## _wp1317

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---

### I. Introduction and motivation
- Context: evaluation of fiscal policy actions during the recent financial crisis, focusing on whether fiscal stimulus (i) moderates the output collapse and (ii) boosts job creation given a "jobless recovery".
- Empirical patterns motivating the model:
  - Private investment positively co-moves with output, is considerably more volatile, and leads cyclical deviations of private capital, which lies well below trend after the Great Recession.
  - Hours worked per employee co-move positively with real output; unemployment co-moves negatively with output.
  - Since Q2 2009, hours per employee and output recovered while the unemployment rate remained persistently well above average and vacancy posting below average.
  - Overtime hours per employee increased quickly during recovery and recently deviate positively from trend more than pre-recession.
- Data sources for business-cycle statistics: ALFRED, Federal Reserve Bank of St. Louis and authors’ calculations.

### II. Model features and setup
- Model type and key ingredients:
  - Dynamic stochastic general equilibrium (DSGE) with Mortensen-Pissarides labour market frictions (MPMF).
  - External deep habits in private and public consumption (Ravn et al. (2006) style).
  - Investment adjustment costs; CES production function; adjustments in employment at the intensive and extensive margins.
- Labour matching technology:
  - M_t = κ u_t^ω v_t^(1−ω); worker finding p_t = k θ_t^(1−ω); firm filling q_t = k θ_t^(−ω); employment law n_{t+1} = M_t + (1−λ) n_t.
- Deep habit in private consumption:
  - (X^c_t)_j = [ ∫_0^1 (C_{it}^j − θ_c S^c_{it−1})^(1−1/η) di ]^(1/(1−1/η)), S^c_{it} = ρ_c S^c_{it−1} + (1−ρ_c) C_{it}.
- Government consumption with deep habits and shock process:
  - X^g_t analogous to private; log(G_t / ̄G) = ρ_G log(G_{t−1} / ̄G) + ε^g_t, ε^g_t i.i.d. mean zero, st.dev. σ_G.
- Firms: differentiated goods, hiring costs HC_{it} = g(z_{it}) n_{it}, vacancy/job-creation condition derived from expected surplus.
- Wages: Nash bargaining with firm power ε ∈ [0;1]; surplus-splitting S^w_{it} = (1−ε)/ε (1−τ^w_t) S^f_{it}; hours chosen privately efficiently (hours independent of hourly wage when τ^w_t = 0).
- CES production:
  - F = [ α_K ((Z^K)_t K_t)^( (σ−1)/σ ) + α_N ((Z^N)_t n_t h_t)^( (σ−1)/σ ) ]^( σ/(σ−1) ); σ is elasticity of substitution between capital and labour.

### III. Baseline calibration and steady-state targets
- Time period: one quarter.
- Key parameter values (from Table 1 and text):
  - Discount factor β = 0.99
  - Capital share of income S_K = 1 =3 (text uses notation "1 =3")
  - Capital depreciation rate δ = 0.025
  - Relative risk aversion σ_c = 2
  - Elasticity of substitution in production function σ = 0.4
  - Elasticity across varieties η = 6
  - Investment adjustment cost parameter γ = 3.24
  - Degree of deep habit formation θ_c = 0.86
  - Habit persistence ρ_c = 0.85
  - Job separation rate λ = 0.103
  - Elasticity of matching to unemployment ω = 0.5
  - Firms’ bargaining power ε = 0.5
  - Share of government spending in output ̄g = ̄y = 0.2
  - Persistence of government spending shock ρ_g = 0.90
  - Persistence of tax shocks ρ_X = 0.90
  - Convexity in hiring cost ψ = 0
  - Elasticity of substitution leisure/consumption ρ set to target ̄h = 0:33
  - Scaling factor in hiring cost χ set to target ̄p = 0:95
  - Scaling factor in matching function κ set to target ̄q = 0:70
  - Unemployment benefit w_u set to target ̄Θ = 0:70
- Steady-state "great ratios" resulting from calibration:
  - Consumption/output ratio = 61%
  - Investment/output ratio = 18%
  - Hiring costs/output ratio = 1%
  - Steady-state unemployment rate ≈ 9.5%

### IV. Main quantitative and qualitative results
- Baseline fiscal experiment: government spending expansion sized 1% of output, financed by lump-sum taxes (balanced budget); unemployment responses reported in absolute percentage points; output responses interpreted as fiscal multipliers.
- Primary findings:
  - (i) Output multipliers in the high range of empirical estimates can be obtained even without nominal rigidities.
  - (ii) The model can reproduce a fiscal expansion with low job creation ("jobless" fiscal expansion).
  - (iii) A fiscal stimulus can mitigate output collapse while containing the rise in unemployment only marginally—consistent with post-Great Recession observations.
- Quantitative examples (selected):
  - Neoclassical benchmark (Cobb-Douglas, no deep habits): output multiplier slightly above 0.5; almost negligible negative effects on unemployment.
  - Cobb-Douglas with deep habits: output multiplier ≈ 2; peak unemployment multiplier ≈ −0.6 percentage points.
  - CES production with σ = 0.4: output multiplier ≈ 1.6 (81% of CD case); unemployment multiplier ≈ −0.35 percentage points (≈ 66% of CD case).
- Jobless recovery experiment:
  - Recession via negative technology shock: CES yields peak output contraction ≈ 7.5% from steady state; Cobb-Douglas yields ≈ 6%.
  - Peak unemployment increase: CES ≈ more than 4 percentage points; CD ≈ 2.5 percentage points.
  - Fiscal stimulus scenario: government spending expansion of 5% of output (lump-sum taxes; balanced budget, approximating ARRA).
  - With stimulus: output contraction is ~50% of no-policy scenario under CD and ~30% under CES; rise in unemployment with stimulus is ~50% less pronounced under CD and ~20% less pronounced under CES.
  - Conclusion: under empirically plausible σ<1, stimulus raises output mainly via intensive margin (hours) rather than extensive margin (job creation), producing a more jobless recovery.

### V. Mechanisms behind the "jobless" outcome
- Key interaction:
  - Deep habits magnify macro responses to fiscal stimulus by inducing a countercyclical mark-up (via intra-temporal price-elasticity effect and inter-temporal locking-in effect), crowding in private consumption and raising real wages.
  - CES production with lower σ (capital and labour more complementary) reduces firms' incentives to create new jobs because capital is a slow-adjusting stock (investment adjustment costs), so expansion relies more on increasing hours of existing employees.
- Economic intuition:
  - Lower σ → closer to Leontief → capital and labour complements → constrained capital adjustment → vacancy posting less attractive → smaller unemployment decline.
  - Negative wealth effect and substitution of leisure with consumption still increase hours per employee, supporting output growth without much job creation.

### VI. Sensitivity exercises and robustness
- Common fiscal experiment: government spending expansion 1% of output (lump-sum taxes, balanced budget).
- Bargaining power and σ (Figure B.1):
  - If σ = 0:10 (almost Leontief) real wage can decline despite deep habits; firms post fewer vacancies; output multiplier < 1; unemployment small or positive.
  - As σ increases to Cobb-Douglas (σ→1), bargaining power ε affects real wage and unemployment responses.
- Hagedorn and Manovskii effect (replacement ratio Θ):
  - Higher ̄Θ increases magnitudes of both output and unemployment multipliers; output multiplier changes marginally, unemployment multiplier can be considerably higher.
- Quantitative implications (Table B.1 selected entries):
  - Θ = 0:7, ε = 0:5: ∆Y/∆G = 1.95 (σ→1) and 1.57 (σ=0:4); ∆u/∆G = −0.53 and −0.35.
  - Θ = 0:9, ε = 0:5: ∆Y/∆G = 1.91 and 1.54; ∆u/∆G = −1.29 and −0.79.
  - General conclusion: dropping σ from 1 to 0.4 reduces output multipliers to around 4/5 and unemployment multipliers to around or below 2/3 of CD case.
- Sensitivity to investment adjustment costs and leisure/consumption elasticity (Table B.2):
  - No investment adjustment costs (γ = 0): ∆Y/∆G = 2.50 (σ→1) and 1.81 (σ=0:4); ∆u/∆G = −0.40 and −0.25. Investment adjustment costs are important to deliver a jobless stimulus.
  - Varying steady-state hours (̄h = 0:25 or 0:40) affects ∆Y/∆G; unemployment multiplier relatively insensitive to ρ.
- Debt-financed fiscal policy and distortionary taxation:
  - Introducing distortionary taxes with steady-state rates τc = 0:05, τw = 0:24, τk = 0:32 and tax feedback ρXB = 0:02 reduces absolute output and unemployment multipliers; unemployment affected more due to tax distortions on employment.
- New-Keynesian (NK) extension with sticky prices (Appendix C):
  - Price-adjustment costs ξ and Taylor rule with output gap coefficient ρy matter: an aggressive monetary response to output gap (ρy sufficiently large) can offset fiscal expansion effects, potentially producing crowding-out and rising unemployment.
  - Empirically plausible low ρy values imply robustness of crowding-in and countercyclical mark-up results in the NK extension.

### VII. Policy implications and conclusions
- Combination of deep habits and CES technology is crucial to reconcile:
  - Large output multipliers (deep habits) with limited job creation (σ<1 → capital-labour complementarity).
- Policy recommendation:
  - To achieve stronger reductions in unemployment, fiscal stimulus should prioritize measures that enhance the economy’s stock of capital (e.g., incentives for private investment and direct government investment in public infrastructure).
- Further research:
  - Model suitable for designing optimal fiscal and monetary rules; investigate optimised Taylor rules and monetary-fiscal interactions (sensitivity to monetary response documented).

### VIII. Appendix — business-cycle statistics and steady-state relationships (selected numerical facts)
- Business-cycle standard deviations relative to real output (1995-2007 and 1995-2011):
  - Private investment: 4.16 and 4.90
  - Private capital: 1.04 and 1.14
  - Hours per employee: 0.55 and 0.53
  - Overtime hours per employee: 4.20 and 5.84
  - Job openings: 3.72 and 3.19
  - Unemployment rate: 10.06 and 17.38
- Autocorrelations (1995-2007 and 1995-2011) include GDP = 0.93 and 0.91; private investment = 0.89 and 0.87; unemployment rate = 0.96 and 0.98.
- Correlations with real output (1995-2007 and 1995-2011):
  - Private investment: 0.88 and 0.86
  - Private capital: 0.59 and 0.41
  - Hours per employee: 0.88 and 0.91
  - Unemployment rate: −0.74 and −0.60

*Source: extracted content from _wp1317 (IMF working paper chapter/section).*

### References................................................................................................28

### _wp1317 - References................................................................................................28

### I. Introduction and motivation
- Context: fiscal policy actions during the recent financial crisis and the consequent focus on whether fiscal stimulus (i) moderates the output collapse and (ii) boosts job creation, in the light of a "jobless recovery".
- Empirical patterns motivating the analysis:
  - Private investment positively co-moves with output, exhibits considerably greater volatility and leads cyclical deviations of the stock of capital, which lies well below trend in the quarters following the Great Recession.
  - Hours worked per employee co-move positively with real output; unemployment co-moves negatively with output.
  - Since Q2 2009 (the trough of the great recession), hours worked per employee and output have been on a recovery path, while the unemployment rate has persistently remained well above average and vacancy posting below average.
  - Overtime hours per employee have quickly increased during the recovery and recently deviate positively from trend more than in the pre-recession period.
- Data sources cited in Figure 1: ALFRED, Federal Reserve Bank of St. Louis and authors’ calculations.
- Measurement notes from Figure 1:
  - Percentage deviations from HP-trend for GDP, private investment, private capital, hours per employee, and overtime hours per employee.
  - Percentage deviations from the sample mean for vacancies and the unemployment rate.

### II. Model features and setup
- Model type: dynamic stochastic general equilibrium (DSGE) model with:
  - Mortensen-Pissarides labour market frictions (MPMF);
  - Deep habits in private and public consumption;
  - Investment adjustment costs;
  - Constant-elasticity-of-substitution (CES) production function;
  - Adjustments in employment at both the intensive and extensive margin.
- Model is calibrated/assessed to be consistent with four empirical regularities:
  - (a) private consumption increases following a public spending expansion;
  - (b) real wage increases following a public spending expansion;
  - (c) the mark-up is countercyclical and falls following a government spending shock;
  - (d) factor shares are time-varying at business cycle frequencies and capital and labour are gross complements in production.

### III. Main quantitative and qualitative results
- Primary results:
  - (i) Output multipliers in the high range of empirical estimates can be obtained even in the absence of nominal rigidities.
  - (ii) The model can reproduce a fiscal expansion with low job creation ("jobless" fiscal expansion).
  - (iii) The model can simulate a fiscal stimulus that mitigates the output collapse in a recession but contains the rise in unemployment only marginally—consistent with post-Great Recession observations.
- Mechanisms behind the "jobless" outcome:
  - The combination of deep habits and CES technology is crucial.
  - If the elasticity of substitution between capital and labour approaches one (production approximates Cobb-Douglas), deep habits magnify macroeconomic responses to fiscal stimulus.
  - When the elasticity of substitution drops to values in the range of available estimates (i.e., capital and labour become more complements), then:
    - The output multiplier falls only marginally;
    - The unemployment multiplier experiences a sizeable contraction.
  - Economic intuition: Lower elasticity (closer to Leontief) implies capital and labour are more complements; because capital cannot change instantaneously (stock nature and investment adjustment costs), firms have smaller incentives to create new jobs via costly vacancy posting. Negative wealth effect and substitution of leisure with consumption (from mark-down decline due to deep habits) still increase hours of work per employee. Hence output expansion is driven relatively more by increased hours of current employees than by new job creation.

### IV. Relationship to empirical and theoretical literature
- Empirical literature on fiscal multipliers:
  - VAR estimates of the output multiplier generally range from 0.7 to 2.5.
  - References to specific findings (as summarized in the text): pessimistic estimates around 0.7; some around 1; others report values above 1; Auerbach and Gorodnichenko (2012) report output multiplier of up to 2.5 during recessions.
  - Caldara and Kamps (2008) find that differences in sample selection and VAR specification explain many disagreements; after harmonizing identification, private consumption and the real wage increase after a government spending shock.
  - Caldara and Kamps (2012) provide further evidence in favour of the spending multiplier being larger than one.
  - Monacelli and Perotti (2008) and Canova and Pappa (2011) provide evidence of a countercyclical mark-up in response to government spending shocks.
- Theoretical literature:
  - In models with rational expectations, government spending multipliers are typically small due to negative wealth effects crowding out private consumption and investment.
  - Woodford (2011) summarized multiplier properties:
    - Below one in a neoclassical RBC model and exactly the same in an RBC with monopolistic competition and in a sticky-price NK model with strict inflation targeting;
    - Exactly one in an NK model with fixed real interest rate;
    - Between the two values in a model featuring a Taylor rule.
  - New-Keynesian multipliers depend on monetary policy accommodation; larger multipliers possible at the ZLB (e.g., Christiano et al. (2011) find multipliers may reach 10 at the ZLB under specific timing).
  - To study fiscal effects on unemployment, many studies embed MPMF frictions into DSGE models; standard issue: the "unemployment volatility puzzle"—difficulty matching observed unemployment volatility.
  - Responses to the unemployment volatility puzzle have included staggered nominal wages (criticized by Pissarides (2009)) and the introduction of deep habits in consumption (Di Pace and Faccini (2012) following Ravn et al. (2006)).

### V. Structure and supplementary material
- Paper structure (as stated):
  - Section II: literature context and relation to jobless recovery literature.
  - Section III: model description.
  - Section IV: parameter choice.
  - Section V: results, isolation of features, simulation of a scenario compatible with the jobless recovery.
  - Section VI: conclusions and agenda for future research.
- Online appendix contents (high level):
  - Battery of sensitivity exercises;
  - Sticky prices extension of the model;
  - Symmetric equilibrium and steady state of the baseline model.
- Appendix and figures/tables index excerpts in the source:
  - Appendix sections and subsections listed (e.g., A Business Cycle Statistics; B Sensitivity Exercises including Bargaining Power, Hagedorn and Manovskii Effect, etc.).
  - Tables referenced: Table 1 Baseline Calibration; Table A.1 Business Cycle Properties; Table B.1 The Impact of the Fiscal Stimulus in Different Scenarios; Table B.2 The Impact of the Fiscal Stimulus in Different Scenarios.
  - Figures referenced: Figure 1 A Jobless Recovery; Figures 2–5 and Appendix figures B.1–C.2 with various sensitivity and comparison plots.

*Source: content unit _wp1317 - References................................................................................................28 (source PDF content provided).*

### introduction

### introduction

### Main findings on deep habits and fiscal expansions
- Inclusion of deep habits in a DSGE model implies that a government spending expansion, even with flexible prices, reduces the mark-up, fosters the real wage, and crowds in private consumption.
- With deep habits and Mortensen-Pissarides matching and bargaining (MPMF) frictions, the model magnifies the amplitude of unemployment responses to shocks and matches three of four empirical regularities:
  - (i) private consumption is typically crowded in by a government spending expansion, rather than crowded out as a canonical DSGE model predicts;
  - (ii) the real wage increases after a government spending expansion, rather than falling as in the canonical Real Business Cycle (RBC) model;
  - (iii) the mark-up is typically countercyclical.
- Introducing a CES production function completes the picture by matching:
  - (iv) the evidence of gross complementarity between capital and labour in production and time-varying factor shares,
  - and allowing the model to reproduce a scenario compatible with the jobless recovery.

### Jobless recovery — interpretation and literature positioning
- The paper does not claim exclusivity in explaining jobless outcomes of fiscal stimulus; the jobless recovery remains controversial.
- Structural explanations for delayed unemployment responses in recoveries include:
  - more flexible labour force (temporary workers and offshoring),
  - temporary increase in the natural rate of unemployment (Daly et al., 2011),
  - mismatch between job-seekers and vacancies (Sahin et al., 2011),
  - increases in health benefits and faster sectoral reallocation (Groshen and Potter (2003); Andolfatto and MacDonald (2004); Schreft et al. (2005)).
- Some studies find little support for structural change hypotheses (Aaronson et al. (2004a); Aaronson et al. (2004b)).
- Cyclical explanations include negative labour supply shocks (Aaronson et al. (2004a)), sluggish aggregate demand (Bernanke (2003)), adjustment costs to the extensive margin (Bachmann (2011)), and combinations of wage rigidities and labour frictions (Shimer (2012)).
- This paper links the jobless outcome of fiscal stimulus to factor complementarity, while acknowledging that structural and cyclical mechanisms can reinforce the mechanism presented.

### Model: labour market search-match technology
- Matches M_t follow a Cobb-Douglas function:
  - M_t = κ u_t^ω v_t^(1−ω),
  - κ represents matching efficiency,
  - ω ∈ (0;1) is elasticity of matches to unemployment.
- Worker finding probability:
  - p_t = M_t / u_t = k θ_t^(1−ω), where θ_t ≡ v_t / u_t.
- Firm filling probability:
  - q_t = M_t / v_t = k θ_t^(−ω).
- Employment law of motion:
  - n_{t+1} = M_t + (1−λ) n_t,
  - λ is exogenous job destruction rate.

### Households: preferences, habits, and constraints
- Households indexed j ∈ [0;1]; members can be employed or unemployed; total population normalized to one.
- External deep habit formation in consumption (Ravn et al. (2006)) with habit-adjusted composite (per household):
  - (X^c_t)_j = [ ∫_0^1 (C_{it}^j − θ_c S^c_{it−1})^(1−1/η) di ]^(1/(1−1/η))
  - θ_c ∈ (0;1) degree of deep habit formation;
  - S^c_{it} evolves: S^c_{it} = ρ_c S^c_{it−1} + (1−ρ_c) C_{it}.
- Household instantaneous utility:
  - U((X^c_t)_j; n^j_t; 1−h^j_t) = n^j_t U((X^c_t)_j; 1−h^j_t) + (1−n^j_t) U((X^c_t)_j; 1).
- Beveridge curve for household j:
  - n^j_{t+1} = (1−λ) n^j_t + p(θ_t) (1−n^j_t).
- Capital accumulation:
  - K^j_{t+1} = (1−δ) K^j_t + I^j_t [1−S(I^j_t / I^j_{t−1})], with S(1)=S′(1)=0, S′′(1)>0.
- Investment demand for variety i:
  - I^j_{it} = (P_{it} / P_t)^(−η) I^j_t.
- Expenditure minimisation yields demand for variety i:
  - C^j_{it} = (P_{it} / P_t)^(−η) (X^c_t)_j + θ S^c_{it−1}.
- Real consumption relates to composite and habit stock:
  - C^j_t = (X^c_t)_j + Ω_t, Ω_t ≡ θ_c ∫_0^1 (P_{it} / P_t) S^c_{it−1} di.
- Household lifetime maximisation subject to budget constraint (taxes τ^C_t, τ^W_t, τ^K_t; lump-sum τ_t):
  - (1+τ^C_t) ((X^c_t)_j + Ω_t) + I^j_t + τ_t + B^j_t = (1−τ^W_t) n^j_t h^j_t w_{it} + (1−n^j_t) w_u + (1−τ^K_t) R^K_t K^j_t + R_t B^j_{t−1} + ∫_0^1 J_{it} di.
- Euler equation for capital (Tobin’s Q):
  - Q^j_t = E_t { D^j_{t;t+1} [ (1−τ^k_{t+1}) R^K_{t+1} + (1−δ) Q^j_{t+1} ] }.
- First-order condition for investment:
  - Q^j_t (1−S(I^j_t/I^j_{t−1})−S′(I^j_t/I^j_{t−1}) I^j_t/I^j_{t−1}) + E_t( D^j_{t;t+1} Q^j_{t+1} S′(I^j_{t+1}/I^j_t) (I^j_{t+1}/I^j_t)^2 ) = 1.
- Bond pricing:
  - 1 = E_t[ D^j_{t;t+1} R_{t+1} ].
- Household surplus from employment:
  - (S^w_t)_j = (1−τ^W_t) w_{kt} h^j_t − [ w_u − U^n_t / U^x_t ] + (1−λ−p(θ_t)) E_t[ D_{t;t+1} (S^w_{t+1})_j ].
- Hours chosen to make bargain privately efficient.

### Government: habits, spending process, and fiscal rule
- Government consumption exhibits deep habits analogously to private consumption:
  - X^g_t = [ ∫_0^1 (G_{it} − θ_c S^g_{it−1})^(1−1/η) di ]^(1/(1−1/η))
  - S^g_{it} = ρ_c S^g_{it−1} + (1−ρ_c) G_{it}.
- Government minimises expenditure subject to ∫_0^1 P_{it} G_{it} di ≤ P_t G_t; optimum:
  - G_{it} = (P_{it} / P_t)^(−η) X^g_t + θ_c S^g_{it−1}.
- Aggregate real government consumption process:
  - log(G_t / ̄G) = ρ_G log(G_{t−1} / ̄G) + ε^g_t, ε^g_t i.i.d. mean zero, st.dev. σ_G.
- Government budget constraint:
  - B_t = R_t B_{t−1} + G_t + (1−n_t) w_u − τ_t − τ^C_t C_t − τ^W_t w_t n_t h_t − τ^K_t R^k_t K_t.
- Taxes follow feedback rule for each tax X ∈ {τ, τ^c, τ^w, τ^k}:
  - log(X_t / ̄X) = ρ_X log(X_{t−1} / ̄X) + ρ_X^B B_{t−1} / Y_{t−1} + ε^X_t.
- Benchmark balanced-steady-state scenario: steady-state government debt and tax rates set equal to zero implies B_t = τ^C_t = τ^W_t = τ^K_t = 0 and τ_t = G_t + (1−n_t) w_u for fully financed lump-sum taxation.

### Firms: production, costs, pricing, and job creation
- Firms indexed i ∈ [0;1] produce differentiated goods Y_{it} with technology F((Z^K)_t K_{it}; (Z^N)_t n_{it} h_{it}), selling at price p_{it} ≡ P_{it} = P_t.
- Employment law of motion at firm i:
  - n_{i,t+1} = (1−λ) n_{it} + q(θ_t) v_{it}.
- Hiring costs per firm:
  - HC_{it} = g(z_{it}) n_{it}, with z_{it} ≡ v_{it} / n_{it}, g′; g′′ ≥ 0.
- Firm profit maximisation subject to resource and demand constraints yields first-order conditions including:
  - R^K_t = MC_t F_{K;it},
  - μ_{it} = (MC_t F_{N;it} − w_{it}) h_{it} + g′(z_{it}) z_{it} − g(z_{it}) + (1−λ) E_t[ D_{t;t+1} μ_{it+1} ],
  - g′(z_{it}) = q(θ_t) E_t[ D_{t;t+1} μ_{it+1} ],
  - ν^c_t = p_{it} − MC_t + (1−ρ_c) λ^c_t, with analogous conditions for ν^g_t.
- Symmetric equilibrium: p_{it} = 1, mark-up is inverse of marginal cost.
- Vacancy/job-creation condition (iterated and combined):
  - g′(z_{it}) / q(θ_t) = E_t[ D_{t;t+1} μ_{it+1} ] = E_t{ D_{t;t+1} [ (MC_t F_{N;it+1} − w_{it+1}) h_{it+1} + g′(z_{it+1}) z_{it+1} − g(z_{it+1}) + (1−λ) g′(z_{it+1}) / q(θ_{t+1}) ] }.
- Competitive labour market outcome (no hiring costs): MC_t F_{N;it} = w_{it}.

### Wage bargaining and hours
- Nash bargaining with firm bargaining power ε ∈ [0;1]:
  - max_{w_{it}} (S^w_{it})^(1−ε) (S^f_{it})^ε.
- Surplus-splitting rule:
  - S^w_{it} = (1−ε)/ε (1−τ^w_t) S^f_{it}.
- Wage equation:
  - w_{it} h_{it} = (1−ε) [ MC_t F_{N;it} h_{it} − g(z_{it}) + g′(z_{it}) z_{it} + θ_t g′(z_{it}) ] + ε [ w_u − U^n_t / U^x_t 1−τ^w_t ].
- Hours determined privately efficiently (maximising joint surplus), yielding:
  - MC_t F_{N;it} = − U_{nh;it} / U^x_{i;t} + τ^w_t w_t.
  - With τ^w_t = 0 ∀t (as in most experiments), hours are independent of the hourly wage.

### Equilibrium and resource constraint
- Market clearing and resource constraint:
  - Y_t = C_t + I_t + G_t + g(z_t) n_t.
- Full equilibrium system and steady state summarised in online appendix (Section D) and steady state in Section E (as noted in the text).

### CES production function and re-parameterization
- CES production function specified:
  - F = [ α_K ((Z^K)_t K_t)^( (σ−1)/σ ) + α_N ((Z^N)_t n_t h_t)^( (σ−1)/σ ) ]^( σ/(σ−1) ),
  - σ is elasticity of substitution between capital and labour;
  - α_K and α_N distribution parameters (not dimensionless outside Cobb-Douglas).
- As σ → 1, CES collapses to Cobb-Douglas iff α_K + α_N = 1.
- Marginal products:
  - F_{K;t} = α_K (Z^K)^{(σ−1)/σ}_t (Y_t / K_t)^{1/σ},
  - F_{N;t} = α_N (Z^N)^{(σ−1)/σ}_t (Y_t / (n_t h_t))^{1/σ}.
- Re-parameterization expressing α_K and α_N as functions of calibrated capital share S_K and endogenous variables:
  - α_K = S_K ( Y / (Z^K · K) )^{(σ−1)/σ},
  - α_N = (1−S_K) ( Y / (Z^N n h) )^{(σ−1)/σ}.
- Note: with non-Walrasian labour market, (1−S_K) does not equal labour share S_N, because it includes search-matching wage costs S_SM ≡ g(z) n / Y; in equilibrium S_K + S_N + S_SM = 1.

### Additional functional forms
- Instantaneous utility specialised:
  - U(X^c_t; n_t; 1−h_t) = n_t [ X^c_t^(1−ρ) (1−h_t)^ρ ]^(1−σ_c) − 1 / (1−σ_c) + (1−n_t) X^c_t^(1−ρ)(1−σ_c) − 1 / (1−σ_c).
  - σ_c > 0 coefficient of relative risk aversion; ρ elasticity of substitution between leisure and consumption.
- Investment adjustment costs quadratic:
  - S(I_t / I_{t−1}) = γ / 2 ( I_t / I_{t−1} − 1 )^2; μ>0.
- Hiring cost function (possibly convex):
  - g(z_t) = χ / (1+ψ) z_t^(1+ψ); ψ ≥ 0.

### Parameter choice (baseline calibration)
- Time period: one quarter.
- Parameter values (as listed in Table 1):
  - Discount factor β 0.99
  - Capital share of income S_K 1 =3
  - Capital depreciation rate δ 0.025
  - Relative risk aversion σ_c 2
  - Elasticity of substitution in production function σ 0.4
  - Elasticity of substitution across varieties η 6
  - Investment adjustment cost parameter γ 3.24
  - Degree of deep habit formation θ_c 0.86
  - Habit persistence ρ_c 0.85
  - Job separation rate λ 0.103
  - Elasticity of matching to unemployment ω 0.5
  - Firms’ bargaining power ε 0.5
  - Share of government spending in output ̄g = ̄y 0.2
  - Persistence of government spending shock ρ_g 0.90
  - Persistence of tax shocks ρ_X 0.90
  - Convexity in hiring cost ψ 0
  - Elasticity of subst leisure/consumption ρ set to target ̄h = 0:33
  - Scaling factor in hiring cost function χ set to target ̄p = 0:95
  - Scaling factor in matching function κ set to target ̄q = 0:70
  - Unemployment benefit w_u set to target ̄Θ = 0:70

*Source: _wp1317 - introduction*

### 0.025 and 2, respectively, while the capital share of income,S

### _wp1317 - 0.025 and 2, respectively, while the capital share of income,S

### Calibration and model setup
- Preference, technology and market parameters:
  - Capital share of income, S_K, takes the conventional value of 1=3.
  - Elasticity of substitution across varieties, η, set to 6 (implies steady-state mark-up ≈ 20% in absence of deep habits).
  - When production is CES, elasticity of substitution, σ, set to 0.40 (close to empirical estimates in León-Ledesma et al. (2012)). Cobb-Douglas obtained as σ→1.
  - Investment adjustment cost parameter = 3.24 (Christiano et al., 2005).
  - Deep habit formation degree, θ_c = 0.86; habit persistence, ρ_c = 0.85 (values from Ravn et al. (2006)).
  - Convexity parameter in hiring cost function, ψ = 0 (hiring cost linear).
  - Firms’ bargaining power, ε = 0.5.
  - Elasticity of matching to unemployment, ω = 0.5.
  - Job separation rate, λ = 0.103 (implies jobs last on average 2 years and a half; aligns with Shimer (2005)).
  - Persistence of fiscal shocks = 0.90 (approximate empirical value; see Monacelli et al., 2010).
- Parameters chosen to match steady-state targets by setting: (i) elasticity of substitution between leisure and consumption, ρ; (ii) scaling factor in hiring cost, χ; (iii) scaling factor in matching function, κ; and (iv) unemployment benefit, w_u, so as to match:
  - Steady-state share of hours worked over total hours, h̄ = 33%.
  - Steady-state job finding probability, p̄ = 95% (as in Gertler et al. (2008)).
  - Vacancy filling probability, q̄ = 70% (as in Trigari (2009)).
  - Replacement ratio, Θ̄ ≡ (w_u − Ū_n)/(Ū_c F_n) = 70% (close to point estimate of 72% by Sala et al. (2008)); sensitivity to this parameter shown in online appendix, Section B.2.
- Resulting steady-state "great ratios":
  - Consumption/output ratio = 61%.
  - Investment/output ratio = 18%.
  - Hiring costs/output ratio = 1%.
  - Steady-state unemployment rate ≈ 9.5% (from Beveridge curve given chosen λ and p̄; close to Hall (2005)).

### Results — overview
- Base experiment: government spending expansion sized 1% of output, financed by lump-sum taxes. Output responses interpreted as fiscal multipliers; unemployment responses reported in absolute percentage points.
- General findings:
  - Baseline RBC with search-match frictions and Cobb-Douglas production (no deep habits) yields output multipliers well below many empirical estimates and nearly negligible negative effects on unemployment.
  - Introducing deep habits magnifies macroeconomic responses to fiscal stimulus: mark-up falls, real wages rise, consumption is crowded in.
  - Introducing CES production with lower σ (capital and labour more complementary) shifts the expansion to be sustained relatively more by intensive margin (hours worked) than extensive margin (job creation). Factor shares exhibit cyclical fluctuations.
  - Fiscal stimulus at recession time fosters a jobless recovery under CES technology.
  - Monetary accommodation and price stickiness materially affect stabilisation properties of fiscal stimulus (shown in online appendix).

### A. Neoclassical benchmark with search-match frictions
- In Cobb-Douglas, no deep habits:
  - Fiscal expansion generates negative wealth effect (higher tax obligations) → curbs consumption, boosts labour supply.
  - Labour market: households’ reservation wage falls more than firms’ reservation wage → more vacancies posted, tighter labour market, equilibrium unemployment falls, real wage falls.
  - Private investment is crowded out; real interest rate rises.
  - Price mark-up remains constant; capital and labour shares of income remain constant.
- Quantitative result:
  - Output multiplier slightly above 0.5 for this calibration (government spending expansions yield output multipliers well below one).
  - Almost negligible negative effects on unemployment.

### B. Deep habits
- Mechanism:
  - Deep habits induce counter-cyclical mark-up even with fully flexible prices via two effects:
    - Intra-temporal (price-elasticity) effect: demand AD_it = C_it + G_it + I_it = (P_it/P_t)^−η (X^c_t + X^g_t + I_t) + θ_c (S^c_it−1 + S^g_it−1). Habit-adjusted aggregate demand rise increases effective price elasticity ˜η_it relative to η when θ_c>0.
    - Inter-temporal effect: firms lower current mark-up to lock-in consumers (anticipating higher future sales), reducing current profits to gain future mark-ups.
- Outcome under deep habits (Cobb-Douglas baseline calibration):
  - Mark-up falls; future sales expected to be higher → more vacancy posting → higher labour market tightness → larger fall in unemployment.
  - Intensive margin (hours worked) increases and real wage increases due to higher firm reservation wage leading to higher bargained wage.
  - Leisure becomes relatively more expensive → substitution towards consumption that more than offsets negative wealth effect → consumption rises.
  - Quantitative:
    - Output multiplier ≈ 2 in Cobb-Douglas with deep habits.
    - Peak unemployment multiplier ≈ −0.6 percentage points.

### C. CES production function (role of σ)
- Empirical motivation:
  - Estimates of σ between 0.3 and 0.6 (Klump et al., 2007; Chirinko, 2008; Cantore et al., 2012a, 2010b; León-Ledesma et al., 2012).
- Key results as σ declines from 1 to lower values:
  - σ = 0.4: output multiplier ≈ 1.6 (which is 81% of the CD case value); unemployment multiplier ≈ −0.35 percentage points (≈ 66% of CD case).
  - Lower σ makes capital and labour closer to Leontief (gross complements); capital cannot adjust instantaneously → firms have smaller incentives to create new jobs via vacancy posting.
  - Despite smaller job creation, negative wealth effect and substitution of leisure with consumption still generate substantial increases in hours supplied (intensive margin).
  - Effect stronger with investment adjustment costs (see online appendix, Section B.4).
- Sensitivity:
  - When σ drops from 1 to σ = 0.3 (lower bound of empirical estimates), peak elasticity of unemployment rate to output drops by around 15% (Figure 4).
- Aggregate implication:
  - As σ falls, output growth from government spending is sustained more by increased hours per worker (intensive margin) rather than new job creation (extensive margin). Factor shares exhibit cyclical responses.

### D. Jobless recovery (fiscal stimulus during recession)
- Recession simulated via negative technology shock calibrated so CES case yields peak output contraction ≈ 7.5% from steady state.
  - Same technology shock yields 6% peak output contraction when production is Cobb-Douglas.
  - Peak unemployment increase: CES ≈ more than 4 percentage points; CD ≈ 2.5 percentage points.
- Fiscal stimulus experiment: government spending expansion of 5% of output (lump-sum taxes; balanced budget), approximating ARRA spending increases.
- Findings:
  - Fiscal stimulus produces similar output stabilisation under CD and CES, but unemployment stabilisation is much less pronounced under CES.
  - Quantitative illustrative ratios:
    - With fiscal stimulus, output contraction is ~50% of no-policy scenario under CD and ~30% under CES.
    - Rise in unemployment with fiscal stimulus is ~50% less pronounced under CD and ~20% less pronounced under CES.
  - Conclusion: at recession time, CES technology predicts a considerably more jobless recovery—the stimulus raises output mainly via intensive margin rather than job creation.

### Policy implications and concluding remarks
- Combination of deep habits and CES technology is crucial:
  - Deep habits magnify macroeconomic responses to fiscal stimulus.
  - Elasticity of substitution between capital and labour in empirically plausible range (σ<1) produces jobless recovery by emphasizing intensive margin adjustments.
- Policy implication:
  - To have stronger impact on reducing unemployment, fiscal stimulus should prioritize measures that enhance the economy’s stock of capital (e.g., incentives for private investment and direct government investment in public infrastructure).
- Further research avenues:
  - Model is suitable for designing optimal fiscal and monetary rules.
  - Sensitivity to monetary response suggests examining optimised Taylor rules would be useful (references to Cantore et al. (2012b) and Levine et al. (2008) noted).

*Italic: Source — extracted content from the provided PDF chapter/section.*

### References

### _wp1317 - References

### References (selected bibliographic entries)
- Aaronson, D., Rissman, E., and Sullivan, D. G. (2004a). Assessing the jobless recovery. Economic Perspectives, (Q II):2–21.
- Aaronson, D., Rissman, E., and Sullivan, D. G. (2004b). Can sectoral reallocation explain the jobless recovery? Economic Perspectives, (Q II):36–39.
- Acconcia, A., Corsetti, G., and Simonelli, S. (2011). Mafia and public spending: Evidence on the fiscal multiplier from a quasi-experiment. CEPR Discussion Papers 8305, C.E.P.R. Discussion Papers.
- Auerbach, A. J., Gale, W. G., and Harris, B. H. (2010). Activist fiscal policy. Journal of Economic Perspectives, 24(4):141–64.
- Auerbach, A. J. and Gorodnichenko, Y. (2012). Measuring the output responses to fiscal policy. American Economic Journal: Economic Policy, 4(2):1–27.
- Barro, R. J. and Redlick, C. J. (2011). Macroeconomic effects from government purchases and taxes. The Quarterly Journal of Economics, 126:51–102.
- Blanchard, O. and Perotti, R. (2002). An empirical characterization of the dynamic effects of changes in government spending and taxes on output. The Quarterly Journal of Economics, 117(4):1329–1368.
- Christiano, L., Eichenbaum, M., and Rebelo, S. (2011). When is the government spending multiplier large? Journal of Political Economy, 119(1):pp. 78–121.
- Gali, J. (2011). Unemployment Fluctuations and Stabilization Policies: A New Keynesian Perspective. The MIT Press.
- Ramey, V. A. (2009). Identifying government spending shocks: It’s all in the timing. NBER Working Papers 15464, National Bureau of Economic Research, Inc.
- Woodford, M. (2011). Simple analytics of the government expenditure multiplier. American Economic Journal: Macroeconomics, 3(1):1–35.
- (Additional bibliographic entries appear in the supplied content.)

### Appendix A — Business Cycle Statistics (Table A.1)
- Standard deviations relative to real output (two sample periods reported: 1995- 2007 and 1995 - 2011):
  - Private investment: 4.16 and 4.90
  - Private capital: 1.04 and 1.14
  - Hours per employee: 0.55 and 0.53
  - Overtime hours per employee: 4.20 and 5.84
  - Job openings: 3.72 and 3.19
  - Unemployment rate: 10.06 and 17.38
- Autocorrelations (1995- 2007 and 1995 - 2011):
  - GDP: 0.93 and 0.91
  - Private investment: 0.89 and 0.87
  - Private capital: 0.97 and 0.96
  - Hours per employee: 0.95 and 0.92
  - Overtime hours per employee: 0.87 and 0.87
  - Job openings: 0.94 and 0.96
  - Unemployment rate: 0.96 and 0.98
- Correlations with real output (1995- 2007 and 1995 - 2011):
  - Private investment: 0.88 and 0.86
  - Private capital: 0.59 and 0.41
  - Hours per employee: 0.88 and 0.91
  - Overtime hours per employee: 0.34 and 0.35
  - Job openings: 0.88 and 0.75
  - Unemployment rate: -0.74 and -0.60
- Note: The series of job openings starts from 2001. Source: ALFRED, Federal Reserve Bank of St. Louis and authors’ calculations. Quarterly data. Percentage deviations from HP-trend for GDP, private investment, private capital, hours per employee, and overtime hours per employee; percentage the sample mean for vacancies and the unemployment rate.

### Appendix B — Sensitivity exercises (summary of key results and parameter experiments)
- Common fiscal experiment across exercises: government spending expansion (1% of output, lump-sum taxes, balanced budget).
- Model: RBC with Mortensen-Pissarides Matching Friction (MPMF) and deep habits in consumption (θc=0:86 and ρc=0:85) unless otherwise stated.
- A. Bargaining power (Figure B.1)
  - Parameters varied: elasticity of substitution σ ∈ {0:10, 0:40, →1} and firms’ bargaining power ε ∈ {0:10, 0:50, 0:90}.
  - Key qualitative findings:
    - If σ=0:10 (almost Leontief) the real wage declines despite deep habits, firms post fewer vacancies, output multiplier < 1, unemployment small or positive.
    - As σ increases toward Cobb-Douglas (σ→1), outcomes change: bargaining parameter ε affects real wage and unemployment responses.
- B. Hagedorn and Manovskii effect (Figure B.2)
  - Parameter: replacement ratio Θ varied across values including 0.40, 0.50, 0.60, 0.70, 0.80, 0.90, 0.95.
  - Findings:
    - Higher steady-state replacement ratio ̄Θ increases magnitudes of both output and unemployment multipliers; output multiplier changes marginally, unemployment multiplier can be considerably higher when Θ is high.
- C. Quantitative implications (Table B.1)
  - Scenarios compared: σ→1 (CD) vs σ=0:4; Θ values 0:7 and 0:9; ε values 0:1, 0:5, 0:9.
  - Selected reported impact multipliers (impact output multiplier ∆Y/∆G and peak unemployment multiplier ∆u/∆G):
    - Θ=0:7, ε=0:1: ∆Y/∆G = 1.78 (σ→1) and 1.46 (σ=0:4); ratio (B)/(A)=0.82. ∆u/∆G = -0.65 and -0.42; ratio 0.65.
    - Θ=0:7, ε=0:5: ∆Y/∆G = 1.95 and 1.57; ratio 0.81. ∆u/∆G = -0.53 and -0.35; ratio 0.66.
    - Θ=0:7, ε=0:9: ∆Y/∆G = 2.02 and 1.61; ratio 0.80. ∆u/∆G = -0.49 and -0.33; ratio 0.67.
    - Θ=0:9, ε=0:1: ∆Y/∆G = 1.63 and 1.38; ratio 0.85. ∆u/∆G = -1.27 and -0.75; ratio 0.59.
    - Θ=0:9, ε=0:5: ∆Y/∆G = 1.91 and 1.54; ratio 0.81. ∆u/∆G = -1.29 and -0.79; ratio 0.61.
    - Θ=0:9, ε=0:9: ∆Y/∆G = 2.05 and 1.62; ratio 0.79. ∆u/∆G = -1.01 and -0.65; ratio 0.64.
  - Conclusion: dropping σ from 1 to 0.4 reduces output multipliers to around 4/5 and unemployment multipliers to around or below 2/3 of CD case; jobless stimulus robust to Θ and ε calibration.
- D. Sensitivity to investment adjustment costs and leisure/consumption elasticity (Table B.2)
  - Scenarios: baseline vs ̄h=0:25 and ̄h=0:40 (via ρ), and No invest. adj. costs (γ=0); σ→1 vs σ=0:4.
  - Selected reported impact multipliers:
    - Baseline: ∆Y/∆G = 1.95 and 1.57 (B)/(A)=0.81. ∆u/∆G = -0.53 and -0.35; ratio 0.66.
    - ρ s.t. ̄h=0:25: ∆Y/∆G = 2.18 and 1.74; ratio 0.80. ∆u/∆G = -0.52 and -0.35; ratio 0.67.
    - ρ s.t. ̄h=0:40: ∆Y/∆G = 1.74 and 1.43; ratio 0.82. ∆u/∆G = -0.51 and -0.34; ratio 0.67.
    - No invest. adj. costs: ∆Y/∆G = 2.50 and 1.81; ratio 0.72. ∆u/∆G = -0.40 and -0.25; ratio 0.63.
  - Findings:
    - Lower elasticity between leisure and consumption (higher steady-state hours) reduces output multiplier; unemployment multiplier insensitive to ρ.
    - Removing investment adjustment costs increases output multiplier and lowers unemployment multiplier; investment adjustment costs important to deliver a jobless stimulus.
- E. Debt-financed fiscal policy and distortionary taxation
  - Steady-state tax rates set to τc=0:05, τw=0:24, τk=0:32 (as in Christiano et al. (2010)).
  - Tax rule: log(Xt/ ̄X) = ρX log(Xt−1/ ̄X) + ρXB Bt−1/yt−1; Xt = (τ;τc;τw;τk).
  - Tax responsiveness parameter used: ρXB = 0:02 (Monacelli et al. (2010) value).
  - Result: introduction of distortionary taxes and debt reduces (in absolute value) output and unemployment multipliers, with unemployment affected more—due to increased tax distortions on employment and dynamics implied by the feedback rule.
- F. Sensitivity to tax persistence and tax responsiveness to debt (Figures B.4 and B.5)
  - Persistence ρX varied {0:30, 0:60, 0:90}: output and unemployment multipliers very robust to ρX changes.
  - Responsiveness ρXB varied {0:005, 0:02, 0:10}: multipliers robust up to ~2 years; beyond two years larger ρXB implies faster fall of output and more pronounced rise in unemployment.

### Appendix C — The Fiscal Stimulus in a New-Keynesian (NK) extension
- Sticky prices introduced via Rotemberg quadratic price-adjustment costs ξ/2 (Pt/Pt−1 − 1)2; pricing equation (C.3) generalizes flexible-price equation when ξ>0.
- Monetary policy: Taylor rule log(Rnt/ ̄Rn) = ρr log(Rnt−1/ ̄Rn) + (1−ρr)[ρπ log(Πt/ ̄Π) + ρy log(Yt/ ̄Y)] (equation C.4). Fisher equation R t+1 = Et[R n t / Π t+1] (equation C.5).
- Calibration notes:
  - Deep habit parameters θc and ρc as in baseline (θc=0:86 and ρc=0:85) unless stated otherwise.
  - Example ξ values mapped to implied Calvo durations: ξ=29:41 corresponds to ~3 quarters; ξ=9:90 corresponds to ~2.0 quarters; ξ=58:25 corresponds to ~4.0 quarters.
- Key findings:
  - Price stickiness can soften fiscal expansion effects via reduced inflation, but with empirically plausible deep habits and price stickiness, consumption and investment crowding-ins, decline in mark-up, increase in real wage, and sizes of output and unemployment multipliers are quite robust.
  - An aggressive monetary response to the output gap (ρy sufficiently large) can offset fiscal expansion effects—possibly producing crowding-out and even rising unemployment.
  - Surface experiments (Figure C.2) show:
    - If ρy above ~0.4, unemployment may rise and consumption may be crowded out.
    - If ρy above ~0.6, the output multiplier can fall below one.
  - Empirical NK literature typically finds low ρy (e.g., Smets and Wouters (2007) posterior ~0.08); optimal-policy literature also finds weak long-run response to output gap (examples: Schmitt-Grohe and Uribe (2007) find ρy=0:1).
  - Conclusion: it is the strength of monetary response to the output gap, not price stickiness per se, that can subvert fiscal expansion effects.

### Appendix D — Symmetric equilibrium (model equations and definitions)
- CES production function:
  - F((ZK)t Kt;(ZN)t nt ht) = [αK((ZK)t Kt)σ−1/σ + αN((ZN)t nt ht)σ−1/σ]σ/(σ−1) (D.1)
  - FK,t and FN,t marginal products given in (D.2) and (D.3).
- Utility with deep habits and marginal utilities in equations (D.4)–(D.9), including:
  - Scarring: Sc t = ρc Sc t−1 + (1−ρc) Ct (D.8); Ct = Xc t + θc Sc t−1 (D.9).
- Investment dynamics and adjustment costs:
  - Kt+1 = (1−δ)Kt + It[1 − S(It/It−1)] (D.10)
  - S(It/It−1) = γ/2 (It/It−1 − 1)2 (D.11)
- Pricing, hiring, and bargaining conditions include equations (D.16)–(D.31), e.g. wage equation (D.21):
  - wt ht = (1−ε)[MCt FN,t ht − g(zt) + g′(zt) zt + θt g′(zt)] + ε[wu − Un,t/Ux,t (1−τw t)]
- Matching and vacancy functions: qt = k θt −ω (D.26); pt = θt qt (D.27); θt = vt/ut (D.24).
- Government budget and fiscal rules: Bt = Rt Bt−1 + Gt + (1−nt) wu − τt − τC t Ct − τW t wt nt ht − τK t Rk t Kt (D.33). Fiscal processes (D.34)–(D.37).
- Resource constraint including price-adjustment cost: Yt = Ct + It + Gt + gt nt + ξ/2 (Pt/Pt−1 − 1)2 (D.38).
- Sticky-price Taylor rule and Fisher equation: log(Rnt/ ̄Rn) = ρπ log(Πt/ ̄Π) + ρy log(Yt/ ̄Y) (D.39); Rt+1 = Et[Rnt/Πt;t+1] (D.40).

### Appendix E — Steady state (brief)
- Steady-state values of employment rate n, hours worked h, and marginal cost MC solve simultaneously the wage equation (D.21), resource constraint (D.38), and pricing equation (D.32). Remaining steady-state unknowns follow from the system of equations reported in the symmetric equilibrium.

*Source: _wp1317 - References*

### Appendix A can be found recursively by using the following relationships:

### _wp1317 - Appendix A can be found recursively by using the following relationships:

### Definitions and Recursive Relationships
- Z
- N=(ZN)
  0
  (E.1)
- ZK=(ZK)
  0
  (E.2)
- Y=Y
  0
  (E.3)
- D=β(E.4)
- Q=1(E.5)
- Π=1(E.6)

### Production and Capital
- R
  K
  =
  R+δ
  1−τ
  K
  (E.7)
- (
  K
  Y
  )=MC
  S
  K
  R
  K
  (E.8)
- K=
  (
  K
  Y
  )
  Y(E.9)
- I=δK(E.10)

### Government and Public Goods
- 51
- G=
  (
  G
  Y
  )
  Y(E.11)
- S
  g
  =G(E.12)
- X
  g
  =(1−θ
  c
  )
  G(E.13)

### Labor and Prices
- u=1−n(E.14)
- p=
  λn
  1−n
  (E.15)
- ̄
  θ=
  (
  p
  κ
  )
  1
  1−ω
  (E.16)
- q=κθ
  −ω
  (E.17)
- v=uθ(E.18)
- z=
  v
  n
  (E.19)
- g=
  χ
  1+ψ
  z
  1+
  ψ
  (E.20)
- g
  z
  =χz
  ψ
  (E.21)

### Goods, Inputs, and Factor Functions
- F
  N=
  α
  N
  M
  C(
  Z
  N)
  σ−1
  σ
  (
  Y
  nh
  )
  1
  σ
  (E.22)
- F
  n
  =F
  N
  h(E.23)

### Consumption, Utility, and Aggregates
- 52
- X
  c
  =
  1−ρ
  ρ
  F
  N
  1+n
  (
  (
  1−
  h
  )
  ρ(1−σ
  c
  )
  −1
  )
  (
  1−
  h
  )
  ρ(1−σ
  c
  )−1
  (E.24)
- C=
  X
  c
  1−θ
  c
  (E.25)
- S
  c
  =C(E.26)
- U
  n
  =
  (
  X
  c
  )
  (1−
  ρ)(1−σ
  c
  )
  (
  (
  1−
  h
  )
  ρ(1−σ
  c
  )
  −1
  )
  1−σ
  c
  (E.27)
- U
  x
  =
  (1−
  ρ)
  (
  X
  c
  )
  (1−ρ)(1−σ
  c
  )−1
  (
  1+n
  (
  (
  1−h
  )
  ρ(1−σ
  c
  )
  −1
  ))
  (E.28)

### Wage Expression
- w=
  1
  Dh
  [
  −
  g
  z
  q
  +D
  (
  F
  n
  −g+zg
  z
  +(1−λ)
  g
  z
  q
  )
  ]
  (E.29)

*Source: _wp1317 - Appendix A can be found recursively by using the following relationships:*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1317.pdf_