## _wp07217

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---

### Consensus on monetary policy and cross-country differences
- Strong consensus: achieve a low and stable inflation rate as the main goal of monetary policy.
- Inflation has converged to single digit rates during the last 10 years in both emerging and industrialized economies.
- Implementation environments differ: most emerging countries exhibit underdeveloped financial markets that prevent efficient consumption and saving decisions.
- Financial depth indicator (ratio of M3 and private credit to GDP) is lower in developing countries relative to developed countries.

### Financial imperfections and monetary policy role
- Financial imperfection modeled as asset market segmentation: only a fraction of the population has access to financial markets; excluded households can only save via real money balances.
- Monetary policy can reallocate consumption across states:
  - Increase consumption in bad states and reduce it in good states to improve intertemporal allocation for households with limited financial access.
  - Such insurance entails inducing some inflation volatility, which is distortionary under price stickiness.
- Central questions:
  - To what extent is price stability an optimal policy criterion for emerging economies?
  - Is there a role for monetary policy to correct financial imperfections?

### Model overview
- Small open economy with two sectors: tradable (perfect competition, flexible prices) and non-tradable (monopolistic competition, price stickiness via quadratic adjustment cost à la Rotemberg (1982)).
- Financial friction: asset market segmentation with fraction λ participating in financial markets and fraction (1 − λ) excluded.
- Excluded households save only through accumulation of real money balances; money reduces transaction costs.
- Fiscal policy: balanced budget; government expenditure gNt financed with lump-sum taxes Tt levied on both household types.
- Money injections Xt engineered in financial markets and absorbed only by traders.
- Exogenous shocks: productivity shocks zTt, zNt and government expenditure shocks gNt.

### Ramsey approach and commitment
- Ramsey planner maximizes weighted expected household utility subject to resource and equilibrium constraints (primal approach).
- Monetary authority implements policy under full commitment (credible policy, prevents inflation-bias).

### Main conclusions
- Optimal allocation is quantitatively similar to the flexible-price economy; Ramsey policy minimizes nominal-distortion impacts.
- Under optimal policy the volatility of non-tradable inflation is close to zero:
  - Ramsey planner reproduces the flexible-price equilibrium by stabilizing the price level, allowing stable markups and flexible-price allocation.
  - Under segmented asset markets, monetary policy’s goal of improving risk-sharing conflicts with price stability; calibration shows the trade-off is resolved in favor of undoing price stickiness (prioritizing price stability).
- Achieving low non-tradable inflation volatility is optimal for any degree of asset market segmentation:
  - Sensitivity analysis shows near-zero inflation volatility remains optimal across different segmentation assumptions.
  - Correcting asset market segmentation with monetary instruments, even under extreme segmentation, is highly distortionary.

### Relation to literature
- Closed-economy Ramsey results: Friedman’s rule in flexible-price perfect-competition (Lucas and Stokey (1983); Chari et al. (1991); Chari and Kehoe (1999)); near-zero inflation volatility under monopolistic competition with sticky prices (Khan et al.(2003); Schmitt-Grohé and Uribe (2004); Siu (2004)).
- Open-economy: Galí and Monacelli (2005) — optimal policy fully stabilizes domestic price level; Lahiri et al. (2006) — with flexible prices and segmentation optimal policy may induce high inflation volatility for insurance.
- Contribution: characterize interaction between sticky prices and asset market segmentation in optimal monetary policy design for a small open economy, solved via Ramsey under full commitment.

### Model structure and key household conditions
- Two household types: traders (size λ) with access to contingent domestic bond d_{t+1}(tr) and international bond b*_t(tr), and non-traders (size (1−λ)) who cannot hold bonds and save via money.
- Transaction costs on consumption purchases s(v_t(i)) depend on money velocity v_t(i) = (p_Tt c_Tt(i) + p_Nt c_Nt(i)) / M_t(i).
- Traders’ budget constraint (exact expression preserved):
  - (1 + s(v_t(tr)))(p_Tt c_Tt(tr) + p_Nt c_Nt(tr)) + E_t[q_t,t+1 d_t+1(tr)] + e_t b*_t(tr) + M_t(tr) =
    W_t l_t(tr) + d_t(tr) + e_t R*_t−1 b*_t−1(tr) + M_t−1(tr) + Π_t/λ + X_t/λ − T_t
- Traders’ first-order conditions (selected exact relations):
  - u_cT,t(tr) / u_cN,t(tr) = p_Tt / p_Nt
  - −u_l,t(tr) / u_cT,t(tr) = W_t / p_Tt h(v_t(tr))
  - u_cN,t(tr) / [p_Nt h(v_t(tr)) (1 − s'(v_t(tr))(v_t(tr))^2)] = β E_t[ u_cN,t+1(tr) / (p_Nt+1 h(v_t+1(tr))) ]
  - u_cT,t(tr) e_t / [p_Tt h(v_t(tr))] = β R*_t E_t[ u_cT,t+1(tr) e_t+1 / (p_Tt+1 h(v_t+1(tr))) ]
  - q_t,t+1 = β [u_cN,t+1(tr) / u_cN,t(tr)] [p_Nt h(v_t(tr)) / p_Nt+1 h(v_t+1(tr))]
  - h(v_t(tr)) = 1 + s(v_t(tr)) + s'(v_t(tr)) v_t(tr)
- Money demand and interest parity implications (exact forms):
  - Combining leads to R_t (1 − s'(v_t(tr))(v_t(tr))^2) = 1 and M_t(tr)/p_t = c_t(tr) G^−1((R_t − 1)/R_t) where G(v) = s'(v) v^2.
  - Interest parity: R_t / R*_t = E_t[e_t+1 / e_t] + R_t cov(q_t,t+1, e_t+1 / e_t).
- Non-traders’ budget constraint (exact expression preserved):
  - (1 + s(v_t(nt)))(p_Tt c_Tt(nt) + p_Nt c_Nt(nt)) + M_t(nt) = W_t l_t(nt) + M_t−1(nt) − T_t
- Non-traders’ first-order conditions mirror traders’ analogous conditions but their money demand does not depend on the nominal interest rate.

### Firms in the tradable sector (2.2)
- Tradable production: yTt = zTt fT(lTt, Nt).
- First-order conditions:
  - Wt = zTt pTt fT lT,t
  - pNt = zTt pTt fT N,t
- Tradable-sector input demands link marginal products to household marginal rates of substitution and enter market-clearing (constraints (22), (23), (24), (37)).

### Firms in the non-tradable sector (2.3)
- Final non-tradable output: CES aggregator over j ∈ [0,1]:
  - yNt = (∫0 1 yNt(j)^(ε−1)/ε dj)^(ε/(ε−1))
  - yNt(j) = yNt · (pNt(j)/pNt)^−ε
  - pNt = [∫0 1 pNt(j)^(1−ε) dj]^(1/(1−ε))
- Intermediate producers face Rotemberg price adjustment cost (in units of non-tradable final good):
  - κ/2 · [pNt(j)/pN,t−1(j) − 1]^2 with κ measuring degree of price stickiness.
- Production: yNt(j) = zNt lNt(j).
- Symmetric equilibrium yields an expectation-augmented Phillips curve:
  - πNt (πNt − 1) = Et[q t,t+1 (πN,t+1)^2 (πN,t+1 − 1)] − (ε/κ) zNt lNt [ (ε−1)/ε − mc t ]
  - πNt = pNt / pN,t−1, mc t = Wt / (pNt zNt)
- Symmetric final production: yNt = zNt lNt.

### Government law of motion and international transactions (2.4–2.5)
- Money supply law: Ms t = Ms t−1 + X t (Xt are per capita money injections).
- Consolidated government budget constraint:
  - Ms t = Ms t−1 + pNt gNt + X t − Tt
- Law of one price for tradables: pTt = e_t p*_t with p*_t = 1.
- Upward-sloping supply of funds for stationarity:
  - R*_t = R* · (B*_t / B*)^ν with ν = 10^−5 in calibration.

### Market clearing and equilibrium definition (2.6–2.7)
- Labor market clearing:
  - λ l t (tr) + (1 − λ) l t (nt) = lTt + lNt
- Non-tradable goods market clearing (exact expression preserved):
  - λ cNt (tr) + (1 − λ) cNt (nt) + λ s(vt (tr)) [ cNt (tr) + (pTt / pNt) cTt (tr) ] + (1 − λ) s(vt (nt)) [ cNt (nt) + (pTt / pNt) cTt (nt) ] + Nt + gNt + κ/2 (πNt − 1)^2 = yNt
- Tradable goods market clearing (exact expression preserved):
  - λ cTt (tr) + (1 − λ) cTt (nt) + B*t e t pTt = yTt + R*t−1 B*t−1 e t pTt
- Aggregation of foreign assets: B*t = λ b* t (tr)
- No domestic contingent debt: d t (tr) = 0
- Money market: Ms t = λ Mt (tr) + (1 − λ) Mt (nt)
- Equilibrium: set of prices {e_t, p_Tt, p_Nt, q_t,t+1, W_t, R*_t, mc_t} and allocations satisfying equations (2)–(13) and (17)–(27) given policies {X_t, T_t}, exogenous processes {zTt, zNt, gNt}, and initial conditions.

### Ramsey problem setup (section 3)
- Planner maximizes E0[∑ β^t (λ u(cTt (tr), cNt (tr), l t (tr)) + (1 − λ) u(cTt (nt), cNt (nt), l t (nt)))] subject to competitive equilibrium constraints.
- Primal approach: rewrite constraints in allocation terms; classify constraints as intratemporal, predetermined, or expectational.
- Expectational constraints require Marcet and Marimon (1998) solution: treat Lagrange multipliers on expectational constraints as additional state variables to obtain recursive saddle-point functional equation.
- Implementation assumes a "timeless perspective" optimal policy (Woodford, 2003).

### Calibration, solution method, and exogenous shocks (section 4)
- Numerical solution: second-order approximation algorithm of Schmitt-Grohé and Uribe (2004).
- Time unit: one quarter.
- Utility: u(c, l) = ln c + ψ log(1 − l) with ψ chosen to yield steady-state labor supply 0.22.
- Composite consumption CES: c = [ θ (cT)^μ + (1 − θ) (cN)^μ ]^(1/μ) with intratemporal elasticity 1/(1 − μ) = 0.50 and θ chosen to match Chilean CPI tradables share.
- Tradable production: yTt = zTt [ αT (lTt)^(1−φ) + (1 − αT) (Nt)^(1−φ) ]^(1/(1−φ)) with αT = 0.40 and φ = 1.5.
- Markup: ε = 6 (steady-state markup 20 percent).
- Transactions technology (Schmitt-Grohé and Uribe 2004 specification):
  - s(v) = ω v + ξ v^(−2) p ξ/ω leading to reduced-form v_t^2 = ξ/ω + (1/ω) (R_{t−1} / R_t).
  - Empirical estimates (OLS on Chilean data): ω = 0.06, ξ = 0.17; estimated relation v_t^2 = 2.68 + 15.64 (R_{t−1} / R_t) with t-statistics 20.82 and 15.72; R^2 = 0.82.
- Price-adjustment cost κ calibration:
  - GMM estimate (ε − 1) (κ / cmc) = 0.084; implied κ = 13.16 (≈ 4 quarters Calvo analog).
- Asset market segmentation baseline: λ = 0.5 (50 percent excluded).
- Foreign supply-of-funds elasticity ν = 10^−5.
- Steady-state B* chosen to match net exports to GDP ratio 2.3 percent.
- Discount factor β = 0.99 (β equals inverse of steady-state gross foreign interest rate R*).
- Parameter summary (calibrated values):
  - Discount Factor β = 0.99
  - Tradable weight in consumption θ = 0.07
  - Intratemporal Elasticity of Substitution 1/(1 − μ) = 0.50
  - Parameter Transaction Cost Function ω = 0.06
  - Parameter Transaction Cost Function ξ = 0.17
  - Markup ε/(ε − 1) = 1.20
  - Price adjustment cost κ = 13.16
  - Labor share in the tradable sector αT = 0.40
  - Elasticity of substitution for tradable firms φ = 1.5
  - Foreign interest rate elasticity ν = 10^−5
  - Asset Market Segmentation λ = 0.5
- Exogenous AR(1) shocks with estimated standard errors:
  - zTt = 0.65 (0.11) zTt−1 + εTt, εTt ∼ N(0, σT^2), σT = 0.027
  - zNt = 0.84 (0.08) zNt−1 + εNt, εNt ∼ N(0, σN^2), σN = 0.021
  - gNt = 0.76 (0.10) gN,t−1 + εGt, εGt ∼ N(0, σG^2), σG = 0.026

### Optimal monetary policy with sticky prices (section 4.2) — key quantitative findings
- Case λ = 1 (no segmentation): Ramsey allocations nearly coincide with the real model; Ramsey achieves output gap close to zero.
- Ramsey steady state generates an annual deflation rate of 0.05 percent.
- Impulse responses to a 1 percent tradable-sector productivity shock under Ramsey:
  - Interest rate increases by slightly more than 100 basis points.
  - Real exchange rate appreciates by 1 percent.
  - Output, labor supply, markups, and real exchange rate dynamics closely replicate the real model.
- Mechanism:
  - Tradable productivity shock raises wages in non-tradable sector → nominal and real marginal cost shifts → sticky non-tradable prices reduce markups.
  - Optimal monetary authority contracts money supply to reduce aggregate demand, offset markup reductions, and limit non-tradable allocation distortions.
  - Non-tradable inflation largely stabilized; nominal exchange rate absorbs most of real shock.
- Non-tradable inflation targeting (π^N_t = π with π = Ramsey steady state inflation) closely mimics Ramsey allocations; Ramsey smooths nominal interest rate more due to monetary transaction costs.
- Responses to a non-tradable-sector productivity shock:
  - Ramsey again close to flexible-price benchmark; monetary policy is procyclical (expands aggregate demand) to stabilize marginal costs and markups.
  - Real exchange rate depreciates in response to non-tradable productivity increase.
- Sensitivity to price stickiness κ:
  - For wide range of κ, Ramsey policy implies non-tradable inflation volatility close to zero.
  - As κ decreases (less price rigidity), optimal policy shifts toward stabilizing the nominal interest rate, converging toward Friedman rule as flexible prices approached.
- Asset market segmentation (section 4.3) effects:
  - Introducing segmentation does not substantially change Ramsey policy properties; allocation remains close to benchmark real model.
  - Monetary policy is inefficient to correct segmented asset markets when price stickiness exists; correcting segmentation via monetary policy requires departures from flexible-price allocations and is welfare-reducing.
- Heterogeneous-agent dynamics:
  - Traders respond mainly via substitution effects; non-traders face offsetting wealth effects and maintain more stable non-tradable consumption.
  - Labor supply: traders increase labor effort following tradable productivity shock; non-traders' labor supply remains stable.
  - Ramsey policy only marginally reduces IMRS deviations for non-traders relative to non-tradable inflation targeting.
- Sensitivity figures summarized (exact qualitative relationships preserved):
  - Figure 13: Non-tradable inflation standard deviation robustly close to zero across λ and κ for baseline κ.
  - Figure 14: Nominal interest rate volatility increases with asset market segmentation for any κ; reducing price stickiness leads optimal policy to stabilize nominal interest rate.
  - Figure 15: Increasing planner weight on non-traders raises inflation and nominal interest volatility slowly; volatility remains well below flexible-price outcomes even for substantial weight changes.

### Policy implications and recommendations (concluding remarks)
- Principal recommendation: prioritize stabilizing non-tradable inflation to stabilize markups, reduce firms’ incentives to change prices, and approximate flexible-price allocations.
- Two policy implications:
  - Target the appropriate price index: optimal policy should target the subset of prices that display stickiness (empirically non-tradable sector) rather than the overall CPI.
  - Avoid using monetary policy to correct asset market segmentation: stabilizing non-tradable inflation is welfare-improving irrespective of financial structure; using monetary policy for risk-sharing is distortionary because it induces non-tradable price and markup fluctuations and resource misallocation.
- Caveats and extensions:
  - Monetary-policy inefficiency in correcting segmentation does not imply financial imperfections should be ignored; fiscal policy may be more appropriate to address segmentation.
  - Extensions could include additional real and nominal frictions or borrowing constraints and foreign-currency debt to evaluate exchange rate redistribution effects.

*italic: _wp07217 - 1. Introduction (PDF chapter/section).*

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

### _wp07217 - 1. Introduction ................................................................................................

### Consensus on monetary policy and cross-country differences
- Strong consensus over the last two decades about the importance of achieving a low and stable inflation rate as the main goal of monetary policy.
- Inflation has converged to single digit rates during the last 10 years in both emerging and industrialized economies (Figure 1 referenced in source).
- The implementation environment differs significantly across economies: most emerging countries exhibit underdeveloped financial markets that prevent efficient consumption and saving decisions.
- Financial depth indicator (ratio of M3 and private credit to GDP) is lower in developing countries relative to developed countries (Figure 2 referenced in source).

### Financial imperfections and potential role for monetary policy
- If financial imperfections matter, monetary policy may correct distortions by reallocating consumption across states:
  - Monetary policy that increases consumption in bad states and reduces it in good states can improve intertemporal allocation for households with limited access to financial markets.
  - Such insurance via monetary policy entails inducing some inflation volatility, which is distortionary under price stickiness.
- Central questions examined:
  - To what extent is price stability an optimal policy criterion for emerging economies?
  - Is there a role for monetary policy to correct financial imperfections?

### Model overview (as described in the introduction)
- Two-sector small open economy: tradable and non-tradable sectors.
  - Tradable sector: perfectly competitive, flexible prices.
  - Non-tradable sector: monopolistic competition, price stickiness modeled as a quadratic adjustment cost à la Rotemberg (1982).
- Financial imperfection modeled as asset market segmentation:
  - Only a fraction of the population has access to financial markets.
  - Households excluded from financial markets can only save through accumulation of real money balances.
  - Fraction of households participating in financial markets is fixed over time.
- Money introduced to reduce household transaction costs.
- Fiscal policy: balanced budget; government expenditure financed with lump-sum taxes levied on both household types.
- Money injections engineered in financial markets and absorbed only by traders.
- Exogenous shocks: productivity shocks in tradable and non-tradable sectors, and government expenditure shocks.

### Ramsey approach and monetary commitment
- Ramsey planner chooses allocations that maximize household welfare subject to resource constraints and equilibrium constraints (primal approach per Ramsey (1927), Lucas and Stokey (1983)).
- Monetary authority implements policy under full commitment (monetary policy is credible and prevents inflation-bias).

### Main conclusions (threefold)
- First: Allocation under the optimal policy is quantitatively similar to the one in an economy with flexible prices — the Ramsey policy successfully minimizes the impact of nominal distortions.
- Second: Under the optimal policy the volatility of non-tradable inflation is close to zero.
  - With sticky prices, the Ramsey planner aims to reproduce the flexible price equilibrium; this is achieved by a monetary policy that stabilizes the price level, allowing stable markups and ensuring a flexible-price allocation.
  - Under segmented asset markets, monetary policy’s goal is to improve risk-sharing between agents (the intertemporal marginal rate of substitution of households excluded from financial markets should be as close as possible to that of agents with access). Improving risk-sharing requires stabilizing consumption paths of excluded agents, which entails allowing some price level variation — conflicting with price stability under sticky prices.
  - Calibration for plausible values of asset market segmentation and price stickiness shows the trade-off is resolved in favor of undoing the effects of price stickiness (i.e., prioritizing price stability).
- Third: Achieving low non-tradable inflation volatility is optimal for any degree of asset market segmentation.
  - Sensitivity analysis shows bringing inflation volatility close to zero remains optimal across different assumptions of asset market segmentation.
  - Correcting asset market segmentation with monetary instruments, even under extreme segmentation, is highly distortionary.

### Relation to literature
- Closed-economy Ramsey results:
  - Lucas and Stokey (1983), Chari et al. (1991), Chari and Kehoe (1999): in flexible-price, perfect-competition closed economies the Ramsey optimal monetary policy sets the nominal interest rate to zero (Friedman’s rule).
  - Khan et al.(2003), Schmitt-Grohé and Uribe (2004), Siu (2004): in monopolistic competition with sticky prices, the Ramsey prescription is to reduce inflation volatility close to zero.
- Open-economy studies:
  - Galí and Monacelli (2005): small open economy with sticky prices — optimal policy fully stabilizes the domestic price level.
  - Lahiri et al. (2006): small open economy with flexible prices and segmented asset markets — optimal monetary policy induces high inflation volatility to provide insurance to agents excluded from asset markets.
- Contribution: Characterize interaction between sticky prices and asset market segmentation in optimal monetary policy design for a small open economy, solved via the Ramsey problem under full commitment.

### Model structure (summary of Sections 2 and 2.1 as presented)
- Infinite-horizon production economy with sticky prices and segmented asset markets.
- Two household types: traders (size λ) with access to financial assets; non-traders (size (1−λ)) without access to financial markets and can only save via money.
- Transaction costs on consumption purchases modeled via a proportional transaction cost s(v_t(i)) that depends on money velocity:
  - v_t(i) = (p_Tt c_Tt(i) + p_Nt c_Nt(i)) / M_t(i)
- Traders:
  - Traders can hold a domestic one-period contingent bond d_t+1(tr) and an international one-period non-contingent bond b*_t(tr) denominated in foreign currency with exchange rate e_t.
  - Budget constraint (equation (1) in source):
    (1 + s(v_t(tr)))(p_Tt c_Tt(tr) + p_Nt c_Nt(tr)) + E_t[q_t,t+1 d_t+1(tr)] + e_t b*_t(tr) + M_t(tr) =
    W_t l_t(tr) + d_t(tr) + e_t R*_t−1 b*_t−1(tr) + M_t−1(tr) + Π_t/λ + X_t/λ − T_t
    - X_t are per-capita money injections absorbed only by traders; Π_t/λ are dividends per trader.
  - Traders’ first-order conditions include:
    - u_cT,t(tr) / u_cN,t(tr) = p_Tt / p_Nt (equation (2))
    - −u_l,t(tr) / u_cT,t(tr) = W_t / p_Tt h(v_t(tr)) (equation (3))
    - u_cN,t(tr) / [p_Nt h(v_t(tr)) (1 − s'(v_t(tr))(v_t(tr))^2)] = β E_t[ u_cN,t+1(tr) / (p_Nt+1 h(v_t+1(tr))) ] (equation (4))
    - u_cT,t(tr) e_t / [p_Tt h(v_t(tr))] = β R*_t E_t[ u_cT,t+1(tr) e_t+1 / (p_Tt+1 h(v_t+1(tr))) ] (equation (5))
    - q_t,t+1 = β [u_cN,t+1(tr) / u_cN,t(tr)] [p_Nt h(v_t(tr)) / p_Nt+1 h(v_t+1(tr))] (equation (6))
    - where h(v_t(tr)) = 1 + s(v_t(tr)) + s'(v_t(tr)) v_t(tr)
  - Interpretation:
    - (2) relative demand of tradable vs non-tradable goods.
    - (3) labor supply equating MRS with real wage in terms of tradable goods; transaction costs introduce wedge h(v_t(tr)).
    - (4)-(6) define money demand function, interest parity, and nominal interest rate indirectly.
    - Gross nominal interest R_t = (E_t[q_t,t+1])^−1. Combining with (4) yields R_t (1 − s'(v_t(tr))(v_t(tr))^2) = 1 and money demand M_t(tr)/p_t = c_t(tr) G^−1((R_t − 1)/R_t) where G(v) = s'(v) v^2 (as derived in the source).
    - Interest parity: R_t / R*_t = E_t[e_t+1 / e_t] + R_t cov(q_t,t+1, e_t+1 / e_t).
- Non-traders:
  - Budget constraint (equation (7) in source):
    (1 + s(v_t(nt)))(p_Tt c_Tt(nt) + p_Nt c_Nt(nt)) + M_t(nt) = W_t l_t(nt) + M_t−1(nt) − T_t
  - First-order conditions:
    - u_cT,t(nt) / u_cN,t(nt) = p_Tt / p_Nt (equation (8))
    - −u_l,t(nt) / u_cT,t(nt) = W_t / p_Tt h(v_t(nt)) (equation (9))
    - u_cN,t(nt) / [p_Nt h(v_t(nt)) (1 − s'(v_t(nt))(v_t(nt))^2)] = β E_t[ u_cN,t+1(nt) / (p_Nt+1 h(v_t+1(nt))) ] (equation (10))
    - h(v_t(nt)) = 1 + s(v_t(nt)) + s'(v_t(nt)) v_t(nt)
  - These conditions mirror traders’ (2), (3), (4), but implicit money demand for non-traders does not depend on the nominal interest rate because non-traders do not participate in asset markets.

*Italic source attribution: _wp07217 - 1. Introduction (PDF chapter/section).*

### 2.2  Firms in the Tradable Sector

### 2.2  Firms in the Tradable Sector

### Firm technologies and profit-maximization
- Production of tradable goods:
  - yTt = zTt fT(lTt, Nt)
  - Inputs: lTt (labor), Nt (non-tradable inputs); zTt is an exogenous productivity shock.
- First-order conditions for input demand (profit maximization) imply:
  - Wt = zTt pTt fT lT,t
  - pNt = zTt pTt fT N,t

### Role in equilibrium and linkages to households
- Tradable-sector input demands link marginal products to household marginal rates of substitution through subsequent equilibrium conditions (see constraints (28)–(31) in the model).
- Tradable production and input choices enter the market-clearing conditions that determine allocations across tradable and non-tradable goods (see equations (22), (23), (24), and (37)).

---

### 2.3  Firms in the Non-tradable Sector

### Retailers (final non-tradable goods)
- Final non-tradable output is a CES aggregator over a continuum of intermediate varieties j ∈ [0,1]:
  - yNt = (∫0 1 yNt(j)^(ε−1)/ε dj)^(ε/(ε−1))
- Demand for intermediate input j:
  - yNt(j) = yNt · (pNt(j)/pNt)^−ε
- Aggregate non-tradable price index:
  - pNt = [∫0 1 pNt(j)^(1−ε) dj]^(1/(1−ε))

### Intermediate good producers (monopolistic competitors with price adjustment costs)
- Price adjustment cost (Rotemberg 1982 form), expressed in units of the non-tradable final good:
  - κ/2 · [pNt(j)/pN,t−1(j) − 1]^2
  - Parameter κ measures degree of price stickiness in the non-tradable sector.
- Production technology:
  - yNt(j) = zNt lNt(j), where lNt(j) is labor and zNt is an exogenous productivity shock.
- Intermediate producers maximize expected present value of profits:
  - max {pNt(j)} E0[∑ q0,t (pNt(j) yNt(j) − Wt lNt(j) − pNt(j) κ/2 [pNt(j)/pN,t−1(j) − 1]^2 )]
  - q0,t is the price of a nominal contingent security.
- Symmetric equilibrium (pNt(j) = pNt, lNt(j) = lNt for all j) leads to an expectation-augmented Phillips curve (from first-order condition):
  - πNt (πNt − 1) = Et[q t,t+1 (πN,t+1)^2 (πN,t+1 − 1)] − (ε/κ) zNt lNt [ (ε−1)/ε − mc t ]
  - Gross inflation: πNt = pNt / pN,t−1
  - Real marginal cost: mc t = Wt / (pNt zNt)
- Symmetric final production: yNt = zNt lNt

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### 2.4  Government (monetary and fiscal law of motion)
- Money supply law of motion:
  - Ms t = Ms t−1 + X t
  - X t denotes per capita money injections.
- Fiscal policy:
  - Lump-sum tax Tt levied on both household types to finance government expenditure gNt.
  - No marginal taxes or government debt issuance allowed.
  - Consolidated government budget constraint:
    - Ms t = Ms t−1 + pNt gNt + X t − Tt

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### 2.5  International Transactions
- Law of one price for tradable goods:
  - pTt = et p*t
  - Foreign price p*t assumed constant and equal to 1.
- Upward-sloping supply of funds to induce stationarity of international bonds:
  - R*t = R* · (B*t / B*)^ν
  - R* and B* are steady-state values of the international interest rate and net foreign assets; ν is the elasticity of the supply of funds.

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### 2.6  Market Clearing Conditions
- Labor market:
  - λ l t (tr) + (1 − λ) l t (nt) = lTt + lNt
- Non-tradable goods market (including transaction costs and price adjustment resource use):
  - λ cNt (tr) + (1 − λ) cNt (nt) + λ s(vt (tr)) [ cNt (tr) + (pTt / pNt) cTt (tr) ] + (1 − λ) s(vt (nt)) [ cNt (nt) + (pTt / pNt) cTt (nt) ] + Nt + gNt + κ/2 (πNt − 1)^2 = yNt
  - Note: transaction costs are treated as deadweight losses; Nt is the non-tradable input used in tradable production.
- Tradable goods market:
  - λ cTt (tr) + (1 − λ) cTt (nt) + B*t e t pTt = yTt + R*t−1 B*t−1 e t pTt
- Aggregation of foreign assets (traders only):
  - B*t = λ b* t (tr)
- No borrowing/lending in domestic contingent bonds (traders identical):
  - d t (tr) = 0
- Money market:
  - Ms t = λ Mt (tr) + (1 − λ) Mt (nt)

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### 2.7  Definition of Equilibrium
- An equilibrium is a set of prices {et, pTt, pNt, qt,t+1, Wt, R*t, mc t} and allocations {cTt (tr), cNt (tr), cTt (nt), cNt (nt), b* t (tr), B* t, dt, Mt (tr), Mt (nt), Ms t, l t (tr), l t (nt), lTt, lNt, Nt, yTt, yNt} such that equations (2)–(13) and (17)–(27) hold, given policies {Xt, Tt}, exogenous processes {zTt, zNt, gNt}, and initial conditions (M−1 (tr), M−1 (nt), b*−1 (tr), pN−1).

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### 3  Ramsey Problem — Optimal monetary policy setup
- Objective: monetary authority maximizes weighted expected household utility under full commitment.
  - Maximize E0[∑ β^t (λ u(cTt (tr), cNt (tr), l t (tr)) + (1 − λ) u(cTt (nt), cNt (nt), l t (nt)))] subject to the competitive equilibrium constraints.
- Primal approach: rewrite prices and policy instruments in terms of allocations; use allocations as control variables.
- Key constraints recast into compact forms (examples):
  - Marginality conditions linking tradable/non-tradable consumption and marginal productivity in tradable sector:
    - zTt fT N(lTt, Nt) = ucN,t (tr) / ucT,t (tr)
    - zTt fT N(lTt, Nt) = ucN,t (nt) / ucT,t (nt)
  - Labor marginal productivity relations:
    - zTt fT l (lTt, Nt) = − u l,t (tr) h(vt (tr)) / ucT,t (tr)
    - zTt fT l (lTt, Nt) = − u l,t (nt) h(vt (nt)) / ucT,t (nt)
  - Money/portfolio expectational conditions:
    - ucN,t (tr) h(vt (tr)) [1 − s0(vt (tr))(vt (tr))^2] = β Et[ ucN,t+1 (tr) πN,t+1^−1 h(vt+1 (tr)) ]
    - ucT,t (tr) h(vt (tr)) = β R* (B*t / B*)^ν Et[ ucT,t+1 (tr) h(vt+1 (tr)) ]
  - Expectation-augmented Phillips curve expressed in allocation terms (example):
    - ucN,t (tr) h(vt (tr)) πNt (πNt − 1) = β Et[ ucN,t+1 (tr) h(vt+1 (tr)) πN,t+1 (πN,t+1 − 1) ] − (ε−1)/κ · [ ε/(ε−1) u l,t (tr) + ucN,t (tr) h(vt (tr)) / zNt ] · [ λ l t (tr) + (1 − λ) l t (nt) − lTt ]
- Classification of constraints:
  - Intratemporal: constraints dated t only (e.g., (28), (29), (30), (31), (36)).
  - Predetermined: include t and t−1 (e.g., (37), (38)).
  - Expectational: contain Et[·] terms (e.g., (32), (33), (34), (35)).
- Solution method:
  - Expectational constraints prevent a standard Bellman formulation.
  - Use Marcet and Marimon (1998) approach: introduce Lagrange multipliers on expectational equations as additional state variables to obtain a recursive saddle-point functional equation (details in Appendix A).
- Implementation assumption:
  - Analysis abstracts from the initial-time optimal policy problem; assumes the government has been conducting the optimal policy "from a timeless perspective" (Woodford, 2003).

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### 4  Dynamics under the Optimal Policy — Calibration and solution method
- Numerical solution:
  - Model solved using second-order approximation algorithm of Schmitt-Grohé and Uribe (2004).
  - Second-order approximation used because first-order may mis-measure welfare in optimal policy problems.
- Calibration targets and functional forms:
  - Time unit: one quarter.
  - Household utility:
    - u(c, l) = ln c + ψ log(1 − l)
    - ψ chosen to yield steady-state labor supply of 0.22.
  - Composite consumption (C.E.S.):
    - c = [ θ (cT)^μ + (1 − θ) (cN)^μ ]^(1/μ)
    - Intratemporal elasticity of substitution 1/(1 − μ) set to 0.50.
    - θ chosen to match share of tradables in Chilean CPI.
  - Tradable-sector production (C.E.S. between labor and non-tradable input):
    - yTt = zTt [ αT (lTt)^(1−φ) + (1 − αT) (Nt)^(1−φ) ]^(1/(1−φ))
    - αT = 0.40 (labor weight in tradable sector).
    - φ = 1.5 (elasticity of substitution parameter).
  - Markup:
    - ε = 6 → steady-state markup of 20 percent.
- Transactions technology (Schmitt-Grohé and Uribe 2004 specification):
  - s(v) = ω v + ξ v^(−2) p ξ/ω
  - This specification implies a satiation point of real money balances (needed for well-defined money demand at zero nominal interest rate, i.e., the Friedman rule).
  - Reduced-form money-demand implied relation estimated:
    - v_t^2 = ξ/ω + (1/ω) (R_{t−1} / R_t)
  - Empirical estimates for transaction function parameters (aggregate money demand, OLS on Chilean data):
    - ω = 0.06
    - ξ = 0.17
    - Estimated equation: v_t^2 = 2.68 + 15.64 (R_{t−1} / R_t)
      - t-statistics: first coefficient 20.82, second coefficient 15.72; R^2 = 0.82.
- Price-adjustment cost calibration:
  - Estimate log-linearized expectational Phillips curve:
    - bπN t = β Et[bπN t+1] + (ε − 1) (κ / cmc)  (reduced form)
  - GMM estimate of marginal cost coefficient (ε − 1) (κ / cmc) = 0.084.
  - Given steady-state labor supply and ε, implied quadratic cost κ = 13.16.
  - κ = 13.16 corresponds to price stickiness of about 4 quarters (Calvo analog); US typical estimate ~3 quarters (Sbordone, 2002).
- Asset market segmentation:
  - Baseline assumption: 50 percent of the population excluded from asset markets (λ = 0.5).
  - Rationale: developing economies likely have greater asset market segmentation; sensitivity analysis conducted.
- International supply-of-funds elasticity:
  - ν = 0.00001 (highly elastic; purpose is stationarity, not modeling risk-premium behavior).
- Steady-state calibration:
  - B* chosen consistent with net exports to GDP ratio of 2.3 percent.
  - Discount factor β equals inverse of steady-state gross foreign interest rate R*.
- Parameter summary (Table 1 reproduced as calibrated values):
  - Discount Factor β = 0.99
  - Tradable weight in consumption θ = 0.07
  - Intratemporal Elasticity of Substitution 1/(1 − μ) = 0.50
  - Parameter Transaction Cost Function ω = 0.06
  - Parameter Transaction Cost Function ξ = 0.17
  - Markup ε/(ε − 1) = 1.20
  - Price adjustment cost κ = 13.16
  - Labor share in the tradable sector αT = 0.40
  - Elasticity of substitution for tradable firms φ = 1.5
  - Foreign interest rate elasticity ν = 10^−5
  - Asset Market Segmentation λ = 0.5

### Exogenous shock processes (AR(1) specifications and estimated standard errors)
- Tradable-sector productivity:
  - zTt = 0.65 (0.11) zTt−1 + εTt, εTt ∼ N(0, σT^2), σT = 0.027
- Non-tradable-sector productivity:
  - zNt = 0.84 (0.08) zNt−1 + εNt, εNt ∼ N(0, σN^2), σN = 0.021
- Government expenditure process:
  - gNt = 0.76 (0.10) gN,t−1 + εGt, εGt ∼ N(0, σG^2), σG = 0.026

*Source: _wp07217 - 2.2  Firms in the Tradable Sector (IMF Working Paper PDF content).*

### 4.2  Optimal Monetary Policy in a Model with Sticky Prices

### 4.2  Optimal Monetary Policy in a Model with Sticky Prices

### Overview
- Analysis of the Ramsey optimal policy problem in a small open economy with sticky prices, focusing on the case λ = 1 (all agents have access to international and domestic bonds; no asset market segmentation).
- Comparison is made with a benchmark two-sector real model with monopolistic competition to quantify how well Ramsey policy mitigates distortions from sticky prices and monetary transaction costs.
- Given the parametrization in section 4.1, quantitatively small deviations of the Ramsey solution from the benchmark real model indicate the optimal policy largely reproduces the flexible-price equilibrium.

### Ramsey allocations versus benchmark real model
- If the optimal policy eliminates all nominal frictions, Ramsey allocations coincide with the real model; calibrated results show near-coincidence.
- The Ramsey policy achieves an output gap close to zero because it replicates flexible-price potential output in New Keynesian terms.
- Steady state under the Ramsey policy generates an annual deflation rate of 0.05 percent.

### Responses to a tradable-sector productivity shock (impulse responses summarized)
- Shock: one percent increase in tradable-sector productivity.
- Key outcomes under Ramsey policy (Figures 3 and 4 referenced):
  - Output, labor supply, markups, and real exchange rate dynamics closely replicate the real model.
  - Interest rate response: increases by slightly more than 100 basis points (countercyclical monetary policy).
  - Real exchange rate: real appreciation of 1 percent in response to a 1 percent increase in tradable-sector productivity (Balassa-Samuelson effect).
- Mechanism and rationale:
  - Tradable productivity shock bids up wages in the non-tradable sector → shift in nominal and real marginal costs → sticky non-tradable prices reduce markups.
  - Optimal monetary authority contracts the money supply to reduce aggregate demand, offsetting markup reductions and limiting non-tradable sector misallocation.
  - Under optimal policy non-tradable inflation is largely stabilized, so nominal exchange rate absorbs most of the real shock (consistent with Friedman’s argument favoring flexible exchange rates).
- Comparison with non-tradable inflation targeting rule (π^N_t = π, with π set equal to Ramsey steady state inflation):
  - Non-tradable inflation targeting closely mimics Ramsey allocations.
  - Ramsey policy smooths the nominal interest rate more than the monetary rule due to monetary transaction costs.
  - Trade-offs faced by the Ramsey planner: Friedman rule corrects money-distortion; stabilizing non-tradable inflation corrects sticky-price distortion. Baseline calibration implies sticky-price costs dominate, so Ramsey approximated by non-tradable inflation targeting.
  - Interest rate smoothing under Ramsey induces a small departure of the markup from the flexible-price equilibrium.

### Responses to a non-tradable-sector productivity shock
- Ramsey policy again produces allocations close to the flexible-price benchmark; non-tradable inflation targeting closely resembles Ramsey policy.
- Differences relative to tradable shock:
  - Monetary policy is procyclical (expands aggregate demand) to stabilize marginal costs and markups when non-tradable productivity rises.
  - Real exchange rate depreciates in response to a non-tradable productivity increase (opposite sign from tradable shock).
  - Optimal policy stabilizes non-tradable inflation and allows nominal exchange rate depreciation to avoid welfare-reducing non-tradable price fluctuations.

### Sensitivity analysis: price stickiness (κ) and monetary transactions
- Figure 7: Standard deviation of non-tradable inflation plotted for different κ.
  - Under Ramsey policy, non-tradable inflation volatility is close to zero for a wide range of κ.
  - Even with price stickiness several times lower than empirical estimates, optimal policy targets stable non-tradable prices.
- Figure 8: As κ decreases (price rigidity reduced), the optimal policy shifts to stabilizing nominal interest rate to mitigate monetary transaction costs.
  - As flexible prices are approached, the optimal policy converges toward the Friedman rule, eliminating money distortions.

### Optimal monetary policy with asset market segmentation (section 4.3 summary)
- Introducing asset market segmentation (agents without access to international bonds) does not substantially change Ramsey policy properties:
  - Ramsey allocation remains quantitatively close to benchmark real model.
  - Optimal policy can still be approximated by non-tradable inflation targeting.
- Interpretation:
  - Monetary policy is an inefficient tool to correct segmented asset markets when price stickiness exists; correcting sticky prices yields larger welfare gains than correcting asset market segmentation.
  - To achieve full risk-sharing via monetary policy would require deviations from flexible-price allocations, producing larger welfare costs than leaving segmentation uncorrected.
- Heterogeneous-agent implications (Figures 9–12):
  - Traders versus non-traders: traders’ allocations primarily reflect substitution effects (wealth effects absorbed by the rest of the world via net foreign assets); non-traders face both substitution and wealth effects.
  - Example dynamics: in response to tradable productivity shock, traders reduce non-tradable consumption (substitution toward tradables); non-traders maintain stable non-tradable consumption because wealth effects offset substitution effects.
  - Labor supply responses differ by access to international financial markets: traders increase labor effort; non-traders’ substitution and wealth effects largely cancel, leaving labor supply stable.
  - Intertemporal marginal rate of substitution (IMRS) for non-traders deviates from the efficiency condition; Ramsey policy marginally reduces this deviation compared to non-tradable inflation targeting.
- Sensitivity results with segmentation (Figures 13–15):
  - Figure 13: Standard deviation of non-tradable inflation robustly close to zero for baseline κ across degrees of asset market segmentation; stabilizing non-tradable inflation is optimal regardless of segmentation except for very small κ.
  - Figure 14: Standard deviation of nominal interest rate shows:
    - For any level of segmentation, reducing price stickiness leads the optimal policy to stabilize nominal interest rate (to correct monetary distortions).
    - For any κ, increasing asset market segmentation raises nominal interest rate volatility because domestic bond prices (and thus nominal interest rates) are determined by traders’ transactions; fewer traders imply higher asset price volatility.
  - Figure 15: Varying social welfare weights (increasing weight on non-traders):
    - Higher weight on non-traders increases importance of consumption smoothing, raising volatility of inflation and nominal interest.
    - Inflation volatility increases slowly as planner departs from population-weighted welfare; volatility remains well below that observed under flexible prices even for substantial changes in welfare weights.
    - Note: "λ-Ramsey Planner" used to vary planner weights while holding λ (population size of traders) constant.

### Policy recommendations and implications (Concluding remarks)
- Principal finding: Optimal monetary policy prioritizes stabilizing non-tradable inflation (volatility close to zero) to stabilize markups, reduce firms’ incentives to change prices, and approximate flexible-price allocations.
- Two implications for policymakers:
  - Target the appropriate price index: optimal policy should target only the subset of prices that display stickiness (empirically non-tradable sector shows more stickiness); targeting a price index that places more weight on non-tradables is welfare-improving compared with stabilizing the overall CPI.
  - Stabilizing non-tradable inflation is optimal irrespective of financial structure (important for developing countries with shallow financial markets); using monetary policy to smooth consumption and correct financial frictions is distortionary because it causes non-tradable price fluctuations, markup fluctuations, and resource misallocation.
- Caveats and extensions:
  - The fact that using monetary policy to correct asset market segmentation is welfare-reducing does not imply financial imperfections should be ignored; fiscal policy may be a more appropriate instrument to address segmentation.
  - Possible extensions: include additional real and nominal frictions to better match data (as in Schmitt-Grohé and Uribe (2005)), or model borrowing constraints and foreign-currency debt in emerging markets (as in Monacelli (2006)) to evaluate the role of the nominal exchange rate in redistributing resources.

*Document: _wp07217 - 4.2  Optimal Monetary Policy in a Model with Sticky Prices*

### References

### _wp07217 - References

### References list
- Comprehensive bibliography including (selected entries as presented):
  - Adão, B., I. Correia and P. Teles, 2003, “Gaps and Triangles,” Review of Economic Studies, vol. 70, 699-713.
  - Aiyagari, R., A. Marcet, T. Sargent and J. Seppälä, 2002, “Optimal Taxation without State-Contingent Debt," Journal of Political Economy, vol. 110, 1220-1254.
  - Alvarez, F and A. Atkeson, 1997, “Money and Exchange Rate in the Grossman-Weiss-Rotemberg Model,” Journal of Monetary Economics, vol. 40, 619-640.
  - Alvarez, F., A. Atkeson and P. Kehoe, 2002, “Money, Interest Rates, and Exchange Rates with Endogenously Segmented Asset Markets,” Journal of Political Economy, vol. 110, 73-112.
  - Burstein, A., M. Eichenbaum and S. Rebelo, 2005,“Large Devaluations and the Real Exchange Rate," Journal of Political Economy, vol. 113, 742-784.
  - Chari, V.V. and P. Kehoe, 1999, “Optimal Fiscal and Monetary Policy,” Handbook of Macroeconomics, vol. III, edited by J. Taylor and M. Woodford, 1999.
  - Galí, J. and M. Gertler, 1999, “Inflation Dynamics: A Structural Econometric Analysis,” Journal of Monetary Economics, vol. 44, 195-222.
  - Galí, J. and T. Monacelli, 2005, “Monetary Policy and Exchange Rate Volatility in a Small Open Economy,” Review of Economic Studies, vol. 72, 707-734.
  - Goodfriend, M. and R. King, 2001, “The Case for Price Stability,” NBER Working Paper 8423.
  - Lucas, R. and N. Stokey, 1983, “Optimal Fiscal and Monetary Policy in an Economy without Capital,” Journal of Monetary Economics, vol. 12, 55-93.
  - Marcet, A. and R. Marimon, 1998, “Recursive Contracts,” mimeo, Universitat Pompeu Fabra.
  - Monacelli, T., 2006, “Optimal Monetary Policy with Collateralized Household Debt and Borrowing Constraints," NBER Working Paper 12470.
  - Ramsey, F., 1927, “A Contribution to the Theory of Taxation,” Economic Journal, vol. 37, 47-61.
  - Rotemberg, J., 1982 and 1984, on monopolistic price adjustment and monetary equilibrium with transactions costs, Review of Economic Studies and Journal of Political Economy.
  - Schmitt-Grohé, S. and M. Uribe, multiple entries (2001–2006) on stabilization policy, closing small open economy models, optimal policy under sticky prices, solution methods, and implementable rules.
  - Woodford, M., 2002, “Inflation Stabilization and Welfare,” Contributions to Macroeconomics, vol, issue 1, article 1; 2003, "Interest and Prices," Princeton University Press.

### Appendix A — Lagrangian of the Ramsey Problem: structure and key elements
- State vectors and multipliers:
  - d_t = [c_Tt(tr), c_Nt(tr), l_t(tr), v_t(tr), c_Tt(nt), c_Nt(nt), l_t(nt), v_t(nt), l_Tt, N_t, B*_t, π_Nt]'.
  - μ_t = [μ_1,t, ..., μ_11,t]'.
  - μ_1 - μ_11 are Lagrange multipliers associated with constraints (28) - (38).
- Saddle-point optimization problem:
  - min_{ {μ_t} } max_{ {d_t} } E_0 { sum_{t=0}^∞ β^t [ ... Lagrangian terms ... ] } given B*_{-1}, c_T_{-1}(nt), c_N_{-1}(nt), and v_{-1}(nt).
- Key terms (as presented, preserving notation and exact expressions where shown):
  - Objective includes λ u(c_Tt(tr), c_Nt(tr), l_t(tr)) + (1−λ) u(c_Tt(nt), c_Nt(nt), l_t(nt)).
  - Multiple constraint terms multiplied by μ_i,t, e.g., μ_1,t [ μ z_Tt f^T_N(l^T_t, N_t) − u_{c_N,t}(tr) / u_{c_T,t}(tr) ], and analogous μ_2,t through μ_11,t terms as printed.
  - Intertemporal expectations and discounting appear via β E_t[·] and terms like −β E_t[ u_{c_N,t+1}(tr) π_N,t+1 h(v_{t+1}(tr)) ] inside constraints multiplied by μ_5,t and μ_6,t.
  - Inclusion of μ_5 - μ_8 as state variables is explicitly noted, and initial conditions μ_5,−1 = μ_6,−1 = μ_7,−1 = μ_8,−1 = 0 are given.
- Remark and transformation:
  - Remark 1 identity used to rearrange terms: E_0[ Σ_{t=0}^∞ β^t χ_t( g_1(d_t) + β E_t[g_2(d_{t+1})] ) ] = E_0[ Σ_{t=0}^∞ β^t E_t( χ_t g_1(d_t) + χ_{t−1} g_2(d_t) ) ].
  - Lagrangian rewritten using this identity; resulting expression shows terms with μ_{i,t} − μ_{i,t−1} combinations (for i where applicable).
- Value function and state partitioning (Marcet and Marimon framework):
  - W(d^x_{t−1}, μ^x_{t−1}, z_t) = min_{μ_t} max_{d_t} { ... + β E_t[ W(d^x_t, μ^x_t, z_{t+1}) | z_t ] }.
  - Where d^x_t = [B*_t, c_Tt(nt), c_Nt(nt), v_t(nt)]', μ^x_t = [μ_5,t, μ_6,t, μ_7,t, μ_8,t]', z_t = [z_Tt, z_Nt, g_Nt]'.
  - x_t = [d^x_t, μ^x_t]; remaining endogenous variables collected in y_t.

### Appendix B — Benchmark Two-sector Real Model: equilibrium conditions
- Model assumptions:
  - Monopolistic competition and segmented asset markets; abstracts from sticky prices and monetary transaction costs.
- Equilibrium conditions (equations preserved as numbered):
  1. u_{c_N,t}(tr) / u_{c_T,t}(tr) = p_t
  2. − u_{l,t}(tr) / u_{c_T,t}(tr) = w_t
  3. u_{c_T,t}(tr) = β R^*_t E_t[ u_{c_T,t+1}(tr) ]
  4. u_{c_N,t}(nt) / u_{c_T,t}(nt) = p_t
  5. − u_{l,t}(nt) / u_{c_T,t}(nt) = w_t
  6. z_Tt f^T_{l} l^T_{t} = w_t
  7. z_Tt f^T_{N,t} = p_t
  8. p_t ( μ^{ε−1} / ε ) z_Nt = w_t
  9. g_Nt = T_t
  10. R^*_t = R^* ( B^*_t / B^* )^{ν}
  11. λ l_t(tr) + (1−λ) l_t(nt) = l^T_t + l^N_t
  12. λ c_Nt(tr) + (1−λ) c_Nt(nt) + N_t + g_Nt = y_Nt
  13. λ c_Tt(tr) + (1−λ) c_Tt(nt) + B^*_t = y_Tt + R^*_t−1 B^*_t−1
  14. c_Tt(nt) + p_t c_Nt(nt) = w_t l_t(nt) − T_t
- Interpretation notes as given:
  - (1)–(3): first-order conditions for traders (allocation, labor supply, consumption-saving).
  - (4)–(5): first-order conditions for non-traders (allocation and labor supply).
  - (6)–(8): demand functions for labor and non-tradable intermediate input.
  - (9)–(10): government budget constraint and upward-sloping supply of funds.
  - (11)–(13): market clearing conditions.
  - (14): budget constraint for non-traders.
- Equilibrium definition:
  - An equilibrium is a set of Prices: {p_t, w_t, R^*_t}, and Allocations: {c_Tt(tr), c_Nt(tr), c_Tt(nt), c_Nt(nt), B^*_t(tr), l_t(tr), l_t(nt), l^T_t, l^N_t, N_t, y_Tt, y_Nt} such that (1)-(14) hold, given policy {T_t}, exogenous process {z_Tt, z_Nt, g_Nt}, and initial condition B^*_{−1}.

### Figures and empirical/quantitative displays (captions preserved)
- Figure 1: Annual Inflation Rate (Q1 1994 - Q3 2006) — series shown for Australia, Canada, New Zealand, Chile, Colombia, Peru. Source: International Financial Statistics.
- Figure 2: Credit to the Private Sector and M3 as Percentage of GDP (2005) — bars for New Zealand, Australia, Canada, Chile, Colombia, Peru. Source: World Development Indicators.
- Figures 3–15: Multiple impulse response and sensitivity analysis figures:
  - Figures 3–6: Impulse Responses to a Productivity Shock in the Tradable Sector and Non-tradable Sector. Model with Sticky Prices. Panels include Productivity Shock, Nominal Interest Rate (Basis points dev. from SS), Nontradable Inflation, Real Exchange Rate, Tradable Output, Non-tradable Output, Labor Supply, Mark Up. Labels preserve axis units: % dev. from SS, Basis points dev. from SS.
  - Figures 7–8: Sensitivity Analysis of Non-tradable Inflation Rate and Nominal Interest Rate. Horizontal axis: κ (Standard Deviation in %).
  - Figures 9–12: Impulse Responses to Productivity Shocks in models with Sticky Prices and Asset Market Segmentation; panels include consumption and labor supply for traders and non-traders, mark up, intertemporal marginal rate of substitution, etc. Axes and units preserved as in source.
  - Figures 13–14: Sensitivity Analysis of Non-tradable Inflation Rate and Nominal Interest Rate as functions of λ and κ (Standard Deviation in %).
  - Figure 15: Sensitivity Analysis of Nominal Interest Rate and Non-tradable Inflation Rate with horizontal axes values including λ in [0.15, 0.5] and resulting Standard Deviation levels as shown.
- All figure captions and axis labels are preserved as printed (e.g., "% dev. from SS", "Basis points dev. from SS", "Standard Deviation in %", specific panel numbering).

*Content derived from _wp07217 - References (PDF).*

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