## wp1740

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---

### 2.1 Households — preferences, consumption and labor
- Representative household lifetime utility:
  - E0 Σt=0^∞ e^{B_t} U(X_{i,t}, N_{i,t}) (equation (1))
  - Subjective discount factor: β ∈ [0,1]
- Consumption aggregator:
  - X_{i,t} = Z_{i,t}^{1−α} D_{i,t}^{α}, a Cobb-Douglas aggregator of nondurable (Z_{i,t}) and durable (D_{i,t}) goods
  - α ∈ [0,1] is the durables consumption share on total expenditure
- Habit-adjusted nondurable consumption and habit stock:
  - Z_{i,t} = C_{i,t} − ζ S_{t−1} (equation (2))
  - S_{t} = ρ_c S_{t−1} + (1−ρ_c) C_{t} (equation (3))
  - ζ ∈ (0,1) is the degree of external habit formation
  - ρ_c ∈ (0,1) is the persistence of habit formation
  - C_{t} denotes average (aggregate) consumption
- Labor supply and imperfect mobility across sectors:
  - N_{i,t} = [ C̄ (N^C_{i,t})^{(1+ξ)/ξ} + (1−C̄) (N^D_{i,t})^{(1+ξ)/ξ} ]^{ξ/(1+ξ)} (equation (4), CES specification)
  - ξ > 0 is the intra-temporal elasticity of substitution governing labor mobility
    - ξ → 0 denotes case of labor immobility
    - as ξ → 1 labor can be freely reallocated and all workers earn the same marginal wage
    - For ξ < 1 the economy displays a limited degree of labor mobility and sectoral wages are not equal
  - C̄ ≡ N^C/N is the steady-state share of labor in the nondurables sector
  - CES aggregator captures sectoral labor mobility while retaining representative-agent framework

### 2.1 Households — durables, budget and wage adjustment
- Durables accumulation and investment adjustment costs:
  - D_{i,t+1} = (1−δ) D_{i,t} + e^I_t I^D_{i,t} [ 1 − S( I^D_{i,t} / I^D_{i,t−1} ) ] (equation (5))
  - δ is the depreciation rate of durables
  - I^D_{i,t} is investment in durables subject to adjustment costs
  - e^I_t is an investment-specific shock
  - Adjustment costs S(·) satisfy: S(1) = S'(1) = 0 and S''(1) > 0
- Budget constraint, nominal bonds and wage adjustment costs:
  - Q_t ≡ P_{D,t} / P_{C,t} (relative price of durables)
  - Period real budget constraint (per household):
    - C_{i,t} + Q_t I^D_{i,t} + (ψ_W / 2) ( (w_{i,t}/w_{i,t−1}) Π_{C,t} − Π_{C} )^2 w_t N_t + B_{i,t} / P_{C,t} = W_{i,t}/P_{C,t} N_{i,t} + R_{t−1} B_{i,t−1}/P_{C,t} + Ω_t − T_t (equation (6))
  - ψ_W parameterizes quadratic nominal wage adjustment costs à la Rotemberg (1982)
  - Households choose sequences {Z_{i,t}, B_{i,t}, D_{i,t+1}, I^D_{i,t}, w_{i,t}, N^C_{i,t}, N^D_{i,t}} to maximize utility subject to constraints (2)–(6)

### 2.1 Households — first-order conditions and key features
- First-order (optimality) conditions at symmetric equilibrium:
  - Euler equation for nominal bonds: 1 = E_t [ Λ_{t,t+1} R_t / Π_{C,t+1} ] (equation (7))
    - Λ_{t,t+1} ≡ U_{Z,t+1} / U_{Z,t} e^{B_{t+1}} / e^{B_t} is the stochastic discount factor
  - Asset price of durables: Q_t τ_t = U_{D,t} / U_{Z,t} + (1−δ) E_t [ Λ_{t,t+1} Q_{t+1} τ_{t+1} ] (equation (8))
  - Investment optimality (durables): 1 = τ_t e^I_t [ 1 − S( I^D_t / I^D_{t−1} ) − S'( I^D_t / I^D_{t−1} ) ( I^D_t / I^D_{t−1} ) ] + E_t [ Λ_{t,t+1} τ_{t+1} Q_{t+1} / Q_t e^I_{t+1} S'( I^D_{t+1} / I^D_t ) ( I^D_{t+1} / I^D_t )^2 ] (equations (9) and (10))
  - Sectoral labor supplies: N^C_t = C̄ ( w^C_t / w_t )^{−ξ} N_t (equation (11)); N^D_t = (1−C̄) ( w^D_t / w_t )^{−ξ} N_t (equation (12))
  - Wage-setting condition with Rotemberg wage adjustment costs and wage markup:
    - 0 = [ 1 − e^W_t η ] Λ + e^W_t η μ̃_t − ψ_W ( Π_{W,t} / Π_{C} − Π_{W,t} ) + E_t [ Λ_{t,t+1} ψ_W ( Π_{W,t+1} / Π_{C} − Π_{W,t+1} ) w_{t+1} N_{t+1} / (w_t N_t) ] (equation (13))
    - μ̃_t ≡ w_t / MRS_t is the wage markup; MRS_t ≡ − U_{N,t} / U_{Z,t} is the marginal rate of substitution between consumption and leisure
- Key model features summarized:
  - Utility: additively separable and logarithmic in consumption
  - Habit formation in nondurable consumption introduces persistence (ζ, ρ_c)
  - Investment-specific shock e^I_t and quadratic investment adjustment costs S(·)
  - Imperfect labor mobility via CES aggregator governed by ξ; limited mobility implies sectoral wage differentials
  - Nominal wage adjustment costs (Rotemberg) via ψ_W; wage markups and wage-setting dynamics captured by μ̃_t and equation (13)

### 3.2 Estimation results — estimated parameters (posterior means and 90% probability intervals)
- Structural
  - Labor mobility λ: 1.1769 [1.0327;1.3228]
  - Inverse Frisch elasticity ' : 0.572 [0.4137;0.7233]
  - Habit in nondurables consumption ζ: 0.7931 [0.7573;0.8293]
  - Habit persist. nondurables consumption ρ_c: 0.4306 [0.3315;0.5321]
  - Price stickiness nondurables #_c: 37.4735 [26.6634;47.5440]
  - Price stickiness durables #_d: 35.9731 [26.0116;45.6557]
  - Wage stickiness #_W: 102.6213 [86.5891;119.363]
  - Invest. adjust. costs durable goods κ_I: 2.2812 [1.6753;2.8612]
  - Share of durables inflation in aggregator τ: 0.2001 [0.1179;0.2838]
  - Inflation - Taylor rule ρ_π: 1.6388 [1.4600;1.8302]
  - Output - Taylor rule ρ_y: 0.0646 [0.0307;0.0953]
  - Output growth - Taylor rule ρ_Δy: 0.2904 [0.0867;0.4756]
  - Interest rate smoothing ρ_r: 0.7015 [0.6543;0.7481]
- Averages
  - Trend growth rate ḡ: 0.3988 [0.3794;0.4192]
  - Inflation rate nondurables ̄π_C: 1.0298 [0.9396;1.1210]
  - Inflation rate durables ̄π_D: 0.5330 [0.4414;0.6288]
  - Interest rate ̄r: 1.6225 [1.5002;1.7420]
- Exogenous processes (selected)
  - Technology ρ_eA: 0.9258 [0.8943;0.9563]; var ̄_eA: 0.0070 [0.0064;0.0077]
  - Monetary Policy ρ_eR: 0.1355 [0.0423;0.2217]; var ̄_eR: 0.0029 [0.0025;0.0032]
  - Investment Durables ρ_eI: 0.2653 [0.1342;0.4042]; var ̄_eI: 0.0585 [0.0413;0.0752]
  - Preference ρ_eB: 0.6450 [0.5536;0.7388]; var ̄_eB: 0.0192 [0.0159;0.0222]
  - Price mark-up nondurables ρ_eC: 0.9454 [0.9091;0.9822]; var ̄_eC: 0.0201 [0.0151;0.0251]
  - Price mark-up durables ρ_eD: 0.9797 [0.9641;0.9951]; var ̄_eD: 0.0556 [0.0444;0.0657]
  - Wage mark-up ρ_eW: 0.9694 [0.9495;0.9922]; var ̄_eW: 0.0408 [0.0341;0.0477]
  - Government spending ρ_eG: 0.9398 [0.9077;0.9743]; var ̄_eG: 0.0355 [0.0322;0.0387]
- Log-marginal likelihood: -1378.927

### Optimal monetary policy — setup, scenarios and welfare measure
- Interest rate rule reparametrized as equation (33), with parameters:
  - α_π ≡ (1−ρ_r)ρ_π
  - α_y ≡ (1−ρ_r)ρ_y
  - α_Δy ≡ (1−ρ_r)ρ_Δy
- Support constraints for optimization:
  - ρ_R ∈ [0,1] (unless superinertial extension), α_π, α_y, α_Δy ∈ [0,5]
  - Weight on durables inflation τ ∈ [0,1]
- Welfare objective:
  - Maximize present value of households’ utility (34) using second-order approximation around Ramsey steady state
- ZLB treatment:
  - Impose approximate ZLB by penalizing deviations of nominal interest rate from steady state with weight w_r; set Pr(ZLB)=0.01 which corresponds to w_r=80
- Optimization scenarios for labor mobility:
  - λ = 0.25 (quasi labor immobility)
  - λ = 1 (perfect labor mobility)
  - λ = 1.1769 (estimated imperfect labor mobility)

### Impulse responses to a durables price-markup shock (Ramsey policy, sticky durables prices)
- Higher mark-ups in durables sector:
  - Increase durables inflation
  - Decrease investment and employment in durables for any degree of labor mobility
- Low labor mobility (labor prevented from moving):
  - Output and employment fall in nondurables as well; aggregate output persistently below steady state
- High/perfect labor mobility:
  - Consumption in nondurables increases as labor flows from durables sector
  - Output in two sectors shows negative comovement; durables response more gradual
  - Aggregate output increases in first seven quarters (positive nondurables effect dominates), then falls as durables decline dominates
  - Optimal nominal interest rate response can be accommodative (procyclical) under perfect mobility, stimulating nondurables consumption and facilitating supply adjustment via labor reallocation

### Optimized interest-rate rules and welfare (selected results; baseline = parameters at posterior means)
- Main qualitative findings (baseline)
  - Central bank response to output gap and output gap growth is absent (α_y and α_Δy = 0 in optimized rules)
  - Stronger reaction to inflation (α_π > 0)
  - Interest rate smoothing ρ_R systematically hits upper bound 1 → policy characterized as a price-level rule
  - A negative relationship between labor mobility λ and optimal weight on durables inflation τ:
    - As λ increases (labor becomes more mobile) optimal τ decreases (less weight on durables inflation)
    - With perfect labor mobility (λ = 1), optimal τ = 0 in several specifications
  - Wage stickiness induces positive optimal τ even when durables prices are flexible, via pass-through to marginal costs
- Selected optimized rule outcomes — Sticky durables prices (#_d > 0)
  - λ = 0.25: ρ_R = 1, α_π = 0.0633, α_y = 0.0000, α_Δy = 0.0109, τ = 0.4831, welfare loss ! = 0.0716
  - λ = 1.1769: ρ_R = 1, α_π = 0.1104, α_y = 0.0000, α_Δy = 0.0191, τ = 0.1612, welfare loss ! = 0.0846
- Selected optimized rule outcomes — Flexible durables prices (#_d = 0)
  - λ = 0.25: ρ_R = 1, α_π = 0.0404, α_y = 0.0000, α_Δy = 0.0135, τ = 0.1581, ! = 0.0791
  - λ = 1.1769: ρ_R = 1, α_π = 0.1000, α_y = 0.0000, α_Δy = 0.0187, τ = 0.0000, ! = 0.0920

### Relative volatilities (selected)
- Table 4: Relative standard deviations of inflation, wages and output (relative to nondurables)
  - For λ = 0.25: (Δπ, wages, output) = 1.9089, 6.4104, 1.6591
  - For λ = 1.1769: (Δπ, wages, output) = 1.9611, 4.7002, 4.3077
- Insights:
  - Volatility of durables sector prices, wages and output > nondurables in all cases
  - Under perfect labor mobility wages identical across sectors → no difference in wage volatility
  - Under limited mobility, wage volatility in durables > nondurables (about 4.7 times at estimated λ = 1.1769)
  - Relative volatilities of prices and output in durables increase with labor mobility

### Robustness checks and alternative rule forms
- Flexible wages and flexible durables prices/wages (optimized rules)
  - Flexible wages (#_W = 0)
    - λ = 0.25: ρ_R = 1, α_π = 0.1570, α_y = 0.0000, α_Δy = 0.0000, τ = 0.1765, ! = 0.0274
    - λ = 1.1769: ρ_R = 1, α_π = 0.2254, α_y = 0.0000, α_Δy = 0.0000, τ = 0.0214, ! = 0.0095
  - Flexible durables prices and wages (#_d = #_W = 0)
    - λ = 0.25: ρ_R = 1, α_π = 0.1298, α_y = 0.0000, α_Δy = 0.0000, τ = 0.0216, ! = 0.0286
    - λ = 1.1769: ρ_R = 1, α_π = 0.2426, α_y = 0.0000, α_Δy = 0.0000, τ = 0.0000, ! = 0.0101
  - Robustness conclusions:
    - τ and λ remain negatively related: as labor mobility increases, optimal τ decreases
    - With flexible wages τ falls substantially; with both frictions removed τ can be zero
    - Central bank still finds it optimal to implement a price-level rule (ρ_R = 1) in these calibrations
- Superinertial rule (allow ρ_R ∈ [0,5]) — top panel Table 6
  - λ = 0.25: ρ_R = 1.2987, α_π = 0.0313, α_y = 0.0022, α_Δy = 0.0166, τ = 0.3405, ! = 0.0491
  - λ = 1.1769: ρ_R = 1.515372, α_π = 0.0150, α_y = 0.9426, α_Δy = 0.0700, τ = 0.0196, ! = 0.4024
  - Findings:
    - For any degree of labor mobility the central bank chooses a superinertial rule (ρ_R hits upper bound)
    - Welfare losses relative to Ramsey are smaller than under price-level rules with ρ_R = 1
    - Negative relationship between λ and τ survives
- Implementable rule (responds only to inflation and deviation of output from steady state) — lower panel Table 6
  - Results:
    - Negative relationship between λ and τ still holds
    - Choice toward price-level behavior and welfare-loss patterns similar to baseline

### Key policy-relevant conclusions
- The degree of sectoral labor mobility is crucial for the optimal design of monetary policy.
- Main empirical/quantitative regularities:
  - Estimated labor mobility λ = 1.1769 indicates limited but non-negligible frictions.
  - Central bank optimally places more weight on the sector with higher price stickiness (nondurables), i.e., τ < 0.5 in optimizations.
  - Inverse relationship: as labor mobility increases (labor reallocates more easily), optimal weight on durables inflation τ decreases.
  - Wage stickiness raises the optimal τ (via pass-through to marginal costs), so even fully flexible durables prices can lead to positive τ if wages are sticky.
  - Optimal policy displays strong history dependence: interest-rate inertia tends to be chosen at high levels (ρ_R → 1 or higher under superinertial extensions), effectively implementing price-level or superinertial rules.
- Robustness:
  - Results robust to alternative calibrations of price and wage stickiness and to alternative interest-rate rule forms (superinertial and implementable rules)

### Appendix A — data construction and observables
- Sector definitions:
  - Durables sector = composite of durable goods and residential investments
  - Nondurables sector = nondurables goods and services
- Series mnemonics (selected):
  - DUR_N: Nominal Durable Goods — BEA Table 2.3.5 Line 3
  - RI_N: Nominal Residential Investment — BEA Table 1.1.5 Line 13
  - P_DUR: Price Deflator, Durable Goods — BEA Table 1.1.9 Line 4
  - P_RI: Price Deflator, Residential Investment — BEA Table 1.1.9 Line 13
  - ND_N: Nominal Nondurable Goods — BEA Table 2.3.5 Line 8
  - S_N: Nominal Services — BEA Table 2.3.5 Line 13
  - P_ND: Price Deflator, Nondurable Goods — BEA Table 1.1.9 Line 5
  - P_S: Price Deflator, Services — BEA Table 1.1.9 Line 6
  - Y_N: Nominal GDP — BEA Table 1.1.5 Line 1
  - P_Y: Price Deflator, GDP — BEA Table 1.1.9 Line 1
  - FFR: Effective Federal Funds Rate — FRED FEDFUNDS
  - POP: Civilian Non-institutional Population, over 16 — FRED CNP16OV
  - CE: Civilian Employment, 16 over — FRED CE16OV
- Construction steps (selected)
  - Durables and Residential Investments:
    - DR_N = DUR_N + RI_N
    - !D = DUR_N / DR_N; !RI = RI_N / DR_N
    - P_D = !D P_DUR + !RI P_RI
    - D = (DUR_N + RI_N) / P_D
  - Nondurables and Services:
    - NS_N = ND_N + S_N
    - !ND = ND_N / NS_N; !S = S_N / NS_N
    - P_C = !ND P_ND + !S P_S
    - C = (ND_N + S_N) / P_C
- Observables for Bayesian estimation (Table A.2)
  - POP index: POP / POP_2009:1
  - CE index: CE / CE_2009:1
  - Y_o: Real per capita GDP ln( (Y_N / P_Y) / POP_index ) 100
  - I_o_D: Real per capita consumption: durables ln( (D / POP_index) ) 100
  - C_o: Real per capita consumption: nondurables ln( (C / POP_index) ) 100
  - W_o: Real wage ln( (W / P_Y) ) 100
  - N_o: Hours worked per capita ln( (H * CE_index / POP_index) ) 100
  - ⇧_o_C: Inflation: nondurables sector (ln P_C) 100
  - ⇧_o_D: Inflation: durables sector (ln P_D) 100
  - R_o: Quarterly Federal Funds Rate FFR / 4

### Appendix C — steady state (selected equalities)
- In steady state x_t = x_{t+1} = x and stochastic shocks are absent.
- Key steady-state relationships (exact equalities preserved):
  - ⇧_D = ⇧_C^{opt} (68)
  - ̃⇧ = ⇧_C^{opt} (69)
  - ⇧_W = ⇧_C^{opt} (70)
  - ⇤ =  (71)
  - R = 1 / (1) (72)
  - MC_C = (✏_c  1) / ✏_c (73)
  - w_C = MC_C e^{A} (74)
  - w_D = w_C (75)
  - w = w_D (76)
  - MC_D = (✏_d  1) / ✏_d (77)
  - Q = w_D / MC_D e^{A} (78)
  - S = 0 (79)
  - S_0 = 0 (80)
  - = 1 (81)
  - ̃μ = ⌘ / (⌘  1) (82)
  - S = C (83)
  - Z = (1⇣) C (84)
  - U_Z = (1↵) Z (85)
  - U_D = U_z Q [1(1) ] (86)
  - D = ↵ / U_D (87)
  - U_N =  U_Z ̃μ / w (88)
  - N =  ( U_N / ⌫ )^{1/'} (89)
  - Y_D = D (90)
  - N_D = Y_D (91)
  - N_C = N  N_D (92)
  - C = N_C / N (93)
  - e_G = g_y Y (94)
  - Y_C = N_C (95)
  - X = Z^{1↵} D^{↵} (96)

*Source: wp1740 - IMF Working Paper (selected sections: 2.1 Households; 3.2 Estimation results; Appendices A–C)*

### 2.1  Households  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7

### 2.1  Households

### Preferences and lifetime utility
- Representative household lifetime utility:
  - E0 Σt=0^∞ e^{B_t} U(X_{i,t}, N_{i,t})  (equation (1))
  - Subjective discount factor: β ∈ [0,1]
- Consumption aggregator:
  - X_{i,t} = Z_{i,t}^{1−α} D_{i,t}^{α}, a Cobb-Douglas aggregator of nondurable (Z_{i,t}) and durable (D_{i,t}) goods
  - α ∈ [0,1] is the durables consumption share on total expenditure

### Nondurable consumption and habit formation
- Habit-adjusted nondurable consumption:
  - Z_{i,t} = C_{i,t} − ζ S_{t−1}  (equation (2))
- Aggregate habit stock:
  - S_{t} = ρ_c S_{t−1} + (1−ρ_c) C_{t}  (equation (3))
- Parameters and domains:
  - ζ ∈ (0,1) is the degree of external habit formation
  - ρ_c ∈ (0,1) is the persistence of habit formation
  - C_{t} denotes average (aggregate) consumption

### Labor supply and imperfect mobility across sectors
- Household supplies labor to both sectors with CES aggregation:
  - N_{i,t} = [ (C̄ (N^C_{i,t})^{(1+ξ)/ξ} + (1−C̄) (N^D_{i,t})^{(1+ξ)/ξ} ) ]^{ξ/(1+ξ)}  (equation (4), CES specification)
  - ξ > 0 is the intra-temporal elasticity of substitution governing labor mobility
    - ξ → 0 denotes case of labor immobility
    - as ξ → 1 labor can be freely reallocated and all workers earn the same marginal wage
    - For ξ < 1 the economy displays a limited degree of labor mobility and sectoral wages are not equal
  - C̄ ≡ N^C/N is the steady-state share of labor in the nondurables sector
- CES aggregator captures sectoral labor mobility while retaining representative-agent framework

### Durables accumulation and investment adjustment costs
- Durables law of motion:
  - D_{i,t+1} = (1−δ) D_{i,t} + e^I_t I^D_{i,t} [ 1 − S( I^D_{i,t} / I^D_{i,t−1} ) ]  (equation (5))
  - δ is the depreciation rate of durables
  - I^D_{i,t} is investment in durables subject to adjustment costs
  - e^I_t is an investment-specific shock
- Adjustment costs S(·) satisfy:
  - S(1) = S'(1) = 0 and S''(1) > 0

### Budget constraint, nominal bonds and wage adjustment costs
- Relative price of durables:
  - Q_t ≡ P_{D,t} / P_{C,t}
- Period real budget constraint (per household):
  - C_{i,t} + Q_t I^D_{i,t} + (ψ_W / 2) ( (w_{i,t}/w_{i,t−1}) Π_{C,t} − Π_{C} )^2 w_t N_t + B_{i,t} / P_{C,t} = W_{i,t}/P_{C,t} N_{i,t} + R_{t−1} B_{i,t−1}/P_{C,t} + Ω_t − T_t  (equation (6))
  - ψ_W parameterizes quadratic nominal wage adjustment costs à la Rotemberg (1982)
- Households choose sequences {Z_{i,t}, B_{i,t}, D_{i,t+1}, I^D_{i,t}, w_{i,t}, N^C_{i,t}, N^D_{i,t}} to maximize utility subject to constraints (2)–(6)

### First-order (optimality) conditions at symmetric equilibrium
- Euler equation for nominal bonds:
  - 1 = E_t [ Λ_{t,t+1} R_t / Π_{C,t+1} ]  (equation (7))
  - Λ_{t,t+1} ≡ U_{Z,t+1} / U_{Z,t} e^{B_{t+1}} / e^{B_t} is the stochastic discount factor
- Asset price of durables:
  - Q_t τ_t = U_{D,t} / U_{Z,t} + (1−δ) E_t [ Λ_{t,t+1} Q_{t+1} τ_{t+1} ]  (equation (8))
- Investment optimality (durables):
  - 1 = τ_t e^I_t [ 1 − S( I^D_t / I^D_{t−1} ) − S'( I^D_t / I^D_{t−1} ) ( I^D_t / I^D_{t−1} ) ] + E_t [ Λ_{t,t+1} τ_{t+1} Q_{t+1} / Q_t e^I_{t+1} S'( I^D_{t+1} / I^D_t ) ( I^D_{t+1} / I^D_t )^2 ]  (equations (9) and (10))
- Sectoral labor supplies:
  - N^C_t = C̄ ( w^C_t / w_t )^{−ξ} N_t  (equation (11))
  - N^D_t = (1−C̄) ( w^D_t / w_t )^{−ξ} N_t  (equation (12))
- Wage-setting condition with Rotemberg wage adjustment costs and wage markup:
  - 0 = [ 1 − e^W_t η ] Λ + e^W_t η μ̃_t − ψ_W ( Π_{W,t} / Π_{C} − Π_{W,t} ) + E_t [ Λ_{t,t+1} ψ_W ( Π_{W,t+1} / Π_{C} − Π_{W,t+1} ) w_{t+1} N_{t+1} / (w_t N_t) ]  (equation (13))
  - μ̃_t ≡ w_t / MRS_t is the wage markup; MRS_t ≡ − U_{N,t} / U_{Z,t} is the marginal rate of substitution between consumption and leisure

### Key model features summarized
- Utility: additively separable and logarithmic in consumption
- Habit formation in nondurable consumption introduces persistence (ζ, ρ_c)
- Investment-specific shock e^I_t and quadratic investment adjustment costs S(·)
- Imperfect labor mobility via CES aggregator governed by ξ; limited mobility implies sectoral wage differentials
- Nominal wage adjustment costs (Rotemberg) via ψ_W; wage markups and wage-setting dynamics captured by μ̃_t and equation (13)

*Source: wp1740 - 2.1  Households (excerpt from IMF working paper wp1740)*

### 3.2  Estimation results

### 3.2  Estimation results

### Estimated parameters (posterior means and 90% probability intervals)
- Structural
  - Labor mobility λ: 1.1769 [1.0327;1.3228]
  - Inverse Frisch elasticity ' : 0.572 [0.4137;0.7233]
  - Habit in nondurables consumption ζ: 0.7931 [0.7573;0.8293]
  - Habit persist. nondurables consumption ρ_c: 0.4306 [0.3315;0.5321]
  - Price stickiness nondurables #_c: 37.4735 [26.6634;47.5440]
  - Price stickiness durables #_d: 35.9731 [26.0116;45.6557]
  - Wage stickiness #_W: 102.6213 [86.5891;119.363]
  - Invest. adjust. costs durable goods κ_I: 2.2812 [1.6753;2.8612]
  - Share of durables inflation in aggregator τ: 0.2001 [0.1179;0.2838]
  - Inflation - Taylor rule ρ_π: 1.6388 [1.4600;1.8302]
  - Output - Taylor rule ρ_y: 0.0646 [0.0307;0.0953]
  - Output growth - Taylor rule ρ_Δy: 0.2904 [0.0867;0.4756]
  - Interest rate smoothing ρ_r: 0.7015 [0.6543;0.7481]

- Averages
  - Trend growth rate ḡ: 0.3988 [0.3794;0.4192]
  - Inflation rate nondurables ̄π_C: 1.0298 [0.9396;1.1210]
  - Inflation rate durables ̄π_D: 0.5330 [0.4414;0.6288]
  - Interest rate ̄r: 1.6225 [1.5002;1.7420]

- Exogenous processes (selected)
  - Technology ρ_eA: 0.9258 [0.8943;0.9563]; var ̄_eA: 0.0070 [0.0064;0.0077]
  - Monetary Policy ρ_eR: 0.1355 [0.0423;0.2217]; var ̄_eR: 0.0029 [0.0025;0.0032]
  - Investment Durables ρ_eI: 0.2653 [0.1342;0.4042]; var ̄_eI: 0.0585 [0.0413;0.0752]
  - Preference ρ_eB: 0.6450 [0.5536;0.7388]; var ̄_eB: 0.0192 [0.0159;0.0222]
  - Price mark-up nondurables ρ_eC: 0.9454 [0.9091;0.9822]; var ̄_eC: 0.0201 [0.0151;0.0251]
  - Price mark-up durables ρ_eD: 0.9797 [0.9641;0.9951]; var ̄_eD: 0.0556 [0.0444;0.0657]
  - Wage mark-up ρ_eW: 0.9694 [0.9495;0.9922]; var ̄_eW: 0.0408 [0.0341;0.0477]
  - Government spending ρ_eG: 0.9398 [0.9077;0.9743]; var ̄_eG: 0.0355 [0.0322;0.0387]

- Log-marginal likelihood: -1378.927

### Interpretation of estimation results
- Labor mobility parameter λ = 1.1769 implies a non-negligible degree of friction in the labor market (comparison points cited: Horvath 0.999; Iacoviello and Neri 1.51 and 1.03).
- Prices estimated stickier in the nondurables sector; durables show non-negligible stickiness but are generally regarded as more flexible in the literature.
- Wages are found to be sticky (high #_W), but flexible-wage scenarios are explored.
- Results indicate relevance of real frictions: investment adjustment costs in durable goods and habits in nondurables consumption.
- Monetary policy estimated to respond more strongly to inflation than to output and displays a high degree of inertia (ρ_r ~ 0.7015).

---

### Optimal monetary policy: setup and welfare measure
- Interest rate rule reparametrized as equation (33), with parameters:
  - α_π ≡ (1−ρ_r)ρ_π
  - α_y ≡ (1−ρ_r)ρ_y
  - α_Δy ≡ (1−ρ_r)ρ_Δy
- Support constraints for optimization:
  - ρ_R ∈ [0,1] (unless superinertial extension), α_π, α_y, α_Δy ∈ [0,5]
  - Weight on durables inflation τ ∈ [0,1]
- Welfare objective: maximize present value of households’ utility (34) using second-order approximation around Ramsey steady state.
- ZLB treatment: impose approximate ZLB by penalizing deviations of nominal interest rate from steady state with weight w_r; set Pr(ZLB)=0.01 which corresponds to w_r=80.

### Optimization scenarios for labor mobility
- Three cases considered:
  - λ = 0.25  (quasi labor immobility)
  - λ = 1  (perfect labor mobility)
  - λ = 1.1769  (estimated imperfect labor mobility)

---

### Impulse responses to a durables price-markup shock (Ramsey policy, sticky durables prices)
- Higher mark-ups in durables sector:
  - Increase durables inflation
  - Decrease investment and employment in durables for any degree of labor mobility
- Low labor mobility (labor prevented from moving):
  - Output and employment fall in nondurables as well; aggregate output persistently below steady state
- High/perfect labor mobility:
  - Consumption in nondurables increases as labor flows from durables sector
  - Output in two sectors shows negative comovement; durables response more gradual
  - Aggregate output increases in first seven quarters (positive nondurables effect dominates), then falls as durables decline dominates
  - Optimal nominal interest rate response can be accommodative (procyclical) under perfect mobility, stimulating nondurables consumption and facilitating supply adjustment via labor reallocation

---

### Optimized interest-rate rules and welfare (selected results; baseline = parameters at posterior means)
- Main qualitative findings (baseline)
  - Central bank response to output gap and output gap growth is absent (α_y and α_Δy = 0 in optimized rules)
  - Stronger reaction to inflation (α_π > 0)
  - Interest rate smoothing ρ_R systematically hits upper bound 1 → policy characterized as a price-level rule
  - A negative relationship between labor mobility λ and optimal weight on durables inflation τ:
    - As λ increases (labor becomes more mobile) optimal τ decreases (less weight on durables inflation)
    - With perfect labor mobility (λ = 1), optimal τ = 0 in several specifications
  - Wage stickiness induces positive optimal τ even when durables prices are flexible, via pass-through to marginal costs

- Table 3: Optimized monetary policy rule — Sticky vs Flexible Durables Prices (baseline calibration except where optimized)
  - Sticky durables prices (#
d
>0)
    - λ = 0.25: ρ_R = 1, α_π = 0.0633, α_y = 0.0000, α_Δy = 0.0109, τ = 0.4831, welfare loss ! = 0.0716
    - λ = 1.1769: ρ_R = 1, α_π = 0.1104, α_y = 0.0000, α_Δy = 0.0191, τ = 0.1612, welfare loss ! = 0.0846
    - (line with "1 10.4611..." appears in source; baseline interpretation focuses on rows above)
  - Flexible durables prices (#_d = 0)
    - λ = 0.25: ρ_R = 1, α_π = 0.0404, α_y = 0.0000, α_Δy = 0.0135, τ = 0.1581, ! = 0.0791
    - λ = 1.1769: ρ_R = 1, α_π = 0.1000, α_y = 0.0000, α_Δy = 0.0187, τ = 0.0000, ! = 0.0920
    - (final line in panel includes "1 10.5055..." as in source)

- Table 4: Relative standard deviations of inflation, wages and output (relative to nondurables)
  - For λ = 0.25: (Δπ, wages, output) = 1.9089, 6.4104, 1.6591
  - For λ = 1.1769: (Δπ, wages, output) = 1.9611, 4.7002, 4.3077
  - Insights:
    - Volatility of durables sector prices, wages and output > nondurables in all cases
    - Under perfect labor mobility wages identical across sectors → no difference in wage volatility
    - Under limited mobility, wage volatility in durables > nondurables (about 4.7 times at estimated λ = 1.1769)
    - Relative volatilities of prices and output in durables increase with labor mobility

---

### Robustness checks: alternative calibrations and alternative interest-rate rules
- Table 5: Flexible wages and flexible durables prices/wages (optimized rules)
  - Flexible wages (#
W
=0)
    - λ = 0.25: ρ_R = 1, α_π = 0.1570, α_y = 0.0000, α_Δy = 0.0000, τ = 0.1765, ! = 0.0274
    - λ = 1.1769: ρ_R = 1, α_π = 0.2254, α_y = 0.0000, α_Δy = 0.0000, τ = 0.0214, ! = 0.0095
  - Flexible durables prices and wages (#
d
=#
W
=0)
    - λ = 0.25: ρ_R = 1, α_π = 0.1298, α_y = 0.0000, α_Δy = 0.0000, τ = 0.0216, ! = 0.0286
    - λ = 1.1769: ρ_R = 1, α_π = 0.2426, α_y = 0.0000, α_Δy = 0.0000, τ = 0.0000, ! = 0.0101
  - Robustness conclusions:
    - τ and λ remain negatively related: as labor mobility increases, optimal τ decreases
    - With flexible wages τ falls substantially; with both frictions removed τ can be zero
    - Central bank still finds it optimal to implement a price-level rule (ρ_R = 1) in these calibrations

- Superinertial rule (allow ρ_R ∈ [0,5]) — top panel Table 6
  - Optimized superinertial rules (baseline model)
    - λ = 0.25: ρ_R = 1.2987, α_π = 0.0313, α_y = 0.0022, α_Δy = 0.0166, τ = 0.3405, ! = 0.0491
    - λ = 1.1769: ρ_R = 1.515372, α_π = 0.0150, α_y = 0.9426, α_Δy = 0.0700, τ = 0.0196, ! = 0.4024
  - Findings:
    - For any degree of labor mobility the central bank chooses a superinertial rule (ρ_R hits upper bound)
    - Welfare losses relative to Ramsey are smaller than under price-level rules with ρ_R = 1
    - Negative relationship between λ and τ survives

- Implementable rule (responds only to inflation and deviation of output from steady state) — lower panel Table 6
  - Reparametrized as equation (36)
  - Results:
    - Negative relationship between λ and τ still holds
    - Choice toward price-level behavior and welfare-loss patterns similar to baseline

---

### Key policy-relevant conclusions
- The degree of sectoral labor mobility is crucial for the optimal design of monetary policy.
- Main empirical/quantitative regularities:
  - Estimated labor mobility λ = 1.1769 indicates limited but non-negligible frictions.
  - Central bank optimally places more weight on the sector with higher price stickiness (nondurables), i.e., τ < 0.5 in optimizations.
  - Inverse relationship: as labor mobility increases (labor reallocates more easily), optimal weight on durables inflation τ decreases.
  - Wage stickiness raises the optimal τ (via pass-through to marginal costs), so even fully flexible durables prices can lead to positive τ if wages are sticky.
  - Optimal policy displays strong history dependence: interest-rate inertia tends to be chosen at high levels (ρ_R → 1 or higher under superinertial extensions), effectively implementing price-level or superinertial rules.
- Robustness:
  - Results robust to alternative calibrations of price and wage stickiness and to alternative interest-rate rule forms (superinertial and implementable rules).

*Source: IMF Working Paper — section "3.2 Estimation results" and subsequent optimal policy analysis (tables and figures reproduced in source).*

### References

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*Source: wp1740 - References (IMF Working Paper PDF)*

### Appendix A — Data

### Definitions and sector composition
- The durables sector is defined as the a composite of durable goods and residential investments whereas the nondurables sector comprises nondurables goods and services.

### Series definitions and source mnemonics (Table A.1)
- DUR_N: Nominal Durable Goods — BEA Table 2.3.5 Line 3
- RI_N: Nominal Residential Investment — BEA Table 1.1.5 Line 13
- ND_N: Nominal Nondurable Goods — BEA Table 2.3.5 Line 8
- S_N: Nominal Services — BEA Table 2.3.5 Line 13
- P_DUR: Price Deflator, Durable Goods — BEA Table 1.1.9 Line 4
- P_RI: Price Deflator, Residential Investment — BEA Table 1.1.9 Line 13
- P_ND: Price Deflator, Nondurable Goods — BEA Table 1.1.9 Line 5
- P_S: Price Deflator, Services — BEA Table 1.1.9 Line 6
- Y_N: Nominal GDP — BEA Table 1.1.5 Line 1
- P_Y: Price Deflator, GDP — BEA Table 1.1.9 Line 1
- FFR: Effective Federal Funds Rate — FRED FEDFUNDS
- N: Nonfarm Business Sector: Average Weekly Hours — FRED PRS85006023
- W: Nonfarm Business Sector: Compensation Per Hour — FRED COMPNFB
- POP: Civilian Non-institutional Population, over 16 — FRED CNP16OV
- CE: Civilian Employment, 16 over — FRED CE16OV

### A.1 Durables and Residential Investments (construction steps)
- 1. Sum nominal series: DUR_N + RI_N = DR_N
- 2. Calculate sectoral weights of deflators: !D = DUR_N / DR_N; !RI = RI_N / DR_N
- 3. Calculate Deflator: P_D = !D P_DUR + !RI P_RI
- 4. Calculate Real Durable Consumption: D = (DUR_N + RI_N) / P_D

### A.2 Nondurables and Services (construction steps)
- 1. Sum nominal series: ND_N + S_N = NS_N
- 2. Calculate sectoral weights of deflators: !ND = ND_N / NS_N; !S = S_N / NS_N
- 3. Calculate Deflator: P_C = !ND P_ND + !S P_S
- 4. Calculate Real Nondurable Consumption: C = (ND_N + S_N) / P_C

### A.3 Data transformation for Bayesian estimation (Table A.2 observables)
- POP index: POP / POP_2009:1
- CE index: CE / CE_2009:1
- Y_o: Real per capita GDP ln( (Y_N / P_Y) / POP_index ) 100
- I_o_D: Real per capita consumption: durables ln( (D / POP_index) ) 100
- C_o: Real per capita consumption: nondurables ln( (C / POP_index) ) 100
- W_o: Real wage ln( (W / P_Y) ) 100
- N_o: Hours worked per capita ln( (H * CE_index / POP_index) ) 100
- ⇧_o_C: Inflation: nondurables sector (ln P_C) 100
- ⇧_o_D: Inflation: durables sector (ln P_D) 100
- R_o: Quarterly Federal Funds Rate FFR / 4

### Appendix B — Symmetric equilibrium (key model relationships)
- Aggregate composite: X_t = Z_t^(1↵) D_t^↵ (equation (37))
- Z_t = C_t  ⇣ S_{t-1} (38)
- S_t = ⇢_c S_{t-1} + (1⇢_c) C_t (39)
- Utility: U(X_t,N_t) = log(X_t)  ⌫ N_t^{1+'} / (1+') (40)
- U_Z,t = (1↵) Z_t (41)
- U_D,t = ↵ D_t (42)
- U_N,t = ⌫ N_t^' (43)
- Labor aggregator: N_t = [ ... ]^{1/(1+)} (44)  (full functional form preserved in source)
- Sectoral labor allocations:
  - N_Ct = C ( w_Ct / w_t ) N_t (45)
  - N_Dt = D ( w_Dt / w_t ) N_t (46)
- Intertemporal marginal rates and pricing Q:
  - ⇤_{t,t+1} ≡ U_{Z,t+1} / U_{Z,t} e^{B_{t+1}} / e^{B_t} (47)
  - ̃μ_t defined in terms of ⇤ and wage/inflation terms (48–49)
  - Q_t^t = U_{D,t} / U_{Z,t} + (1) E_t[⇤_{t,t+1} Q_{t+1}^{t+1}] (50)
- Investment adjustment and convexity functions S(·) and S'(·) (51–53)
- Euler equation: 1 = E_t[⇤_{t,t+1} R_t / ⇧_{C,t+1}] (54)
- Price relationships:
  - ⇧_{D,t} = ⇧_{C,t} Q_t / Q_{t-1} (55)
  - ⇧_{W,t} = w_t / w_{t-1} e^{⇧_t} (56)
- Production:
  - Y_{C,t} = e^{A_t} N_{C,t} (57)
  - Y_{D,t} = e^{A_t} N_{D,t} (58)
- Marginal costs and pricing markups for consumption and durables sectors (59–62)
- CPI inflation aggregation: e^{⇧_t} = (⇧_{C,t})^{1⌧} (⇧_{D,t})^{⌧} (63)
- Monetary policy (generalized Taylor rule): log( R_t / ̄R ) = _⇢⇡ log( ⇧_t / ̄⇧ ) + ⇢_y log( Y_t / ̄Y ) + e_Mt (64)
- Resource constraints and definitions for aggregate output components (65–67)

### Appendix C — Steady state (deterministic)
- In steady state x_t = x_{t+1} = x and stochastic shocks are absent.
- The steady-state inflation rate in the nondurables sector is the optimal under the Ramsey policy and is denoted by ⇧_C^{opt}. C solves equation (65).
- Key steady-state relationships (exact equalities preserved):
  - ⇧_D = ⇧_C^{opt} (68)
  - ̃⇧ = ⇧_C^{opt} (69)
  - ⇧_W = ⇧_C^{opt} (70)
  - ⇤ =  (71)
  - R = 1 / (1) (72)
  - MC_C = (✏_c  1) / ✏_c (73)
  - w_C = MC_C e^{A} (74)
  - w_D = w_C (75)
  - w = w_D (76)
  - MC_D = (✏_d  1) / ✏_d (77)
  - Q = w_D / MC_D e^{A} (78)
  - S = 0 (79)
  - S_0 = 0 (80)
  - = 1 (81)
  - ̃μ = ⌘ / (⌘  1) (82)
  - S = C (83)
  - Z = (1⇣) C (84)
  - U_Z = (1↵) Z (85)
  - U_D = U_z Q [1(1) ] (86)
  - D = ↵ / U_D (87)
  - U_N =  U_Z ̃μ / w (88)
  - N =  ( U_N / ⌫ )^{1/'} (89)
  - Y_D = D (90)
  - N_D = Y_D (91)
  - N_C = N  N_D (92)
  - C = N_C / N (93)
  - e_G = g_y Y (94)
  - Y_C = N_C (95)
  - X = Z^{1↵} D^{↵} (96)

*Source: wp1740 - References (IMF Working Paper PDF)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp1740.pdf_