## _wp10182

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---

### Introduction and research question
- Small open economies experience recurrent episodes of exchange rate appreciation in response to shocks (demand shocks such as discovery of natural resources or supply shocks such as higher productivity relative to main trade partners).
- Paper evaluates exchange rate stabilization as a policy to counteract a Dutch Disease episode using a dynamic stochastic general equilibrium (DSGE) model with nominal rigidities and a learning-by-doing externality in the tradable (manufacturing) sector.
- Empirical motivation (Canada case):
  - Period 2002-07: terms of trade improved 25 percent and real effective exchange rate appreciated around 25 percent.
  - The period of exchange rate appreciation coincides with a contraction of 3 percentage points of GDP in the share of manufacturing production.
  - In the last 20 years Canada experienced a maximum contraction of 2 percentage points of GDP every time it enters into a recession related to the U.S. business cycle; the recent episode shows a larger decline in manufacturing production unrelated to the U.S. cycle, pointing to Dutch Disease effects.

### Main finding and policy trade-off
- Main result: exchange rate intervention is a welfare-reducing policy to counteract the effects of the Dutch Disease.
  - Exchange rate stabilization can prevent tradable production from falling below the efficient level (by depreciating the real exchange rate through monetary injections).
  - But stabilizing the exchange rate exacerbates aggregate demand effects generated by an improvement of the terms of trade and increases macroeconomic volatility.
  - In the calibrated model to the Canadian economy, costs in terms of macroeconomic volatility and misallocation of resources far exceed benefits from a depreciated exchange rate.
- Policy intuition:
  - Learning-by-doing externality implies output reductions today lower future productivity and future production; absent internalization by firms, there is an inefficient loss of tradable production.
  - Exchange rate intervention through monetary policy is a blunt instrument: it expands tradable output but also stimulates all sectors in tandem, causing distortions when terms of trade improve.
- Policy implication:
  - Stabilizing the nominal exchange rate (via FX interventions or domestic open-market operations) is not an effective instrument to correct a learning-by-doing externality given nominal rigidities across sectors.

### Model overview (structure and key ingredients)
- Economy: multi-sector small open economy with three sectors:
  - Manufactured tradable good (H) — subject to learning-by-doing externality.
  - Non-tradable good (N).
  - Commodity good — exported entirely at a given international price.
- Households and assets:
  - Expected discounted utility Ut = Et[ sum_{i=0}^1 β^i u(C_{t+i} - h H_{t+i}; 1 - L_{t+i}) ], with habit H_{t+i} = C_{t+i-1}.
  - Assets include money Mt, one-period non-contingent foreign bonds B*_t, one-period domestic contingent bonds D_{t+1}.
  - Budget constraint includes foreign interest rate i*_t and a risk premium Π(:) as a function of net foreign asset positions.
- Production and price setting:
  - Final good Y^F_t aggregates home-produced inputs Y^DH_t, imports Y^M_t, and non-tradable inputs Y^DN_t via CES with parameters β_Y, σ_Y.
  - Tradable inputs Y^T_t combine domestic tradable Y^DH_t and imports Y^M_t via CES with parameter ω_Y.
  - Assemblers combine differentiated intermediate goods j ∈ [0,1] with elasticity of substitution ε_J to produce Y^J_t (J = H, N).
  - Calvo (1983) price-setting: fraction (1 - ξ_J) of retailers in sector J set prices optimally each period.
- Intermediate good technologies:
  - Non-tradable firms: Y^N_t(n) = A^N_t [L^N_t(n)]^{φ_N} [K^N_t(n)]^{1-φ_N}.
  - Tradable (manufacturing) firms subject to learning-by-doing:
    - Y^H_t(h) = A^H_t [H_t(h)]^{χ_H} [L^H_t(h)]^{φ_H} [K^H_t(h)]^{1-φ_H}.
    - Organizational capital law of motion: H_{t+1}(h) = [H_t(h)]^{(1-δ_H)} [Y^H_t]^{γ_H}.
    - Organizational capital interpreted as agglomeration of technologies and practices; higher current production increases organizational capital and improves future sector efficiency.

### Capital producers, commodity sector, and monetary policy
- Representative capital firm solves:
  - V^J_t = max_{K^J_{t+i}, I^J_{t+i}} E_t ( sum_{i=0}^{\infty} β^{t;t+i} [ Z^J_{t+i} K^J_{t+i} - P^C_{t+i} I^J_{t+i} ] ).
- Physical capital law of motion:
  - K^J_{t+1} = (1 - ϕ) K^J_t + S(I^J_t / I^J_{t-1}) I^J_t.
  - Investment adjustment cost S(:) satisfies S(1) = 1, S'(1) = 0, S''(1) = -κ_S < 0.
- Commodity processes (exogenous):
  - X_t = [X_{t-1}]^{ζ_x} [X_0]^{1-ζ_x} exp(ε^x_t), ε^x_t ~ N(0, σ^2_x).
  - P^x_t = (P^x_{t-1})^{ζ_{px}} [P^x_0]^{1-ζ_{px}} exp(ε^{px}_t), ε^{px}_t ~ N(0, σ^2_{px}).
- Monetary policy (Taylor-type rule):
  - (1 + i_t)/(1 + \bar i) = ( (1 + i_{t-1})/(1 + \bar i) )^{ρ_i} × (Y_t/\bar Y)^{(1-ρ_i) α_y} × (π_{C,t}/\bar π)^{(1-ρ_i) α_π} × (s_t/\bar s)^{(1-ρ_i) α_s}.

### Market clearing and equilibrium conditions
- Labor market: L_t = ∫_0^1 L^N_t(n) dn + ∫_0^1 L^H_t(h) dh.
- Capital market: K^J_t = ∫_0^1 K^J_t(j) dj, J = H, N.
- Home and non-tradable intermediate goods:
  - Y^{DN}_t = Y^N_t.
  - Y^{DH}_t + C^{H*}_t = Y^H_t, where C^{H*}_t = (P_{H,t}/E_t P^*_{F,t})^{-θ} C^*_t (preserved expression).
- Final good and external asset flow:
  - Y^F_t = C_t + I^H_t + I^N_t.
  - E_t B^*_t = (1 + i^*_t-1)^{-1} E_t B^*_{t-1} + P^M_t Y^M_t - P_{H,t} C^{H*}_t - P^x_t X_t.

### Calibration (quarterly) — key calibrated parameter values (Table 1, baseline)
- Discount Factor β = 0.99
- Habit Persistence h = 0.65
- Labor Supply Elasticity 1/ϕ = 0.60
- Share of Tradable Inputs - Final Good Sector ρ_Y = 0.50
- Elasticity of Substitution - Final Good Sector θ_Y = 0.50
- Share of Home Inputs - Tradable Good Sector ρ^Y = 0.50
- Elasticity of Substitution - Tradable Good Sector ε_Y = 0.50
- Depreciation Rate ϕ = 0.02
- Capital Share - Non-tradable Sector α_N = 0.30
- Labor Share - Non-tradable Sector 1 - α_N = 0.70
- Capital Share - Home Goods Sector α_H = 0.20
- Labor Share - Home Goods Sector 1 - α_H = 0.55
- Calvo Parameter - Home Goods Sector θ_H = 0.75
- Calvo Parameter - Non-tradable Sector θ_N = 0.75
- Elasticity of Substitution - Home Goods Sector "H = 6
- Elasticity of Substitution - Non-tradable Sector "N = 6
- Foreign Interest Rate Elasticity ((ζ_0 = ζ) B_t) = 0.001
- Foreign Demand Elasticity ω^* = 0.50
- Depreciation Rate - Organizational Capital 1 - δ_H = 0.37
- Output Elasticity - Organizational Capital κ_H = 0.37
- Learning Rate ψ_H = 0.25
- Interest Rate Smoothing Coefficient - Taylor Rule ρ_i = 0.70
- Inflation Coefficient - Taylor Rule α_π = 1.30
- Output Coefficient - Taylor Rule α_y = 0.23
- Depreciation Coefficient - Taylor Rule α_s = 0.14
- Investment adjustment cost elasticity κ_S = 2.5
- Organizational capital share ψ_H = 0.25 corresponds to a 20 percent learning rate in the literature.
- Depreciation of organizational capital 1 - δ_H = 0.37.
- Elasticity of risk premium with respect to debt (ζ_0 = ζ) B_t = 0.001.

### Estimated shock processes (Canada data 1980:Q1 - 2008:Q4)
- a^H_t = 0.96 a^H_{t-1} + ε^H_t; ε^H_t ~ N(0, σ^2_H); σ_H = 0.015
- a^N_t = 0.97 a^N_{t-1} + ε^N_t; ε^N_t ~ N(0, σ^2_N); σ_N = 0.005
- x_t = 0.86 x_{t-1} + ε^X_t; ε^X_t ~ N(0, σ^2_X); σ_X = 0.017
- p^X_t = 0.94 p^X_{t-1} + ε^{PX}_t; ε^{PX}_t ~ N(0, σ^2_{PX}); σ_{PX} = 0.15
- c^*_t = 0.99 c^*_{t-1} + ε^*_t; ε^*_t ~ N(0, σ^2_{C^*}); σ_{C^*} = 0.006

### Learning-by-doing: mechanisms and calibrated impulse responses
- Two channels of the learning-by-doing externality:
  - Decline in home goods production → lower organizational capital via law of motion → lower productivity in home goods sector → exacerbates contraction.
  - Lower home goods productivity → higher domestic price level → real exchange rate appreciation → reallocation from tradable to non-tradable.
- Impulse response to one standard deviation increase in commodity prices:
  - Without learning-by-doing (New Keynesian baseline): reallocation from tradable to non-tradable; higher demand for tradable and non-tradable goods; real exchange rate appreciation; higher imports satisfy tradable demand; GDP, consumption, investment expand; trade balance deteriorates (trade balance shown excluding commodity exports).
  - With learning-by-doing: gradual decline of GDP over time driven by lower manufacturing/tradable production; deterioration of trade balance due to lower tradable output and higher imports.
- Aggregate effects of learning-by-doing:
  - Decline of tradable production.
  - Decline of GDP.
  - Deterioration of the trade balance.
  - Reduction in household welfare.

### Policy rules evaluated and qualitative effects
- Four policy rules:
  - Taylor-type rule (restricted, s = 0).
  - Empirical monetary policy rule (s = 0.15; includes some exchange rate intervention).
  - Fixed exchange rate (s → ∞; effectively pegged).
  - Optimized rule that maximizes unconditional welfare (parameters chosen to maximize welfare).
- Benchmarks:
  - Flexible prices with internalized learning-by-doing (first-best real allocation at business cycle frequency).
- Qualitative findings:
  - Taylor-type rule (s = 0): home goods production below efficient level; monetary expansion could close gap but risks overstimulating non-tradable and imports.
  - Empirical rule (s = 0.15): reduces appreciation of real exchange rate; home goods production closer to efficient level; non-tradable production and imports exceed efficient level; increases in GDP, consumption, investment.
  - Fixed exchange rate (s → ∞): limits exchange rate appreciation; quantities overshoot benchmark allocations; generates higher macroeconomic volatility.
  - Optimized rule: allocation remains very close to benchmark; allows for appreciation and reallocation from tradable to non-tradable.
- Two main results summarized:
  - Monetary policy can increase tradable production and prevent inefficiencies in tradable sector.
  - Leaning against an appreciated exchange rate increases macroeconomic volatility; exchange rate stabilization amplifies aggregate demand expansion during commodity price shocks and may worsen volatility. Welfare comparison required to choose policy.

### Welfare calculations and simulated business cycle moments (Table 2 summary)
- Welfare measured as fraction of consumption agents would give up to eliminate excess volatility (Lucas (1987) approach); computed using second-order approximation.
- Canadian Data (1980-2008) empirical moments:
  - Std. Dev. GDP = 1.51
  - Std. Dev. NX/GDP = 0.89
  - Std. Dev. x / Std. Dev. GDP = 0.57
  - Std. Dev. C = 2.73
  - Std. Dev. INV = 0.71
  - Std. Dev. LRER = 2.46
  - Welfare = -
- Empirical Rule:
  - Std. Dev. GDP = 1.49
  - Std. Dev. NX/GDP = 0.93
  - Std. Dev. x / Std. Dev. GDP = 0.78
  - Std. Dev. C = 2.65
  - Std. Dev. INV = 0.76
  - Std. Dev. LRER = 2.81
  - Welfare = 0.04
- Taylor-Type Rule:
  - Std. Dev. GDP = 1.58
  - Std. Dev. NX/GDP = 1.32
  - Std. Dev. x / Std. Dev. GDP = 0.77
  - Std. Dev. C = 2.70
  - Std. Dev. INV = 0.79
  - Std. Dev. LRER = 2.97
  - Welfare = 0.04
- Fixed Exchange Rate:
  - Std. Dev. GDP = 2.52
  - Std. Dev. NX/GDP = 1.22
  - Std. Dev. x / Std. Dev. GDP = 0.79
  - Std. Dev. C = 2.49
  - Std. Dev. INV = 0.98
  - Std. Dev. LRER = 1.50
  - Welfare = 0.19
- Optimized Rule:
  - Std. Dev. GDP = 1.49
  - Std. Dev. NX/GDP = 1.33
  - Std. Dev. x / Std. Dev. GDP = 0.78
  - Std. Dev. C = 2.70
  - Std. Dev. INV = 0.48
  - Std. Dev. LRER = 3.30
  - Welfare = 0.00
- Interpretation of moments:
  - Consumption and employment are less volatile than GDP; investment and the real exchange rate are more than twice as volatile as output.
  - Real exchange rate exhibits greater volatility than consumption or output ("exchange rate disconnect").
- Welfare ranking:
  - Optimized rule yields Welfare = 0.00 (closest to benchmark).
  - Empirical and Taylor-type rules generate small welfare costs (Welfare = 0.04).
  - Fixed exchange rate generates largest welfare cost (Welfare = 0.19).

### Policy experiments, welfare costs, and Figure 9 interpretation
- Empirical rule generates a welfare cost equivalent to 0.04 percent of lifetime consumption.
- Taylor-type rule results are very close to the empirical rule; trade balance has a standard deviation 50 percent larger than under the empirical rule.
- Fixed exchange rate policy:
  - Increases volatility of GDP by 60 percent.
  - Increases volatility of labor by 25 percent.
  - Reduces real exchange rate volatility by 50 percent.
  - Generates welfare cost equivalent to 0.19 percent of the stream of lifetime consumption.
  - Welfare loss mainly from increase in labor supply volatility by 25 percent.
- Optimized rule: by construction does not generate substantial welfare costs; allocation close to flexible-price internalized-learning benchmark.
- Figure 9 (text description): welfare costs are an increasing function of the exchange rate intervention parameter s; for s = 0 the plotted ratio equals 1. Greater central bank intervention in foreign exchange markets moves allocation away from efficient equilibrium and increases macroeconomic volatility, lowering household welfare.

### Sensitivity analysis (Section VI)
- Four alternative specifications tested:
  - Larger learning-by-doing externalities (increase organizational capital share to H = 0:5 implying a 40 percent learning rate).
  - Learning-by-doing in the non-tradable sector.
  - Incomplete pass-through for imported goods (import prices set with Calvo mechanism every four quarters).
  - Financial autarky (modeled by setting (0 =)B_t = 100).
- Key sensitivity findings:
  - Higher learning-by-doing increases home goods sensitivity to exchange rate appreciations; empirical rule may leave home goods production below efficient level; higher LBD rates generate greater costs during a commodity boom.
  - LBD in both sectors exacerbates reallocation from tradable to non-tradable; with LBD in non-tradable sector there are welfare gains from appreciating the currency, and exchange rate intervention induces deviations from efficient allocation.
  - Limited pass-through weakens expenditure switching; depreciation generates lower substitution between imported and home goods; leaning against appreciation remains distortionary.
  - Financial autarky amplifies commodity price shock effects; despite larger externality effects monetary policy remains inefficient to correct distortion; exchange rate stabilization generates significant deviations from benchmark allocation.
- Overall conclusion: alternative frictions do not overturn the basic trade-off; exchange rate stabilization to correct LBD is highly distortionary across specifications.

### Conclusions and policy implications (Section VII)
- Framework: New Keynesian small open economy with learning-by-doing externality in tradable sector creates a trade-off:
  - LBD externality calls for exchange rate stabilization to bring tradable production toward efficient level.
  - Standard New Keynesian logic: higher terms of trade call for real exchange rate appreciation to achieve efficient allocation.
- Under calibration consistent with Canadian data:
  - Trade-off favors allowing a real exchange rate appreciation.
  - Stabilizing the nominal exchange rate with monetary policy is a blunt, distortionary instrument; attempts to lean against appreciation result in misallocation and reduced welfare.
  - Monetary policy should focus on stabilizing prices subject to nominal rigidities to allow smooth adjustment to higher commodity prices.
- Policy recommendation:
  - Use alternative instruments (not monetary exchange rate intervention) to address learning-by-doing problems in a Dutch Disease episode: fiscal policy, a commodity fund if government owns the resource, or structural reforms in goods and factor markets to facilitate reallocation.
- Empirical note:
  - The paper provides theoretical support for the Bank of Canada’s policy of not intervening in the foreign exchange market; the bank has not intervened in the foreign exchange rate market in the last ten years.
- Suggested extensions:
  - Evaluate effectiveness of alternative instruments (fiscal, commodity fund, structural reforms).
  - Explore alternative frictions affecting tradable sector during commodity booms (e.g., financial frictions).
  - Conduct similar quantitative exercise for a developing small open economy where exchange rate stabilization may better anchor inflation expectations.

*Source: _wp10182 - References and sections 3–4 (IMF chapter PDF).*

### References .............................................................................................................

### _wp10182 - References .............................................................................................................

### Introduction and research question
- Small open economies experience recurrent episodes of exchange rate appreciation in response to shocks (demand shocks such as discovery of natural resources or supply shocks such as higher productivity relative to main trade partners).
- Paper evaluates the merits of exchange rate stabilization as a policy to counteract a Dutch Disease episode using a dynamic stochastic general equilibrium (DSGE) model with nominal rigidities and a learning-by-doing externality in the tradable (manufacturing) sector.
- Empirical motivation (Canada case):
  - Period 2002-07: terms of trade improved 25 percent and real effective exchange rate appreciated around 25 percent.
  - The period of exchange rate appreciation coincides with a contraction of 3 percentage points of GDP in the share of manufacturing production.
  - In the last 20 years Canada experienced a maximum contraction of 2 percentage points of GDP every time it enters into a recession related to the U.S. business cycle; the recent episode shows a larger decline in manufacturing production unrelated to the U.S. cycle, pointing to Dutch Disease effects.

### Main finding and policy trade-off
- Main result: exchange rate intervention is a welfare-reducing policy to counteract the effects of the Dutch Disease.
  - Exchange rate stabilization can prevent tradable production from falling below the efficient level (by depreciating the real exchange rate through monetary injections).
  - But stabilizing the exchange rate exacerbates aggregate demand effects generated by an improvement of the terms of trade and increases macroeconomic volatility.
  - In calibrated model to the Canadian economy, costs in terms of macroeconomic volatility and misallocation of resources far exceed benefits from a depreciated exchange rate.
- Policy intuition:
  - Learning-by-doing externality implies output reductions today lower future productivity and future production; absent internalization by firms, there is an inefficient loss of tradable production.
  - Exchange rate intervention through monetary policy is a blunt instrument: it expands tradable output but also stimulates all sectors in tandem, causing distortions when terms of trade improve.
- Policy implication:
  - Stabilizing the nominal exchange rate (via FX interventions or domestic open-market operations) is not an effective instrument to correct a learning-by-doing externality given nominal rigidities across sectors.

### Model overview (structure and key ingredients)
- Economy: multi-sector small open economy with three sectors:
  - Manufactured tradable good (H) — subject to learning-by-doing externality.
  - Non-tradable good (N).
  - Commodity good — exported entirely at a given international price.
- Preferences and financial assets:
  - Households: expected discounted utility Ut = Et[ sum_{i=0}^1 β^i u(C_{t+i} - h H_{t+i}; 1 - L_{t+i}) ], with habit H_{t+i} = C_{t+i-1}.
  - Assets: money Mt, one-period non-contingent foreign bonds B*_t, one-period domestic contingent bonds D_{t+1}.
  - Budget constraint (nominal terms) given, including foreign interest rate i*_t and risk premium Π(:) as a function of net foreign asset positions.
- Final good producers:
  - Final good Y^F_t aggregates home-produced inputs Y^DH_t, imports Y^M_t, and non-tradable inputs Y^DN_t via CES with parameters β_Y, σ_Y (equations (2) and (3)).
  - Tradable inputs Y^T_t combine domestic tradable Y^DH_t and imports Y^M_t via CES with parameter ω_Y.
- Retailers:
  - Assemblers combine differentiated intermediate goods j ∈ [0,1] with elasticity of substitution ε_J to produce Y^J_t (J = H, N) (equations (4)–(6)).
  - Price setting: Calvo (1983) framework — fraction (1 - ξ_J) of retailers in sector J set prices optimally each period.
  - Each retailer maximizes expected discounted profits E_t[ sum_{i=0} (ξ_J)^i Λ_{t,t+i} (P^*_{J,t}(j) - P^W_{J,t+i}) Y^J_{t+i}(j) ].
- Intermediate good producers:
  - Non-tradable firms: production Y^N_t(n) = A^N_t [L^N_t(n)]^{φ_N} [K^N_t(n)]^{1-φ_N} (equation (8)).
  - Tradable sector (manufacturing) subject to learning-by-doing:
    - Firm production Y^H_t(h) = A^H_t [H_t(h)]^{χ_H} [L^H_t(h)]^{φ_H} [K^H_t(h)]^{1-φ_H} (equation (9) — notation preserved).
    - Organizational capital H_t(h) evolves: H_{t+1}(h) = [H_t(h)]^{(1-δ_H)} [Y^H_t]^{γ_H} (equation (10) — parameters (1 - ε_H) as depreciation, γ_H elasticity of organizational capital w.r.t. current output).
    - Organizational capital interpreted as agglomeration of technologies and practices that raise productivity; higher current production increases organizational capital and improves future sector efficiency.

### Calibration, quantitative exercises, and analysis (components referenced)
- The document lists:
  - Tables:
    - Table 1: Baseline Parameter Values.
    - Table 2: Business Cycles Moments and Welfare Costs.
  - Figures (selected):
    - Figure 1: Real Effective Exchange Rate and Terms of Trade in Canada, 1981–2008.
    - Figure 2: Real Effective Exchange Rate and Share of Tradable Production in Canada, 1981–2008.
    - Figures 3–10: Effects of Learning-by-Doing; Learning-by-Doing and Exchange Rate Intervention; Relative Welfare Cost and Exchange Rate Intervention; Sensitivity Analysis.
- Model follows Christiano et al. (2005), Smet and Wouters (2007), Adolfson et al. (2007), Chang et al. (2002), Cooper and Johri (2002); introduces learning-by-doing in manufacturing akin to Cooper and Johri (2002) and empirical evidence cited (Cooper and Johri (2002); Chang et al. (2002); Clarke (2008) for Canada).
- Stationarity device: risk premium Π(:) as function of net foreign asset positions (Schmitt-Grohé and Uribe, 2003).

### Welfare assessment and robustness
- Welfare analysis compares costs and benefits of exchange rate stabilization policy:
  - Stabilization can preserve tradable output but increases macroeconomic volatility and causes misallocation due to nominal rigidities.
  - Calibrated results for Canada indicate that volatility and misallocation costs dominate benefits from preventing tradable contraction.
- Sensitivity analysis is included (Figure 10 referenced) to test robustness of results.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10182.pdf*

### 4.  Capital Producers

### 4.  Capital Producers

### Model: Capital producers and physical capital
- Representative firm of capital type J solves:
  - V^J_t = max_{K^J_{t+i}, I^J_{t+i}} E_t ( sum_{i=0}^{\infty} β^{t;t+i} [ Z^J_{t+i} K^J_{t+i} - P^C_{t+i} I^J_{t+i} ] )
- Law of motion of physical capital:
  - K^J_{t+1} = (1 - ϕ) K^J_t + S(I^J_t / I^J_{t-1}) I^J_t
  - where Z^J_t is the rental rate of physical capital, ϕ is the depreciation rate, and S(:) characterizes the investment adjustment cost.
- Investment adjustment cost assumptions (following Christiano et al. (2005)):
  - S(1) = 1, S'(1) = 0, S''(1) = -κ_S < 0, generating investment inertia consistent with time-to-build.

### Commodity sector (exogenous processes)
- Commodity export process:
  - X_t = [X_{t-1}]^{ζ_x} [X_0]^{1-ζ_x} exp(ε^x_t)
  - ε^x_t ~ N(0, σ^2_x), ζ_x measures persistence.
- Commodity price process:
  - P^x_t = (P^x_{t-1})^{ζ_{px}} [P^x_0]^{1-ζ_{px}} exp(ε^{px}_t)
  - ε^{px}_t ~ N(0, σ^2_{px}), ζ_{px} measures persistence.
- Note: exports and prices are modeled exogenously to simplify the model; the commodity sector hiring physical capital and labor is abstracted from.

### Monetary policy rule
- Taylor-type rule specified as:
  - (1 + i_t)/(1 + \bar i) = ( (1 + i_{t-1})/(1 + \bar i) )^{ρ_i} × (Y_t/\bar Y)^{(1-ρ_i) α_y} × (π_{C,t}/\bar π)^{(1-ρ_i) α_π} × (s_t/\bar s)^{(1-ρ_i) α_s}
  - where i_t = nominal interest rate, Y_t = GDP, π_{C,t} = CPI inflation, s_t = depreciation rate; α_y, α_π, α_s are weights; ρ_i indicates interest rate smoothing.

### Market clearing and equilibrium conditions
- Labor market:
  - L_t = ∫_0^1 L^N_t(n) dn + ∫_0^1 L^H_t(h) dh
- Capital market:
  - K^J_t = ∫_0^1 K^J_t(j) dj, J = H, N
- Home and non-tradable intermediate goods:
  - Y^{DN}_t = Y^N_t
  - Y^{DH}_t + C^{H*}_t = Y^H_t
  - where C^{H*}_t = (P_{H,t}/E_t P^*_{F,t})^{-θ} C^*_t is foreign demand for home goods (expression preserved as in source).
- Final good and international bonds:
  - Y^F_t = C_t + I^H_t + I^N_t
  - E_t B^*_t = (1 + i^*_t-1)^{-1} E_t B^*_{t-1} + P^M_t Y^M_t - P_{H,t} C^{H*}_t - P^x_t X_t
  - (equations presented in source form).

### Calibration (quarterly frequency)
- Utility: u(C_t - h H_t, 1 - L_t) = log(C_t - h H_t) - χ_L (1+ϕ)^{-1} L_t^{1+ϕ}
- Price adjustment frequency: four quarters.
- Nominal rigidities present in home goods and non-tradable goods sectors in baseline; sensitivity analysis for incomplete pass-through considered.
- Key calibrated parameter values (Table 1, baseline):
  - Discount Factor β = 0.99
  - Habit Persistence h = 0.65
  - Labor Supply Elasticity 1/ϕ = 0.60
  - Share of Tradable Inputs - Final Good Sector ρ_Y = 0.50
  - Elasticity of Substitution - Final Good Sector θ_Y = 0.50
  - Share of Home Inputs - Tradable Good Sector ρ^Y = 0.50
  - Elasticity of Substitution - Tradable Good Sector ε_Y = 0.50
  - Depreciation Rate ϕ = 0.02
  - Capital Share - Non-tradable Sector α_N = 0.30
  - Labor Share - Non-tradable Sector 1 - α_N = 0.70
  - Capital Share - Home Goods Sector α_H = 0.20
  - Labor Share - Home Goods Sector 1 - α_H = 0.55
  - Calvo Parameter - Home Goods Sector θ_H = 0.75
  - Calvo Parameter - Non-tradable Sector θ_N = 0.75
  - Elasticity of Substitution - Home Goods Sector "H = 6
  - Elasticity of Substitution - Non-tradable Sector "N = 6
  - Foreign Interest Rate Elasticity ((ζ_0 = ζ) B_t) = 0.001
  - Foreign Demand Elasticity ω^* = 0.50
  - Depreciation Rate - Organizational Capital 1 - δ_H = 0.37
  - Output Elasticity - Organizational Capital κ_H = 0.37
  - Learning Rate ψ_H = 0.25
  - Interest Rate Smoothing Coefficient - Taylor Rule ρ_i = 0.70
  - Inflation Coefficient - Taylor Rule α_π = 1.30
  - Output Coefficient - Taylor Rule α_y = 0.23
  - Depreciation Coefficient - Taylor Rule α_s = 0.14
- Additional calibration notes:
  - Investment adjustment cost elasticity κ_S = 2.5 (from Christiano et al. (2005)).
  - Organizational capital share ψ_H = 0.25 corresponds to a 20 percent learning rate in the literature.
  - Depreciation of organizational capital 1 - δ_H = 0.37.
  - Elasticity of risk premium with respect to debt (ζ_0 = ζ) B_t = 0.001 to induce stationarity.
- Estimated shock processes (using Canada data 1980:Q1 - 2008:Q4):
  - a^H_t = 0.96 a^H_{t-1} + ε^H_t; ε^H_t ~ N(0, σ^2_H); σ_H = 0.015
  - a^N_t = 0.97 a^N_{t-1} + ε^N_t; ε^N_t ~ N(0, σ^2_N); σ_N = 0.005
  - x_t = 0.86 x_{t-1} + ε^X_t; ε^X_t ~ N(0, σ^2_X); σ_X = 0.017
  - p^X_t = 0.94 p^X_{t-1} + ε^{PX}_t; ε^{PX}_t ~ N(0, σ^2_{PX}); σ_{PX} = 0.15
  - c^*_t = 0.99 c^*_{t-1} + ε^*_t; ε^*_t ~ N(0, σ^2_{C^*}); σ_{C^*} = 0.006

### Learning-by-doing: mechanisms and calibrated impulse responses
- Two channels of the learning-by-doing externality:
  - Decline in home goods production → lower organizational capital via law of motion → lower productivity in home goods sector → exacerbates contraction.
  - Lower home goods productivity → higher domestic price level → real exchange rate appreciation → reallocation from tradable to non-tradable.
- Impulse response to one standard deviation increase in commodity prices:
  - Without learning-by-doing (New Keynesian baseline): reallocation from tradable to non-tradable; higher demand for tradable and non-tradable goods; real exchange rate appreciation; higher imports satisfy tradable demand; GDP, consumption, investment expand; trade balance deteriorates (trade balance shown excluding commodity exports).
  - With learning-by-doing: gradual decline of GDP over time driven by lower manufacturing/tradable production; deterioration of trade balance due to lower tradable output and higher imports.
- Aggregate effects of learning-by-doing:
  - Decline of tradable production.
  - Decline of GDP.
  - Deterioration of the trade balance.
  - Reduction in household welfare.

### Exchange rate intervention and alternative monetary policy rules
- Four policy rules evaluated:
  - Taylor-type rule (restricted, s = 0).
  - Empirical monetary policy rule (s = 0.15; includes some exchange rate intervention).
  - Fixed exchange rate (s → ∞; effectively pegged).
  - Optimized rule that maximizes unconditional welfare (parameters chosen to maximize welfare; specification same as equation (14)).
- Benchmark for comparison:
  - Flexible prices with internalized learning-by-doing (first-best real allocation at business cycle frequency).
- Findings across rules (qualitative):
  - Taylor-type rule (s = 0): home goods production below efficient level (negative output gap in home goods). Possible closing of gap via monetary expansion that depreciates exchange rate, but risks stimulating non-tradable and imports excessively.
  - Empirical rule (s = 0.15): reduces appreciation of real exchange rate; home goods production closer to efficient level; non-tradable production and imports exceed efficient level; real exchange rate more depreciated; increases in GDP, consumption, investment.
  - Fixed exchange rate (s → ∞): effectively limits exchange rate appreciation; quantities overshoot benchmark allocations; generates higher macroeconomic volatility compared to other rules.
  - Optimized rule (parameters chosen to maximize unconditional welfare):
    - Parameters of optimized Taylor-type rule (reported from related literature Schmitt-Grohé and Uribe (2007)): α_π = 2.23; ρ_i = 0.56; α_y = -0.76; α_s = -0.25 (parameters given in source).
    - Allocation under optimized rule remains very close to benchmark; allows for appreciation and reallocation from tradable to non-tradable.
- Two main results:
  - Monetary policy is potent to increase tradable production and prevent inefficiencies in tradable sector.
  - Leaning against an appreciated exchange rate increases macroeconomic volatility; exchange rate stabilization amplifies aggregate demand expansion during commodity price shocks and may worsen volatility. Trade-offs between boosting tradable production and higher macro volatility require welfare comparison.

### Welfare calculations and costs of rules
- Welfare measured following Lucas (1987) as fraction of consumption agents would give up to eliminate excess volatility.
- Welfare of benchmark model (B) and regime R:
  - U_B = E[ sum_{t=0}^∞ β^t u(C^B_t - h C^B_{t-1}, 1 - L^B_t) ]
  - U_R = E[ sum_{t=0}^∞ β^t u(C^R_t - h C^R_{t-1}, 1 - L^R_t) ]
- Welfare cost θ solves:
  - E[ sum_{t=0}^∞ β^t u((1 - θ)(C^B_t - h C^B_{t-1}), 1 - L^B_t) ] = E[ sum_{t=0}^∞ β^t u(C^R_t - h C^R_{t-1}, 1 - L^R_t) ]
  - Computed using second-order approximation (as in Schmitt-Grohé (2005)). θ > 0 indicates welfare loss due to policy.
- Simulated business cycle moments and welfare costs (Table 2; columns: Std. Dev. GDP, Std. Dev. NX/GDP, Std. Dev. C, Std. Dev. INV, Std. Dev. LRER, x/ Std. Dev. GDP, Welfare):
  - Canadian Data (1980-2008):
    - Std. Dev. GDP = 1.51
    - Std. Dev. NX/GDP = 0.89
    - Std. Dev. x / Std. Dev. GDP = 0.57
    - Std. Dev. C = 2.73
    - Std. Dev. INV = 0.71
    - Std. Dev. LRER = 2.46
    - Welfare = -
  - Empirical Rule:
    - Std. Dev. GDP = 1.49
    - Std. Dev. NX/GDP = 0.93
    - Std. Dev. x / Std. Dev. GDP = 0.78
    - Std. Dev. C = 2.65
    - Std. Dev. INV = 0.76
    - Std. Dev. LRER = 2.81
    - Welfare = 0.04
  - Taylor-Type Rule:
    - Std. Dev. GDP = 1.58
    - Std. Dev. NX/GDP = 1.32
    - Std. Dev. x / Std. Dev. GDP = 0.77
    - Std. Dev. C = 2.70
    - Std. Dev. INV = 0.79
    - Std. Dev. LRER = 2.97
    - Welfare = 0.04
  - Fixed Exchange Rate:
    - Std. Dev. GDP = 2.52
    - Std. Dev. NX/GDP = 1.22
    - Std. Dev. x / Std. Dev. GDP = 0.79
    - Std. Dev. C = 2.49
    - Std. Dev. INV = 0.98
    - Std. Dev. LRER = 1.50
    - Welfare = 0.19
  - Optimized Rule:
    - Std. Dev. GDP = 1.49
    - Std. Dev. NX/GDP = 1.33
    - Std. Dev. x / Std. Dev. GDP = 0.78
    - Std. Dev. C = 2.70
    - Std. Dev. INV = 0.48
    - Std. Dev. LRER = 3.30
    - Welfare = 0.00
- Interpretation of moments:
  - Consumption and employment are less volatile than GDP; investment and the real exchange rate are more than twice as volatile as output (pattern consistent with industrialized countries).
  - Real exchange rate exhibits greater volatility than consumption or output ("exchange rate disconnect").
- Welfare ranking:
  - Optimized rule yields Welfare = 0.00 (closest to benchmark).
  - Empirical and Taylor-type rules generate small welfare costs (Welfare = 0.04).
  - Fixed exchange rate generates largest welfare cost (Welfare = 0.19).

*Italic: Source: _wp10182 - 4.  Capital Producers (IMF chapter PDF).*

### section 3.  Overall, the model with the empirical rule matches the main

### _wp10182 - section 3.  Overall, the model with the empirical rule matches the main

### Policy experiments and quantitative results
- Empirical rule:
  - Generates a welfare cost equivalent to 0.04 percent of life time consumption.
  - Matches the main features of the data; consumption is more volatile in the model than in the data but remains less volatile than GDP.
- Taylor-type rule (without exchange rate intervention):
  - Results are very close to the empirical rule.
  - Allows for greater real exchange rate volatility, which spills over to the trade balance: the trade balance has a standard deviation 50 percent larger than in the case of the empirical rule.
- Fixed exchange rate policy:
  - Increases the volatility of GDP by 60 percent.
  - Increases the volatility of labor by 25 percent.
  - Reduces real exchange rate volatility by 50 percent.
  - Generates a welfare cost equivalent to 0.19 percent of the stream of lifetime consumption.
  - The welfare loss is about two times the measure of costs of business cycles estimated by Lucas (1987).
  - The welfare loss is generated mainly by the increase in labor supply volatility by 25 percent.
  - Intuition: a fixed exchange rate policy exacerbates the business cycle in the presence of sticky prices because stabilizing the nominal exchange rate in response to an increase in commodity prices requires increasing the money supply, providing an additional stimulus that amplifies volatility.
- Optimized rule:
  - By construction, does not generate substantial welfare costs compared to the benchmark model.
  - Brings the allocation close to the one with flexible prices and internalized learning-by-doing.
  - Generates relatively low volatility of output and labor and a higher volatility of the real exchange rate.
  - Consistent with Friedman (1953): a flexible exchange rate can insulate the economy from external shocks by enabling faster adjustment of relative prices.

### Welfare costs and exchange rate intervention (Figure 9)
- Welfare costs are an increasing function of the exchange rate intervention parameter s.
  - The horizontal axis in Figure 9 is the depreciation rate coefficient in the Taylor rule (ψ_dep), and the vertical axis is the relative value of the welfare costs.
  - The graph describes how welfare costs rise as a central bank intervenes more in the foreign exchange market: intervention moves allocation away from the efficient equilibrium and increases macroeconomic volatility, negatively affecting households’ welfare.
  - Technical note from the text: The plotted ratio is the ratio of welfare for a rule with exchange rate intervention such as equation (13) compared with the welfare with no exchange rate intervention. For s = 0 this ratio is equal to 1.

### Main interpretation and policy trade-off
- Main result: a policy of exchange rate intervention results in welfare losses.
  - Although exchange rate stabilization can correct inefficiencies in the home goods sector due to learning-by-doing, it is costly in terms of macroeconomic volatility.
  - Correcting the learning-by-doing externality with monetary policy is highly distortionary.
  - During a Dutch disease episode there are welfare gains if the monetary authority allows a real exchange rate appreciation to ensure a reallocation of resources across sectors.

### Sensitivity analysis (Section VI)
- Purpose: assess robustness of results under four alternative model specifications:
  - Larger learning-by-doing externalities.
  - Learning-by-doing in the non-tradable sector.
  - Incomplete pass-through for imported goods.
  - Financial autarky.
- High organizational capital (higher learning-by-doing):
  - Increased the share of organizational capital to H = 0:5, which implies a learning rate of 40 percent, two times the empirical estimate.
  - Home goods production becomes more sensitive to exchange rate appreciations.
  - With the empirical rule, contraction of output is below the efficient level of production (contrast to baseline).
  - For a given policy rule, a higher learning-by-doing rate generates greater costs during a commodity boom; currency depreciation to correct the externality causes macroeconomic variables to overshoot efficient levels.
- Learning-by-doing present in both sectors (home goods and non-tradable):
  - Assuming the same parameter values for both sectors exacerbates the reallocation from tradable to non-tradable.
  - Real exchange rate appreciation stimulates non-tradable output, and via LBD increases future non-tradable production, reallocating factors away from tradables.
  - With the externality in the non-tradable sector there are welfare gains from appreciating the currency; greater exchange rate intervention induces deviations from the efficient allocation.
- Limited pass-through (import prices set with Calvo mechanism every four quarters):
  - Expenditure switching effect is weaker because import prices are less sensitive to exchange rate fluctuations.
  - A currency depreciation generates a lower substitution between imported and home goods.
  - Even with incomplete pass-through, leaning against an appreciated exchange rate remains distortionary.
- Financial autarky:
  - Modeled by setting(0 =)B_t = 100. Under this assumption the small open economy finds it extremely costly to issue foreign debt and is limited in intertemporal consumption smoothing, although it still trades goods with the rest of the world.
  - Effects of a commodity price shock are much larger: higher aggregate demand from unsmoothed consumption leads to greater reallocation across sectors and a larger contraction of home goods production.
  - Despite larger externality effects, monetary policy remains an inefficient instrument to correct the distortion; exchange rate stabilization generates significant deviations from the benchmark real allocation.
- Overall sensitivity conclusion:
  - Introducing these alternative frictions does not change the basic trade-off for exchange rate intervention.
  - Results remain valid: correcting LBD with exchange rate stabilization is highly distortionary despite larger LBD effects in some specifications.

### Conclusions and policy implications (Section VII)
- Setup: New Keynesian small open economy model with learning-by-doing externality in the tradable sector creates a trade-off for monetary policy:
  - LBD externality calls for exchange rate stabilization to bring tradable production toward the efficient level.
  - Standard New Keynesian logic: higher terms of trade call for an appreciation of the real exchange rate to achieve efficient allocation across sectors.
- Under calibration consistent with the Canadian data:
  - The trade-off favors allowing a real exchange rate appreciation.
  - Stabilizing the nominal exchange rate with monetary policy is a blunt and distortionary instrument to correct the externality; attempts to lean against an appreciated exchange rate result in misallocation and reduced welfare.
  - Monetary policy should focus on stabilizing prices subject to nominal rigidities to ensure smooth adjustment toward a situation of higher commodity prices.
- Policy recommendation:
  - Governments wishing to address learning-by-doing problems in a Dutch Disease episode should use alternative instruments (not monetary exchange rate intervention). Suggested alternatives include fiscal policy, adoption of a commodity fund if the government owns the resource, or structural reforms in goods and factor markets to facilitate reallocation.
- Empirical and policy note:
  - The paper provides theoretical support for the Bank of Canada’s policy of not intervening in the foreign exchange market. In fact, this central bank has not intervened in the foreign exchange rate market in the last ten years.
- Suggested extensions for future research:
  - Evaluate effectiveness of alternative instruments to correct the LBD externality (fiscal, commodity fund, structural reforms).
  - Explore alternative frictions affecting the tradable sector during commodity booms (e.g., financial frictions as in Caballero and Lorenzoni (2009)).
  - Conduct a similar quantitative exercise for a developing small open economy where the monetary authority may lack credibility; in such contexts stabilizing the exchange rate could be better to stabilize prices and anchor inflation expectations, as shown empirically in Rogoƒ et al. (2004).

*Italic line: Source: _wp10182 - section 3.  Overall, the model with the empirical rule matches the main*

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