of substitution across varieties produced within a country.
- Proposition 4 (when ε = α = φ = 0, σ_c = 1, international asset markets are complete and τ = 1/σ):
  - Welfare loss function approximated up to second order:
    - W_DCP ≈ E_0 ∑_{t=0}^{∞} β^t [ 1/2 ̃y_t^2 + γ/(σ 2 λ_p) π_HH,t^2 + γ(1−γ)/2 ̃m_t^2 ] + t.i.p  (36)
- Contrast with PCP (Galí and Monacelli (2005)):
  - W_PCP ≈ E_0 ∑_{t=0}^{∞} β^t γ_H [ 1/2 ̃y_t^2 + σ/(2 λ_p) π_HH,t^2 ] + t.i.p  (37)
- Key differences and interpretation:
  - Both loss functions include variance of inflation and output gap.
  - Under DCP there is an additional misalignment term ̃m_t arising from failure of the law of one price of H’s good across domestic and export markets.
  - Under PCP, m_t = 0.
  - Under DCP, fluctuations in m_t lead to real exchange rate fluctuations as long as there is home-bias in consumption; real exchange rate appreciates (depreciates) when m_t declines (increases).
  - Fluctuations in the real exchange rate affect home consumption through the complete markets condition, generating welfare losses relative to the flexible price allocation.
  - Misalignment term under DCP is similar to Engel (2011) under LCP but differs in that:
    - Under DCP there is a single misalignment term tied to the dominant-currency exchange rate (policy-relevant), regardless of the share of exports to U.
    - Under LCP bilateral exchange rates matter for misalignment.
    - Under DCP the terms-of-trade cannot be influenced by monetary policy, while under LCP the relative price of imports to home goods is independent of monetary policy (under same parameter restrictions).

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### 6.3  Monetary policy trade-offs

### Main findings
- Unlike PCP, no “divine coincidence” — cannot attain zero PPI inflation and zero output gap simultaneously.
- Optimal monetary policy targets deviations from the law of one price arising from dominant currency fluctuations, in addition to inflation and output gap.
- In absence of cost-push shocks, optimal policy calls for domestic producer price inflation targeting while the output gap fluctuates with the terms of trade.

### Discretionary policy
- Monetary authority minimizes eq. (36) subject to implementation constraints eq. (33) and eq. (35), taking future inflation as given.
- Optimal discretionary trade-off:
  - ̃y_t + (1−γ) ̃m_t = −σ π_HH,t  (38)
- Contrast with PCP optimal trade-off:
  - ̃y_t = −σ π_HH,t  (39)
- Interpretation:
  - Under PCP the authority leans against inflation by reducing the output gap.
  - Under DCP the authority can lean against inflation by lowering the output gap ̃y_t and/or the misalignment term ̃m_t.

### Role of ̃m_t and marginal cost
- Real marginal cost relation:
  - ̃mc_t − ̃p_HH,t = ̃w_t − ̃p_HH,t = ̃p_t − ̃p_HH,t + ̃c_t = (1−γ) ̃s_t + (1−γ) ̃m_t + ̃c_t  (40)
  - Labor supply: ̃w_t = ̃p_t + ̃c_t.
  - Price index definition: ̃p_t − ̃p_HH,t = (1−γ)(̃s_t + ̃m_t).
- An increase in ̃m_t raises the price of the consumption basket relative to H’s goods, raising real marginal cost and inflationary pressure in H good prices; the effect scales with openness (1−γ).

### Optimal policy implications (discretionary replaced into constraints)
- Optimal policy under DCP yields:
  - π_HH,t = 0  (41)
  - ̃y_t = (1−γ) ̃s_t  (42)
- Thus, optimal policy calls for producer price inflation targeting (π_HH,t = 0). Unlike PCP, under DCP the output gap fluctuates with the terms of trade gap, increasing with openness.

### Example: domestic productivity shocks
- Consider only shocks are to domestic productivity a_t; a positive productivity shock:
  - Under flexible price allocation, domestic prices decline relative to imported goods and terms of trade deteriorate.
  - Under DCP the dollar exchange rate moves one to one with productivity: e_U,t = a_t.
  - This generates same movement in relative price of imports as under flexible prices but does not generate desired movement in terms of trade (̃s_t < 0), producing a negative output gap.
  - Note: import prices and world prices in dollars unchanged due to SOE assumption, so real exchange rate behaves as in flexible price allocation yielding ̃c_t = ̃q_t = 0.

### Commitment vs. discretion
- Commitment solution (as derived in Appendix A.3.10) at t = 0:
  - ̃y_t = (1−γ) ̃s_t − γσ (p_HH,t − p_HH,−1)
    - (p_HH,t − p_HH,−1) = (p_HH,t−1 − p_HH,−1) + βE_t (p_HH,t+1 − p_HH,−1) all over 1 + λ_p σ + β
    - where p_HH,−1 is the given initial price level from period t = −1.
- Two notes:
  - Optimal price level under commitment follows same path as PCP case; desired output gap still fluctuates with terms of trade under DCP.
  - Solution under discretion and commitment are the same in absence of cost-push shocks because under discretion optimal policy calls for zero inflation in each period; no gains to smoothing inflation over time via commitment.

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### Conclusion (selected implications from sections 6.1–6.3 and summary)
- The dominant currency paradigm (DCP) features pricing in a dominant currency, strategic complementarities in pricing, and imported input use.
- Model predicts stability in the terms-of-trade while price of imported goods relative to domestic goods is volatile; this volatility is driven by exchange rate fluctuations with respect to the dominant currency.
- Following a depreciation of the exchange rate, imports from all origins will decrease.
- Exports to dominant-currency destinations will not be responsive to currency movements; exports to other destinations depend on co-movement of the destination country’s exchange rate with the dominant currency.
- DCP implies that weakening of emerging market currencies relative to the dominant currency (e.g., dollar) will be associated with a decline in world trade (exports plus imports) relative to PCP or LCP.
- Empirical comparison (Colombia) shows DCP predictions outperform PCP and LCP in the data.
- Optimal monetary policy for a small open economy deviates from Mundell-Fleming benchmarks and involves targeting, alongside output and inflation, deviations from the law of one price against the dominant currency; in absence of cost-push shocks this implies domestic PPI inflation targeting while the output gap fluctuates with the terms of trade.

*Source: IMF Working Paper (section 6.1–6.3 content as provided).*

### 451. Elsevier.

### wp17264 - 451. Elsevier.

### Online Appendix: Proofs, derivations, and auxiliary results
- A.1 Proof of Proposition 1
  - First-order approximation of the flexible-price FOC yields:
    - ∆pHi,t = 1/(1 + Γ) ∆mc t + Γ/(1 + Γ) (∆pii,t + ∆ei,t)
    - ∆mc t = (1−α)∆wt + α∆pt − ∆at
    - ∆pt = γH ∆pHH,t + γU ∆pUH,t + γR ∆pRH,t = γH ∆mc t + ∑i∈U,R γi (∆mc ii,t + ∆ei,t)
    - Final expression when setting ∆mc ii,t = 0 follows from substitution.
  - Import-side price relation:
    - ∆piH,t = 1/(1 + Γ) (∆mc ii,t + ∆ei,t) + Γ/(1 + Γ) (∆pt)
  - Simple substitution for ∆p and ∆mc yields equation (19) in the text.

- A.2 Derivation of Equation 24 (optimal reset price for exports from H into currency j)
  - Calvo pricing framework:
    - ̄pjHi,t = (1−βδp)/(1 + Γ) (mcjH,t + Γpj i,t + μ) + βδp Et ̄p j Hi,t+1
    - πjHi,t = (1−δp)[̄p j Hi,t − pjHi,t−1] and algebraic manipulation (shown step-by-step in the source) yields equation 24.

### Functional forms, utility, and labor
- A.3.1 Functional forms
  - Parameter restrictions: ε=α=φ= 0, σc = 1 imply:
    - U = ln(C) − N
    - C = CγHH CγU UH CγR RH
    - Cij = (∫0 1 Cij(ω)σ−1/σ dω)σ/σ−1
    - Y = ea N
  - Home bias parameter γ; remaining share 1−γ divided between regions with relative sizes γU/(1−γ) and γR/(1−γ).

- A.3.2 Utility (second-order approximation)
  - U = ln(C) − N
  - U − ̄U ≈ ˆct − ̄N ˆnt − 1/2 ̄N ˆn2 t, where ̄U = ln(̄C) − ̄N.

- A.3.3 Labor demand
  - Lemma 2 (dollar pricing): Under dollar pricing and Γ = 0 (ϵ = 0), each firm ω in H sets the same export price in dollars for all destinations; P U Hj,t is same for all j ∈ U ∪ R and all t. Implies PU HU,t = PU HR,t.
  - Aggregate labor:
    - Nt = ∫0 1 Nt(ω) dω = ∫0 1 Yt(ω) ea t dω
    - Yt(ω) = YHH,t(ω) + YHU,t(ω) + YHR,t(ω) with standard CES aggregation and destination-invariant dollar prices under Γ = 0.
  - Second-order approximation yields:
    - ˆnt + ˆat = ˆyt + γ σ2 varω{pHH,t(ω)} + (1−γ) σ2 varω{pUHU,t(ω)} (equation A.3).

### Market clearing, complete markets, and terms-of-trade
- A.3.4 Market clearing condition
  - Goods market clearing (general expression in the source) combined with complete markets (under ex-ante symmetry and NFA = 0):
    - EU,t PU,t CU,t = Pt Ct, ER,t PR,t CR,t = Pt Ct (equation A.4)
  - Define misalignment term (a la Engel):
    - MU t = MR t = Mt = EU,t PU HU,t / P HH,t = EU,t PU HR,t / P HH,t (equation A.5)
  - Trade-weighted TOT S and component definitions:
    - S = SU γU/(1−γ) SU γR/(1−γ) R (notation in source)
    - SU = PU UH / PU HU , SR = PU RH / PU HR
  - Second-order approximation of market clearing yields:
    - ˆyt = (1−γ) ˆst + ˆct + γ(1−γ)/2 ˆm t2 (from simplification shown in the source).

- A.3.5 Complete markets identities
  - ct = cU,t + qU,t, ct = cR,t + qR,t; aggregated:
    - ct = c∗t + qt where c∗t = γU/(1−γ) cU,t + γR/(1−γ) cR,t, qt = γU/(1−γ)H qU,t + γR/(1−γ)H qR,t
  - CPI relations and TOT definitions lead to:
    - ct = c∗t + γ(st + mt) (exact in logs).

- A.3.6 Terms-of-Trade (Lemma 3)
  - Lemma 3: When φ = 0 and asset markets are complete then the terms-of-trade evolves independently of policy.
  - Proof sketch: With small-open-economy assumption and φ = 0, H firms' marginal cost in dollars MC H,t / E U,t = Wt / E U,t ea,t = PU,t CU,t / ea,t evolves exogenously; hence export prices in dominant currency are exogenous to H monetary policy.

### Natural (flexible-price) allocation and steady state
- A.3.7 Natural/Flexible price allocation
  - Under flexible prices law of one price holds: Pn HH,t = EU,t PU HU,t = ER,t PU HU,t.
  - Pricing decision:
    - Pn HH,t = σ/(σ−1) (1−τ) Wn t ea t = σ/(σ−1) (1−τ) Pn t Cn t ea t
    - With market-clearing and Yn t = Nn t ea t, derive (1−τ) Nn t = (σ−1)/σ ⇒ Nn t = 1 when (1−τ) = (σ−1)/σ.
  - Remark on steady-state approximation:
    - Paper approximates around constrained efficient steady state where (1−τ) = (σ−1)/σ and ̄N = 1 to obtain a linear-quadratic representation; contrasts with the efficient steady state (1−τ) = (1/γ) (σ−1)/σ and ̄N = γ.

### Second-order welfare loss and constraints
- A.3.8 Second order welfare loss function (deviations from natural allocation)
  - Welfare approximation:
    - W ≈ E0 ∑0∞ βt (ˆct − ̄N ˆnt − 1/2 ̄N ˆn2 t)
  - Define deviations ̃xt ≡ ˆxt − ˆx n t, and using ̄N = 1:
    - W ≈ E0 ∑0∞ βt (̃ct − ̃nt − 1/2 ̃n2 t) + t.i.p.
  - Market clearing in deviations (̃ variables) and using varω p relations:
    - ∞∑0 βt varω pHH,t = 1/λp ∞∑0 βt π2 HH,t (equation A.6) and analogous for exports.
  - Result: Second-order loss function in Proposition 4 expressed in deviations from natural allocation; constraints need only first-order NKPC and relation between ̃y t and ̃m t.

- A.3.9 PCP (producer-currency pricing) case
  - Market clearing linear in logs; complete markets condition implies ̃ct = γ ̃st and ̃nt = ̃yt + σ2 varω(pH HH,t) (equation A.7).
  - Loss function simplifies (see source) with single constraint NKPC where (̃mcHH,t − ̃pHH,t) = ̃yt.

### Optimal policy under commitment (DCP and PCP)
- A.3.10 Optimal policy under commitment — DCP (dominant currency pricing)
  - Rewritten minimization problem:
    - min E0 ∑t=0∞ βt [1/2 ̃y2 t + γ σ2 λp π2 HH,t + γ(1−γ)/2 (̃yt − ̃st/γ)2] + t.i.p (equation A.8)
    - s.t. πHH,t = λp/γ [̃yt − (1−γ) ̃st] + β Et πHH,t+1
  - Lagrangian with multiplier θt (equation A.9) yields FOCs:
    - ̃yt + (1−γ)(̃yt − ̃st/γ) − λp θt/γ = 0 ⇒ ̃yt − (1−γ) ̃st − λp θt = 0
    - γσ/λp πHH,t + θt − θt−1 = 0
  - Combining conditions:
    - [̃yt − (1−γ) ̃st] = [̃yt−1 − (1−γ) ̃st−1] − γσ πHH,t
    - [̃y0 − (1−γ) ̃s0] = −γσ πHH,0
  - Level-targeting policy expression:
    - ̃yt = (1−γ) ̃st − γσ (pHH,t − pHH,−1)
  - Price-level follows a second-order difference equation:
    - (pHH,t − pHH,−1) = (pHH,t−1 − pHH,−1) + β Et (pHH,t+1 − pHH,−1) / (1 + λp σ + β)
  - Interpretation: Output-gap target depends on exogenous terms-of-trade under DCP; price-level path same as PCP case.

- A.3.10 Optimal policy under commitment — PCP
  - FOCs yield:
    - ̃yt − λp θt = 0
    - σ/λp πHH,t + θt − θt−1 = 0
  - Implies:
    - ̃yt = ̃yt−1 − σ πHH,t ; ̃y0 = −σ πHH,0
    - Level targeting: ̃yt = −σ (pHH,t − pHH,−1)
    - Price-level dynamics identical to DCP expression above.

### Dynamic IS curve (DCP, simplified parametrization)
- A.4 Dynamic IS curve under DCP (simplified parametrization)
  - Final form:
    - ̃yt = Et ̃yt+1 − (it − Et πHH,t+1 − ρ) + (1−γ) Et (∆ ̃m t+1) + Et ∆at+1
  - Alternative expression using natural interest rate rn t:
    - ̃yt = Et ̃yt+1 − (it − Et πHH,t+1 − rn t) + (1−γ) Et (∆ ̃m t+1)
  - Derivation highlights:
    - Euler condition with σc = 1 and approximation (1 + it) ≈ it yields:
      - ˆct = Et ˆct+1 − (it − Et πt+1 − ρ)
    - Combining market clearing, CPI relations p t − pHH,t = (1−γ)(st + mt), and log-linearization gives the IS curve with misalignment/misalignment-change term (1−γ) Et(∆ ̃m t+1) appearing under DCP.

_Online Appendix: Not for publication — contains A.1–A.4 proofs and derivations as presented in the source._

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17264.pdf_