## Home hours worked

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---

### Key findings and main contributions
- Service sector now accounts for nearly 80 percent of the economy-wide value-added.
- Consumer services grew from one-third of total private consumption expenditures in 1950 to about two-thirds in 2008.
- Embedding household production into a two-sector sticky-price New Keynesian model (market services and nondurables) yields:
  - An “Augmented” New Keynesian Phillips Curve (NKPC) for the service sector with an extra endogenous term in addition to the standard output gap term.
  - The extra term increases with the elasticity of substitution between market- and home-produced services (ν) and co-moves negatively with the output gap, amplifying nominal rigidity endogenously in the service sector.
  - Firms in the service sector face additional competition from household production, making them more reluctant to change prices and therefore adjust quantities more — amplifying real effects of monetary shocks in services but not in nondurables.
- VAR evidence (U.S., 1967:Q1–2007:Q4, quarterly, 4 lags and a constant) finds consumer services more interest-rate sensitive than consumer nondurable goods:
  - Immediate drop in real spending on services of 0.12% after a one-standard-deviation federal funds rate innovation, versus 0.02% and 0.01% for durables and nondurables in the first post-shock quarter.
  - Services effects last 9 quarters, durables 6 quarters, nondurables 7 quarters.

### Empirical evidence on household production and sectoral responses
- Household production quantitative importance and substitutability:
  - U.S. Gross Household Production (GHP) estimated at 37.5% of the extended Gross National Product (GNP).
  - American Time Use Survey (ATUS) indicates nearly 26 hours a week devoted to discretionary home activities.
  - Literature estimates the elasticity of substitution between market aggregate and home production in the range 1.5 to 2.5; these provide a likely lower bound for substitution between market services and home production.
- Time use (2003 annual average, persons aged 15 years and over):
  - Total hours of home production: 26.39 (All), 20.09 (Men), 31.15 (Women).
  - Market hours worked: 25.83 (All), 31.99 (Men), 20.09 (Women).
  - Leisure and Sports: 33.46 (All), 35.00 (Men), 32.06 (Women).
- Child care costs (families with employed mothers, percent of monthly income, 1991–2005):
  - Fall 1991: 7.12 (All), 26.6 (Below poverty), 6.9 (Above poverty)
  - Fall 1993: 7.32 (All), 21.1 (Below poverty), 7.0 (Above poverty)
  - Spring 1997: 6.92 (All), 20.0 (Below poverty), 6.6 (Above poverty)
  - Spring 1999: 6.73 (All), 33.3 (Below poverty), 6.4 (Above poverty)
  - Winter 2002: 7.12 (All), 25.7 (Below poverty), 6.9 (Above poverty)
  - Spring 2005: 6.42 (All), 29.2 (Below poverty), 6.1 (Above poverty)
- Evidence on substitutability during business cycles:
  - De-trended home and market hours worked series (1950–2005) show home hours are counter-cyclical and increase in most recessions.
  - Food at home and food away from home (HP-de-trended series) co-move negatively over the business cycle, suggesting substitution between cooking at home and going out.
  - Shopping evidence: doubling shopping frequency lowers prices by 7 to 10%; estimated elasticity of substitution between time and money due to shopping nearly 1.8.

### Model structure and mechanism
- Economy and frictions:
  - Continuum of identical infinitely-lived households.
  - Three sectors: nondurable goods market, market services, home services produced for self-consumption.
  - Monopolistic competition within sectors; Calvo(1983) price stickiness.
  - Taylor-type monetary policy; monetary policy shocks only source of aggregate uncertainty.
- Preferences and production:
  - Utility: expected discounted sum with log(C_t) + Φ L_t, Φ>0, β ∈ (0,1).
  - Consumption aggregator: C = C_g^ω C_S^(1−ω), 0<ω<1.
  - Service aggregate CES: C_S = [ γ C_h^(ν−1)/ν + (1−γ) C_s^(ν−1)/ν ]^(ν/(ν−1)), ν>0, 0≤γ<1.
  - Household time constraint: N_h,t + N_s,t + N_g,t + L_t = 1.
  - Home production constraint: C_h,t ≤ Γ(N_h,t).
- Representative household first-order conditions (selected):
  - MU_g,t / MU_s,t = P_g,t / P_s,t.
  - MRS_k,t = Φ MU_k,t = W_t / P_k,t, k∈{g,s}.
  - W_t / P_k,t MU_k,t = (∂Γ/∂N_h) MU_h,t, k∈{g,s}.
  - Λ_t = β R_t E_t(Λ_t+1).
- Firms and pricing:
  - Final good producers: Dixit-Stiglitz aggregation Y_k,t = (∫_0^1 Y_k,t(z)^{(ε_k−1)/ε_k} dz)^{ε_k/(ε_k−1)}.
  - Intermediate firm technology linear: Y_k,t(z)=A N_k,t(z), A normalized to one.
  - Calvo pricing: fraction 1−θ_k re-optimize prices each period; κ_j = (1−θ_j)(1−β θ_j)/θ_j.
- Sectoral inflation dynamics:
  - Nondurable goods sector NKPC: π_g,t = κ_g y_g,t + β E_t(π_g,t+1).
  - Service sector “Augmented” NKPC (Proposition 1): π_s,t = κ_s y_s,t + (1−1/ν) κ_s (y_S,t − y_s,t) + β E_t(π_s,t+1), where y_S,t = c_S.
  - Extra term vanishes if ν = 1 (Cobb-Douglas separability), reproducing standard NKPC.
- Aggregate NKPC (Corollary 1) with equal price stickiness across sectors:
  - π_t = κ y_t + (1−χ)(1−1/ν) κ (y_S,t − y_s,t) + β E_t(π_t+1), where 1−χ is steady-state share of services in total market consumption; κ = (1−θ)(1−β θ)/θ.

### Intuition: supply and demand channels
- Supply-side channel:
  - Household production functions as an additional competitor for market services; this raises endogenous nominal rigidity in the service sector because firms face less incentive to change prices and instead adjust quantities.
- Demand-side channel:
  - An expansionary monetary shock raises real wages (nominal wages rise while sticky prices lag), increasing substitution from home production to market services, reinforcing larger real effects in the service sector.
- Sectoral asymmetry:
  - Nondurable goods lack home substitutes; household production does not amplify monetary shocks in that sector.

### Calibration (quarterly) and parameter values
- Discount factor: β = 1.04^(−1/4) = 0.99 (implies annualized real interest rate of 4%).
- Elasticity of substitution between market and home services: ν = 2.3 (Chang and Schorfheide (2003) estimate used).
- Time-use targets: calibration uses Table 1 time allocations implying people spend 30% of reported discretionary time on market work and 31% on homework.
- Markups: ε_g = ε_s = 11 (imply a 10% steady-state markup in each market sector).
- Price stickiness: θ_g = θ_s = 2/3 (prices last 3 quarters on average).
- Taylor rule: φ_π = 1.5, φ_y = 0, interest-rate shock persistence ρ_r = 0.7.
- Consumption and other parameters: γ = 0.5, Φ = 1.62, χ = 0.4, ω = 0.24.
- Full calibrated vector listed: β ν Φ γ χ ω ε_g ε_s θ_g θ_s φ_π φ_y ρ_r = 0.99 2.3 1.62 0.5 0.4 0.24 11 11 2/3 2/3 1.5 0 0.7.

### Simulation results and quantitative implications
- Scenarios considered:
  - Baseline without household production and with equally sticky prices across sectors (“No H.P.”).
  - With household production and flexible prices (H.P., flexible prices).
  - With household production and sticky prices equally across sectors (H.P., preferred specification).
- Core quantitative outcomes (responses to a negative percentage point innovation in the nominal interest rate, i.e., a 1% cut):
  - With sticky prices and household production:
    - Initial response of consumption of market-provided services: 2.4% (with H.P.) versus 1.4% (No H.P.).
    - Initial response of real spending on nondurables: 1.4% in both models (no home substitutes).
    - Initial real wage increase: 1.4% (drives greater market hours and substitution from home to market).
  - Aggregate effects:
    - Aggregate output response to a monetary shock is about 50% larger when household production is accounted for compared to a model without household production (composition effect given large service share).
- Sensitivity and robustness:
  - Household production has no implications under fully flexible prices (real effects vanish).
  - Results robust to alternative small φ_y > 0 in Taylor rule (qualitatively similar).

### Policy implications
- Accounting for household production provides an endogenous source of nominal rigidity in the service sector, reducing reliance on large exogenous Calvo price stickiness to generate realistic real responses to monetary shocks.
- Monetary policy transmission:
  - Monetary expansions have stronger real effects when household production exists and market/home services are substitutable (higher ν).
  - Service-sector dynamics are central for aggregate monetary policy effects given services’ large share of output.
- Model suggests sectoral targeting and consideration of household production complementarities could matter for monetary policy design and the interpretation of sectoral responses to interest-rate changes.

*Source: _wp12206 — https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12206.pdf*

### 1.    Home hours worked  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .9

### Home hours worked

### Contents overview
- 1.    Home hours worked  . . . . . . . . . . . . . . . . . . . . . .9
- 2.    Households and the production of services  . . . . . . . . .11
- C. Household and Market Production Over the Business Cycle   . .11
  - 1.    Fluctuations of home and market hours worked    . . . . . . . .11
  - 2.    Substitutability between home and market services over the business cycle13

### The Model Economy (structure)
- III.    The Model Economy   . . . . . . . . . . . . . . . . . . . . . .15
- A.    The Economic Environment  . . . . . . . . . . . . . . . . . . . .15
- B. The Representative Household . . . . . . . . . . . . . . . . . . .16
- C. Final Goods Producers  . . . . . . . . . . . . . . . . . . . . . . .19
- D.    Intermediate Goods producers  . . . . . . . . . . . . . . . . .19
- E. Sectoral and Aggregate New Keynesian Phillips Curves   . . . . . . .20
- F. Monetary Policy   . . . . . . . . . . . . . . . . . . . . . . . . . .24
- G.    Aggregation   . . . . . . . . . . . . . . . . . . . . . . . . . .24

### Calibration and Results
- IV. Calibration and Results   . . . . . . . . . . . . . . . . . . . . .26
- A.    Parameter Values  . . . . . . . . . . . . . . . . . . . . . . .26
- B. Simulation Results  . . . . . . . . . . . . . . . . . . . . . . . .27

### Conclusion
- V. Conclusion    . . . . . . . . . . . . . . . . . . . . . . . . . . .29

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12206.pdf*

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

### _wp12206 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Key findings and main contributions
- Service sector now accounts for nearly 80 percent of the economy-wide value-added.
- Consumer services grew from one-third of total private consumption expenditures in 1950 to about two-thirds in 2008.
- Embedding household production into a two-sector sticky-price New Keynesian model (market services and nondurables) yields:
  - An “Augmented” New Keynesian Phillips Curve (NKPC) for the service sector with an extra endogenous term in addition to the standard output gap term.
  - The extra term increases with the elasticity of substitution between market- and home-produced services (ν) and co-moves negatively with the output gap, amplifying nominal rigidity endogenously in the service sector.
  - Firms in the service sector face additional competition from household production, making them more reluctant to change prices and therefore adjust quantities more — amplifying real effects of monetary shocks in services but not in nondurables.
- VAR evidence (U.S., 1967:Q1–2007:Q4, quarterly, 4 lags and a constant) finds consumer services more interest-rate sensitive than consumer nondurable goods:
  - Immediate drop in real spending on services of 0.12% after a one-standard-deviation federal funds rate innovation, versus 0.02% and 0.01% for durables and nondurables in the first post-shock quarter.
  - Services effects last 9 quarters, durables 6 quarters, nondurables 7 quarters.

### Empirical evidence on household production and sectoral responses
- Household production quantitative importance and substitutability:
  - U.S. Gross Household Production (GHP) estimated at 37.5% of the extended Gross National Product (GNP).
  - American Time Use Survey (ATUS) indicates nearly 26 hours a week devoted to discretionary home activities.
  - Literature estimates the elasticity of substitution between market aggregate and home production in the range 1.5 to 2.5; these provide a likely lower bound for substitution between market services and home production.
- Time use (2003 annual average, persons aged 15 years and over):
  - Total hours of home production: 26.39 (All), 20.09 (Men), 31.15 (Women).
  - Market hours worked: 25.83 (All), 31.99 (Men), 20.09 (Women).
  - Leisure and Sports: 33.46 (All), 35.00 (Men), 32.06 (Women).
- Child care costs (families with employed mothers, percent of monthly income, 1991–2005):
  - Fall 1991: 7.12 (All), 26.6 (Below poverty), 6.9 (Above poverty)
  - Fall 1993: 7.32 (All), 21.1 (Below poverty), 7.0 (Above poverty)
  - Spring 1997: 6.92 (All), 20.0 (Below poverty), 6.6 (Above poverty)
  - Spring 1999: 6.73 (All), 33.3 (Below poverty), 6.4 (Above poverty)
  - Winter 2002: 7.12 (All), 25.7 (Below poverty), 6.9 (Above poverty)
  - Spring 2005: 6.42 (All), 29.2 (Below poverty), 6.1 (Above poverty)
- Evidence on substitutability during business cycles:
  - De-trended home and market hours worked series (1950–2005) show home hours are counter-cyclical and increase in most recessions.
  - Food at home and food away from home (HP-de-trended series) co-move negatively over the business cycle, suggesting substitution between cooking at home and going out.
  - Shopping evidence: doubling shopping frequency lowers prices by 7 to 10%; estimated elasticity of substitution between time and money due to shopping nearly 1.8.

### Model structure and mechanism
- Economy: continuum of identical infinitely-lived households, three sectors (nondurable goods market, market services, home services produced for self-consumption), monopolistic competition, Calvo(1983) price stickiness, Taylor-type monetary policy, monetary policy shocks only source of aggregate uncertainty.
- Preferences and production:
  - Utility: expected discounted sum with log(Ct)+ΦLt, Φ>0, β∈(0,1).
  - Consumption aggregator: C = Cg^ω C_S^(1−ω), 0<ω<1.
  - Service aggregate: C_S = [γ C_h^(ν−1)/ν + (1−γ) C_s^(ν−1)/ν]^(ν/(ν−1)), ν>0, 0≤γ<1 (CES to allow varying ν).
  - Household time constraint: N_h,t + N_s,t + N_g,t + L_t = 1.
  - Home production constraint: C_h,t ≤ Γ(N_h,t).
- Key first-order conditions (representative household):
  - MU_g,t / MU_s,t = P_g,t / P_s,t.
  - MRS_k,t = Φ MU_k,t = W_t / P_k,t, k∈{g,s}.
  - W_t / P_k,t MU_k,t = (∂Γ/∂N_h) MU_h,t, k∈{g,s}.
  - Λ_t = β R_t E_t(Λ_t+1).
- Firms and pricing:
  - Final good producers: Dixit-Stiglitz aggregation Y_k,t = (∫_0^1 Y_k,t(z)^{(ε_k−1)/ε_k} dz)^{ε_k/(ε_k−1)}.
  - Intermediate firms: linear technology Y_k,t(z)=A N_k,t(z), A normalized to one.
  - Calvo pricing: fraction 1−θ_k reset prices each period; κ_j = (1−θ_j)(1−β θ_j)/θ_j.
- Sectoral NKPCs (Proposition 1):
  - Nondurable goods sector: π_g,t = κ_g y_g,t + β E_t(π_g,t+1).
  - Service sector (Augmented): π_s,t = κ_s y_s,t + (1−1/ν) κ_s (y_S,t − y_s,t) + β E_t(π_s,t+1).
    - y_S,t = c_S is aggregate of market and home services.
    - Extra term vanishes if ν = 1 (Cobb-Douglas separability), reproducing standard NKPC.
- Aggregate NKPC (Corollary 1), with equal price stickiness across sectors:
  - π_t = κ y_t + (1−χ)(1−1/ν) κ (y_S,t − y_s,t) + β E_t(π_t+1),
    - where 1−χ is steady-state share of services in total market consumption; κ = (1−θ)(1−β θ)/θ.

### Intuition: supply and demand channels
- Supply-side: household production acts as an additional competitor for market services. The extra endogenous NKPC term reduces incentive for service firms to change prices, making them adjust quantities instead — increasing nominal rigidity endogenously in services.
- Demand-side: an expansionary monetary shock raises real wages (nominal wages rise while sticky prices lag), increasing the incentive to substitute from home production to market services, reinforcing larger real effects in the service sector.
- Sectoral asymmetry: nondurable goods lack home substitutes, so household production does not amplify monetary shocks in that sector.

### Calibration (quarterly) and numerical values
- Discount factor: β = 1.04^(−1/4) = 0.99 (implies annualized real interest rate of 4%).
- Elasticity of substitution between market and home services: ν = 2.3 (Chang and Schorfheide (2003) estimate used).
- Steady-state and time-use targets:
  - Calibration uses Table 1 time allocations implying people spend 30% of reported discretionary time on market work and 31% on homework.
- Markups: ε_g and ε_s set to imply a 10% steady-state markup in each market sector; ε_g = ε_s = 11.
- Price stickiness: θ_g = θ_s = 2/3 (prices last 3 quarters on average).
- Taylor rule: φ_π = 1.5, φ_y = 0, interest-rate shock persistence ρ_r = 0.7.
- Consumption share and related parameters: γ = 0.5, Φ = 1.62, χ = 0.4, ω = 0.24 (as summarized in calibration table).
  - Full calibrated vector listed: β ν Φ γ χ ω ε_g ε_s θ_g θ_s φ_π φ_y ρ_r = 0.99 2.3 1.62 0.5 0.4 0.24 11 11 2/3 2/3 1.5 0 0.7.

### Simulation results and quantitative implications
- Scenarios considered:
  - Baseline without household production and with equally sticky prices across sectors (“No H.P.”).
  - With household production and flexible prices (H.P., flexible prices).
  - With household production and sticky prices equally across sectors (H.P., preferred specification).
- Core quantitative outcomes (responses to a negative percentage point innovation in the nominal interest rate, i.e., a 1% cut):
  - With sticky prices and household production:
    - Initial response of consumption of market-provided services: 2.4% (with H.P.) versus 1.4% (No H.P.).
    - Initial response of real spending on nondurables: 1.4% in both models (no home substitutes).
    - Initial real wage increase: 1.4% (drives greater market hours and substitution from home to market).
  - Aggregate effects:
    - Aggregate output response to a monetary shock is about 50% larger when household production is accounted for compared to a model without household production (composition effect given large service share).
- Sensitivity and robustness:
  - Household production has no implications under fully flexible prices (real effects vanish).
  - Results robust to alternative small φ_y > 0 in Taylor rule (qualitatively similar).

### Policy implications
- Accounting for household production provides an endogenous source of nominal rigidity in the service sector, reducing reliance on large exogenous Calvo price stickiness to generate realistic real responses to monetary shocks.
- Monetary policy transmission:
  - Monetary expansions have stronger real effects when household production exists and market/home services are substitutable (higher ν).
  - Service-sector dynamics are central for aggregate monetary policy effects given services’ large share of output.
- Model suggests sectoral targeting and consideration of household production complementarities could matter for monetary policy design and the interpretation of sectoral responses to interest-rate changes.

*Source: _wp12206 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .*

### REFERENCES

### _wp12206 - REFERENCES

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### Appendix A — Proof of Proposition 1 (summary of analytical result)
- Objective: Derive an “Augmented” New Keynesian Phillips Curve (NKPC) for the market-service sector by using the pricing equation of re-optimizing monopolistically competitive firms, consumer first order conditions, and equilibrium conditions; result obtained by first order Taylor expansion around the deterministic steady state.
- Key notation preserved:
  - ̄X denotes deterministic steady state value of X; x denotes deviation.
  - Πs,t = Ps,t / Ps,t−1 (gross inflation rate in market-service sector).
  - θs is the exogenous probability that a firm in the market-service sector does not re-optimize price in a given period.
  - εs is the constant elasticity of substitution among varieties in that sector.
- Aggregate market-service price identity:
  - Ps,t = [ (1−θs)P?s,t^(1−εs) + θs Ps,t−1^(1−εs) ]^(1/(1−εs))
  - Implication (Equation (35)): P?s,t / Ps,t = ( (1−θs) Πs,t^(εs−1) / (1−θs) )^(1/(1−εs))
- Pricing rule for firms resetting prices (Equation (36)):
  - P?s,t = (εs/(εs−1)) ( E_t Σ_{i=0}^∞ (β θs)^i MU_{s,t+i} Ψ_{s,t+i} P_{s,t+i}^{εs−1} Y_{s,t+i} ) / ( E_t Σ_{i=0}^∞ (β θs)^i MU_{s,t+i} P_{s,t+i}^{εs−1} Y_{s,t+i} )
- Consumer first-order condition linking marginal utility and wages:
  - MRS_{s,t} = Φ / MU_{s,t} = W_t / P_{s,t}
  - Hence MU_{s,t+i} Ψ_{s,t+i} = Φ P_{s,t+i} / A for all i. (Equation (38))
- The dependence of MU_s on household-produced services enters through the CES aggregate:
  - C_S = [ γ C_h^{(ν−1)/ν} + (1−γ) C_s^{(ν−1)/ν} ]^{ν/(ν−1)}
  - MU_s = ∂U/∂C_s = (∂U/∂C_S)(∂C_S/∂C_s)
- Recursive representations (definitions of V1,t and V2,t) and their linearized forms (Equations (41)a and (41)b):
  - V1,t = A^(−1) C_{s,t} + β θs E_t( Π_{s,t}^{εs} V1,t+1 )
  - V2,t = (C_{s,t}/C_{S,t})^{1−1/ν} + β θs E_t( Π_{s,t}^{εs−1} V2,t+1 )
  - Linearizations:
    - v1,t = (1−β θs) c_{s,t} + β θs εs E_t(π_{s,t+1}) + β θs E_t(v1,t+1)
    - v2,t = (1−1/ν)(1−β θs)(c_{s,t} − c_{S,t}) + β θs (εs−1) E_t(π_{s,t+1}) + β θs E_t(v2,t+1)
- Key linearized relation (Equation (43)):
  - v1,t − v2,t = θs/(1−θs) π_{s,t}
- Final “Augmented” NKPC (result of algebra):
  - π_{s,t} = κs y_{s,t} + (1−1/ν) κs (y_{S,t} − y_{s,t}) + β E_t(π_{s,t+1})
  - κs = (1−θs)(1−β θs) / θs
- Market clearing and definitions used: c_s = y_s and y_S = c_S.

### Appendix B — Proof of Corollary 1 (aggregation across sectors)
- Aggregate inflation linearization:
  - π_t = χ π_{g,t} + (1−χ) π_{s,t}, where χ = ̄P_g ̄C_g / ̄Y.
- Assuming equal price stickiness across sectors (θg = θs = θ ⇒ κg = κs = κ), substitution into sectoral inflation formulas yields:
  - π_t = κ ( χ y_{g,t} + (1−χ) y_{s,t} ) + (1−χ)(1−1/ν) κ ( y_{S,t} − y_{s,t} ) + β E_t(π_{t+1})
- Using y_t = χ y_{g,t} + (1−χ) y_{s,t} gives the aggregated result stated in the text.

### Appendix C — Reduced set of linearized model equations
- Euler/consumption intratemporal relations and identities (as presented):
  - c_{i,t} = E_t c_{i,t+1} − ( r_t − E_t π_{i,t+1} ), i ∈ {g, s}
  - c_i = y_i, i ∈ {g, h, s, S}
  - π_{g,t} = κ_g y_{g,t} + β E_t(π_{g,t+1}), κ_i = (1−θ_i)(1−β θ_i)/θ_i, i ∈ {g, s}
  - π_{s,t} = κ_s y_{s,t} + (1−1/ν) κ_s ( y_{S,t} − y_{s,t} ) + β E_t(π_{s,t+1})
  - r_t = φ_π π_t + φ_y y_t + ln ξ_t
  - ln ξ_t = ρ_r ln ξ_{t−1} + ζ_t
  - y_t = χ y_{g,t} + (1−χ) y_{s,t}
  - π_t = χ π_{g,t} + (1−χ) π_{s,t}
  - c_{S,t} = γ ( ̄C_h / ̄C_S )^{1−1/ν} c_{h,t} + (1−γ) ( ̄C_s / ̄C_S )^{1−1/ν} c_{s,t}
  - c_{h,t} = (1−ν) c_{S,t}
  - w_t − p_{g,t} = c_{g,t}

### Appendix D — Dynamic responses to an expansionary monetary shock (figures described)
- Figures included in the source (names and captions preserved as in the text):
  - Figure 5. Responses of real sectoral consumption to a 1% interest-rate cut.
  - Figure 6. Responses of sectoral inflation and real aggregates to a 1% interest-rate cut.

*Source: _wp12206 - REFERENCES (appendices and bibliography) — content as provided in the PDF chapter.*

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