## wpiea2021208-print-pdf

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---

### 2.1 The labour market — labour market structure and matching
- Newly formed firm–worker matches: m_t = ̄m u_t^ζ v_t^(1−ζ), with ̄m >0 and ζ∈(0,1).
- Searching workers at beginning of period: u_t = 1 − (1−ρ) N_{t−1}, where ρ is the fraction of employment relationships destroyed each period.
- Vacancy fill probability: q_t = m_t / v_t = ̄m θ_t^(−ζ), where θ_t = v_t / u_t (labor market tightness).
- Job-finding probability for a worker: f_t = m_t / u_t = θ_t q(θ_t).
- After-hiring unemployment (fraction of searching workers remaining unemployed after hiring): ur_t = 1 − N_t.
- Employment law of motion: N_t = (1−ρ) N_{t−1} + f_t u_t, with f_t u_t = m_t.

### 2.1 The labour market — households, production, and matching frictions
- Representative household preferences: maximize E_0 ∑_{t=0}^∞ β^t log C_t; C_t is a Dixit–Stiglitz consumption bundle with elasticity ε; β subjective discount factor.
- Household budget constraint components:
  - C_t + I_t + B_t / P_t = w_t N_t + r^K_t z_t k_t + b_t (1−N_t) + (R_{t−1}/π_{t−1}) B_{t−1} / P_{t−1} + D_t.
  - z_t denotes capital utilization; %_t is a risk premium á la Smets and Wouters (2007).
- Physical capital law of motion:
  - k_{t+1} = (1 − δ_K(z_t)) k_t + [1 − Θ_I / 2 (I_t / I_{t−1} − g)^2] I_t,
  - Θ_I > 0, g steady state growth, δ_K(z_t) = δ z_t^{φ_k}.
- Tobin’s Q:
  - Q^K_t = E_t β λ_{t+1}/λ_t { r^K_{t+1} z_{t+1} + Q^K_{t+1} (1 − δ_K(z_{t+1})) }.
- Value of employment for the family:
  - V^E_t = w_t − b_t + β(1−ρ) E_t [λ_{t+1}/λ_t { (1−f_{t+1}) V^E_{t+1} } ].
- Search and matching generate surplus; hiring costs and wage bargaining determine equilibrium outcomes.

### 2.1 The labour market — production, prices, and patents
- Four sectors: intermediate goods, specialized patent producers, retailers, and innovation (R&D).
- Final good aggregator:
  - Y_t = [ ∫_0^1 (Y^i_t)^{(ε−1)/ε} di ]^{ε/(ε−1)}, and Y^i_t = (P^i_t / P_t)^{−ε} Y_t.
- Retailer production from specialized goods:
  - Y^i_t = [ Z_t ∫_0^{Z_t} (Y^S_j,t)^v dj ]^{1/v}, with v < 1.
- Retailer nominal rigidities: Rotemberg adjustment costs Γ^i_Pt = ψ / 2 (π^i_t − π^R_t)^2; reference π^R_t cases described in text.
- Phillips curve (retail level):
  - Γ'_Pt π_t = ε (p_{S,t} + Γ_t) − (ε − 1) + β E_t [ (λ_{t+1}/λ_t) (Y_{t+1}/Y_t) Γ'_{Pt+1} π_{t+1} ].
- Specialized producers: Y^S_{j,t} = X_{j,t}; mark-up possibly constrained by imitator entry μ_{I,C} ≤ μ_{I,U} = 1/ν; p^S_{j,t} = μ_{I,C} p_{I,t}.
- Patent value:
  - V^j_t = Π^j_t + (1 − δ_Z) E_t β_{t,t+1} V^j_{t+1}, with δ_Z patent obsolescence.

### 2.1 The labour market — hiring costs, wages, and downward nominal rigidity
- Intermediate production: X_t = A_t N_t^{1−α} K_t^{α}, with K_t = z_t k_t.
- Hiring rate and convex hiring costs:
  - x_t ≡ q_t v_t / N_{t−1}; hc_t = κ_t γ (x_t)^γ N_{t−1}, κ_t = κ Ψ_t; γ determines convexity.
  - Convex hiring costs smooth vacancy creation and produce persistent hiring dynamics.
- Nominal wage adjustment cost:
  - c^W_t = φ_w Ψ_t / 2 (π^w_t − g^R_t)^2, where π^w_t = W_t / W_{t−1}.
- Hiring condition:
  - κ_t (x_t)^{γ−1} = p_{I,t} (1−α) X_t / N_t − (w_t + c^W_t) + E_t β_{t,t+1} [ (1−ρ) κ_{t+1} (x_{t+1})^{γ−1} + (γ−1)/γ κ_{t+1} (x_{t+1})^γ ].
- Value of new worker after sunk adjustment:
  - J_t = p_{I,t} (1−α) X_t / N_t − (w_t + c^W_t) + E_t β_{t,t+1} [ (1−ρ + x_{t+1}) J_{t+1} − κ_{t+1} / γ (x_{t+1})^γ ].
- Downward nominal wage rigidity (DWR) specification:
  - W_t = max( ι W_{t−1}, W^*_t ), ι∈(0,1). In real terms: w_t = max( ι w_{t−1} / π_t, w^*_t ).
  - Desired real wage from Nash bargaining yields expression for w^*_t involving $ _t and continuation values CV^F_t and CV^W_t.
- Effective bargaining power:
  - $ _t = η / [ η + (1−η) (1 + τ^W_{t,t+1}) ], with τ^W_{t,t+1} defined via wage adjustment costs; state-dependent and increases during declining wages.

### 2.1 The labour market — innovation and aggregation
- Patent accumulation:
  - Z_{t+1} = θ_t S^{RD}_t + (1 − δ_Z) Z_t.
- R&D productivity:
  - θ_t = χ Z_t [ (Ψ_t)^τ (S^{RD}_t)^{1−τ} ]^{−1}, τ∈[0,1]; captures congestion (∂θ/∂S^{RD} < 0) and spillovers (∂θ/∂Z > 0).
- Free entry / marginal condition:
  - 1 / θ_t = E_t β_{t,t+1} V_{t+1}.
- Market clearing and resource constraint:
  - X_t = Z_t ∫_0^{Z_t} X_{j,t} dj = Z_t X_{j,t}.
  - Resource constraint: Y_t (1 − Γ_{Pt}) − c^W_t N_t = C_t + I_t + S^{RD}_t + κ_t / γ (x_t)^γ N_{t−1}.
- Aggregated production and TFP measures:
  - Y_t = Z_t^{(1/v − 1)} A_t N_t^{1−α} (z_t k_t)^α.
  - TFP_t = Y_t / (N_t^{1−α} k_t^α) = Z_t^{(1/v − 1)} A_t z_t^α.
  - TFP^{util}_t = Z_t^{(1/v − 1)} A_t.
- Balanced growth scaling: Ψ_t = Z_t^Υ, where Υ = (1/v − 1) / (1 − α).

---

### 3.1 Calibration — targets, parameters, and solution method
- Empirical targets: Euro Area, 1999q1 to 2019q4.
- Long run inflation target: 1.8 percent (annualized).
- χ chosen to match average annual TFP growth g_TFP = 0.84 percent.
- Implied average output growth rate g = 1.2 percent; data value g = 1.11.
- Preferences and markups:
  - β = 0.997; ε = 6 → steady state retail markup ≈ 20 percent.
- Price/wage adjustment and DWR:
  - φ_p corresponds to Calvo 0.58 → mean price duration ≈ 7 months.
  - φ_w = 10.5.
  - Occasionally binding wage parameter ι = 1 in baseline; robustness includes negative effective lower bound on wage inflation.
- Labour market calibration:
  - UR = 9 percent; quarterly job separation rate = 0.06; implied f = 0.378; q = 0.6.
  - η = 0.5; ζ = 0.5.
  - Replacement rate b/w = 0.62 in steady state.
  - Hiring costs quadratic with γ = 2; implied total hiring costs = 1.31 percent of total output.
- R&D and production:
  - δ_Z = 0.03; τ = 0.85; α = 0.3; v = 0.74 → elasticity between specialized goods = 3.85.
  - μ_{I,C} = 1.18; φ_k = 1.30 (steady state z = 1); δ = 0.02 (quarterly); Θ_I = 0.17.
- Monetary policy:
  - φ_π = 1.5; φ_y = 0.1; φ_r = 0.85.
- Shock processes (baseline):
  - ρ_% = 0.92; σ_% = 0.2 percent; ρ_z = 0.95; σ_z = 0.45 percent; σ_m = 0.1 percent.
- Solution method:
  - Model solved at second order using DynareOBC for occasionally binding constraints.
  - Simulations: 101000 periods simulated, first 1000 discarded; long-run means annualized; second moments after HP(1600) filtering.
- Model fit highlights:
  - Steady-state vs Data long-run means (annualized percent): UR = 9.00 vs 9.41; Δy = 1.20 vs 1.11; ΔTFP = 0.84 vs 0.83; π = 1.80 vs 1.68; π_w = 3.00 vs 2.07.
  - Business cycle moments: σ(y) = 1.20 in both Data and Model; unemployment volatility and correlations reported (e.g., Unemployment σ(x)/σ(y): Data 4.55, Model 4.67).
- Implications from calibration:
  - Endogenous growth + DWR generate permanent asymmetric responses to demand shocks and output/TFP hysteresis through depressed R&D and intangible capital.
  - Simulations: long-run simulations use 100000 periods; probability of hitting wage lower bound in baseline calibration: 11 percent on average; conditional persistence implies probability of being at the lower bound = 22 percent.
  - Quantitative long-run effects across 100 years:
    - Endogenous growth without DWR implies lower average output growth by around 0.02 percentage points per year relative to exogenous growth → more than 2 percent loss over 100 years.
    - Adding DWR increases this loss to almost 6 percentage points over 100 years.

---

### 4.3 Trade-offs for the long-run Phillips curve — main findings
- Increasing the inflation target (experiments from 0.5 to 6 percent) yields two opposing effects:
  - Reduces probability of hitting the occasionally binding wage constraint.
  - Increases distortions from quadratic price and wage adjustment costs, reducing output, consumption and employment.
- At low inflation levels the model exhibits a negative relationship between inflation and unemployment (positively sloped Phillips curve in authors' terminology) because DWR binds more frequently and dominates price-adjustment efficiency losses.
- Benchmark models without DWR display an almost vertical, monotonically upward sloping Phillips curve where only price distortions matter.
- Endogenous growth effects:
  - Without DWR, higher inflation target lowers output growth (downward sloping inflation–growth relationship) because distortions reduce R&D and long-run growth.
  - With DWR, long-run Phillips curve is highly non-linear: positively sloped for low inflation but turning downward sloping above 4 percent inflation, as unemployment–inflation trade-offs transmit through R&D into output growth.
- Quantitative importance depends on:
  - (i) long-run growth rate (steady-state productivity growth determines wage growth and distance from wage floor),
  - (ii) size and persistence of demand (risk premium) shocks.
- Higher steady-state productivity growth increases the wage buffer, reducing DWR relevance and rotating unemployment–inflation curve left; via endogenous growth this shifts the inflation–output curve right and makes it steeper.
- Higher risk premium volatility shifts the Phillips curve between inflation and growth left, implying higher output loss for a given inflation rate.

### 4.3 Trade-offs — welfare, optimal inflation, and decomposition
- Welfare metric and computation:
  - EV_0 approximated as EV_0 = 1/(1−β) [ E log(C_0) + β/(1−β) E log(g_{C,t}) ] (equation (15) in source).
  - Consumption-equivalent difference γ_CE = exp((1−β) (EV_M − EV_A)) − 1.
  - Welfare comparisons computed from long simulations over inflation targets from zero to six percent.
- Four main welfare results:
  - (1) Models without DWR: welfare cost minimized at zero inflation.
  - (2) Endogenous growth amplifies losses from adjustment inefficiencies; even absent DWR, endogenous growth increases CE welfare losses by more than 2 percent.
  - (3) With DWR, welfare loss is U-shaped in the inflation target: DWR costs dominate at low targets (so raising target reduces loss), but above the optimum price-distortion costs dominate. Welfare-maximizing inflation in model with endogenous growth and DWR > 3 percent.
  - (4) Combined DWR and endogenous growth imply substantial welfare gains from adopting the optimal inflation target:
    - In exogenous-growth no-DWR, increasing target from 1% to 4% raises welfare losses by < 0.5 percentage points.
    - In endogenous growth + DWR, the same range yields loss difference > 5 percentage points.
- Decomposition (selected Table 5 rows):
  - Baseline (endogenous growth + search frictions + DWR): Optimal π* = 3.76; Welfare Loss at π = π* = 4.34.
  - Baseline without endogenous growth: Optimal π* = 3.16; Welfare Loss at π = π* = 1.19.
  - Lower search frictions + (1): Optimal π* = 1.10.
  - No DWR + (2): Optimal π* = 0.00; Welfare Loss at π = π* = 2.13.
- Interpretation: endogenous growth strongly affects welfare costs of business cycles; search frictions amplify DWR costs and raise the optimal inflation target.

### 4.3 Trade-offs — sensitivity analysis (selected Table 6 results)
- Baseline: Optimal π* = 3.76; Welfare Loss at π = π* = 4.34.
- Parameter variations (selected):
  - b. Higher growth (g = 1.6): Optimal π* = 3.44; Welfare Loss = 4.31.
  - c. Higher wage rigidity: Optimal π* = 3.52; Welfare Loss = 4.41.
  - d. Higher price rigidity: Optimal π* = 3.16; Welfare Loss = 5.67.
  - e. Calvo pricing (recalibrated): Optimal π* = 3.30; Welfare Loss = 4.75.
  - f. Lower OBC on DWR (−1.0%): Optimal π* = 3.00; Welfare Loss = 3.72.
  - g. Lower R&D diffusion: Optimal π* = 3.50; Welfare Loss = 3.40.
- Shock variations:
  - h. Lower risk premium volatility (σ = 0.15): Optimal π* = 3.06; Welfare Loss = 3.05.
  - i. Smaller technology shocks (σ = 0.4): Optimal π* = 3.68; Welfare Loss = 4.12.
- Key takeaways:
  - Higher long-term growth enlarges wage buffer, reducing DWR binding and lowering optimal target.
  - Greater wage rigidity increases welfare losses but can lower optimal inflation because it avoids hitting wage bound.
  - Greater price rigidity increases welfare costs from price distortions, calling for a lower optimal target.
  - Calvo pricing raises welfare losses for a given inflation and implies a slightly lower optimal target.
  - Relaxing DWR (allowing some negative wage inflation) reduces welfare costs and lowers optimal target but optimal remains ≈ 3 percent in cited exercises.
  - Weakening R&D-growth linkage lowers welfare losses and optimal target.
  - Greater risk premium volatility raises optimal inflation target; technology shock volatility has smaller effect.

---

### 6.1 Zero Lower Bound — modeling and quantitative findings
- ZLB modeled with shadow (notional) rate R* that evolves inertially; actual rate Rt = max{R* t, 1}.
- Findings in exogenous-growth benchmark:
  - ZLB shifts optimal inflation target to 1.7 percent; CE welfare loss at this target < 1 percent.
  - ZLB binds about 8 percent of the time.
  - Adding DWR shifts optimal inflation slightly above 3 percent and reduces frequency of hitting the ZLB; welfare gains limited under exogenous growth.
- In endogenous growth model:
  - Accounting for endogenous growth shifts optimal inflation target proportionally up relative to exogenous-growth counterparts.
  - DWR reduces frequency of hitting ZLB but wage-bound frequency remains unchanged and average unemployment higher than without DWR.
  - Optimal inflation target and welfare gains are higher than in baseline without ZLB.
- Mechanisms:
  - In exogenous-growth models, limited wage adjustment reduces downward pressure on marginal costs and inflation, lowering ZLB likelihood; higher real wages have small long-run effects.
  - In endogenous growth with search frictions, higher real wages dampen job creation, reduce firms’ profitability and value of R&D, producing larger hysteresis on unemployment and output growth.
  - Symmetric wage rigidity decreases optimal inflation (higher price-distortion costs); asymmetric DWR calls for higher optimal inflation to minimize unemployment and permanent output growth costs.
- Summary implication:
  - ZLB episodes costly but infrequent; raising inflation target to mitigate ZLB must be balanced against persistent costs of higher inflation.
  - With production-side frictions and endogenous growth, asymmetric wage rigidities amplify persistent output and unemployment effects, increasing the optimal inflation target and potential welfare gains from a higher target.

---

*wpiea2021208-print-pdf — sections 2.1, 3.1, 4.3, 6.1 (excerpts).*

### 2.1  The labour market

### 2.1  The labour market

### Labour market structure and matching
- Newly formed firm–worker matches: m_t = ̄m u_t^ζ v_t^(1−ζ), with ̄m >0 and ζ∈(0,1).
- Searching workers at beginning of period: u_t = 1 − (1−ρ) N_{t−1}, where ρ is the fraction of employment relationships destroyed each period.
- Vacancy fill probability: q_t = m_t / v_t = ̄m θ_t^(−ζ), where θ_t = v_t / u_t (labor market tightness).
- Job-finding probability for a worker: f_t = m_t / u_t = θ_t q(θ_t).
- After-hiring unemployment (fraction of searching workers remaining unemployed after hiring): ur_t = 1 − N_t.

### Households: preferences, budget and labor supply
- Representative household maximizes E_0 ∑_{t=0}^∞ β^t log C_t, where C_t is a Dixit–Stiglitz consumption bundle with elasticity of substitution ε and β is the subjective discount factor.
- Per period budget constraint:
  - C_t + I_t + B_t / P_t = w_t N_t + r^K_t z_t k_t + b_t (1−N_t) + (R_{t−1}/π_{t−1}) B_{t−1} / P_{t−1} + D_t.
  - z_t denotes the utilization rate of capital.
  - %_t is a risk premium á la Smets and Wouters (2007).
- Income components: real wage income w_t N_t, unemployment benefits b_t (1−N_t), return on capital r^K_t z_t k_t, family share of aggregate profits D_t.
- Employment law of motion: N_t = (1−ρ) N_{t−1} + f_t u_t, with f_t u_t = m_t.
- Physical capital law of motion:
  - k_{t+1} = (1 − δ_K(z_t)) k_t + [1 − Θ_I / 2 (I_t / I_{t−1} − g)^2] I_t,
  - Θ_I > 0 is scale parameter, g is steady state growth rate, δ_K(z_t) = δ z_t^{φ_k}.
- First order conditions (selected):
  - λ_t = 1 / C_t.
  - 1 = β E_t [λ_{t+1} / λ_t R_t / π_{t+1} %_t].
  - Q^K_t conditions and r^K_t = Q^K_t δ'_K(z_t) as in text.
- Tobin’s Q: Q^K_t = E_t β λ_{t+1}/λ_t { r^K_{t+1} z_{t+1} + Q^K_{t+1} (1 − δ_K(z_{t+1})) }.
- Value of employment for the family:
  - V^E_t = w_t − b_t + β(1−ρ) E_t [λ_{t+1}/λ_t { (1−f_{t+1}) V^E_{t+1} } ].
- Net value of additional employed worker equals wage net of unemployment benefits plus expected continuation value.

### Production: sectors and price setting
- Four sectors: intermediate goods, specialized patent producers, retailers, and innovation (R&D).
- Final good aggregator (retail level):
  - Y_t = [ ∫_0^1 (Y^i_t)^{(ε−1)/ε} di ]^{ε/(ε−1)}, with ε elasticity of substitution between retail goods.
  - Demand for each retailer: Y^i_t = (P^i_t / P_t)^{−ε} Y_t.
- Retailer production of final good from specialized goods:
  - Y^i_t = [ Z_t ∫_0^{Z_t} (Y^S_j,t)^v dj ]^{1/v}, where Z_t is number of patents in use and v < 1 governs substitution between patents.
- Nominal rigidities for retailers: Rotemberg-style quadratic price adjustment costs:
  - Γ^i_Pt = ψ / 2 (π^i_t − π^R_t)^2, where π^i_t = P^i_t / P^i_{t−1}, and π^R_t is the reference inflation rate (cases: π^R_t = π^*, π^R_t = 1, or π^R_t = π^ς_{t−1} with ς∈(0,1)).
- Retail profit maximization subject to demand and production yields a Phillips curve:
  - Γ'_Pt π_t = ε (p_{S,t} + Γ_t) − (ε − 1) + β E_t [ (λ_{t+1}/λ_t) (Y_{t+1}/Y_t) Γ'_{Pt+1} π_{t+1} ].
  - Price inflation dynamics depend on marginal cost p_{S,t} and evolution of price adjustment costs.

### Specialized goods and patent values
- Specialized producer j: Y^S_{j,t} = X_{j,t}, transforming one unit of intermediate good X_{j,t} into one unit of patented good.
- Desired mark-up may be constrained by imitator entry: μ_{I,C} (constrained) ≤ μ_{I,U} = 1/ν.
- Price: p^S_{j,t} = μ_{I,C} p_{I,t}.
- Profits: Π^j_t = (μ_{I,C} − 1) p_{I,t} Y^S_{j,t}.
- Value of owning patent j:
  - V^j_t = Π^j_t + (1 − δ_Z) E_t β_{t,t+1} V^j_{t+1}, where δ_Z is patent obsolescence rate and β_{t,t+1} = β λ_{t+1}/λ_t.

### Intermediate production and hiring costs
- Intermediate firm production: X_t = A_t N_t^{1−α} K_t^{α}, with K_t = z_t k_t and A_t exogenous technology.
- Hiring rate x_t ≡ q_t v_t / N_{t−1}.
- Convex hiring costs:
  - hc_t = κ_t γ (x_t)^γ N_{t−1}, where κ_t = κ Ψ_t and Ψ_t ensures balanced growth; γ determines convexity.
  - Convex hiring costs smooth vacancy creation, produce persistent hiring dynamics, and reflect post-match training costs.
  - Convexity ensures hiring rate non-negativity and wage staying within bargaining set.
- Nominal wage stickiness via quadratic adjustment cost:
  - c^W_t = φ_w Ψ_t / 2 (π^w_t − g^R_t)^2, where π^w_t = W_t / W_{t−1}, φ_w ≥ 0, and g^R_t is wage inflation reference (cases analogous to price indexation).
- Representative firm maximizes expected profits subject to N_t = (1−ρ + x_t) N_{t−1}.
- Capital demand: r^K_t = p_{I,t} α X_t / K_t.
- Hiring condition:
  - κ_t (x_t)^{γ−1} = p_{I,t} (1−α) X_t / N_t − (w_t + c^W_t) + E_t β_{t,t+1} [ (1−ρ) κ_{t+1} (x_{t+1})^{γ−1} + (γ−1)/γ κ_{t+1} (x_{t+1})^γ ].
- Value of having a new worker after adjustment costs sunk:
  - J_t = p_{I,t} (1−α) X_t / N_t − (w_t + c^W_t) + E_t β_{t,t+1} [ (1−ρ + x_{t+1}) J_{t+1} − κ_{t+1} / γ (x_{t+1})^γ ].

### Wage determination and downward nominal rigidity
- Search frictions generate surplus; without further assumptions wage not uniquely pinned down.
- Downward nominal wage rigidity introduced via wage rule:
  - W_t = max( ι W_{t−1}, W^*_t ), with ι∈(0,1). ι = 1 implies nominal wages cannot fall; ι∈(0,1) allows partial downward adjustment.
  - In real terms: w_t = max( ι w_{t−1} / π_t, w^*_t ), so positive inflation can reduce real wages even if nominal wages are constrained ("greasing the wheels").
- Desired real wage w^*_t from Nash bargaining:
  - arg max_{W^*_t} [ (J_t)^{1−η} (V^E_t)^η ], with η bargaining power of workers.
- Desired real wage expression:
  - w^*_t = (1 − $ _t) { b_t − CV^W_t } + $ _t { p_{I,t} (1−α) X_t / N_t − c^W_t + CV^F_t }.
  - CV^F_t = E_t [ β_{t,t+1} { (1−ρ + x_{t+1}) J_{t+1} − κ_{t+1} / γ (x_{t+1})^γ } ].
  - CV^W_t = E_t β_{t,t+1} (1−ρ) (1−f_{t+1}) V^E_{t+1}.
- Effective bargaining power:
  - $ _t = η / [ η + (1−η) (1 + τ^W_{t,t+1}) ], with τ^W_{t,t+1} = ∂c^W_t / ∂W_t P_t + E_t β_{t+1} [ (1−ρ + x_{t+1}) ∂c^W_{t+1} / ∂W_t P_t ].
  - Wage adjustment costs (∂c^W_t / ∂W_t > 0) make $ _t state-dependent and increase $ _t during periods of declining wages, damping nominal wage fluctuations and reducing probability of hitting the occasionally binding constraint.

### Innovation sector and R&D
- New patents evolve: Z_{t+1} = θ_t S^{RD}_t + (1 − δ_Z) Z_t, with S^{RD}_t R&D expenditure and θ_t R&D productivity.
- Functional form of R&D productivity:
  - θ_t = χ Z_t [ (Ψ_t)^τ (S^{RD}_t)^{1−τ} ]^{−1}, where τ∈[0,1] and Ψ_t ensures balanced growth.
  - Specification captures congestion externality (∂θ/∂S^{RD} < 0) and knowledge spillover (∂θ/∂Z > 0).
- Free entry condition (payoff to innovation equals R&D cost):
  - E_t β_{t,t+1} V_{t+1} ( Z_{t+1} − (1 − δ_Z) Z_t ) = S^{RD}_t.
  - Marginal condition: 1 / θ_t = E_t β_{t,t+1} V_{t+1}.
  - This pins down R&D investment and equilibrium growth rate of TFP and output.

### Monetary policy
- Central Bank rule:
  - R_t = (R_{t−1})^{φ_r} [ r (π_t / π^*)^{φ_π} ( ˆY_t / ˆY_{ss} )^{φ_y} ]^{1−φ_r} ε^m_t,
  - where ˆY_t = Y_t / Ψ_t is detrended output, φ_π and φ_y are response coefficients, and φ_r captures interest rate smoothing.

### Market clearing, aggregation and TFP
- Market clearing for intermediate goods: X_t = Z_t ∫_0^{Z_t} X_{j,t} dj = Z_t X_{j,t} (symmetry).
- Resource constraint (final output net of adjustment costs used for expenditures):
  - Y_t (1 − Γ_{Pt}) − c^W_t N_t = C_t + I_t + S^{RD}_t + κ_t / γ (x_t)^γ N_{t−1}.
- Aggregated production relation:
  - Y_t = Z_t^{(1/v − 1)} A_t N_t^{1−α} (z_t k_t)^α.
- Two TFP measures:
  - Aggregate non-adjusted TFP: TFP_t = Y_t / (N_t^{1−α} k_t^α) = Z_t^{(1/v − 1)} A_t z_t^α.
  - Utilization-adjusted TFP: TFP^{util}_t = Z_t^{(1/v − 1)} A_t.
- Even after adjusting for capital utilization, TFP^{util}_t varies with exogenous A_t and endogenous intangible capital Z_t.
- Balanced growth scaling: Ψ_t = Z_t^Υ, where Υ = (1/v − 1) / (1 − α).

### Calibration and solution method
- The section closes by stating that the next part discusses calibration strategy, solution method, and model fit to euro area data for variables of interest.

*wpiea2021208-print-pdf - 2.1  The labour market*

### 3.1  Calibration

### 3.1  Calibration

### Calibration targets and long-run parameters
- Empirical targets correspond to the Euro Area in the period 1999q1 to 2019q4.
- Long run inflation target: 1.8 percent (annualized), consistent with the ECB objective of "below, but close to 2 percent".
- Scale parameter χ chosen to match average annual TFP growth rate g_TFP = 0.84 percent.
- Implied average output growth rate g = 1.2 percent; data value g = 1.11.

### Preferences and markups
- Discount factor β = 0.997.
- Elasticity of substitution of retail goods ε = 6 → steady state markup in retail sector ≈ 20 percent.

### Price and wage adjustment, occasionally binding constraint
- Degree of price rigidities φ_p set to correspond to a Calvo parameter of 0.58 → mean price duration ≈ 7 months.
- Wage adjustment cost parameter φ_w = 10.5 (chosen to match observed relative volatility of employment).
- Occasionally binding constraint on nominal wages parameter ι = 1 in baseline; robustness includes negative effective lower bound on wage inflation.

### Labour market calibration
- Steady state unemployment rate UR = 9 percent.
- Quarterly job separation rate = 0.06.
- Implied job finding rate f = 0.378.
- Quarterly job filling rate q = 0.6.
- Workers’ bargaining power η = 0.5.
- Elasticity of job matches wrt vacancies ζ = 0.5.
- Real unemployment benefits b set to target replacement rate b/w = 0.62 in steady state.
- Hiring costs assumed quadratic with γ = 2.
- Implied total hiring costs = 1.31 percent of total output.

### R&D sector
- Patent obsolescence rate δ_Z = 0.03.
- Elasticity of new technologies to R&D τ = 0.85.

### Production and capital
- Elasticity of intermediate production to capital α = 0.3.
- Parameter v = 0.74 → elasticity of substitution between specialized goods = 3.85.
- Markup on specialized goods μ_{I,C} = 1.18.
- Utilization-related parameter φ_k = 1.30 (steady state utilization z = 1).
- Quarterly capital depreciation rate δ = 0.02 (annual depreciation = 8 percent).
- Investment adjustment cost Θ_I = 0.17.
- Steady state value of technology A chosen so that output in the steady state of the detrended system = 1.

### Monetary policy rule
- Sensitivity of interest rate to inflation φ_π = 1.5.
- Sensitivity to output φ_y = 0.1.
- Interest rate smoothing φ_r = 0.85.

### Shock processes (baseline)
- Risk premium shock persistence ρ_% = 0.92.
- Risk premium shock volatility σ_% = 0.2 percent.
- Exogenous technology shock persistence ρ_Z = 0.95.
- Exogenous technology shock volatility σ_z = 0.45 percent.
- Standard deviation of monetary policy shocks σ_m = 0.1 percent.

### Selected calibrated parameter table (selected entries preserved)
- Inflation target: 1.8 (ECB objective)
- SS growth rate: 1.2 (Avg TFP growth rate = 0.84%)
- Price rigidity φ_p: 16.37 (Calvo param. 0.58)
- Wage rigidity φ_w: 10.5
- Discount factor β: 0.997 (Amano&Gnocchi (2020))
- Elast. new patents to R&D τ: 0.85 (Kung&Schmidt (2015))
- Patent obsolescence rate δ_Z: 0.03 (Guerron-Quintana&Jinnai (2019))
- Std.dev. interest rate σ_m: 0.1% (Christoffel et al.(2008))
- Autocorr. exog. tech. ρ_z: 0.95 (Sahuc&Smets(2008))
- Std.dev. exog. tech. σ_z: 0.45% (Matchstd(y))
- Autocorr. of risk premium ρ_%: 0.92 (Anzoategui el al.(2019); Andrade et al. (2021))
- Std. dev. risk premium σ_%: 0.2% (Anzoategui el al.(2019); Andrade et al. (2021))

### Benchmark models used for comparison
- Endo. growth - no DWR: model without downward wage rigidities (abstracts from the occasionally binding constraint on wage adjustment).
- Exo. growth - no DWR: shuts down endogenous growth channel via constant R&D investment intensity (equivalent to specifying an exogenous trend growth component in productivity).
- Calibration of benchmark models identical to baseline where applicable; all benchmark models include search and matching unemployment.

### Solution method and model fit
- Model solved at second order using DynareOBC toolkit for occasionally binding constraints (treating bounds in a perfect-foresight manner for computational tractability).
- Simulations for moments: 101000 periods simulated, first 1000 discarded; long-run means annualized and expressed in percentage terms; second moments obtained after HP(1600) filtering.
- Tables of long-run means and business cycle moments indicate:
  - Steady-state (model) vs Data long-run means (annualized percent): UR = 9.00 vs 9.41; Δy = 1.20 vs 1.11; ΔTFP = 0.84 vs 0.83; π = 1.80 vs 1.68; π_w = 3.00 vs 2.07.
  - Business cycle moments (HP-filtered): σ(y) = 1.20 in both Data and Model; selected ratios and correlations preserved in Table 3 (e.g., Unemployment σ(x)/σ(y): Data 4.55, Model 4.67; Unemployment ρ(x,y): Data -0.88, Model -0.69).
- Overall: model reproduces closely long-run means of unemployment, output and TFP growth; long-run mean of price and especially wage inflation slightly larger than in data.

### Implications highlighted in calibration and fit
- The combination of endogenous growth and downward wage rigidities (DWR) is central to the model’s capacity to generate:
  - Permanent asymmetric responses to demand shocks (e.g., risk premium shocks).
  - Output and TFP hysteresis through depressed R&D and intangible capital following negative demand shocks that bind the wage floor.
- Simulation notes:
  - Long-run simulations use 100000 periods; second moments computed after HP(1600) filtering.
  - A large negative or positive risk premium shock (≈ three percent annualized change) is used to illustrate hitting the occasionally binding constraint.
  - Probability of hitting the wage lower bound in baseline calibration: 11 percent on average; conditional persistence implies probability of being at the lower bound = 22 percent in baseline calibration.
- Quantitative long-run effects (across 100 years of simulated annual data):
  - Endogenous growth without DWR implies lower average output growth by around 0.02 percentage points per year relative to exogenous growth counterpart → more than 2 percent loss over 100 years.
  - Adding DWR increases this loss significantly to almost 6 percentage points over 100 years, driven by periods when the wage constraint binds and delays recovery.

*Source: wpiea2021208-print-pdf - 3.1  Calibration*

### 4.3  Trade-offs for the long-run Phillips curve

### 4.3 Trade-offs for the long-run Phillips curve

### Long-run Phillips curve: main findings
- An increase of the inflation target (from 0.5 to 6 percent in the experiments) produces two opposing effects:
  - Reduces the probability of hitting the occasionally binding wage constraint, potentially improving long-run equilibrium.
  - Increases distortions from quadratic price and wage adjustment costs, which tend to reduce output, consumption and employment.
- At low levels of inflation the model exhibits a negative relationship between inflation and the unemployment rate (a positively sloped Phillips curve for low inflation in the authors' terminology), because the occasionally binding downward wage constraint binds more frequently and dominates efficiency losses from price adjustment costs.
- Benchmark models without downward wage rigidities (DWR) display an almost vertical, monotonically upward sloping Phillips curve where only price distortions matter.
- Endogenous growth breaks long-run neutrality:
  - In endogenous-growth models without DWR, a higher inflation target lowers output growth (downward sloping relationship between inflation target and output growth) because price and wage distortions reduce R&D investment and long-run growth.
  - Adding DWR creates a highly non-linear long-run Phillips curve: positively sloped for low inflation but turning downward sloping for inflation rates above 4%. This is because unemployment–inflation trade-offs transmit through investment in R&D into output growth effects.
- The quantitative importance of the long-run inflation–output trade-off depends crucially on:
  - (i) the economy’s long-run growth rate (steady-state productivity growth determines steady-state wage growth and the distance from the wage floor), and
  - (ii) the size and persistence of demand (risk premium) shocks (which determine the probability of sequences of shocks that push the economy to the wage constraint).
- Higher steady-state productivity growth increases the wage buffer, reducing the relevance of DWR and rotating the unemployment–inflation curve left; via endogenous growth this shifts the inflation–output curve right and makes it steeper.
- Higher volatility of risk premium shocks shifts the Phillips curve between inflation and growth to the left, implying higher output loss for a given inflation rate because asymmetric effects from DWR become larger.

### Implications for monetary policy (context and trends)
- Secular declines in productivity growth and low inflation since the Global Financial Crisis increase the likelihood that the non-vertical part of the long-run Phillips curve is empirically relevant.
- Historical context cited: at the start of the Economic and Monetary Union, long-term output growth and average price inflation were about two percent each; more recently the 10-year averages fell from about 2.25 and 1.9 percent before the Global Financial Crisis to 1.0 and 1.3 percent in the 10 years before the COVID shock, respectively.

### Optimal inflation rate: welfare metric and computation
- Welfare metric (stationary approximation in a non-stationary environment):
  - Define consumption growth g_{C,s} ≡ C_s / C_{s−1} and C_t = (∏_{s=1}^t g_{C,s}) C_0.
  - Lifetime utility V_0 = E_0 ∑_{t=0}^∞ β^t log(C_t) can be rewritten so that unconditional welfare is approximated as:
    EV_0 = 1/(1−β) [ E log(C_0) + β/(1−β) E log(g_{C,t}) ]  (equation (15) in the source).
  - Consumption-equivalent difference γ_CE = exp((1−β) (EV_M − EV_A)) − 1, where EV_M is welfare in the welfare-maximizing steady state (zero inflation target) and EV_A is welfare in the alternative case.
- Welfare comparisons are computed from long simulations for inflation targets on a grid from zero to six percent, allowing the steady state to adjust endogenously.

### Four main welfare results (section 5.2)
- First: For benchmark models without DWR the welfare cost is minimized at an inflation target of zero because price and wage adjustment costs increase in the level of inflation.
- Second: Endogenous growth amplifies losses from price and wage adjustment inefficiencies. Even absent DWR, endogenous growth increases consumption-equivalent welfare losses by more than 2 percent because higher distortions lower average R&D investment and TFP growth and make deviations more persistent.
- Third: Inclusion of DWR generates a trade-off for the optimal inflation target; the consumption-equivalent welfare loss is U-shaped in the inflation target:
  - For low inflation targets, DWR costs prevail so increasing the target reduces welfare losses.
  - Above the optimum, price-distortion costs dominate and higher inflation increases welfare losses.
  - The welfare-maximizing inflation rate in the model with endogenous growth and DWR is more than 3 percent.
- Fourth: Combined DWR and endogenous growth imply substantial welfare gains from adopting the optimal inflation target:
  - In exogenous-growth models without DWR, increasing the inflation target from 1% to 4% increases welfare losses by less than 0.5 percentage points.
  - In the model with both DWR and endogenous growth, the loss difference over the same target range exceeds 5 percentage points because deviations affect average output growth.

### Decomposing drivers of the optimal inflation target (Table 5 results)
- Table 5 (selected rows reproduced):
  - Baseline (endogenous growth + search frictions + DWR): Optimal π* = 3.76; Welfare Loss at π = π* = 4.34; π = 1.8 (π* − 1.8) = 4.34 6.05 −1.72 (table formatting in source).
  - (1) Baseline without endogenous growth: Optimal π* = 3.16; Welfare Loss at π = π* = 1.19; π = 1.8 (π* − 1.8) = 1.35 −0.15.
  - (2) Lower search frictions + (1): Optimal π* = 1.10; Welfare Loss at π = π* = 4.34; π = 1.8 (π* − 1.8) = 4.42 −0.08.
  - (3) No DWR + (2): Optimal π* = 0.00; Welfare Loss at π = π* = 2.13; π = 1.8 (π* − 1.8) = 2.99 −0.86.
- Interpretation from the source: Endogenous growth strongly affects welfare costs of business cycles (reducing them by a factor of 4 when closed, per the authors' comparison), while search frictions have a crucial role in amplifying DWR costs and requiring a higher optimal inflation target because they increase duration and severity of recessions and the share of output devoted to hiring costs.

### Sensitivity analysis: how parameter and shock changes affect the optimal target (Table 6 results)
- Baseline calibration: Optimal π* = 3.76; Welfare Loss at π = π* = 4.34.
- Parameter variations (Table 6, selected rows):
  - b. Higher growth (g = 1.6): Optimal π* = 3.44; Welfare Loss = 4.31.
  - c. Higher wage rigidity: Optimal π* = 3.52; Welfare Loss = 4.41.
  - d. Higher price rigidity: Optimal π* = 3.16; Welfare Loss = 5.67.
  - e. Calvo pricing (baseline re-calibrated): Optimal π* = 3.30; Welfare Loss = 4.75.
  - f. Lower OBC on DWR (−1.0%): Optimal π* = 3.00; Welfare Loss = 3.72.
  - g. Lower R&D diffusion: Optimal π* = 3.50; Welfare Loss = 3.40.
- Shock variations (Table 6, selected rows):
  - h. Lower risk premium volatility (σ = 0.15): Optimal π* = 3.06; Welfare Loss = 3.05.
  - i. Smaller technology shocks (σ = 0.4): Optimal π* = 3.68; Welfare Loss = 4.12.
- Key sensitivity takeaways from the source:
  - Higher long-term growth raises steady-state nominal wage growth, enlarges the wage buffer and makes DWR less binding, lowering welfare losses and the optimal target.
  - Greater wage rigidity increases welfare losses but can lower the optimal inflation target since it helps avoid hitting the wage bound.
  - Greater price rigidity increases welfare costs from price distortions and calls for a lower inflation target.
  - Calvo pricing (versus Rotemberg baseline) raises welfare losses for a given inflation target and implies a slightly lower optimal target.
  - Relaxing DWR (allowing some negative wage inflation) reduces welfare costs and lowers the optimal target, but the optimal target remains around 3 percent in the exercise cited.
  - Weakening the R&D-growth linkage reduces the hysteresis–price-distortion trade-off and lowers welfare losses and the optimal target.
  - Greater volatility of risk premium shocks increases the probability of hitting the wage bound and raises the optimal inflation target; technology shock volatility has a smaller effect.

### Extensions (preview of subsequent analysis)
- The authors note they will examine whether introducing the zero lower bound on the nominal interest rate materially affects the main findings, and evaluate three alternative policy rules outside the traditional Taylor-rule inflation-targeting framework (extensions described in the next section).

*Source: 4.3 Trade-offs for the long-run Phillips curve (excerpt from the supplied IMF PDF content).*

### 6.1  Zero Lower Bound

### 6.1  Zero Lower Bound

### Motivation and literature
- A common motivation for a higher inflation target is the presence of a zero lower bound (ZLB) on interest rates, aside from the "greasing the wheels" effect of downward wage rigidities.
- A large body of recent research incorporating the ZLB into standard New Keynesian models typically finds the optimal inflation target is relatively low, typically below 2 percent, because ZLB episodes are infrequent while the costs of higher steady inflation are paid every period.

### Modeling approach
- Monetary policy in normal times follows an inertial Taylor rule for a shadow (notional) rate R*:
  - R*
t depends on the lagged shadow policy rate R*
t−1 rather than the lagged actual rate Rt−1, implying nominal rates remain lower for longer after ZLB episodes.
- The actual nominal interest rate is constrained by the ZLB as:
  - Rt = max{R*
t, 1}
- The term "effective lower bound" and "zero lower bound" are used interchangeably and modeled as ZLB. Relaxing the bound somewhat into negative territory has no effect on the findings.

### Main quantitative findings
- In the benchmark New Keynesian model with exogenous growth:
  - The ZLB causes the optimal inflation target to shift up to 1.7 percent.
  - The consumption-equivalent welfare loss at this target is less than 1 percent.
  - The ZLB binds about 8 percent of the time.
  - Augmenting the model with downward wage rigidity (DWR) shifts the optimal inflation target slightly above 3 percent, and reduces the frequency of hitting the ZLB, but welfare gains remain limited under exogenous growth because effects are transitory.
- In the endogenous growth model:
  - Accounting for endogenous growth shifts the optimal inflation target proportionally up compared to the exogenous-growth counterparts, reflecting amplification from endogenous growth under asymmetries.
  - The presence of DWR reduces the frequency of hitting the ZLB, but the frequency of being constrained by the wage bound remains unchanged and the average unemployment rate is higher than without DWR.
  - Consequently, the optimal inflation target and the associated welfare gains are higher than in the baseline model without the ZLB.

### Mechanisms and interpretation
- In exogenous growth models without search-and-matching unemployment, limited wage adjustment (symmetric wage rigidity or DWR) reduces downward pressure on marginal labor costs and inflation, lowering the likelihood of hitting the ZLB and aiding policy in fine-tuning the business cycle. Higher real wages in these models have relatively small long-term effects on output, and firm profitability is unaffected in the long run due to return to the exogenous balanced growth path.
- In the endogenous growth model with search-and-matching unemployment:
  - Higher real wages dampen job creation by negatively affecting the hiring rate, producing larger hysteresis effects on unemployment.
  - Limited wage adjustments under adverse shocks reduce firms’ profitability and the value of R&D and future productivity growth, leading to larger and more persistent effects on unemployment and output growth.
  - Symmetric wage rigidity (Rotemberg-type) decreases the optimal inflation rate because it raises price-distortion costs and offsets costs from lower output growth.
  - Asymmetric DWR calls for a higher optimal inflation rate to minimize costs from higher unemployment and lower permanent output growth.

### Summary implications
- ZLB episodes are costly but infrequent; raising the inflation target to mitigate ZLB costs must be balanced against persistent per-period costs of higher inflation.
- When production-side frictions and endogenous growth are present, asymmetric wage rigidities amplify persistent output and unemployment effects, increasing the optimal inflation target and potential welfare gains from a higher target.

*Source: 6.1 Zero Lower Bound, wpiea2021208-print-pdf*

### References

### wpiea2021208-print-pdf - References

### A Sensitivity Analysis: Further Robustness Exercises
- Table 9 reports additional sensitivity analysis on the optimal inflation target implied by variations to the baseline specification in the model with endogenous growth and downward wage rigidity.
- Table 9: Optimal π target and welfare: robustness (columns: Model variation | Optimal π* | Welfare π=π* | Loss at π=1.8 (π* − 1.8))
  - Higher growth (g= 1.6): Optimal π* = 3.44; Welfare π=π* = 4.31; Loss at π=1.8 (π* - 1.8) = 5.44 -1.13
  - Higher wage rigidity (ψw = 15): Optimal π* = 3.52; Welfare π=π* = 4.41; Loss at π=1.8 (π* - 1.8) = 5.81 -1.39
  - Higher price rigidity (ψ = 30): Optimal π* = 3.16; Welfare π=π* = 5.67; Loss at π=1.8 (π* - 1.8) = 6.69 -1.02
  - Calvo pricing: Optimal π* = 3.30; Welfare π=π* = 4.75; Loss at π=1.8 (π* - 1.8) = 5.95 -1.20
  - Lower OBC (min(πw) = −1%): Optimal π* = 3.00; Welfare π=π* = 3.72; Loss at π=1.8 (π* - 1.8) = 4.21 -0.49
  - Price & wage index (ς = 0.5): Optimal π* = 5.40; Welfare π=π* = 2.68; Loss at π=1.8 (π* - 1.8) = 5.92 -3.24
  - Lower discount (β = 0.995): Optimal π* = 3.74; Welfare π=π* = 2.89; Loss at π=1.8 (π* - 1.8) = 3.99 -1.11
  - Lower ε of new patents to R&D (0.6): Optimal π* = 3.50; Welfare π=π* = 3.40; Loss at π=1.8 (π* - 1.8) = 4.25 -0.85
  - Small risk premium shocks (σ = 0.15): Optimal π* = 3.06; Welfare π=π* = 3.05; Loss at π=1.8 (π* - 1.8) = 3.73 -0.68
  - Small technology shocks (σ = 0.4): Optimal π* = 3.68; Welfare π=π* = 4.12; Loss at π=1.8 (π* - 1.8) = 5.69 -1.58
  - No monetary policy shocks: Optimal π* = 3.66; Welfare π=π* = 4.13; Loss at π=1.8 (π* - 1.8) = 5.69 -1.56
  - Convexity vacancy cost (γ = 1.2): Optimal π* = 3.66; Welfare π=π* = 4.81; Loss at π=1.8 (π* - 1.8) = 6.49 -1.68
  - Fixed capital utilization: Optimal π* = 3.68; Welfare π=π* = 5.16; Loss at π=1.8 (π* - 1.8) = 6.82 -1.66
  - Fixed capital & utilization: Optimal π* = 3.54; Welfare π=π* = 4.38; Loss at π=1.8 (π* - 1.8) = 5.84 -1.45
  - Baseline: Optimal π* = 3.76; Welfare π=π* = 4.34; Loss at π=1.8 (π* - 1.8) = 6.05 -1.72

### B Impulse responses: sensitivity to different steady-state growth and inflation rates
- Context: Europe has experienced a secular decline in average inflation and growth rates; in steady state the rate of growth of nominal wages depends on inflation and output growth, affecting the distance between wage inflation and the lower wage bound.
- Table 10: Key macro variables, percentages (10-year historical averages; Source: Eurostat)
  - 2003: GDP growth = 2.17; Price Inflation = 1.99; Nom. Wages = 4.2
  - 2007: GDP growth = 2.34; Price Inflation = 1.91; Nom. Wages = 4.3
  - 2020: GDP growth = 1.02; Price Inflation = 1.31; Nom. Wages = 2.3
- Finding: Figure 8 (IRFs to a positive risk premium shock) shows that following a large risk premium shock, a low inflation-low growth environment strongly affects the probability of hitting the lower wage bound and the length and severity of the recessionary episode.

### C Consumption-equivalent welfare losses
- Purpose: Compare welfare implications of different inflation targets under models with exogenous growth and endogenous growth, and assess role of DWR (downward wage rigidity).
- Finding: Figure 9 shows consumption-equivalent (CE) welfare losses for different inflation targets in models with exogenous and endogenous growth.
  - In both models, DWR would call for a higher inflation target.
  - Introduction of endogenous growth strongly amplifies the welfare costs of business cycle fluctuations and the welfare costs of choosing a suboptimal inflation target.

### D Additional moments
- Table 11: Long-run means: different models (annualized percent)
  - Variables listed: UR, ∆y, ∆TFP, π, πw
  - Steady-state: UR = 9.00; ∆y = 1.20; ∆TFP = 0.84; π = 1.80; πw = 3.00
  - Data: UR = 9.41; ∆y = 1.11; ∆TFP = 0.83; π = 1.68; πw = 2.07
  - Sym. Model - Exo. Growth: UR = 9.19; ∆y = 1.20; ∆TFP = 0.84; π = 2.04; πw = 3.23
  - Sym. Model - End. Growth: UR = 9.15; ∆y = 1.18; ∆TFP = 0.82; π = 1.88; πw = 3.05
  - DWR Model - End. Growth: UR = 9.55; ∆y = 1.14; ∆TFP = 0.80; π = 2.13; πw = 3.27
- Table 12: Business cycle moments: different models (HP(1600)-filtered second moments; columns include σ(x)/σ(y) and ρ(x,y))
  - Unemployment: Data σ(x)/σ(y) = 4.55; Sym Exo = 4.30; Sym Endo = 4.09; DWR Endo = 4.67; Correlations: Data = -0.88; Sym Exo = -0.65; Sym Endo = -0.68; DWR Endo = -0.69
  - Employment: Data σ(x)/σ(y) = 0.52; Sym Exo = 0.43; Sym Endo = 0.41; DWR Endo = 0.52; Correlations: Data = 0.83; Sym Exo = 0.66; Sym Endo = 0.69; DWR Endo = 0.68
  - Investment: Data σ(x)/σ(y) = 2.34; Sym Exo = 3.22; Sym Endo = 2.22; DWR Endo = 2.34; Correlations: Data = 0.86; Sym Exo = 0.93; Sym Endo = 0.88; DWR Endo = 0.88
  - Consumption: Data σ(x)/σ(y) = 0.55; Sym Exo = 0.48; Sym Endo = 0.48; DWR Endo = 0.47; Correlations: Data = 0.84; Sym Exo = 0.88; Sym Endo = 0.89; DWR Endo = 0.90
  - TFP: Data σ(x)/σ(y) = 0.79; Sym Exo = 0.85; Sym Endo = 0.84; DWR Endo = 0.81; Correlations: Data = 0.97; Sym Exo = 0.96; Sym Endo = 0.97; DWR Endo = 0.95
  - Nominal wages: Data σ(x)/σ(y) = 0.16; Sym Exo = 0.60; Sym Endo = 0.57; DWR Endo = 0.46; Correlations: Data = 0.31; Sym Exo = 0.61; Sym Endo = 0.68; DWR Endo = 0.63
  - Prices: Data σ(x)/σ(y) = 0.50; Sym Exo = 0.90; Sym Endo = 0.86; DWR Endo = 0.80; Correlations: Data = 0.21; Sym Exo = 0.54; Sym Endo = 0.58; DWR Endo = 0.53
  - Real wages: Data σ(x)/σ(y) = 0.46; Sym Exo = 0.64; Sym Endo = 0.60; DWR Endo = 0.73; Correlations: Data = -0.13; Sym Exo = 0.17; Sym Endo = 0.09; DWR Endo = -0.08
  - σ(y): Data = 1.20; Sym Exo = 1.06; Sym Endo = 1.16; DWR Endo = 1.20

*Content based on the referenced PDF chapter/section.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021208-print-pdf.pdf_