## wp17201

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### Key findings and research questions (Introduction)
- Research questions:
  - Can the post-2008 dynamics of output and unemployment be reconciled within a standard general equilibrium framework together with the rest of the postwar data?
  - If not, what features must a successful model include to explain observed fluctuations absent in the standard model?
- Observed facts motivating the study:
  - The U.S. unemployment rate remained above 8% for more than 11 quarters following the Great Recession.
  - Real U.S. GDP dropped more than 2% during the crisis and, unlike previous postwar recessions, did not jump back to the pre-recession trend.
- Methodological stance:
  - Remove trend from output non-parametrically based on Kaldor facts to obtain series that do not grow over time (GDP in wage units), and do not detrend the unemployment rate in order to assess whether a standard model can explain actual employment levels rather than deviations from a trend.
- Context from literature:
  - Some researchers interpret persistent post-2008 dynamics as a loss in potential output and an increase in equilibrium unemployment.
  - Other literature considers models with multiple equilibria to explain prolonged high unemployment.
- Implication emphasized:
  - If part of the observed decline in potential output and rise in unemployment reflects a switch to an inefficient, demand-determined equilibrium, then demand-stimulating monetary and fiscal policies are justified, contrasting with the standard supply-determined approach.

### Detrending procedure: GDP in wage units (Section II)
- Definitions and construction:
  - Nominal wage W_t defined as: W_t = (compensation to employees)_t / (number of full-time equivalent employees)_t.
  - GDP in wage units: Z_t = Y_t / W_t · 1/N_t, where Y_t is nominal U.S. GDP and N_t is the civilian labor force in period t.
  - Nominal consumption C_t defined as: C_t = Y_t − I_t, where I_t is nominal investment (sum of nominal private and government investment).
  - Consumption and investment in wage units: C^w_t = C_t / W_t · 1/N_t and I^w_t = I_t / W_t · 1/N_t.
  - National accounts identity preserved in wage units: Z_t = C^w_t + I^w_t.
- Intuition and properties:
  - Dividing by nominal wage removes inflation and the trend in TFP from nominal variables; further dividing by civilian labor force removes growth due to increasing number of workers, yielding a per-capita, wage-deflated series that preserves medium- and long-term comovements across series.
  - The constructed series Z_t can be written as Z_t = (1/b_t)·(1 − u_t), where b_t is labor income share and u_t is unemployment rate.
- Empirical relationships and exact statistics from the data:
  - Correlation between constructed GDP in wage units and the civilian employment rate (one minus unemployment) is -0.88.
  - Correlation between the employment rate defined using full-time equivalent employees and the BLS employment measure is 0.93.
  - Correlation between the constructed employment-rate series and the BLS employment-rate series is close to one - 0.93.
- Observed differences and contributing factors:
  - Divergence of GDP in wage units and the BLS unemployment series after 2009 can potentially be attributed to a fall in the labor share.
  - BLS measures do not distinguish part-time vs full-time workers (FTE measure counts part-time proportionally), contributing to differences in measured employment/unemployment.

### Empirical approach and model overview (Sections II–III)
- Estimation strategy:
  - Carry out a business cycle accounting exercise following Chari, Kehoe, and McGrattan (2007) on data detrended using wage units.
  - Describe a benchmark RBC model (prototype economy) and introduce wedges in the model’s equations to capture deviations from model relationships.
  - Estimate the model using Bayesian techniques to obtain smoothed values for all wedges.
- Novel aspects relative to CKM/RBC literature:
  - By constructing wage series, the labor wedge (residual in the labor-market equilibrium equation) is decomposed into labor supply and labor demand wedges, allowing independent evaluation of labor supply and labor demand predictions.
  - This differs from conventional RBC literature that eliminates wages from equilibrium equations.
- Key empirical reversal relative to standard RBC claims:
  - Contrary to the RBC literature claim that 70% or more of fluctuations are explained by TFP shocks, in the transformed (wage-unit detrended) data most movements in GDP are accounted for by the labor supply wedge.
  - The observed consumption, wage, and unemployment data deviate from the optimal solution implied by household utility maximization, indicating households are not on their labor supply equation.
  - These findings support the view that high unemployment after the Great Recession was inefficient and that models with incomplete labor markets or multiple equilibria may be appropriate.

### Model setup: household, firms, equilibrium (selected equations)
- Preferences and labor:
  - One infinitely-lived household. Employment share l_t ∈ (0,1); non-employed share 1−l_t.
  - One-period utility: u(c_t, l_t) = log(c_t) + ψ·log(1−l_t).
  - Real wage per working member: w_t. Real consumption per member: c_t.
- Budget and capital accumulation:
  - Budget constraint: c_t + i_t = r_t k_t + w_t l_t (Equation (9)).
  - Capital law of motion: k_{t+1} = i_t + (1−δ) k_t (Equation (10)).
  - Discount factor: β (fixed at 0.99 in estimation).
- Optimality conditions (household):
  - Euler: 1/c_t = β E_t[1/c_{t+1} · (1−δ + r_{t+1})] (Equation (11)).
  - Labor supply: ψ c_t /(1−l_t) = w_t (Equation (12)).
- Production and firm behavior:
  - Cobb-Douglas production: y_t = A_t k_t^α l_t^(1−α) (Equation (13)), where 1−α is labor’s share.
  - Factor prices:
    - r_t = α y_t / k_t (Equation (14)).
    - w_t = (1−α) y_t / l_t (Equation (15)).
- General equilibrium and accounting identities:
  - National accounts identity: c_t + i_t = y_t (Equation (16)).
  - Euler with production link: 1/c_t = β E_t[1/c_{t+1} · (1−δ + α y_{t+1} / k_{t+1})] (Equation (17)).
  - State vector: x_t = [c_t, y_t, i_t, k_{t+1}, w_t, l_t] (Equation (18)).

### Accounting procedure: wedges, implementation, and interpretation
- Objective:
  - Back out residuals ("wedges") required for each equation to hold exactly given observed series to evaluate how well the RBC model describes fluctuations.
- Interpretation of wedges:
  - Low-dispersion, white-noise wedges → equation describes data well.
  - High-dispersion or autocorrelated wedges → equation problematic; imply volatile shocks.
- Wedges introduced:
  - ε_{invest} in Euler equation (investment wedge) — interpretable as investment tax.
  - ε_{LS} in labor supply equation (labor supply wedge) — interpretable as labor income tax.
  - ε_{eff} in production (efficiency / TFP wedge).
  - ε_{LD} added to labor demand equation (labor demand wedge) to evaluate labor demand separately and avoid statistical singularity.
- Modified system (wedge-included equations, selected):
  - 1/c_t · exp(ε_{invest,t}) = E_t[ β · 1/c_{t+1} · ((1−δ) exp(ε_{invest,t+1}) + α y_{t+1} / k_{t+1}) ] (Equation (19)).
  - w_t = ψ · exp(ε_{LS,t}) · c_t /(1−l_t) (Equation (20)).
  - A_t = exp(z_t), z_t = ρ z_{t−1} + ε_{eff,t} (Equations (21)-(23)).
  - w_t = exp(ε_{LD,t}) · (1−α) y_t / l_t (Equation (24)).
  - y_t = c_t + i_t (Equation (25)); k_{t+1} = i_t + (1−δ) k_t (Equation (26)).
- Solution and estimation:
  - Log-linearize system (19)-(26) around deterministic steady state.
  - Solve via Sims (2001) methods, cast as X_t = F X_{t−1} + G ε_t; Y_t = H X_t.
  - Parameters estimated by Bayesian maximum likelihood using Kalman filter and random walk Metropolis-Hastings MCMC: 100’000 draws, 50’000 kept.

### Priors and parameterization (as used for estimation)
- Fixed and prior choices:
  - β fixed at 0.99 (corresponds to long-run annual interest rate of 4%).
  - Prior means chosen close to common calibrations; leisure parameter prior mean chosen assuming 5% long-run unemployment.
- Prior distributions and prior means (Table 1 summary):
  - α (Capital share): Beta, prior mean 0.33, Std.Dev 0.15
  - δ (Capital depreciation): Beta, prior mean 0.03, Std.Dev 0.015
  - ρ (Tech. persistence): Beta, prior mean 0.9, Std.Dev 0.05
  - ψ (Leisure parameter): Gamma, prior mean 0.04, Std.Dev 0.02
  - σ_eff (St.dev. of ε_eff): Inv. Gamma, prior mean 0.02, Std.Dev 0.01
  - σ_invest (St.dev. of ε_invest): Inv. Gamma, prior mean 0.02, Std.Dev 0.01
  - σ_LS (St.dev. of ε_LS): Inv. Gamma, prior mean 0.1, Std.Dev 0.05
  - σ_LD (St.dev. of ε_LD): Inv. Gamma, prior mean 0.02, Std.Dev 0.01

### Parameter estimates (posterior results, quarterly data 1948:1–2011:4)
- Posterior means and 90% posterior CI (Table 2 summary):
  - Capital share α: Posterior mean 0.3529; CI 90% [0.3514; 0.3546]
  - Capital depreciation δ: Posterior mean 0.0137; CI 90% [0.0126; 0.0147]
  - Tech. persistence ρ: Posterior mean 0.9774; CI 90% [0.9588; 0.9958]
  - Discount factor β: Fixed 0.99
  - Leisure parameter ψ: Posterior mean 0.0493; CI 90% [0.0478; 0.0509]
  - St.dev. of ε_eff: Posterior mean 0.0093; CI 90% [0.0074; 0.0111]
  - St.dev. of ε_invest: Posterior mean 0.0051; CI 90% [0.0045; 0.0055]
  - St.dev. of ε_LS: Posterior mean 0.3091; CI 90% [0.2846; 0.3341]
  - St.dev. of ε_LD: Posterior mean 0.0265; CI 90% [0.0245; 0.0283]
- Additional estimation diagnostics:
  - MCMC accept. rate 36%
  - 100’000 draws, 50’000 kept
  - log(L) = 1945
- Interpretations from parameter estimates:
  - Labor’s share of total income ≈ 65% (implied by α estimate).
  - Quarterly capital depreciation 0.0137 implies about 5.4% depreciation per year.
  - Long-run unemployment rate implied from ψ ≈ 5.7%.
  - Posterior standard deviation of labor supply wedge (σ_LS = 0.3091) is substantially larger than other wedges — suggests large role in explaining fluctuations.
  - σ_eff ≈ 0.0093, σ_invest ≈ 0.0051, σ_LD ≈ 0.0265.

### Realized wedges: properties and graphical impressions (Table 3 summary, 1948:1–2011:4)
- Variance relative to log output and correlations with log output at lags −2, −1, 0, 1, 2:
  - Efficiency: Variance relative to log output 0.15; correlations: −0.2893, −0.2258, −0.0741, 0.0080, 0.0704
  - Investment: Variance relative to log output 0.07; correlations: 0.0253, −0.0548, −0.1469, −0.0807, −0.0630
  - Labor Supply (LS): Variance relative to log output 8.15; correlations: −0.8801, −0.9137, −0.9136, −0.8721, −0.8087
  - Labor Demand (LD): Variance relative to log output 0.64; correlations: −0.8248, −0.8854, −0.9414, −0.9203, −0.8848
- Cross-correlations between wedges (selected lags −2, −1, 0, 1, 2):
  - Efficiency vs Investment: 0.2425, 0.4758, −0.5250, −0.2658, −0.2369
  - Efficiency vs LS: 0.3128, 0.2536, 0.1562, 0.0448, −0.0495
  - Efficiency vs LD: 0.2339, 0.1787, 0.0217, −0.0262, −0.0552
  - Investment vs LS: −0.0279, 0.0470, 0.0887, 0.1072, 0.0932
  - Investment vs LD: −0.0247, 0.0482, 0.1537, 0.0483, 0.0304
  - LS vs LD: 0.7540, 0.7524, 0.7244, 0.6833, 0.6353
- Statistical threshold:
  - 5% critical value (two-tailed) for correlation coefficients is 0.1225.
- Qualitative observations:
  - Investment wedge series looks like white noise with low dispersion and low correlation with output — Euler equation holds well.
  - Efficiency (TFP) wedge: does not explain GDP in wage units; TFP shocks largely eliminated when GDP deflated by money wage.
  - Labor supply wedge: highly autocorrelated and highly (negatively) correlated with log output and strongly correlated with unemployment rate — very large variance relative to output (8.15).
  - Labor demand wedge: highly (negatively) correlated with output; indicates countercyclical labor income share.
  - Strong positive correlation between LS and LD series at short negative lags indicates aggregate labor wedge (LS − LD) can mask large opposing movements.

### Contribution analysis: one-wedge economies and explained variance (Table 4 summary)
- Method:
  - Feed realized innovations of only one wedge (with others held constant, investment wedge included in all experiments) into estimated policy functions and compare model-generated series to actual data.
- One-wedge economy findings:
  - Investment wedge only:
    - Does not explain output, unemployment rate, or investment — consistent with small σ_invest.
  - Efficiency (TFP) wedge only:
    - Explains approximately 50% of variation in investment.
    - Explains virtually none of variation in GDP (wage units) or in unemployment rate.
  - Labor demand wedge only:
    - Explains substantial share of output.
    - Explains negligible variation in unemployment rate or investment.
  - Labor supply wedge only:
    - Explains almost all variation in unemployment rate.
    - Explains significant shares of variation in output and investment.
- Quantitative shares of variation explained (defined as var(log(x_model_t))/var(log(x_obs_t)) · 100%):
  - Efficiency* (with investment wedge always included): Output 0.20; Unemployment rate 1.32; Investment 49.45
  - Labor demand (LD)*: Output 36.77; Unemployment rate 1.65; Investment 4.91
  - Labor supply (LS)*: Output 23.67; Unemployment rate 114.23; Investment 30.32
  - Efficiency and LD*: Output 36.58; Unemployment rate 2.12; Investment 50.94
  - LS and LD*: Output 100.04; Unemployment rate 98.36; Investment 45.56
- Note:
  - Investment wedge is included in all economies; a number higher than 100 indicates the model-generated series is more volatile than the data for that variable.

### Robustness and alternative employment measures
- Hours worked decomposition:
  - HoursWorked_t = h_t · (1−u_t) · (N_t / P_t) (Equation (31)).
  - Average hours per week h_t and labor force participation rate N_t / P_t are largely acyclical except during the Great Recession; most business-cycle variation in standard hours series originates from employment rate (1−u_t).
- Using hours worked instead of employment rate:
  - Estimation with hours worked leads to unrealistic and unstable posterior estimates.
  - Log-likelihood nearly two times lower than benchmark (1945 versus 1083).
  - Standard deviation of labor demand wedge more than 40 times larger than benchmark.
- Conclusion:
  - Focusing on unemployment (employment rate) is preferable; hours worked do not improve model fit.

### Discussion: why the standard RBC model fails in wage units
- Core reasons:
  - GDP in wage units is more volatile than HP-cyclical GDP; HP filtering masks medium-term variability crucial for fit.
  - Productivity (TFP) innovations explain only 50% of investment fluctuations and virtually none of GDP and unemployment fluctuations in wage units — contradicting standard RBC finding that TFP explains >70% of business cycle fluctuations.
  - Cobb-Douglas production function (constant labor share) is contradicted by strong countercyclicality of labor income share; CES production function (allowing varying labor share) is suggested as an alternative.
  - Disaggregation of labor wedge into LS and LD reveals main problem is with labor supply wedge: it accounts for overwhelming share of variation in unemployment rate (LS explains 114.23% in one-wedge experiment and LS and LD together explain 98.36% of unemployment variance), indicating households are not behaving as if on their labor supply equation in equilibrium during high unemployment spells.
- Implications for modeling:
  - The failure supports assumptions of incomplete labor market models (search, matching, or wage rigidity frameworks) as better representations of unemployment dynamics.
  - Allowing for varying labor share (e.g., CES production) may reduce autocorrelation in labor demand residuals.

### Policy implications and conclusion
- Policy implication emphasized:
  - If part of the decline in measured potential output and the rise in unemployment reflect a switch to an inefficient, demand-determined equilibrium rather than a supply-driven decline, demand-stimulating monetary and fiscal policies are justified.
- Summary of empirical findings:
  - Using wage units preserves medium-term comovements across macro series and reveals that:
    - Standard RBC model struggles to match data in wage units.
    - TFP innovations explain about 50% of investment variation but virtually none of GDP and unemployment variation in wage units.
    - Labor supply wedge has large variance (σ_LS = 0.3091) and explains almost all unemployment variation; labor demand wedge is strongly countercyclical.
    - Cobb-Douglas specification is challenged; incomplete labor market features and varying labor share production functions are promising directions to improve fit.

*Source: wp17201 - References (excerpted content)*

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

### References

### List of tables and figures (excerpts)
- Tables:
  - 1. Prior distributions of parameters of the model (page 16)
  - 2. Estimates of all of the parameters on the quarterly data from 1948:1 to 2011:4 (page 17)
  - 3. Properties of the wedges 1948:1-2011:4 (page 21)
  - 4. Share of variation in output and unemployment explained by different wedges, defined as var(log(x_model_t))/var(log(x_obs_t)) ·100% (page 28)
- Figures (selected):
  - 1. Civilian unemployment rate (percent, left scale, inverted) and GDP in wage units (right scale). Quarterly data 1948:1 - 2011:4. Shaded areas are NBER recession dates. (page 9)
  - 2. Posterior distributions. Grey lines represent prior densities. Dark lines represent posterior densities. Dashed lines are estimated modes of the posterior densities. SE stands for standard error. (page 18)
  - 3–4. Realized wedge series as an outcome of the estimation procedure for the entire postwar period and for 2000:1-2011:4. Grey areas indicate NBER recessions. (pages 19–20)
  - 5. Annualized percent change in TFP: data vs. model. For definitions see equations (29) and (30). (page 22)
  - 6–9. "X wedge only economy" comparisons (Investment, Efficiency, Labor demand, Labor supply). (pages 24–27)
  - 10–12. U.S. GDP detrended in three different ways; components and normalized series of hours worked and employment rate. (pages 29–30)

### Key findings and research questions (Introduction)
- Research questions:
  - Can the post-2008 dynamics of output and unemployment be reconciled within a standard general equilibrium framework together with the rest of the postwar data?
  - If not, what features must a successful model include to explain observed fluctuations absent in the standard model?
- Observed facts motivating the study:
  - The U.S. unemployment rate remained above 8% for more than 11 quarters following the Great Recession.
  - Real U.S. GDP dropped more than 2% during the crisis and, unlike previous postwar recessions, did not jump back to the pre-recession trend.
- Methodological stance:
  - Remove trend from output non-parametrically based on Kaldor facts to obtain series that do not grow over time (GDP in wage units), and do not detrend the unemployment rate in order to assess whether a standard model can explain actual employment levels rather than deviations from a trend.
- Context from literature:
  - Some researchers argue persistent post-2008 dynamics imply a loss in potential output and an increase in equilibrium unemployment (Benes and others (2010), IMF (2009), Ball (2014), Haltmaier (2012) among others).
  - Other literature considers models with multiple equilibria to explain prolonged high unemployment (Kaplan and Menzio (2012); Heathcote and Perri (2012); Farmer (2012); Kashiwagi (2012); Cowen (2012); Guerrazzi (2010); Basu (2009)).
- Implication emphasized:
  - If part of the observed decline in potential output and rise in unemployment reflects a switch to an inefficient, demand-determined equilibrium, then demand-stimulating monetary and fiscal policies are justified, contrasting with the standard supply-determined approach.

### Detrending procedure: GDP in wage units (Section II)
- Definitions and construction:
  - Nominal wage W_t defined as: W_t = (compensation to employees)_t / (number of full-time equivalent employees)_t.
  - GDP in wage units: Z_t = Y_t / W_t · 1/N_t, where Y_t is nominal U.S. GDP and N_t is the civilian labor force in period t.
  - Nominal consumption C_t defined as: C_t = Y_t − I_t, where I_t is nominal investment (sum of nominal private and government investment).
  - Consumption and investment in wage units: C^w_t = C_t / W_t · 1/N_t and I^w_t = I_t / W_t · 1/N_t.
  - National accounts identity preserved in wage units: Z_t = C^w_t + I^w_t.
- Intuition and properties:
  - Dividing by nominal wage removes inflation and the trend in TFP from nominal variables; further dividing by civilian labor force removes growth due to increasing number of workers, yielding a per-capita, wage-deflated series that preserves medium- and long-term comovements across series.
  - The constructed series Z_t can be written as Z_t = (1/b_t)·(1 − u_t), where b_t is labor income share and u_t is unemployment rate.
- Empirical relationships and exact statistics from the data:
  - Correlation between constructed GDP in wage units and the civilian employment rate (one minus unemployment) is -0.88.
  - Correlation between the employment rate defined using full-time equivalent employees and the BLS employment measure is 0.93.
  - Correlation between the constructed employment-rate series and the BLS employment-rate series is close to one - 0.93.
- Observed differences and contributing factors:
  - Divergence of GDP in wage units and the BLS unemployment series after 2009 can potentially be attributed to a fall in the labor share.
  - BLS measures do not distinguish part-time vs full-time workers (FTE measure counts part-time proportionally), contributing to differences in measured employment/unemployment.

### Empirical approach and model overview (Sections II–III)
- Estimation strategy:
  - Carry out a business cycle accounting exercise following Chari, Kehoe, and McGrattan (2007) on data detrended using wage units.
  - Describe a benchmark RBC model (prototype economy) and introduce wedges in the model’s equations to capture deviations from model relationships.
  - Estimate the model using Bayesian techniques to obtain smoothed values for all wedges.
- Novel aspects relative to CKM/RBC literature:
  - By constructing wage series, the labor wedge (residual in the labor-market equilibrium equation) is decomposed into labor supply and labor demand wedges, allowing independent evaluation of labor supply and labor demand predictions.
  - This differs from conventional RBC literature that eliminates wages from equilibrium equations.
- Key empirical reversal relative to standard RBC claims:
  - Contrary to the RBC literature claim that 70% or more of fluctuations are explained by TFP shocks, in the transformed (wage-unit detrended) data most movements in GDP are accounted for by the labor supply wedge.
  - The observed consumption, wage, and unemployment data deviate from the optimal solution implied by household utility maximization, indicating households are not on their labor supply equation.
  - These findings support the view that high unemployment after the Great Recession was inefficient and that models with incomplete labor markets or multiple equilibria may be appropriate (citations in the source).

### Policy implications
- If part of the decline in measured potential output and the rise in unemployment reflect a switch to an inefficient, demand-determined equilibrium rather than a supply-driven decline:
  - Demand-stimulating monetary and fiscal policies are justified, in contrast with policy prescriptions from a standard supply-determined potential-output perspective.
- The analysis suggests re-evaluating policy based on whether unemployment and low output represent inefficiency (welfare loss) rather than only deviations from a trend.

_Italicized source: wp17201 - References (excerpted content)_

### section 5 for the standard alternative specification.

### section 5 for the standard alternative specification.

### Model setup: household, firms, equilibrium
- Preferences and labor:
  - One infinitely-lived household. Employment share lt ∈ (0,1); non-employed share 1−lt.
  - One-period utility: u(ct, lt) = log(ct) + ψ·log(1−lt)
  - Real wage per working member: wt. Real consumption per member: ct.
- Budget and capital accumulation:
  - Budget constraint: ct + it = rt kt + wt lt (Equation (9))
  - Capital law of motion: kt+1 = it + (1−δ) kt (Equation (10))
  - Discount factor: β (fixed at 0.99 in estimation; see Priors).
- Optimality conditions (household):
  - Euler: 1/ct = β Et[1/ct+1 · (1−δ + rt+1)] (Equation (11))
  - Labor supply: ψ ct /(1−lt) = wt (Equation (12))
- Production and firm behavior:
  - Cobb-Douglas production: yt = At kt^α lt^(1−α) (Equation (13)), where 1−α is labor’s share.
  - Factor prices (first-order conditions):
    - rt = α yt / kt (Equation (14))
    - wt = (1−α) yt / lt (Equation (15))
- General equilibrium and accounting identities:
  - National accounts identity: ct + it = yt (Equation (16))
  - Euler with production link: 1/ct = β Et[1/ct+1 · (1−δ + α yt+1 / kt+1)] (Equation (17))
  - State vector determined: xt = [ct, yt, it, kt+1, wt, lt] (Equation (18) listing augmented in text)

### Accounting procedure: concept and implementation
- Objective: evaluate how well the RBC model describes fluctuations in the data by backing out residuals ("wedges") required for each equation to hold exactly given observed series.
- Interpretation:
  - Low-dispersion, white-noise wedges → equation describes data well.
  - High-dispersion or autocorrelated wedges → equation problematic; imply volatile shocks.
  - Contribution analysis: simulate economy with all but one wedge held constant to measure each wedge’s role in matching data.
- Key methodological differences from CKM:
  - Use data that preserves medium and long-term comovements and series in wage units.
  - Use unemployment rate as measure of employment (instead of hours worked).
  - Dataset includes the Great Recession of 2008 (data through 2011:4).
- Wedges introduced (prototype economy):
  - ε_invest in Euler equation (investment wedge) — interpretable as investment tax.
  - ε_LS in labor supply equation (labor supply wedge) — interpretable as labor income tax.
  - ε_eff in production (efficiency / TFP wedge).
  - ε_LD added to labor demand equation (labor demand wedge) to evaluate labor demand separately and avoid statistical singularity.
- Modified system (wedge-included equations, selected):
  - 1/ct · exp(ε_invest,t) = Et[ β · 1/ct+1 · ((1−δ) exp(ε_invest,t+1) + α yt+1 / kt+1) ] (Equation (19))
  - wt = ψ · exp(ε_LS,t) · ct /(1−lt) (Equation (20))
  - At = exp(zt), zt = ρ zt−1 + ε_eff,t (Equations (21)-(23))
  - wt = exp(ε_LD,t) · (1−α) yt / lt (Equation (24))
  - yt = ct + it (Equation (25)); kt+1 = it + (1−δ) kt (Equation (26))
- Estimation approach:
  - Log-linearize system (19)-(26) around deterministic steady state.
  - Solve via Sims (2001) methods, cast as Xt = F Xt−1 + G εt; Yt = H Xt.
  - Parameters estimated by Bayesian maximum likelihood using Kalman filter and random walk Metropolis-Hastings MCMC: 100’000 draws, 50’000 kept.

### Priors and parameterization (as used for estimation)
- Fixed and prior choices:
  - β fixed at 0.99 (corresponds to long-run annual interest rate of 4%).
  - Prior means chosen close to common calibrations; leisure parameter prior mean chosen assuming 5% long-run unemployment.
- Prior distributions and prior means (Table 1 summary):
  - α (Capital share): Beta, prior mean 0.33, Std.Dev 0.15
  - δ (Capital depreciation): Beta, prior mean 0.03, Std.Dev 0.015
  - ρ (Tech. persistence): Beta, prior mean 0.9, Std.Dev 0.05
  - ψ (Leisure parameter): Gamma, prior mean 0.04, Std.Dev 0.02
  - σ_eff (St.dev. of ε_eff): Inv. Gamma, prior mean 0.02, Std.Dev 0.01
  - σ_invest (St.dev. of ε_invest): Inv. Gamma, prior mean 0.02, Std.Dev 0.01
  - σ_LS (St.dev. of ε_LS): Inv. Gamma, prior mean 0.1, Std.Dev 0.05
  - σ_LD (St.dev. of ε_LD): Inv. Gamma, prior mean 0.02, Std.Dev 0.01

### Parameter estimates (posterior results, quarterly data 1948:1–2011:4)
- Posterior means and 90% posterior CI (Table 2 summary):
  - Capital share α: Posterior mean 0.3529; CI 90% [0.3514; 0.3546]
  - Capital depreciation δ: Posterior mean 0.0137; CI 90% [0.0126; 0.0147]
  - Tech. persistence ρ: Posterior mean 0.9774; CI 90% [0.9588; 0.9958]
  - Discount factor β: Fixed 0.99
  - Leisure parameter ψ: Posterior mean 0.0493; CI 90% [0.0478; 0.0509]
  - St.dev. of ε_eff: Posterior mean 0.0093; CI 90% [0.0074; 0.0111]
  - St.dev. of ε_invest: Posterior mean 0.0051; CI 90% [0.0045; 0.0055]
  - St.dev. of ε_LS: Posterior mean 0.3091; CI 90% [0.2846; 0.3341]
  - St.dev. of ε_LD: Posterior mean 0.0265; CI 90% [0.0245; 0.0283]
- Additional estimation diagnostics:
  - MCMC accept. rate 36%
  - 100’000 draws, 50’000 kept
  - log(L) = 1945
- Interpretations from parameter estimates:
  - Labor’s share of total income ≈ 65% (implied by α estimate).
  - Quarterly capital depreciation 0.0137 implies about 5.4% depreciation per year.
  - Long-run unemployment rate implied from ψ ≈ 5.7%.
  - Posterior standard deviation of labor supply wedge (σ_LS = 0.3091) is substantially larger than other wedges — suggests large role in explaining fluctuations.
  - σ_eff ≈ 0.0093 (~1%), σ_invest ≈ 0.0051 (below 1%), σ_LD ≈ 0.0265 (small).

### Realized wedges: properties and graphical impressions
- Summary statistics (Table 3, 1948:1–2011:4):
  - Variance relative to log output and correlations with log output at lags −2, −1, 0, 1, 2:
    - Efficiency: Variance relative to log output 0.15; correlations: −0.2893, −0.2258, −0.0741, 0.0080, 0.0704
    - Investment: Variance relative to log output 0.07; correlations: 0.0253, −0.0548, −0.1469, −0.0807, −0.0630
    - Labor Supply (LS): Variance relative to log output 8.15; correlations: −0.8801, −0.9137, −0.9136, −0.8721, −0.8087
    - Labor Demand (LD): Variance relative to log output 0.64; correlations: −0.8248, −0.8854, −0.9414, −0.9203, −0.8848
  - Cross-correlations between wedges (selected lags −2, −1, 0, 1, 2):
    - Efficiency vs Investment: 0.2425, 0.4758, −0.5250, −0.2658, −0.2369
    - Efficiency vs LS: 0.3128, 0.2536, 0.1562, 0.0448, −0.0495
    - Efficiency vs LD: 0.2339, 0.1787, 0.0217, −0.0262, −0.0552
    - Investment vs LS: −0.0279, 0.0470, 0.0887, 0.1072, 0.0932
    - Investment vs LD: −0.0247, 0.0482, 0.1537, 0.0483, 0.0304
    - LS vs LD: 0.7540, 0.7524, 0.7244, 0.6833, 0.6353
  - 5% critical value (two-tailed) for correlation coefficients is 0.1225.
- Qualitative observations:
  - Investment wedge series looks like white noise with low dispersion and low correlation with output — Euler equation holds well.
  - Efficiency (TFP) wedge: does not explain GDP in wage units; TFP shocks largely eliminated when GDP deflated by money wage.
  - Labor supply wedge: highly autocorrelated and highly (negatively) correlated with log output and strongly correlated with unemployment rate — very large variance relative to output (8.15).
  - Labor demand wedge: highly (negatively) correlated with output; indicates countercyclical labor income share.
  - Strong negative correlation between LS and LD implies the aggregate labor wedge (LS − LD) can mask large opposing movements.

### Contribution analysis: one-wedge economies and explained variance
- Method: feed realized innovations of only one wedge (with others held constant) into estimated policy functions and compare model-generated series to actual data (output, unemployment rate, investment).
- One-wedge economy findings (Figures 6–9 summarized):
  - Investment wedge only:
    - Does not explain output, unemployment rate, or investment — consistent with small σ_invest.
  - Efficiency (TFP) wedge only:
    - Explains approximately 50% of variation in investment.
    - Explains virtually none of variation in GDP (wage units) or in unemployment rate.
    - Intuition: GDP in wage units is almost flat with respect to TFP because real wage reacts almost one-to-one to productivity; investment is more elastic in wage units.
  - Labor demand wedge only:
    - Explains substantial share of output.
    - Explains negligible variation in unemployment rate or investment.
    - Reflects countercyclical labor income share; Cobb-Douglas assumption (constant labor share) is restrictive here.
  - Labor supply wedge only:
    - Explains almost all variation in unemployment rate.
    - Explains significant shares of variation in output and investment.
    - Consistent with large estimated σ_LS.
- Quantitative shares of variation explained (Table 4; all shares defined as var(log(x_model))/var(log(x_obs)) · 100%):
  - Efficiency* (with investment wedge always included): Output 0.20; Unemployment rate 1.32; Investment 49.45
  - Labor demand (LD)*: Output 36.77; Unemployment rate 1.65; Investment 4.91
  - Labor supply (LS)*: Output 23.67; Unemployment rate 114.23; Investment 30.32
  - Efficiency and LD*: Output 36.58; Unemployment rate 2.12; Investment 50.94
  - LS and LD*: Output 100.04; Unemployment rate 98.36; Investment 45.56
  - Note: Investment wedge is included in all economies; a number higher than 100 indicates the model-generated series is more volatile than data for that variable.

### Robustness and alternative employment measures
- Comparison of employment measures:
  - Hours worked decomposition: HoursWorked_t = h_t · (1−u_t) · (N_t / P_t) (Equation (31)).
  - Empirical observation: average hours per week h_t and labor force participation rate N_t / P_t are largely acyclical except during the Great Recession; most business-cycle variation in standard hours series originates from employment rate (1−u_t).
  - Using hours worked instead of employment rate:
    - Estimation with hours worked leads to unrealistic and unstable posterior estimates.
    - Log-likelihood nearly two times lower than benchmark (1945 versus 1083).
    - Standard deviation of labor demand wedge more than 40 times larger than benchmark.
  - Conclusion: focusing on unemployment (employment rate) is preferable; hours worked do not improve model fit.

### Discussion: why the standard RBC model fails in wage units
- Main points:
  - GDP in wage units is more volatile than HP-cyclical GDP; HP filtering masks medium-term variability crucial for fit.
  - Productivity (TFP) innovations explain only 50% of investment fluctuations and virtually none of GDP and unemployment fluctuations in wage units — contradicts standard RBC finding that TFP explains >70% of business cycle fluctuations.
  - Cobb-Douglas production function (constant labor share) is contradicted by strong countercyclicality of labor income share; a CES production function (allowing varying labor share) is suggested as an alternative.
  - Disaggregation of labor wedge into LS and LD reveals main problem is with labor supply wedge: it accounts for 99% (text states "99%" in descriptive conclusion) of variation in unemployment rate (Table 4 and discussion show LS explains overwhelming share). This indicates households are not behaving as if on their labor supply equation in equilibrium during high unemployment spells.
- Implications for modeling:
  - The failure supports assumptions of incomplete labor market models (search, matching, or wage rigidity frameworks) as better representations of unemployment dynamics (references in text: Farmer (2010, 2011); Farmer and Plotnikov (2012); Kocherlakota (2012); Justiniano, Primiceri, and Tambalotti (2010)).
  - Allowing for varying labor share (e.g., CES production) may reduce autocorrelation in labor demand residuals.

### Conclusion (summary of empirical findings)
- Using wage units preserves medium-term comovements across macro series and reveals that:
  - Standard RBC model struggles to match data in wage units.
  - TFP innovations explain about 50% of investment variation but virtually none of GDP and unemployment variation in wage units.
  - Labor supply wedge has large variance (σ_LS = 0.3091) and explains almost all unemployment variation; labor demand wedge is strongly countercyclical.
  - Cobb-Douglas specification is challenged; incomplete labor market features and varying labor share production functions are promising directions to improve fit.

*Source: section 5 for the standard alternative specification. wp17201 - section 5 for the standard alternative specification.*

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*Source: wp17201 - REFERENCES*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17201.pdf_