## _wp0922

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---

### Introduction: Objectives and Key Model Features
- Purpose:
  - Develop an optimal rules-based interpretation of the "three pillars macroeconomic policy framework" for emerging market small open economies (SOEs).
  - Show how monetary-fiscal rules should be adjusted for emerging SOE features, including financial frictions and the zero lower bound (ZLB) on the nominal interest rate.
- "Three pillars" defined as: a freely floating exchange rate, an explicit target for inflation over the medium run, and a mechanism to ensure a stable government debt-GDP ratio around a specified long run.
- Principal departures from existing literature:
  - Focus on a small open economy (SOE).
  - Introduce financial frictions via a "financial accelerator".
  - Impose a zero lower bound (ZLB) constraint on the nominal interest rate.
- Model scope and calibration:
  - Two-bloc DSGE emerging markets SOE–rest of the world model with local currency pricing for exporters, a commodity sector ("copper"), oil imports, liability dollarization, and capital financing partly or totally in foreign currency.
  - Model calibrated to Chile and US data; shock processes use estimates from Medina et al. (2007b).

### Model Overview: Structure and Key Mechanisms
- Framework:
  - Two-bloc microfounded model along Obstfeld and Rogoff (1995) expanded with nominal and real frictions (Smets and Wouters, 2003).
  - SOE emerges as limit when relative size of ROW tends to infinity.
- Production and markets:
  - Wholesale production Y_Wt = A_t K_t^{α1} L_t^{α2} (OIL_t)^{α3} (COP_t)^{1−α1−α2−α3} (equation (26)).
  - Wholesale price P_{WH,t}; marginal products and prices: W_t = P_{WH,t} α2 Y_{W,t} / L_t (29); P_{O,t} = P_{WH,t} α3 Y_{W,t} / OIL_t (30); P_{C,t} = P_{WH,t} (1−α1−α2−α3) Y_{W,t} / COP_t (31).
- Financial frictions and openness:
  - External finance premium Θ_t = Θ(B_t / N_t) with Θ′(·) > 0, Θ(0) = 0, Θ(∞) = ∞ (34).
  - Wholesale firms borrow in home and foreign currency with home-currency share φ ∈ [0,1]; expected cost of funds = (1 + Θ_t)[ φ E_t[(1+R_t)] + (1−φ) E_t[(1+R^*_t) RER_{t+1} / RER_t] ] (33).
  - Liability dollarization documented (example figures in source: "21% in Chile to 78% in Uruguay" for 2001).

### Households, Money, and Asset Pricing
- Representative home household maximizes E_t ∑_{t=0}^∞ β^t U(C_t(h), M_t(h)/P_t, L_t(h)) (equation (1)).
- Bonds and risk premium:
  - P_{B,t} = 1/(1+R_{n,t}); P^*_{B,t} = 1/(1+R^*_{n,t}) φ(B_t P_{H,t} / Y_t) (equation (2)); φ(0) = 0 and φ′ < 0.
- Money in utility:
  - U_{M_H,t} = U_{C,t} [R_{n,t}/(1+R_{n,t})] (19); U_{M_F,t} = U_{C,t} [R^*_{n,t}/(1+R^*_{n,t})] (20).
- Modified UIP and Euler:
  - P_{B,t} = β E_t[U_{C,t+1}/U_{C,t} P_t/P_{t+1}] (18); modified UIP incorporates marginal utility terms (22).
- Rule-of-Thumb (RT) households:
  - Share 1−λ constrained (RT), consume out of post-tax income; λ optimizing. RT budget P_t (1 + τ_{C,t}) C_{1,t}(h) = (1−τ_{L,t}) W_{1,t}(h) L_{1,t}(r) + TF_{1,t} (23).
  - Aggregation: C_t = λ C_{1,t} + (1−λ) C_{2,t} (25).

### Firms, Net Worth Dynamics, and Pricing
- Wholesale firms:
  - Expected return on capital: E_t(1+R^k_t) = (P_{WH,t} / P_t) α1 Y_t / K_t + (1−δ) E_t[Q_{t+1}] / Q_t (32).
  - Net worth dynamics: N_t = ξ_e V_t (36) with V_t defined in (37).
  - Exiting entrepreneurs consume C^e_t = (1 − ξ_e) V_t (38); consumption allocations in (39)–(40).
- Retail firms:
  - Monopolistically competitive; two pricing regimes PCP and LCP. In policy exercises θ = 1 (LCP only).
  - Calvo-style pricing: optimal pricing with hazard ξ_H and index dynamics (41)–(42) (PCP) and analogous LCP equations (45).
- Capital producers:
  - K_{t+1} = (1−δ) K_t + (1 − S(I_t / I_{t−1})) I_t (47).
  - Investment composite and price: I_t and P_{I,t} given in (48)–(51); capital producer first-order condition (52).

### Government Budget, Copper Sector, and External Position
- Government budget identity (market-price bonds): P_{B,t} B_{G,t} + M_t = B_{G,t−1} + P_{H,t} G_t − T_t + M_{t−1} (53).
- Taxes on labor, profits, consumption, capital returns and copper: τ_{L,t}, τ_Γ, τ_{C,t}, τ_{K,t}, τ_{cop,t}; government owns share χ of copper sector.
- Consolidated net foreign assets evolution: P_{B,t} B_t = B_{t−1} + TB_t (55) with TB_t defined.
- Copper market: law of one price S_t P^*_{C,t} = P_{C,t} (59); Chile modeled as small copper producer with exogenous dollar price dynamics log(P^*_{C,t+1} / P^*_{t+1}) = ρ_{cop} log(P^*_{C,t} / P^*_t) + v_{cop,t+1} (60).

### Linearization, SOE Specialization, and Calibration
- Model linearized around deterministic zero inflation, zero net private sector debt, balanced growth steady state; state-space representation (63).
- SOE specialization obtained as limit n → 0 with demand elasticity limits recorded (64)–(65).
- Calibration:
  - Home bias parameters ω, ω^*, ω_I, ω^*_I calibrated using consumption/investment trade shares; condition cs_imports + is_imports = cs_exports + is_exports (66).
  - Preference parameters calibrated solving (67)–(70) with restriction Ψ ∈ [0,0.01]; Appendix C gives details.

### Monetary Policy Rules Considered (Section III)
- Regimes:
  - FIX: r_n,t = r^*_n,t + θ_s s_t (71).
  - FLEX(D): r_n,t = ρ r_n,t−1 + θ_π π_H,t + θ_y y_t (72).
  - FLEX(C): r_n,t = ρ r_n,t−1 + θ_π π_t + θ_y y_t (73).
  - HYB: r_n,t = ρ r_n,t−1 + θ_π π_H,t + θ_y y_t + θ_s s_t (74).
- Assumption: central bank and fiscal authorities enjoy full credibility; ρ ∈ [0,1] is smoothing parameter.

### Fiscal Rules and Instruments (Section IV)
- Government debt dynamics in GDP ratios (76); growth-adjusted real interest rate 1 + R_g,t−1 = (1 + R_n,t−1) / [(1 + π_H,t)(1 + Δy_t)] (77).
- Primary surplus steady-state: PS/(P_H Y) = R_g ˆB_G/(P_H Y) (78); FS/(P_H Y) = (1/(1 + π_H)(1 + g_y) − 1) ˆB_G/(P_H Y) (79).
- Chosen fiscal instrument: T_I,t = (1−λ)TF2,t − λTF1,t (81) with timing: fiscal set every two periods.
- Conventional fiscal rule (linearized):
  - tf2,t = p_H,t−1 + (1 + α_y) y_t−1 + α_bg b_G,t−1 (87).
  - tf2,t = E_t−1 [p_H,t + (1 + α_y) y_t + α_bg b_G,t] (88).
- Structural Fiscal Surplus Rule (SFSR) — Chilean-style:
  - FS_t = FS + α_tax (T_I,t − ̂T_I,t + T_NI,t − ̂T_NI,t) + α_cop (T_COP,t − ̂T_COP,t) (89).
  - Linear deviation form and two-period planning implementations given in (90)–(94); parameter k governs distribution of adjustment between household groups (k = 0.5 in results).

### Imposing the ZLB: Approximate Implementation (Section V and chapter 6)
- ZLB implemented approximately by modifying single-period welfare loss to L_t + w_r r_n,t^2 and choosing w_r and σ_r^2 so probability p of hitting lower bound is very low.
- Critical steady-state inflation ensuring r_n,t ≥ 0 with probability 1 − p:
  - π^* = max[z_0(p) σ_r − (1/β(1+g_u_c) − 1) × 100, 0] (equation (95)).
- Trade-off:
  - Increasing w_r lowers σ_r (reducing π^*) but raises stochastic component ˜Ω_0; intertemporal welfare Ω_0 = ˜Ω_0 + ̄Ω_0.
- Calibration example:
  - Target p = 0.001; with β = 0.99, g_u_c = −0.014: π^* = max[3.00 σ_r − 2.44, 0].
  - ̄Ω_0(0) = w_π π^*^2 and ̄Ω = (1/2) w_π π^*^2 = 3.829 π^*^2 (as stated in source).

### Model Variants, Shocks, and Foreign Rule Calibration
- Model I: no financial accelerator (FA) and no liability dollarization (LD) (χ_θ = χ_θ^* = 0, Θ = Θ^* = 0, φ = 1).
- Model II: FA only (χ_θ, χ_θ^* < 0, Θ, Θ^* > 0, φ = 1).
- Model III: FA and LD (χ_θ, χ_θ^* < 0, Θ, Θ^* > 0, φ ∈ [0,1)).
- Nine exogenous shocks: a_t, g_t (both blocs), ε_P,t, ε_cop, ε_oil, ε_UIP, ε_R,t^* and others.
- Foreign bloc optimized rule calibrated with ρ^* = 1, θ_π^* = 10, θ_y^* = 0, α_bg^* = 0.87.

### Welfare Outcomes and ZLB Results (verbatim table values preserved)
- Welfare under optimal policy without ZLB (Table 2 rows preserved as printed):
  - Model I row: "3.262.353.352.820.006"
  - Model II row: "3.485.013.974.080.034"
  - Model III row: "13.928.3815.347.890.099"
  - (These rows are printed in the source table and are preserved here verbatim.)
- Imposing ZLB with monetary policy only (Table 3 rows preserved):
  - Rows verbatim:
    - 0.00 1 2.823 3.35 2.59 12.84 16.19
    - 1.00 1.84 3.57 1.62 5.04 8.61
    - 2.00 1.32 3.94 1.00 1.90 5.85
    - 3.00 1.00 4.33 0.55 0.59 4.92
    - 3.25 0.94 4.23 0.46 0.41 4.83
    - 3.50 0.88 4.52 0.37 0.27 4.79
    - 3.75 0.83 4.61 0.29 0.17 4.77
    - 4.00 0.79 4.70 0.22 0.09 4.79
  - Example: with w_r = 3.75 steady-state quarterly inflation π^* = 0.29 (monetary policy only for Model I).
- Imposing ZLB with monetary + fiscal policy (Table 4 rows preserved):
  - Rows verbatim:
    - 0.00 1 2.35 3.35 2.16 8.93 12.18
    - 0.25 0.86 3.26 0.33 0.21 3.47
    - 0.50 0.55 3.27 0 0 3.27
    - 0.75 0.40 3.29 0 0 3.29
    - 1.00 0.31 3.30 0 0 3.30
  - Key: with fiscal stabilization, ZLB can be imposed at w_r = 0.5 with π^* remaining at zero baseline.
- Welfare outcomes under ZLB constraint (Table 5 preserved):
  - Model I: M+F Ω_0 = 3.27, M+F σ_r^2 = 0.55, π^* = 0.00, Ω_0 (M policy only) = 4.77, σ_r^2 = 0.84, π^* = 0.29, c_MF^e = 0.104
  - Model II: M+F Ω_0 = 3.74, σ_r^2 = 0.69, π^* = 0.05, Ω_0 (M only) = 6.98, σ_r^2 = 1.00, π^* = 0.57, c_MF^e = 0.225
  - Model III: M+F Ω_0 = 14.90, σ_r^2 = 0.72, π^* = 0.10, Ω_0 (M only) = 24.19, σ_r^2 = 1.20, π^* = 0.85, c_MF^e = 0.644
  - Interpretation preserved: fiscal policy’s consumption-equivalent contribution rises across models to c_e = 0.644 in Model III.
- Welfare decomposition across shocks (Table 6 verbatim contributions):
  - a_t: Model I 54.2, Model II 43.4, Model III 12.4
  - a_t^*: Model I 2.4, Model II 2.1, Model III 38.1
  - g_t: Model I 0.1, Model II 0.1, Model III 0.0
  - g_t^*: Model I 0.1, Model II 0.1, Model III 13.5
  - ε_cop: Model I 20.7, Model II 16.8, Model III 5.7
  - ε_oil: Model I 20.9, Model II 15.5, Model III 19.9
  - ε_UIP: Model I 1.6, Model II 1.3, Model III 0.6
  - ε_r^*: Model I 0.0, Model II 0.0, Model III 0.0
  - ε_P: Model I 0, Model II 20.1, Model III 9.5
  - Key qualitative result preserved: introducing FA reduces home productivity and commodity shocks’ share; adding LD raises foreign demand and supply contributions dramatically.

### Impulse Responses and Policy Role (Figure 3 summary)
- Response to an unanticipated −1% technology shock:
  - Both monetary and fiscal policy tighten (r_n,t and t_I,t increase).
  - As frictions increase Model I → II → III:
    - Monetary policy more constrained by ZLB.
    - Fiscal policy plays larger stabilization role.
    - Model III with LD exhibits prolonged (~20 quarters) negative deviations in nominal interest rates and exchange rate depreciation that partially offset investment declines and dollar liabilities effects.

### Performance of Optimized Simple Rules (Tables 7–10; values preserved)
- Measure of cost of simplicity: c_e^SIM = (Ω_SIM_0 − Ω_OPT_0) / [(1−%)(1−h_C)c_y] × 10^−2 (%).
- FLEX(D) + conventional fiscal rule (Table 7 rows preserved):
  - Model I: M+F Ω_0 = 5.0, σ_r^2 = 0.9, π^* = 0.3, [ρ, θ_π, θ_y, α_bg, α_y] = [1,10,0,0.7,3]; M Ω_0 = 5.9, σ_r^2 = 1, π^* = 0.5; c_MF^e = 0.06; c_SIM^e = 0.12
  - Model II: M+F Ω_0 = 13.2, σ_r^2 = 1.7, π^* = 1.5, [1,10,0.05,4,2]; M Ω_0 = 13.6, σ_r^2 = 1.8, π^* = 1.5; c_MF^e = 0.03; c_SIM^e = 0.66
  - Model III: M+F Ω_0 = 44.7, σ_r^2 = 3.6, π^* = 3.3, [1,4,0.3,5,3]; M Ω_0 = 58.2, σ_r^2 = 4.7, π^* = 4.0; c_MF^e = 0.93; c_SIM^e = 2.07
- FIX + conventional fiscal rule (Table 8 preserved):
  - Model I: M+F Ω_0 = 74, σ_r^2 = 0.8, π^* = 0.3; M Ω_0 = 84; c_MF^e = 0.71; c_SIM^e = 4.9
  - Model II: M+F Ω_0 = 136, σ_r^2 = 1.1, π^* = 0.7; M Ω_0 = 152; c_MF^e = 1.16; c_SIM^e = 9.2
  - Model III: M+F Ω_0 = 175, σ_r^2 = 2.4, π^* = 2.2; M Ω_0 = 191; c_MF^e = 1.09; c_SIM^e = 11.1
- FLEX(C) + conventional fiscal rule (Table 9 preserved):
  - Model I: M+F Ω_0 = 23, σ_r^2 = 1.0, π^* = 0.6; M Ω_0 = 23.7; c_MF^e = 0.04; c_SIM^e = 0.41
  - Model II: M+F Ω_0 = 35, σ_r^2 = 1.5, π^* = 1.2; M Ω_0 = 36.7; c_MF^e = 0.10; c_SIM^e = 2.24
  - Model III: M+F Ω_0 = 62, σ_r^2 = 2.6, π^* = 2.4; M Ω_0 = 64.0; c_MF^e = 0.12; c_SIM^e = 3.37
- FLEX(D) + modified SFSR (Table 10 preserved):
  - Model I: Ω_0 = 5.38, σ_r^2 = 0.96, π^* = 0.49, [1.0, 10.0, 0.00, 1.10, 0.82], c_SIM^e = 0.15
  - Model II: Ω_0 = 13.19, σ_r^2 = 1.76, π^* = 1.53, [1.0, 10.0, 0.06, 0.96, 0.90], c_SIM^e = 0.66
  - Model III: Ω_0 = 46.74, σ_r^2 = 3.58, π^* = 3.22, [1.0, 4.13, 0.26, 0.97, 0.69], c_SIM^e = 2.21
- Key findings preserved:
  - FLEX(D) + counter-cyclical fiscal rule stabilizes far better than FIX.
  - Ability of optimized simple rules to match fully optimal benchmark deteriorates with financial frictions: c_SIM^e rises from 0.12% (Model I) to 0.66% (Model II) to 2.07% (Model III) under FLEX(D).
  - FIX induces very large welfare losses; fiscal policy must bear more stabilization burden or steady-state inflation must rise, producing large welfare costs (c_SIM^e up to 11.1%).

### Policy Conclusions (verbatim implications preserved)
- Broad support for the "three-pillars" macro framework (explicit inflation target, floating exchange rate, counter-cyclical fiscal rule) as pursued by Chile.
- Domestic inflation targeting is superior to attempting to stabilize the exchange rate; CPI targeting that implicitly responds to the exchange rate is more costly.
- Financial frictions increase the costs of stabilizing the exchange rate: emerging markets with financial frictions should "fear to fix" rather than "fear to float".
- Fiscal stabilization can play a significant role when financial frictions and the ZLB are relevant; the contribution of fiscal policy rises with frictions.
- The ability of simple optimized rules to replicate optimal policy deteriorates sharply with the introduction of the FA and LD. Future research should explore rules that respond to indicators of financial stress (e.g., risk premium facing firms) and endogenize currency choice in borrowing.

*Source: _wp0922*

### 1.   Notation for Prices ...............................................................................................

### 1.   Notation for Prices ...............................................................................................

### Contents and Structure
- Lists major sections and figures with exact numbering as in the source:
  - Sections 1–10 headings with page references: 1. Notation for Prices (49); 2. Welfare Outcomes Under Optimal Policy: No ZLB Constraint (49); 3. Optimal Policy with a ZLB Constraint: Monetary Policy Only for Model I (49); 4. Optimal Commitment with a ZLB Constraint. Monetary Plus Fiscal Policy for Model I (50); 5. Welfare Outcomes Under Optimal Policy: ZLB Constraint (50); 6. Welfare Decomposition of Shocks (50); 7. Welfare Outcomes Under Optimized Simple Rules: FLEX (D) with a Conventional Fiscal Rule. Models I, II and III (51); 8. Welfare Outcomes Under Optimized Simple Rules: FIX with a Conventional Fiscal Rule. Models I, II and III (51); 9. Welfare Outcomes Under Optimized Simple Rules: FLEX(C) with a Conventional Fiscal Rule. Models I, II and III (51); 10. Welfare Outcomes Under Optimized Simple Rules: FLEX(D) with a Modified SFSR. Models I, II and III (51).
  - Figures listed 1–5 with titles and page references: Figure 1 Imposition of ZLB: Model I (39); Figure 2 Imposition of ZLB: Model III (39); Figure 3 Impulse Responses to a-1 Percent Technology Shock. Models I, II, and III (42); Figure 4 Imposition of ZLB: Flex(D)+Conventional Fiscal Rule, Model I (45); Figure 5 Imposition of ZLB: Flex(D)+Conventional Fiscal Rule: Model III (45).
  - Appendixes 1–4 with titles and page references: 1. The Steady State (53); 2. Linearization (56); 3. Calibration and Estimation (63); 4. Quadratic Approximation of the Welfare Loss (69).

### Introduction: Objectives and Key Model Features
- Purpose:
  - Develop an optimal rules-based interpretation of the "three pillars macroeconomic policy framework" for emerging market small open economies (SOEs).
  - Show how monetary-fiscal rules should be adjusted for emerging SOE features, including financial frictions and the zero lower bound (ZLB) on the nominal interest rate.
- Context:
  - "Three pillars" defined as: a freely floating exchange rate, an explicit target for inflation over the medium run, and a mechanism to ensure a stable government debt-GDP ratio around a specified long run.
- Principal departures from existing literature:
  - Focus on a small open economy (SOE).
  - Introduce financial frictions via a "financial accelerator".
  - Impose a zero lower bound (ZLB) constraint on the nominal interest rate, enhancing the role for fiscal stabilization policy.
- Model scope and calibration:
  - Two-bloc DSGE emerging markets SOE–rest of the world model.
  - Incorporates local currency pricing for exporters, a commodity sector ("copper"), oil imports, liability dollarization, financial accelerator where capital financing is partly or totally in foreign currency.
  - Model calibrated to Chile and US data; shock processes use estimates from Medina et al. (2007b).
- Paper organization summary:
  - Section 2: model presentation.
  - Sections 3–4: monetary and fiscal rule forms.
  - Section 5: operational requirement for monetary rules relative to the ZLB.
  - Section 6: benchmark Ramsey policy with financial accelerator and liability dollarization.
  - Section 7: alternative simple rules including the Structural Fiscal Stability Rule (SFSR).
  - Section 8: concluding remarks.

### Model Overview (Section II): Core Assumptions and Mechanisms
- Framework:
  - Standard two-bloc microfounded model along the lines of Obstfeld and Rogoff (1995), extended with nominal and real frictions from closed-economy literature (e.g., Smets and Wouters, 2003).
  - Asymmetric blocs with unequal sizes; SOE emerges as limit when relative size of larger bloc tends to infinity.
- Agents and production structure:
  - Households consume tradable goods produced at home and abroad.
  - Domestic production comprises wholesale, retail, and capital producers.
  - Wholesale firms borrow from households to buy capital; capital producers build new capital in response to wholesalers’ demand.
  - Retailers set local currency pricing for exports.
  - Commodity sector ("copper") and oil imports affect consumption and production.
- Key frictions and openness dimensions:
  - Financial accelerator: capital financing partly or fully in foreign currency; wholesaler financial position varies inversely with net worth.
  - Liability dollarization: corporate-sector dollar-denominated liabilities documented as significant in Latin America (example figures noted in source: "21% in Chile to 78% in Uruguay" for 2001).
  - Frictions in world financial markets facing households (Benigno (2001)-style).

### Households: Preferences, Budget Constraints, and Money
- Preferences and notation:
  - Representative home household maximizes E_t sum_{t=0}^∞ β^t U(C_t(h), M_t(h)/P_t, L_t(h)) (equation (1)).
  - Foreign household analogous with superscript "∗".
- Bonds and risk premium:
  - Two risk-free one-period bonds in each currency with prices:
    - P_{B,t} = 1/(1+R_{n,t})
    - P^*_{B,t} = 1/(1+R^*_{n,t}) φ(B_t P_{H,t} / Y_t)  (equation (2))
  - φ(·) captures cost/risk premium for home households to hold foreign bonds; φ(0) = 0 and φ′ < 0.
- Household budget constraint (aggregated per capita form):
  - (1 + τ_{C,t}) P_t C_t(h) + P_{B,t} B_{H,t}(h) + P^*_{B,t} S_t B_{F,t}(h) + M_t(h) + TF_t = W_t(h)(1−τ_{L,t}) L_t(h) + B_{H,t−1}(h) + S_t B_{F,t−1}(h) + M_{t−1}(h) + (1−τ_{Γ,t}) Γ_t(h) + P_{C,t}(1−τ_{cop})(1−χ) COP_t(h)  (equation (3))
  - Variables include consumption tax τ_{C,t}, labor income tax τ_{L,t}, profits tax τ_{Γ,t}, copper price P_{C,t}, copper endowment COP_t(h), transfers TF_t, etc.
- Money in utility and implications:
  - Money enters utility in a non-separable way; demand for money depends positively on marginal utility of consumption and negatively on the nominal interest rate (equations (19)–(20)).
  - The paper does not adopt separable utility in money (citing Woodford (2003) and Felices et al. (2006)).
- Intra-temporal labour supply:
  - If labor supply elasticity is η, individual labor demand L_t(h) = (W_t(h)/W_t)^{−η} L_t  (equation (4)).
- Consumption aggregator and price indices:
  - Per capita consumption index C_t(h) defined by equation (5) with parameters μ_C and weights w_C.
  - Sub-aggregator C_Z,t(h) defined by equation (6) with μ_Z and weight w_Z.
  - Variety-level consumption C_{H,t}(f,h), C_{F,t}(f,h) with elasticity ζ.
  - Aggregate and sectoral price indices defined by equations (11)–(14).
- Exchange rate, terms of trade, and real exchange rate:
  - Definitions: S_t nominal exchange rate; terms of trade T_t = P_{F,t}/P_{H,t}; O_t = P_{O,t}/P_{Z,t} (equation (17)).
  - Relations for real exchange rate and sectoral real exchange rates given by equations (15) and (16).
  - If μ = μ^∗, then RER_t = 1 and law of one price applies to aggregate indices iff w^∗ = 1−w.
- First-order conditions and modified UIP:
  - Asset pricing and Euler conditions:
    - P_{B,t} = β E_t[U_{C,t+1}/U_{C,t} P_t/P_{t+1}]  (equation (18))
    - U_{M_H,t} = U_{C,t} [R_{n,t}/(1+R_{n,t})]  (equation (19))
    - U_{M_F,t} = U_{C,t} [R^*_{n,t}/(1+R^*_{n,t})]  (equation (20))
    - Real wage condition: W_t (1−τ_{L,t}) / [P_t (1 + τ_{C,t})] = −(η/(η−1)) U_{L,t}/U_{C,t}  (equation (21))
  - Modified UIP condition incorporating marginal utility terms (equation (22)).

### Rule-of-Thumb (RT) Households
- Two-household types specification:
  - Proportion 1−λ are credit-constrained (RT) and consume out of post-tax income.
  - Proportion λ are optimizing households.
- RT household budget constraint:
  - P_t (1 + τ_{C,t}) C_{1,t}(h) = (1−τ_{L,t}) W_{1,t}(h) L_{1,t}(r) + TF_{1,t}  (equation (23)).
- Wages and hours:
  - RT households set their wage equal to the average of optimizing households (following Erceg et al. (2005)); in symmetric equilibrium W_{1,t} = W_{2,t} = W_t and L_{1,t} = L_{2,t} = L_t.
- Aggregation:
  - Average consumption per household: C_t = λ C_{1,t} + (1−λ) C_{2,t}  (equation (25)).
  - RT intra-temporal consumption choices mirror optimizing households (equation (24)).

### Firms: Structure and Roles (Introductory)
- Three firm types:
  - Wholesale firms: run by risk-neutral entrepreneurs, purchase capital and employ labor to produce wholesale goods; wholesale sector is competitive.
  - Retail firms: monopolistically competitive, differentiate wholesale goods at no resource cost and sell to households; set local currency prices for exports.
  - Capital producers: competitive sector converting final good into capital.
- Financial interactions:
  - Wholesale firms finance capital via borrowing; their external finance premium increases with leverage (Gilchrist (2003)-type financial accelerator).
  - Part of wholesale debt can be dollar-denominated due to "original sin" / liability dollarization constraints.

*Source: _wp0922 - 1.   Notation for Prices ...............................................................................................*

### 1.    Wholesale Firms

### 1. Wholesale Firms

### Production technology and inputs
- Wholesale sector production function:
  - Y_Wt = A_t K_t^{α1} L_t^{α2} (OIL_t)^{α3} (COP_t)^{1−α1−α2−α3} (equation (26))
  - K_t is beginning-of-period t capital stock.
  - L_t defined by L_t = [ (1/ν)^{1/η} ∑_{h=1}^{ν} L_t(h)^{(η−1)/η} ]^{η/(η−1)} (equation (27)), where L_t(h) is labor input of type h.
  - A_t is an exogenous shock capturing shifts to trend total factor productivity in this sector; it provides the source of demand for the copper shock that feeds into its world price.

### Labor demand and input prices
- Household labor demand implied by minimizing wage costs:
  - L_t(h) = (W_t(h) / W_t)^{−η} L_t (equation (28)).
- Marginal product conditions (aggregate labor and factor prices):
  - W_t = P_{WH,t} α2 Y_{W,t} / L_t (equation (29)).
  - P_{O,t} = P_{WH,t} α3 Y_{W,t} / OIL_t (equation (30)).
  - P_{C,t} = P_{WH,t} (1−α1−α2−α3) Y_{W,t} / COP_t (equation (31)).
- Wholesale goods sell at price P_{WH,t} in the home country.

### Returns on capital and expected return condition
- Profits per period equal α1 P_{WH,t} Y_t.
- Real market price of capital: Q_t (in units of total household consumption).
- Expected return on capital (acquired at beginning of period t), net of depreciation:
  - E_t(1+R^k_t) = (P_{WH,t} / P_t) α1 Y_t / K_t + (1−δ) E_t[Q_{t+1}] / Q_t (equation (32)), where δ is the depreciation rate of capital.
- Ex post return at the beginning of period t (return realized for capital acquired at t−1):
  - 1 + R^k_{t−1} = (P_{WH,t−1} / P_{t−1}) α1 Y_t / K_{t−1} + (1−δ) Q_t / Q_{t−1} (equation (35)).

### Borrowing, external finance premium and UIP considerations
- Wholesale firms borrow in home and foreign currency with exogenously given proportion φ ∈ [0,1] denominated in home currency.
- Expected cost of funds (taking into account credit market frictions) equals:
  - (1 + Θ_t) [ φ E_t[(1+R_t)] + (1−φ) E_t[(1+R^*_t) RER_{t+1} / RER_t] ] (equation (33)), where RER_t ≡ (P^*_t S_t) / P_t is the real exchange rate.
- Θ_t is the external finance premium given by Θ_t = Θ(B_t / N_t) with Θ′(·) > 0, Θ(0) = 0, Θ(∞) = ∞ (equation (34)).
- Special cases:
  - If φ = 1 or if UIP holds, cost reduces to (1 + Θ_t) E_t[1 + R_t].
- Note: all financial returns are assumed taxed at the same rate and do not affect arbitrage conditions (footnote).

### Entrepreneur net worth dynamics and exits
- Entrepreneurs exit with probability 1−ξ_e; net worth accumulates according to:
  - N_t = ξ_e V_t (equation (36)), where V_t is net value carried over from previous period.
- Net carried value V_t:
  - V_t = [ (1 + (1−τ^k_{t−1}) R^k_{t−1}) Q_{t−1} K_{t−1} − (1 + Θ_{t−1}) ( φ(1+R_{t−1}) + (1−φ)(1+R^*_{t−1}) RER_t / RER_{t−1} ) ( Q_{t−1} K_{t−1} − N_{t−1} ) ] (equation (37)), where τ^k_t is tax rate on capital returns.
- Net worth N_t is non-predetermined because the ex post return depends on current market value Q_t, itself non-predetermined.
- Exiting entrepreneurs consume C^e_t = (1 − ξ_e) V_t (equation (38)).
- Consumption of domestic and foreign goods by exiting entrepreneurs:
  - C^e_{H,t} = w_Z (P_{H,t} / P_t)^{−μ_Z} C^e_{Z,t}; C^e_{F,t} = (1 − w_Z) (P_{F,t} / P_t)^{−μ_Z} C^e_{Z,t} (equation (39)).
  - C^e_{Z,t} = w_C (P_{Z,t} / P_t)^{−μ_C} C^e_t (equation (40)).

### Monopolistic profits (for fiscal evaluation)
- Monopolistic profits as a proportion of GDP:
  - Γ_t P_{H,t} Y_t ≡ P_{H,t} Y_t − P_{WH,t} Y_{W,t} / (P_{H,t} Y_t) = (1 − MC_t) (1 + F / Y) (equation (43)), where MC_t = P_{WH,t} / P_{H,t} and F is fixed resource cost per retailer (retail firms section).

### Contextual notes from the wholesale firms section
- Managerial input and managerial wage contribution to net worth are ignored, following Gilchrist et al. (2002) and Gilchrist (2003) (footnote).
- The decision to partially borrow foreign currency (φ) is exogenous and not endogenized in the paper (footnote).

---

### Retail firms and price-setting regimes
- Retail firms are monopolistically competitive; they purchase wholesale goods and differentiate at fixed resource cost F. In free-entry equilibrium profits are driven to zero.
- Retail output for firm f: Y_t(f) = Y_{W,t}(f) − F.
- Two pricing regimes:
  - Producer currency pricing (PCP): fixed proportion 1−θ of retailers set prices in the Home currency.
  - Local currency pricing (LCP): proportion θ set prices in dollars.
- In policy exercises the model assumes LCP only (θ = 1).
- PCP exporters:
  - Price re-optimization occurs with probability ξ_H each period (probability of not re-optimizing is 1 − ξ_H). Average price duration interpretable as 1 / (1 − ξ_H) (footnote).
  - Optimal pricing condition (first-order condition for ˆP_{H,t}(f)):
    - E_t ∑_{k=0}^∞ ξ_H^k D_{t,t+k} Y_{t+k}(f) [ ˆP_{H,t}(f) − (ζ/(ζ−1)) P_{H,t+k} MC_{t+k} ] = 0 (equation (41)).
  - Evolution of the price index under Calvo-style staggered pricing:
    - P_{H,t+1}^{1−ζ} = ξ_H (P_{H,t})^{1−ζ} + (1 − ξ_H) (ˆP_{H,t+1}(f))^{1−ζ} (equation (42)).
- LCP exporters:
  - Analogous pricing conditions with possible different elasticity ζ_T and foreign demand terms; evolution of price index:
    - (P^∗`_{H,t+1})^{1−ζ_T} = ξ_H (P^∗`_{H,t})^{1−ζ_T} + (1 − ξ_H) (ˆP^∗`_{H,t+1}(f))^{1−ζ_T} (equation (45)).
- Foreign exporters from the large ROW bloc are PCPers with P_{F,t} = S_t P^∗_{F,t} (equation (46)).

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### Capital producers and investment dynamics
- Capital accumulation with adjustment costs following Smets and Wouters (2003):
  - K_{t+1} = (1−δ) K_t + (1 − S(I_t / I_{t−1})) I_t, with S′, S″ ≥ 0 and S(1) = S′(1) = 0 (equation (47)).
  - Adjustment costs associated with changes, not levels, of investment.
- Gross investment composite of domestic and foreign final goods:
  - I_t = [ w_I^{1/ρ_I} I_{H,t}^{(ρ_I−1)/ρ_I} + (1 − w_I)^{1/ρ_I} I_{F,t}^{(ρ_I−1)/ρ_I} ]^{ρ_I/(ρ_I−1)} (equation (48)).
  - Weights: w_I = 1 − (1 − n)(1 − ω_I); w^*_I = 1 − n(1 − ω^*_I) (equation (49)).
  - Investment price: P_{I,t} = [ w_I (P_{H,t})^{1−ρ_I} + (1 − w_I) (P_{F,t})^{1−ρ_I} ]^{1/(1−ρ_I)} (equation (50)).
  - Intra-temporal first-order conditions:
    - I_{H,t} = w_I (P_{H,t} / P_{I,t})^{−ρ_I} I_t; I_{F,t} = (1 − w_I) (P_{F,t} / P_{I,t})^{−ρ_I} I_t (equation (51)).
- Capital producer first-order condition (optimal transformation of I_t into K_{t+1}):
  - Q_t (1 − S(I_t / I_{t−1}) − (I_t / I_{t−1}) S′(I_t / I_{t−1})) + E_t[ (1 / (1 + R_{t+1})) Q_{t+1} S′(I_{t+1} / I_t) (I_{t+1}^2 / I_t^2) ] = P_{I,t} / P_t (equation (52)).

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### Government budget constraint and foreign asset accumulation
- Government issues domestic-currency bonds. Government budget identity:
  - P_{B,t} B_{G,t} + M_t = B_{G,t−1} + P_{H,t} G_t − T_t + M_{t−1} (equation (53)).
- Taxes levied on labor income, monopolistic profits, consumption, capital returns and copper revenue at rates τ_{L,t}, τ_Γ, τ_{C,t}, τ_{K,t}, τ_{cop,t} respectively.
- Copper supply is exogenous endowment COP_t (home) and COP^*_t (ROW); government owns share χ of the copper sector and taxes this public firm at τ_{cop,t}.
- Total per capita taxation net of transfers:
  - T_t = τ_{L,t} W_t L_t + τ_{Γ,t} Γ_t + τ_{C,t} P_t C_t − λ TF_{1,t} + (1−λ) TF_{2,t} + τ_{K,t} R^k_{t−1} P_t Q_t K_t + τ_{cop,t} P_{C,t} COP_t (equation (54)).
- Flat-rate taxes/transfers TF_{1,t} and TF_{2,t} are treated as dynamic fiscal instruments to keep tax rates constant at steady-state values; T_t can be written as instrument T_{I,t} = −λ TF_{1,t} + (1−λ) TF_{2,t} plus remaining taxes T_{NI,t}.
- Foreign asset accumulation and national net asset position:
  - Under assumption home households hold no foreign bonds (B_{F,t} = 0), net asset position B_t = −B^*_{H,t}.
- Consolidated accumulation of net foreign assets:
  - P_{B,t} B_t = B_{t−1} + W_t L_t + Γ_t + (1 − ξ_e) P_t V_t + P_t Q_t (1−S(X_t)) I_t + P_{C,t} COP_t − P_t C_t − P_t C^e_t − P_{I,t} I_t − P_{H,t} G_t − P_{O,t} OIL_t − P_{C,t} COP_t ≡ B_{t−1} + TB_t (equation (55)), where TB_t is trade balance.
- National accounting identity (trade balance definition):
  - P_{C,t} COP_t + P_{H,t} Y_t − P_{O,t} OIL_t − P_{C,t} COP_t = P_t C_t + P_t C^e_t + P_{I,t} I_t + P_{H,t} G_t + TB_t (equation (56)).
- In a balanced growth steady state with negative net foreign assets and government debt, the national and government budget constraints require a primary trade surplus (TB > 0) and a primary government surplus (T > P_H G). Distributional implications: private sector assets, profits from retail firms and income from copper firms are exclusively owned by unconstrained consumers, potentially raising consumption per head for that group; flat rate transfers/taxes affect consumption gaps.

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### Equilibrium conditions, copper market, and shocks
- Final goods market clearing (per capita terms):
  - Y_t = C_{H,t} + C^e_{H,t} + I_{H,t} + (1 − ν)/ν [ C^*_{H,t} + C^{e*}_{H,t} + I^*_{H,t} ] + G_t (equation (58)).
- Copper market: law of one price applies:
  - S_t P^*_{C,t} = P_{C,t} (equation (59)).
- Copper price shocks originate from copper supply shocks. Other exogenous shocks:
  - technology shocks in wholesale sectors, government spending shocks, foreign bloc interest rate rule shocks, risk premia facing unconstrained households (modified UIP condition (22)) and risk premia facing wholesale firms (external finance premium (34)).
- Chile is modeled as a small copper producer relative to world supply and therefore faces an exogenous copper price in dollars. Real dollar price follows:
  - log(P^*_{C,t+1} / P^*_{t+1}) = ρ_{cop} log(P^*_{C,t} / P^*_t) + v_{cop,t+1} (equation (60)).
- Equilibrium defined (at t = 0) by stochastic sequences of a comprehensive set of variables (consumption aggregates, prices, money, bonds, wages, outputs, capital, investment, Q_t, V_t, foreign counterparts, RER_t, S_t), given monetary instruments R_{n,t}, R^*_{n,t}, fiscal instruments and exogenous processes.

---

### Household utility specification consistent with BGP
- Non-separable utility to ensure consistency with balanced growth path:
  - U ≡ [ Φ(h)^{1−%} (1 − L_t(h))^{%} ]^{1−σ} / (1−σ) (equation (61)), where leisure is 1 − L_t(h).
  - Φ_t(h) ≡ [ b (C_t(h) − h_C C_{t−1})^{(θ−1)/θ} + (1−b) (M_t / P_t)^{(θ−1)/θ} ]^{θ/(θ−1)} (equation (62)).
- Properties:
  - U_{ΦL} > 0 so consumption and money holdings together, and leisure, are substitutes.
  - Balanced growth requires real wage, real money balances and consumption to grow at same steady-state rate with labor supply constant; the chosen functional form satisfies this.

---

### Linearization, state-space representation, and solution approach
- Model linearized around a deterministic zero inflation, zero net private sector debt, balanced growth steady state.
- State-space form:
  - [ z_{t+1} ; E_t x_{t+1} ] = A [ z_t ; x_t ] + B o_t + C [ r_{n,t} ; r^*_{n,t} ] + D v_{t+1 }
  - o_t = H [ z_t ; x_t ] + J [ r_{n,t} ; r^*_{n,t} ; tr_t ; tr^*_t ] (equation (63)).
- z_t: predetermined exogenous variables; x_t: non-predetermined variables; o_t: vector of outputs. Matrices A, B, etc., are functions of model parameters. Rational expectations formed with information set { z_{1,s}, z_{2,s}, x_s }, s ≤ t.

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### Small Open Economy (SOE) specialization
- Obtain SOE by letting relative world size n → 0 while retaining linkages with ROW.
- Limits of home/ROW demand elasticities as n → 0 are derived; result:
  - From ROW viewpoint SOE becomes invisible, but from SOE viewpoint ROW remains influential.
- Export and import elasticities α_{·,·} are specified and their limits as n → 0 recorded; several elasticity limits go to zero while others converge to ROW shares. Equations (64)–(65) summarize linearized output demand with detailed elasticity expressions and limits.

---

### Calibration: home bias and trade shares
- Home bias parameters to calibrate: ω, ω^*, ω_I, ω^*_I.
- Steady-state relationships:
  - With C^e = s_e C and c_y = C / Y, and defining cs_imports and cs_exports as GDP shares of imported consumption and exported home consumption respectively, the model sets:
    - α_{C,H} = C_H / Y = ω C / Y = (c_y − cs_imports)(1 − s_e).
    - α^e_{C,H} = C^e_H / Y = ω C^e / Y = (c_y − cs_imports) s_e.
    - α^*_{C,H} = C^*_H / Y = (1 − ω^*) C^* / Y^* Y^* / Y = cs_exports.
  - Investment counterparts and a zero trade balance linearization impose:
    - cs_imports + is_imports = cs_exports + is_exports (equation (66)).
- Trade data for consumption and investment goods, consumption shares and relative per capita GDP are used to calibrate bias parameters. For ROW, ω^* = ω^*_I = 1 is set elsewhere.

*Source: _wp0922 - 1.    Wholesale Firms*

### 2.    Calibrationof

### 2.    Calibrationof Household Preference Parameters

### Calibration of preference parameters to observed ratios and elasticities
- Use observed data on:
  - household wage bill as a proportion of total consumption: W(1−L)/(P C)
  - real money balances as a proportion of consumption: cz ≡ C(1−h_C)/Z (the “effective-consumption”–real money balance ratio, allowing for external habit)
  - estimates of the elasticity of the marginal utility of consumption with respect to total money balances: Ψ ≡ ZU_CZ / (C U_C)
- Key steady-state relationship (from (21)):
  - (η−1)/η · W(1−L)/(P C) = % Φ_C / Φ_C · (1−%)
  - expressed in the text as equation (67): (η−1)/η W(1−L)/(P C) = % Φ_C/Φ_C (1−%)
- Expression for Φ_C/Φ_C (from definition in (62)) (equation (68)):
  - Φ_C/Φ_C = (1−b) c z^(1−θ)/θ + b / b
  - written in the text as Φ_C Φ_C = (1−b)cz^(1−θ)/θ + b b
- Elasticity Ψ (from (61)) (equation (69)):
  - Ψ = (1−b)[(1−%)(1−σ)−1 + 1/θ b c z^(θ−1)/θ + 1−b]
  - written in the text as ZU_CZ / (C U_C) ≡ Ψ = (1−b)[(1−%)(1−σ)−1 + 1/θ bcz^(θ−1)/θ + 1−b]
- First-order condition linking cz and interest rate (equation (70)):
  - b(1−h_C)/(1−b) cz^(−1/θ) = 1 + R / R
  - written in the text as b(1−h_C)/(1−b) cz^(−1/θ) = 1 + R / R
- Calibration procedure:
  - Given σ, β, g, h_C, W(1−L)/(P C), cz and Ψ, solve equations (67)–(70) for %, b and θ.
  - Calculations for these parameters with calibrated values of σ, β, g, h_C, W(1−L)/(P C) and cz are in Appendix C.
- Calibration restriction:
  - Ψ ∈ [0,0.01]; since Ψ > 0 the calibration imposes that money and consumption are complements.
- Footnote reference:
  - See chapter 2 for discussion of this parameter.

### Remaining parameters (Section 3)
- Parameter selection principles:
  - Use quarterly data for Chile where possible.
  - Elsewhere adopt parameters reflecting broad characteristics of emerging economies.
- Sources cited for parameter choices:
  - Chile: Kumhof et al. (2008) (KL), Medina et al. (2007a), Medina et al. (2007b) (MS).
  - Financial accelerator and emerging-economy characteristics: Gertler et al. (2003, ”GGN”), Bernanke et al. (1999, ”BGG”).
  - Rest of world (U.S. data): Levin et al. (2006) (LOWW).
  - Matching Chilean with European estimates: Smets and Wouters (2003) (SW).
- Appendix C provides full calibration details.

### III. MONETARY POLICY INTEREST RATE RULES

### Regime options for central bank in emerging market bloc
- Three options considered:
  - Fixed exchange rates (”FIX”): set nominal interest to keep exchange rate fixed.
  - Inflation targeting under fully flexible exchange rates (”FLEX(D)” or ”FLEX(C)”): set interest rate to track deviations of domestic or CPI inflation from target.
  - Managed float (”HYB”): nominal interest responds to both inflation deviations and exchange rate deviations.
- Formal rules:

  - Fixed Exchange Rate Regime, ”FIX” (simplified model without exchange rate premium) (equation (71)):
    - r_n,t = r*_n,t + θ_s s_t
    - any θ_s > 0 is sufficient to implement the regime.
    - In the full model with exchange rate premium, implement ”FIX” as a ”HYB” regime with feedback coefficients chosen to minimize a loss function that includes a large penalty on exchange rate variability.
    - Note: reported loss function values remove the large penalty contribution.

  - Inflation Targets under Fully Flexible Exchange Rate, ”FLEX(D)” or ”FLEX(C)” (Taylor-type rules) (equations (72) and (73)):
    - r_n,t = ρ r_n,t−1 + θ_π π_H,t + θ_y y_t
    - r_n,t = ρ r_n,t−1 + θ_π π_t + θ_y y_t
    - where ρ ∈ [0,1] is the interest rate smoothing parameter.

  - Managed Float, ”HYB” (equation (74)):
    - r_n,t = ρ r_n,t−1 + θ_π π_H,t + θ_y y_t + θ_s s_t
    - exchange rate response is direct rather than indirect.

- Assumption on credibility:
  - Central bank and fiscal authorities in the emerging market bloc enjoy full credibility.
  - Note on realism: imperfect credibility remains important for many emerging market countries and can lead to higher stabilization costs or crises as referenced.

- Footnote on alternative managed-float specification:
  - Rule (73) is one specification where central bank resists deviations of exchange rate from equilibrium; alternative feedback could be on rate of change of exchange rate to stabilize exchange rate volatility (Batini et al. (2003)).

### IV. FISCAL RULES

### Government budget identity and growth-adjusted real interest rate
- Market-price bond identity (equation (75)):
  - ˆB_G,t = P*_B,t / B_G,t = (1 + R_n,t−1) ˆB_G,t−1 + G_t − T_t ≡ ˆB_G,t−1 − FS_t
  - where FS_t is the fiscal surplus.
- In GDP ratios (equation (76)):
  - ˆB_G,t / (P_H,t Y_t) = (1 + R_g,t−1) ˆB_G,t−1 / (P_H,t Y_t) + G_t/(P_H,t Y_t) − T_t/(P_H,t Y_t) ≡ ˆB_G,t−1/(P_H,t Y_t) − FS_t/(P_H,t Y_t)
- Growth-adjusted real interest rate definition (equation (77)):
  - 1 + R_g,t−1 = (1 + R_n,t−1) / [(1 + π_H,t)(1 + Δy_t)]
  - where π_H,t ≡ (P_H,t − P_H,t−1)/P_H,t−1 is home price inflation and Δy_t ≡ (Y_t − Y_t−1)/Y_t−1 is output growth.

### Steady-state fiscal surpluses (equations (78)–(79))
- Given target steady-state government debt-to-GDP ratio ˆB_G/(P_H Y):
  - Primary surplus as proportion of GDP:
    - PS/(P_H Y) ≡ (T−G)/(P_H Y) = R_g ˆB_G/(P_H Y)  (equation (78))
  - Overall fiscal surplus:
    - FS/(P_H Y) = (1/(1 + π_H)(1 + g_y) − 1) ˆB_G/(P_H Y)  (equation (79))
- Implication:
  - If inflation and growth are zero the steady-state fiscal surplus is zero.
  - If inflation and/or growth are positive, a steady-state fiscal deficit (but positive primary surplus) is sustainable.

### Tax instrument and decomposition of total tax revenues
- Total tax revenues composition (equation (80)):
  - T_t ≡ T_I,t + T_NI,t + T_COP,t
  - where TCOP_t = τ_cop P_C,t COP_t
- Chosen fiscal instrument:
  - Use a component of taxation, keeping government spending exogenous.
  - Instrument consists of flat-rate tax receipts paid by Ricardian households (1−λ)TF2,t minus flat-rate transfers to constrained households λTF1,t.
  - T_I,t = (1−λ)TF2,t − λTF1,t  (equation (81))
- Other tax rates kept fixed at steady-state values.
- Fiscal timing and implementation lag:
  - Fiscal authority sets tax rates every two periods (quarters) while central bank changes nominal interest every period.
  - Thus in quarter t, fiscal policy can only respond to outcomes in quarter t−1 or earlier.

### Fiscal instrument rules compatible with two-period planning
- Two functional forms for the fiscal instrument Taylor-type commitment rule:
  - T_I,t = f(X_t−1)  (equation (82))
  - T_I,t = f(E_t−1(X_t))  (equation (83))
  - where X_t is a vector of macro variables defining the simple fiscal rule.
- Linear-deviation form (equation (84)):
  - t_I,t = − [λ TF1 / T_I] t f1,t + [(1−λ) TF2 / T_I] t f2,t
  - where t_I,t = T_I,t − T_I; tf1,t = (TF1,t − TF1)/TF1; tf2,t similarly.
- Assumption on distribution of adjustment between household groups (equations (85)–(86)):
  - tf1,t − p_H,t−1 = − k/(1−k) (tf2,t − p_H,t−1)  (equation (85))
  - tf1,t − E_t−1 p_H,t = − k/(1−k) (tf2,t − E_t−1 p_H,t)  (equation (86))
  - If k = 0 all adjustment borne by unconstrained group 2; if k = 1 by constrained group 1.
  - The authors set k = 0.5 in results.

### A. Conventional fiscal rule
- Conventional form: real tax receipts as proportion of GDP feed back on government debt-to-GDP and output.
- Define b_G,t = ˆB_G,t/(P_H,t Y_t) − ˆB_G/(P_H Y).
- Linearized fiscal rule corresponding to (82) and (83):
  - tf2,t = p_H,t−1 + (1 + α_y) y_t−1 + α_bg b_G,t−1  (equation (87))
  - tf2,t = E_t−1 [p_H,t + (1 + α_y) y_t + α_bg b_G,t]  (equation (88))

### B. Structural Fiscal Surplus Rule (SFSR) — Chilean-style
- Chile’s rule targets structural surplus (equation (89)):
  - FS_t = FS + α_tax (T_I,t − ̂T_I,t + T_NI,t − ̂T_NI,t) + α_cop (T_COP,t − ̂T_COP,t)
  - ̂T_I,t, ̂T_NI,t, ̂T_COP,t are revenues “at potential”; i.e., at current tax rates but steady-state levels.
- Noting ̂T_NI,t = T_NI (but ̂T_I,t ≠ T_I) and ̂T_COP,t = T_COP, combine (76), (80), (89) to obtain fiscal instrument rule (equation (90)):
  - ∆(T_I,t/(P_H,t Y_t)) = α_tax/(1−α_tax) ∆(̂T_I,t/(P_H,t Y_t)) − ∆(T_NI,t/(P_H,t Y_t)) − (1−α_cop/(1−α_tax)) ∆(T_COP,t/(P_H,t Y_t)) + 1/(1−α_tax) ∆(R_n,t−1 ˆB_G,t−1/(P_H,t Y_t) + G_t/(P_H,t Y_t))
  - where ∆X_t ≡ X_t − X denotes deviation about BGP steady state.
- Special case when instrument is lump-sum TF1,t or TF2,t so that ̂T_I,t = T_I,t (equation (91)):
  - ∆(T_I,t/(P_H,t Y_t)) = −(1−α_tax) ∆(T_NI,t/(P_H,t Y_t)) − (1−α_cop) ∆(T_COP,t/(P_H,t Y_t)) + ∆(R_n,t−1 ˆB_G,t−1/(P_H,t Y_t) + G_t/(P_H,t Y_t))
- Interpretation for α_tax ∈ [0,1):
  - Rule adjusts tax instrument negatively in response to rise in non-instrument tax and copper revenues.
  - Adjusts positively to rise in government spending and interest payments on accumulated debt.
  - Appropriate α_tax ∈ [0,1) stabilizes government debt-to-GDP about BGP steady state with FS_t = FS.

### Linear form of SFSR and two-period planning implementation
- Linear form (equation (92)):
  - tr_I,t ≡ T_I/(P_H Y) (t_I,t − p_H,t − y_t) = −(1−α_tax) tr_NI,t − (1−α_cop) tc_t + (1/(β(1+g)) − 1) b_G,t−1 + B_G/(P_H Y) r_g,t−1 + gr_t
  - where tr_I,t, tc_t, b_G,t−1 and gr_t are similarly defined deviations.
- Two-period planning compatible forms substituting (84) and (85) (equations (93)–(94)):

  - Form responding to observable lagged variables (equation (93)):
    - 1/(P_H Y) (k λ TF1/(1−k) + (1−λ) TF2) (tf2,t − p_H,t−1) = T_I/(P_H Y) y_t−1 − (1−α_tax) tr_NI,t−1 − (1−α_cop) tc_t−1 + (1/(β(1+g)) − 1) b_G,t−1 + B_G/(P_H Y) r_g,t−1 + gr_t−1

  - Form responding to expected variables (equation (94)):
    - 1/(P_H Y) (k λ TF1/(1−k) + (1−λ) TF2) (tf2,t − E_t−1 p_H,t) = T_I/(P_H Y) E_t−1 y_t − (1−α_tax) E_t−1 tr_NI,t − (1−α_cop) E_t−1 tc_t + (1/(β(1+g)) − 1) b_G,t−1 + B_G/(P_H Y) r_g,t−1 + E_t−1 gr_t

### V. IMPOSING THE NOMINAL INTEREST RATE ZERO LOWER BOUND

### Approximate ZLB implementation and loss function
- Modify interest-rate rules to approximately impose an interest rate zero lower bound (ZLB) so that this event hardly ever occurs.
- Motivation:
  - Few emerging market countries have experienced deflationary episodes (Peru and Israel in 2007 are examples), but many inflation-targeting emerging-market countries choose low single-digit inflation targets, making ZLB robustness relevant.
- Loss function specification (quadratic approximation):
  - L_t = y'_t Q y_t
  - where y'_t = [z'_t, x'_t]' and Q is a symmetric matrix.
- Reference to methodology:
  - As in Woodford (2003), the quadratic loss approximation is used (text ends at this point).

*Source: _wp0922 - 2.    Calibrationof*

### chapter 6, the ZLB constraint is implemented by modifying the single period welfare

### _wp0922 - chapter 6, the ZLB constraint is implemented by modifying the single period welfare

### Implementation of the ZLB in the LQ framework
- The ZLB is implemented by modifying the single period welfare loss to L_t + w_r r_n,t^2 and choosing w_r and the unconditional distribution for r_n,t (characterized by the steady state variance σ_r^2 = var(r_n)) so the probability, p, of the interest rate hitting the lower bound is very low.
- The steady state nominal interest rate is R_n = 1/β(1+g_u_c) − 1 + π^*. The critical value z_0(p) is defined such that prob(Z ≤ z_0) = p for Z ∼ N(0,1).
- The steady-state positive inflation rate that ensures r_n,t ≥ 0 with probability 1 − p is given by
  - π^* = max[z_0(p) σ_r − (1/β(1+g_u_c) − 1) × 100, 0] (equation (95) in source).
- In practice the approach (following Woodford, 2003 and Levine et al. (2007)) replaces the non-linear i_t ≥ 0 constraint with an interest-rate variability constraint so the ZLB is hardly ever hit.

### Trade-offs in choosing w_r and welfare decomposition
- Intertemporal expected welfare loss at t = 0 is split into stochastic and deterministic components: Ω_0 = ˜Ω_0 + ̄Ω_0.
  - ̄Ω_0 incorporates the new steady state values; main extra term arises from the π^2 term in (D.30).
- Increasing w_r lowers σ_r, decreasing π^* and reducing the deterministic component ̄Ω_0, but increases the stochastic component ˜Ω_0.
- Optimal policy exploits this trade-off to impose the ZLB constraint r_t ≥ 0 with probability 1 − p in the vicinity of the steady state.

### Optimal monetary and fiscal policy with financial frictions — model variants and shocks
- Three model parameterizations (increasing frictions):
  - Model I: no financial accelerator (FA) and no liability dollarization (LD) (χ_θ = χ_θ^* = 0, Θ = Θ^* = 0, φ = 1).
  - Model II: FA only (χ_θ, χ_θ^* < 0, Θ, Θ^* > 0, φ = 1).
  - Model III: FA and LD with firms borrowing fraction 1−φ ∈ [0,1] in dollars (χ_θ, χ_θ^* < 0, Θ, Θ^* > 0, φ ∈ [0,1)).
- Nine exogenous independent shocks considered: a_t, g_t (both blocs), external risk premium facing firms ε_P,t, a copper price shock ε_cop, an oil shock ε_oil, a UIP risk premium shock ε_UIP, and shock to foreign interest-rate rule ε_R,t^*.
- Foreign bloc monetary rule (optimized): r_n,t^* = ρ_r^* r_n,t−1^* + θ_π^* π_F,t^* + θ_y^* y_t^* + ε_r,t^*. Calibration yields ρ^* = 1, θ_π^* = 10, θ_y^* = 0, α_bg^* = 0.87.

### Welfare outcomes without ZLB (Table 2 summary)
- Reported conditional welfare losses and long-run variance σ_r^2 under optimal monetary & fiscal policy (no ZLB):
  - Model I: M+F Ω_MF_0 = 3.26, M+F σ_r^2 = 2.35, M Ω_M_0 = 3.35, M σ_r^2 = 2.82, c_MF^e = 3.2? (Table lists "3.262.353.352.820.006" — interpreted as columns but preserve values presented).
  - Model II: values in table row: "3.485.013.974.080.034".
  - Model III: values in table row: "13.928.3815.347.890.099".
- The high variances indicate frequent violation of the ZLB under these unconstrained optimal policies.

### Imposing the ZLB — calibration example and key numeric results
- Chosen target probability p = 0.001 (quarterly model; once every 250 years).
- With p = 0.001, β = 0.99, g_u_c = −0.014, notation:
  - π^* = max[z_0(p) σ_r − (1/β(1+g_u_c) − 1) × 100, 0] = max[3.00 σ_r − 2.44, 0].
  - ̄Ω_0(0) = w_π π^*^2 and ̄Ω = (1/2) w_π π^*^2 = 3.829 π^*^2 (as stated).
- Optimal policy under ZLB violated with probability p = 0.001 (monetary policy only) for Model I:
  - occurs when w_r = 3.75 and steady state quarterly inflation π^* = 0.29.
- Table 3 excerpt (Optimal Policy with a ZLB Constraint: Monetary Policy Only for Model I) — preserve values as in table:
  - w_r = 0.00 → σ_r^2 = 1.00? (table rows given; reproduce representative rows exactly)
  - Representative rows (from table): 
    - w_r 0.00  σ_r^2 1.00  ˜Ω_0 2.35  π^* 2.16  ̄Ω_0 8.93  Ω_0 12.18  (first table row entries preserved from Table 4 header/rows).
  - Explicit block of Table 3 (rows preserved):  
    - 0.00 1 2.823 3.35 2.59 12.84 16.19  
    - 1.00 1.84 3.57 1.62 5.04 8.61  
    - 2.00 1.32 3.94 1.00 1.90 5.85  
    - 3.00 1.00 4.33 0.55 0.59 4.92  
    - 3.25 0.94 4.23 0.46 0.41 4.83  
    - 3.50 0.88 4.52 0.37 0.27 4.79  
    - 3.75 0.83 4.61 0.29 0.17 4.77  
    - 4.00 0.79 4.70 0.22 0.09 4.79
  - Note: these rows are taken verbatim from Table 3 as printed.
- With monetary + fiscal policy together (Model I), Table 4 gives:
  - Example rows reproduced exactly:
    - w_r 0.00 σ_r^2 1 ˜Ω_0 2.35 π^* 2.16 ̄Ω_0 8.93 Ω_0 12.18 (first row)
    - w_r 0.25 0.86 3.26 0.33 0.21 3.47
    - w_r 0.50 0.55 3.27 0 0 3.27
    - w_r 0.75 0.40 3.29 0 0 3.29
    - w_r 1.00 0.31 3.30 0 0 3.30
  - Key point: with fiscal stabilization, ZLB can be imposed at w_r = 0.5 with π^* remaining at zero baseline.

### Consumption-equivalent contribution of fiscal policy with ZLB (Table 5)
- Table 5 (Welfare Outcomes under Optimal Policy: ZLB Constraint) preserves entries:
  - Model I: M+F Ω_0 = 3.27, M+F σ_r^2 = 0.55, π^* = 0.00, Ω_0 (M policy only) = 4.77, σ_r^2 = 0.84, π^* = 0.29, c_MF^e = 0.104
  - Model II: M+F Ω_0 = 3.74, σ_r^2 = 0.69, π^* = 0.05, Ω_0 (M only) = 6.98, σ_r^2 = 1.00, π^* = 0.57, c_MF^e = 0.225
  - Model III: M+F Ω_0 = 14.90, σ_r^2 = 0.72, π^* = 0.10, Ω_0 (M only) = 24.19, σ_r^2 = 1.20, π^* = 0.85, c_MF^e = 0.644
- Interpretation: with ZLB imposed, fiscal policy’s consumption-equivalent contribution rises from c_e = 0.10% to c_e = 0.64% moving from Model I to Model III (values preserved as printed).

### Welfare decomposition across shocks (Table 6)
- Percent contributions to total welfare loss when all shocks present (Table 6 verbatim):
  - Shock contributions (%):  
    - a_t: Model I 54.2, Model II 43.4, Model III 12.4  
    - a_t^*: Model I 2.4, Model II 2.1, Model III 38.1  
    - g_t: Model I 0.1, Model II 0.1, Model III 0.0  
    - g_t^*: Model I 0.1, Model II 0.1, Model III 13.5  
    - ε_cop: Model I 20.7, Model II 16.8, Model III 5.7  
    - ε_oil: Model I 20.9, Model II 15.5, Model III 19.9  
    - ε_UIP: Model I 1.6, Model II 1.3, Model III 0.6  
    - ε_r^*: Model I 0.0, Model II 0.0, Model III 0.0  
    - ε_P: Model I 0, Model II 20.1, Model III 9.5
- Key qualitative finding preserved: introducing FA reduces home productivity and commodity shocks’ share; adding LD (25% of firms’ finance in dollars) raises foreign demand and supply contributions dramatically (from 2.3% to almost 46%).

### Impulse responses (Figure 3 summary)
- Responses to an unanticipated −1% technology shock:
  - Both monetary and fiscal policy tighten (nominal interest rate r_n,t and flat-rate taxes t_I,t increase).
  - As frictions increase from Model I → II → III:
    - Monetary policy becomes more constrained by ZLB.
    - Fiscal policy plays a larger stabilization role.
    - Nominal and expected interest rates fall with frictions, partially offsetting investment declines.
  - Model III with LD exhibits a prolonged period (~20 quarters) where nominal interest rate drops below baseline (r_n,t becomes negative) causing nominal and real exchange rate depreciation (rer_t > 0), partially offsetting FA-induced investment declines and dollar liabilities effects.

### Performance of optimized simpler rules (Tables 7–10)
- Simple monetary rules examined: FLEX(D) (nominal interest rate responds to domestic inflation π_H,t and output y_t) and FIX (fixed exchange rate). Fiscal simple rule: conventional form with k = 0.5 and variant allowing two-period-ahead tax planning.
- Measure of cost of simplicity:
  - c_e^SIM = (Ω_SIM_0 − Ω_OPT_0) / [(1−%)(1−h_C)c_y] × 10^−2 (%) (equation (105) in source).
- Table 7 (FLEX(D) + conventional fiscal rule) — rows preserved exactly:
  - Model I: M+F Ω_0 = 5.0, σ_r^2 = 0.9, π^* = 0.3, parameters [ρ, θ_π, θ_y, α_bg, α_y] = [1,10,0,0.7,3]; M Ω_0 = 5.9, σ_r^2 = 1, π^* = 0.5, [1,10,0.05]; c_MF^e = 0.06, c_SIM^e = 0.12
  - Model II: M+F Ω_0 = 13.2, σ_r^2 = 1.7, π^* = 1.5, [1,10,0.05,4,2]; M Ω_0 = 13.6, σ_r^2 = 1.8, π^* = 1.5, [1,10,0.01]; c_MF^e = 0.03, c_SIM^e = 0.66
  - Model III: M+F Ω_0 = 44.7, σ_r^2 = 3.6, π^* = 3.3, [1,4,0.3,5,3]; M Ω_0 = 58.2, σ_r^2 = 4.7, π^* = 4.0, [1,7.8,0]; c_MF^e = 0.93, c_SIM^e = 2.07
- Table 8 (FIX + conventional fiscal rule) — rows preserved:
  - Model I: M+F Ω_0 = 74, σ_r^2 = 0.8, π^* = 0.3, [10, −0.01]; M Ω_0 = 84, σ_r^2 = 0.8, π^* = 0.3; c_MF^e = 0.71; c_SIM^e = 4.9
  - Model II: M+F Ω_0 = 136, σ_r^2 = 1.1, π^* = 0.7, [6.3, −0.64]; M Ω_0 = 152, σ_r^2 = 1.0, π^* = 0.6; c_MF^e = 1.16; c_SIM^e = 9.2
  - Model III: M+F Ω_0 = 175, σ_r^2 = 2.4, π^* = 2.2, [7.8, 0.64]; M Ω_0 = 191, σ_r^2 = 2.4, π^* = 2.2; c_MF^e = 1.09; c_SIM^e = 11.1
- Table 9 (FLEX(C) — CPI targeting + conventional fiscal rule) — rows preserved:
  - Model I: M+F Ω_0 = 23, σ_r^2 = 1.0, π^* = 0.6, [1,4,0.3,5,1.5]; M Ω_0 = 23.7, σ_r^2 = 1, π^* = 0.5, [1,2.5,0.01]; c_MF^e = 0.04; c_SIM^e = 0.41
  - Model II: M+F Ω_0 = 35, σ_r^2 = 1.5, π^* = 1.2, [1,2,0.4,5,0.01]; M Ω_0 = 36.7, σ_r^2 = 1.8, π^* = 1.5, [1,2.6,0.4]; c_MF^e = 0.10; c_SIM^e = 2.24
  - Model III: M+F Ω_0 = 62, σ_r^2 = 2.6, π^* = 2.4, [1,1.3,0.1,5,0.01]; M Ω_0 = 64.0, σ_r^2 = 4.7, π^* = 2.4, [1,1,4,0.1]; c_MF^e = 0.12; c_SIM^e = 3.37
- Table 10 (FLEX(D) + modified SFSR) — preserved rows:
  - Model I: Ω_0 = 5.38, σ_r^2 = 0.96, π^* = 0.49, [1.0, 10.0, 0.00, 1.10, 0.82], c_SIM^e = 0.15
  - Model II: Ω_0 = 13.19, σ_r^2 = 1.76, π^* = 1.53, [1.0, 10.0, 0.06, 0.96, 0.90], c_SIM^e = 0.66
  - Model III: Ω_0 = 46.74, σ_r^2 = 3.58, π^* = 3.22, [1.0, 4.13, 0.26, 0.97, 0.69], c_SIM^e = 2.21
- Main findings preserved:
  - FLEX(D) + counter-cyclical fiscal rule stabilizes far better than FIX.
  - The ability of optimized simple rules to match the fully optimal benchmark deteriorates sharply with financial frictions: c_SIM^e rises from 0.12% (Model I) to 0.66% (Model II) to 2.07% (Model III) under FLEX(D).
  - FIX induces very large welfare losses; fiscal policy must bear more stabilization burden or steady-state inflation must rise, leading to large welfare costs (c_SIM^e up to 11.1%).

### Conclusions (preserved wording and key policy implications)
- Broad support for the "three-pillars" macro framework (explicit inflation target, floating exchange rate, counter-cyclical fiscal rule) as pursued by Chile.
- Domestic inflation targeting is superior to attempting to stabilize the exchange rate; CPI targeting that implicitly responds to the exchange rate is more costly.
- Financial frictions increase the costs of stabilizing the exchange rate: emerging markets with financial frictions should "fear to fix" rather than "fear to float".
- Fiscal stabilization can play a significant role when financial frictions and the ZLB are relevant; the contribution of fiscal policy rises with frictions.
- The ability of simple optimized rules to replicate optimal policy deteriorates sharply with the introduction of the FA and LD. Future research should explore rules that respond to indicators of financial stress (e.g., risk premium facing firms) and endogenize currency choice in borrowing.

*Source: _wp0922 - chapter 6, the ZLB constraint is implemented by modifying the single period welfare (IMF working paper PDF content provided).*

### REFERENCES

### _wp0922 - REFERENCES

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*Source: _wp0922 - REFERENCES*

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