## _wp1477

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---

### Introduction — Research questions
- Paper addresses three questions:
  - What caused the Great Recession of 2008? Was it a demand or a supply failure?
  - Why has the unemployment rate been so persistent and why has it remained above 8 percent for more than 18 quarters since the official end of the recession?
  - How was this recession different from the other eleven post-war recessions? Can it be explained within a single framework together with the other postwar recessions?
- Figure 1 (referenced): employment dynamics for all postwar recessions including the Great Recession of 2008; X-axis = number of months from peak employment; Y-axis = employment drop in percent from the peak.

### Employment dynamics and "jobless" recoveries
- Empirical patterns:
  - Prior to 1990 recoveries in employment were quick and took less than 8 quarters.
  - After 1990 employment recoveries have been much more sluggish; the last three recoveries were dubbed “jobless.”
  - Big financial crises (“Big Five”) show very persistent employment declines; example: Finland took over seventeen years to return to pre-recession employment after the 1990 banking crisis.
- Figure 1 (Total Nonfarm Employment, BLS): zero normalized at peak employment; X-axis = months from peak employment; Y-axis = employment drop in percent from the peak.

### Wealth losses and relationship to employment recoveries
- Key empirical facts cited:
  - During the Great Recession, the real net worth of U.S. households and non-profit organizations declined by 20 percent within one year.
  - The decline from peak to trough in the last recession was larger than 25 percent.
- Table 1 summary (Wealth measured as Net Worth of Households and Nonprofit Organizations / CPI, Source: FRED):
  - Recession — EmploymentAverage wealth recovery (months)change (%)
  - 1953 — 81.20
  - 1957 — 80.75
  - 1960 — 61.12
  - 1969 — 60.65
  - 1974 — 60.71
  - 1980 — 40.51
  - 1981 — 90.75
  - 1990 — 11 — -0.15
  - 2001 — 16 — -0.13
  - 2007 — >66 — -3.04
- Summary insight:
  - The last three “jobless” recessions are associated with negative average wealth changes; recessions prior to 1990 had positive average wealth change.
  - Recessions associated with large wealth losses (financial crises, housing busts) show sharp, persistent employment declines.

### Model: core hypotheses and mechanisms
- Model type:
  - General equilibrium rational expectations model with multiple steady state equilibria.
- Central mechanism:
  - Wealth (including beliefs about permanent income) can change for both fundamental and non-fundamental reasons.
  - Belief-driven declines in perceived real assets reduce real wealth, lower aggregate demand, cause layoffs, and can move the economy to a higher-unemployment equilibrium.
- Main implications:
  - Unemployment exhibits hysteresis and path dependence (Blanchard and Summers (1986, 1987)).
  - Demand and supply shocks are both important; persistent demand shocks can have permanent effects on unemployment and the real wage but not on real output and its components.
  - Demand (belief/permanent-income) shocks can generate high persistence in unemployment even if they are i.i.d.
  - TFP shocks cause large increases in real investment, smaller increases in real output, and small increases in real consumption (qualitatively similar but more protracted effects than standard RBC).

### Model structure — key features and equations (high-level)
- Core: real version of incomplete factor market / matching model (based on Farmer (2006)).
- Households:
  - Representative household, no disutility from working, search participation h_t = 1, utility u(c_t) = log(c_t).
  - Budget: c_t + I_t = r_t k_t + w_t l_t.
  - Capital evolution: I_t = k_{t+1} − (1−δ)k_t.
  - Euler equation: 1/c_t = β E_t[1/c_{t+1} (1−δ + r_{t+1})].
- Firms / production:
  - CES production y_t = (a·k_t^ρ + b·s_t^ρ x_t^ρ)^{1/ρ}, a + b = 1.
  - Elasticity of substitution ε_{k,l} = 1/(1−ρ).
  - First order conditions: r_t = a (y_t / k_t)^{1−ρ}, w_t = b s_t^ρ Θ_t^{ρ} (y_t / l_t)^{1−ρ}, where Θ_t = (1 − 1/q_t).
- Labor market / matching:
  - m_t = (Γ v̄_t)^θ h̄_t^{1−θ}; q_t = m_t / v̄_t, ̃q_t = m_t / h̄_t.
  - Under assumptions q_t = Γ l̄_t^{(1−θ)/θ}, ̃q_t = l̄_t.
  - Externality Θ_t = (1 − l̄_t^{(1−θ)/θ} Γ).
- Aggregate production with externality:
  - ȳ_t = (a k̄_t^ρ + b s_t^ρ l̄_t^ρ (1 − l̄_t^{(1−θ)/θ} Γ)^ρ)^{1/ρ}, which for fixed k̄_t is an inverted U-shape in l̄_t.
  - Socially optimal employment l^* = (θ Γ)^{θ/(1−θ)}.
- Model algebra:
  - Six core equilibrium equations (Euler + resource + capital law of motion + production + wage + TFP AR(1)) with seven unknowns → continuum of steady states unless closed by extra condition.

### Closing the model with beliefs / permanent income hypothesis
- Consumption proportional to perceived permanent income: c_t = φ y^P_t.
- Adaptive updating of permanent income (Friedman (1957) inspired):
  - (y^P_t / w_t) = (y^P_{t−1} / w_{t−1})^{χ} (y_t / w_t)^{1−χ} exp(e^b_t), with e^b_t ∼ N(0, σ_b^2).
  - e^b_t interpreted as a belief (non-fundamental) shock (“animal spirits”); 1−χ measures speed of adjustment.
- Under rational expectations and in steady states y^P_ss = y_ss.
- φ must equal steady-state consumption to income ratio:
  - φ ≡ 1−δ (a / (1/β − (1−δ)))^{1/(1−ρ)}.
- The belief function resolves dynamic indeterminacy (selects the saddle path each period) but multiplicity of steady states remains; initial y^P_0 pins down which steady state the economy converges to → hysteresis and path dependence.

### Shocks, dynamics, and estimation strategy
- Two shock types:
  - i.i.d. perceived permanent-income shocks (belief shocks).
  - Persistent productivity (TFP) shocks.
- Estimation:
  - Bayesian MCMC (random walk Metropolis-Hastings); data: quarterly civilian unemployment rate, output in wage units, and investment in wage units for 1948:1–2011:4.
  - Observed variables mapped to model counterparts: 1 − l_t, y_t / w_t, I_t / w_t.
  - Wedge ε^LD_t in wage/output relation to avoid stochastic singularity: 1 = b·exp(ε^LD_t) s_t^ρ (1 − l_t^{(1−θ)/θ} Γ)^ρ (y_t / l_t)^{1−ρ}.
  - Linearized around steady state associated with long-run mean civilian unemployment rate 5.7%.
  - Solution methods: Sims (2001); likelihood via Kalman filter (diffuse Kalman for unit eigenvalue).
  - MCMC draws: 100,000, keep 50,000.

### Key parameter choices, priors and calibration
- Fixed / calibration:
  - β = 0.99 (fixed).
  - u^* (optimal unemployment rate) = 0.045 (fixed); Γ chosen so that l^* matches u^*.
- Prior means / distributions (selected entries):
  - a (≈ Capital share if ρ = 0): beta prior, prior mean 0.33, Std. Dev. 0.15.
  - ε_{k,l} (Elasticity between capital and labor): beta prior, prior mean 0.50, Std. Dev. 0.25.
  - δ (Capital depreciation): beta prior, prior mean 0.03, Std. Dev. 0.015.
  - λ (Labor productivity persistence): beta prior, prior mean 0.90, Std. Dev. 0.05.
  - χ (Expectations persistence): beta prior, prior mean 0.90, Std. Dev. 0.05.
  - θ (Elasticity of matching function): beta prior, prior mean 0.5, Std. Dev. 0.25.
  - σ_p (St.dev. of productivity innovation ε_p): Inv. Gamma prior, prior mean 0.02, Std. Dev. 0.01.
  - σ_b (St.dev. of belief shock ε_b): Inv. Gamma prior, prior mean 0.02, Std. Dev. 0.01.
  - Prior mean for θ set at 0.5 with Sd. 0.25.
  - Prior mean for σ_p and σ_b set to 2 percent (steady state TFP calibration).
  - Prior standard deviations chosen "as large as possible" given means and natural parameter limits.

### Posterior estimates (MCMC outcome)
- MCMC details: log L = 2101; MCMC accept. rate 32.84%; 100000 draws; 50000 kept.
- Posterior means and 90 percent posterior confidence intervals (centered around posterior mode):
  - a ≈ Capital share (= if ρ = 0): 0.4585 [0.3929, 0.5212]
  - ε_{k,l} Elasticity b/w capital and labor: 0.9209 [0.8804, 0.9611]
  - δ Capital depreciation: 0.0082 [0.0079, 0.0086]
  - λ Labor productivity persistence: 0.9175 [0.8784, 0.9531]
  - χ Expectations persistence: 0.9513 [0.9223, 0.9820]
  - θ Elasticity of the matching function: 0.6284 [0.3680, 0.8920]
  - σ_p St.dev. of ε_p: 0.0156 [0.0141, 0.0172]
  - σ_b St.dev. of ε_b: 0.0082 [0.0076, 0.0089]
  - σ_LD St.dev. of ε_LD: 0.0245 [0.0227, 0.0263]
- Interpretations / derived quantities:
  - a = 0.4585 and ε_{k,l} = 0.9209 imply ρ = -0.089; economy approximated by Cobb-Douglas.
  - Estimated quarterly capital depreciation δ = 0.0082 corresponds to approximately 3.3 percent per year.
  - a, ρ, δ together with β imply households consume 84 percent of their permanent income (reference Equation (25)); compares with Friedman (1957) estimate of 90 percent.
  - Persistence: λ = 0.92 and χ = 0.95 (posterior means).
  - θ not well identified; CI contains values from 0.4 to 0.9.
  - Posterior standard deviations relatively small: σ_p = 0.0156, σ_b = 0.0082, σ_LD = 0.0245.
- Identification notes:
  - Posterior densities for all estimated parameters, except θ, differ significantly from priors → identification.
  - Low σ_b implies belief disturbances need not be highly volatile to explain the data.
  - Low σ_p suggests belief shocks explain a share of observed volatility usually attributed to TFP shocks.

### Model performance (Monte-Carlo exercise)
- Simulation design: R = 10000 Monte-Carlo samples, T = 256; report Monte-Carlo average and 90 percent Monte-Carlo confidence interval.
- Series analyzed (log-deviations from statistical means): I_t^w, c_t^w, y_t^w, l_t.
- Standard deviations (Simulations MC Avg. [90% CI] — Data):
  - I_t^w: 14.468 [10.883,18.017] — Data: 12.363
  - c_t^w: 5.589 [2.319,8.849] — Data: 5.648
  - y_t^w: 6.007 [2.831,9.173] — Data: 3.730
  - l_t: 5.972 [2.784,9.149] — Data: 1.767
- Persistence (Corr(X_t, X_{t−1}), Simulations MC Avg. [90% CI] — Data):
  - I_t^w: 0.904 [0.858,0.950] — Data: 0.904
  - c_t^w: 0.981 [0.963,0.999] — Data: 0.988
  - y_t^w: 0.968 [0.942,0.997] — Data: 0.966
  - l_t: 0.971 [0.942,0.997] — Data: 0.970
- Unit root test results (Simulations: share where unit root not rejected at 5% — Data: p-value for unit root null):
  - I_t^w: Simulations share 0.0022 — Data p-value 0.0001 (investment stationary in >99% of simulations)
  - c_t^w: Simulations share 0.7358 — Data p-value 0.3407 (consumption behaves as random walk in 73% of simulations)
  - y_t^w: Simulations share 0.5167 — Data p-value 0.0307 (output borderline)
  - l_t: Simulations share 0.5518 — Data p-value 0.0758 (employment borderline)
- Performance summary:
  - Model generates standard deviations similar to data; Monte-Carlo means close to actual standard deviations except employment rate.
  - Investment most volatile; consumption smooth; GDP in model more volatile than measured GDP partly because data GDP includes government expenditures (excluding government purchases raises data GDP std from 3.730 to 5.204).
  - Model matches persistence of all series almost exactly despite belief shocks being i.i.d. and productivity persistence 0.9175.
  - Simulations reproduce unit-root behavior similar to data: investment mostly stationary, consumption often random walk, output and employment borderline.

### Mechanism and decomposition
- Structural explanation:
  - Estimated policy rule always has an eigenvalue equal to unity → simulated series nonstationary; economy shifts between steady states (hysteresis).
  - Consumption and output nonstationarity driven by belief shocks; investment more stationary because steady-state shifts from belief shocks are small relative to productivity effects.
- Decompositions:
  - Both shocks together: consumption smooth and persistent; investment stationary and less persistent; output volatile and persistent.
  - Belief shocks only: all series highly persistent; belief shocks drive persistence but produce very small investment volatility because consumption share ≈ five times investment share.
  - Productivity shocks only: largest effects on investment; consumption almost flat due to adaptive expectations smoothing; investment fluctuations translate almost one-to-one into output fluctuations.
- Interaction result:
  - Belief shocks dominate consumption dynamics (persistence); productivity shocks dominate investment volatility; combined generate persistent and volatile output as in data.

### Case study: The Great Recession (2007:4–2011:4)
- Experiment setup:
  - Standard RBC model calibrated to same parameter values as benchmark model; ψ calibrated via ψ·c_t = w_t.
  - Raw quarterly TFP series from San Francisco FRB used to back out implied TFP innovations assuming log(s_t) = λ̂·log(s_{t−1}) + ε_p_t with λ̂ = 0.9175 and s_{2007:3} = 1.
  - Economy assumed at steady state prior to 2007:3; linearized policy functions used to generate series.
- Findings:
  - RBC model with TFP shocks fails to explain persistent decline in employment and real output after 2007; predicts recovery by mid-2010.
  - Benchmark multiple-equilibrium model with same TFP shocks produces more persistent effects: recovery only at end of sample; employment more persistent; matches recovery in real output almost exactly.
- Identification of belief shocks from data:
  - Use benchmark model implication: a 1 percent negative belief shock permanently increases real wage by 1 percent and unemployment by 1 percent while real variables negligibly affected.
  - Obtain ε̂_b_t from differences between actual log real wage and log real wage implied by TFP-only model:
    - ε̂_b_t = (log(w_data_t) − log(w_TFP_t)) − (log(w_data_{t−1}) − log(w_TFP_{t−1}))
- Result:
  - Model with both TFP and implied belief shocks can explain almost the entire decline in employment since 2007; real output largely unaffected by belief shocks.

### Impulse response functions (IRFs) — qualitative summary
- IRF to a negative 1 percent TFP shock (≈ 60 percent of σ_p):
  - Productivity autocorrelated with λ = 0.9175; initial effects disappear after ≈ 45 quarters.
  - Investment most volatile; consumption changes by less than 0.25 percent in absolute value.
  - Long run: convergence to same real output level as before shock but with slightly higher investment and slightly lower consumption.
  - Unemployment effect persistent: only 2/3 of initial impact disappears after 50 quarters; after 75 periods 80 percent absorbed and 10 percent permanent.
  - Real wage exhibits small overall change (< 0.2 percent) — endogenous real rigidity.
- Mechanism for real-wage rigidity:
  - Real wage w_t = MPL_t · Θ_t^ρ = b s_t^ρ (y_t / l_t)^{1−ρ} · (1 − l_t^{1−θ}/θ · Γ)^{ρ}.
  - MPL_t falls after negative TFP shock but externality term Θ_t^ρ increases as unemployment rises; these nearly cancel → small net real-wage movement.
- IRF to a one-time negative 1 percent belief shock:
  - Unemployment rate and real wage increase by 1 percent permanently.
  - Real output, investment, consumption change by magnitudes ~20 times less than initial shock (practically unaffected).
  - Belief shocks move economy to new steady state with higher unemployment while leaving real output similar → explains "jobless recoveries."

### Key implications and conclusions
- Model displays hysteresis in unemployment (Blanchard and Summers (1987) type) and can explain both pre-1990 and post-1990 "jobless" recoveries.
- Two driving shocks required to match postwar macro dynamics:
  - TFP shocks: business-cycle frequencies, large effects on investment, transitory on real output but with long-lasting unemployment propagation.
  - Belief (permanent-income) shocks: explain permanent increases in unemployment and jobless recoveries; need not be highly volatile (σ_b posterior mean 0.0082).
- Neither demand (belief) nor supply (TFP) shocks alone suffice:
  - Demand-only generates unemployment volatility puzzle.
  - Supply-only cannot explain investment volatility.
- Contributions:
  - Expands endogenous business cycle models to include investment, consumption and real wages.
  - Provides micro-foundations for endogenous real-wage rigidity with respect to TFP shocks (not imposed).
  - Interprets belief shocks as Keynesian "animal spirits" (Farmer (2012b)); stock market and housing wealth falls can be self-fulfilling belief shocks affecting permanent income expectations and explaining aspects of the Great Recession.

*Source: content unit from _wp1477 (text provided).*

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

### _wp1477 - References .............................................................................................................

### Introduction — Research questions
- This paper addresses three questions.
- First, what caused the Great Recession of 2008? Was it a demand or a supply failure?
- Second, why has the unemployment rate been so persistent and why has it remained above 8 percent for more than 18 quarters since the official end of the recession?
- Third, how was this recession different from the other eleven post-war recessions? Can it be explained within a single framework together with the other postwar recessions?
- Figure 1 presents employment dynamics for all postwar recessions including the Great Recession of

*Source: _wp1477 - References .............................................................................................................*

### 2008. The X-axis corresponds to the number of months from peak employment in the corresponding

### _wp1477 - 2008. The X-axis corresponds to the number of months from peak employment in the corresponding

### Employment dynamics and "jobless" recoveries
- Employment dynamics in postwar recessions varied significantly.
- Prior to 1990 recoveries in employment were quick and took less than 8 quarters.
- After 1990 employment recoveries have been much more sluggish; the last three recoveries were dubbed “jobless.”
- Figure 1 (Total Nonfarm Employment, BLS) normalizes zero at peak employment; X-axis = months from peak employment; Y-axis = employment drop in percent from the peak.
- Big financial crises (“Big Five”) show very persistent employment declines; Figure 2 uses X-axis = years from peak employment and Y-axis = percent from peak. Example: Finland took over seventeen years to return to pre-recession employment after the 1990 banking crisis.

### Wealth losses and their relationship to employment recoveries
- The paper emphasizes wealth changes as a driver of the speed of employment recovery.
- Empirical facts cited:
  - During the Great Recession, the real net worth of U.S. households and non-profit organizations declined by 20 percent within one year.
  - The decline from peak to trough in the last recession was larger than 25 percent.
- Table 1 (Wealth losses and joblessness of recoveries). Wealth measured as Net Worth of Households and Nonprofit Organizations (quarterly data) divided by CPI. Source: FRED.
  - Recession EmploymentAverage wealth recovery (months)change (%)
  - 1953 81.20
  - 1957 80.75
  - 1960 61.12
  - 1969 60.65
  - 1974 60.71
  - 1980 40.51
  - 1981 90.75
  - 1990 11 -0.15
  - 2001 16 -0.13
  - 2007 >66 -3.04
- Summary insight: the last three “jobless” recessions are associated with negative average wealth changes; recessions prior to 1990 had positive average wealth change.
- Recessions associated with large wealth losses (financial crises, housing busts) show sharp, persistent employment declines.

### Model: core hypotheses and mechanisms
- The paper develops and estimates a general equilibrium rational expectations model with multiple steady state equilibria.
- Central mechanism: wealth (including beliefs about permanent income) can change for both fundamental and non-fundamental reasons; belief-driven declines in perceived real assets reduce real wealth, lower aggregate demand, cause layoffs, and can move the economy to a higher-unemployment equilibrium.
- Model implications:
  - The unemployment rate exhibits hysteresis and path dependence (Blanchard and Summers (1986), Blanchard and Summers (1987)).
  - Demand and supply shocks are both important; persistent demand shocks can have permanent effects on unemployment and the real wage but not on real output and its components.
  - Demand shocks (belief/permanent-income shocks) can generate high persistence in unemployment even if they are i.i.d. (self-fulfilling belief shocks).
  - TFP shocks have qualitatively similar but more protracted effects than in standard RBC models: they cause large increases in real investment, smaller increases in real output, and small increases in real consumption.

### Model structure — key features and equations (high-level)
- Core: real version of incomplete factor market / matching model (based on Farmer (2006)), with:
  - Households: representative household, no disutility from working, search participation h_t = 1, utility u(c_t) = log(c_t). Budget: c_t + I_t = r_t k_t + w_t l_t. Capital evolves: I_t = k_{t+1} − (1−δ)k_t. Euler equation (nominal): 1/c_t = β E_t[1/c_{t+1} (1−δ + r_{t+1})].
  - Firms: CES production y_t = (a·k_t^ρ + b·s_t^ρ x_t^ρ)^{1/ρ}, a + b = 1, elasticity of substitution ε_{k,l} = 1/(1−ρ). First order conditions: r_t = a (y_t / k_t)^{1−ρ}, w_t = b s_t^ρ Θ_t^{ρ} (y_t / l_t)^{1−ρ}, where Θ_t = (1 − 1/q_t) is an externality from search frictions.
  - Labor market: matching technology m_t = (Γ v̄_t)^θ h̄_t^{1−θ}; q_t = m_t / v̄_t, ̃q_t = m_t / h̄_t. Under model assumptions q_t = Γ l̄_t^{(1−θ)/θ}, ̃q_t = l̄_t. Externality Θ_t = (1 − l̄_t^{(1−θ)/θ} Γ).
  - Aggregate production with externality: ȳ_t = (a k̄_t^ρ + b s_t^ρ l̄_t^ρ (1 − l̄_t^{(1−θ)/θ} Γ)^ρ)^{1/ρ}, which for fixed k̄_t is an inverted U-shape in l̄_t; socially optimal employment l^* = (θ Γ)^{θ/(1−θ)}.
- The model has six core equilibrium equations (Euler + resource + capital law of motion + production + wage + TFP AR(1)), but seven unknowns, producing a continuum of steady states unless closed by an extra condition.

### Closing the model with beliefs / permanent income hypothesis
- The model is closed with a belief function using adaptive expectations (Friedman (1957) inspired):
  - Consumption proportional to perceived permanent income: c_t = φ y^P_t.
  - Adaptive updating of permanent income: (y^P_t / w_t) = (y^P_{t−1} / w_{t−1})^{χ} (y_t / w_t)^{1−χ} exp(e^b_t), with e^b_t ∼ N(0, σ_b^2).
  - e^b_t interpreted as a belief (non-fundamental) shock (“animal spirits”); 1−χ measures speed of adjustment of permanent income to new information.
- Under rational expectations and in steady states, permanent income equals GDP: y^P_ss = y_ss. The parameter φ must equal steady-state consumption to income ratio:
  - φ ≡ 1−δ (a / (1/β − (1−δ)))^{1/(1−ρ)} (Equation (28)).
- The belief function resolves dynamic indeterminacy (selects the saddle path each period) but does not remove multiplicity of steady states; initial permanent income y^P_0 pins down which steady state the economy converges to — model exhibits hysteresis and path dependence.

### Shocks, dynamics, and estimation strategy
- Two shock types: i.i.d. perceived permanent-income shocks (belief shocks) and persistent productivity (TFP) shocks.
- The model can reproduce jobless recoveries and postwar dynamics when driven by combinations of TFP and belief shocks.
- Estimation:
  - Bayesian MCMC estimation (random walk Metropolis-Hastings); data: quarterly civilian unemployment rate, output in wage units, and investment in wage units for 1948:1–2011:4.
  - Observed variables mapped to model counterparts: 1 − l_t, y_t / w_t, I_t / w_t.
  - To avoid stochastic singularity and accommodate labor income share behavior, a wedge ε^LD_t is inserted in the wage/output relation (Equation (33)): 1 = b·exp(ε^LD_t) s_t^ρ (1 − l_t^{(1−θ)/θ} Γ)^ρ (y_t / l_t)^{1−ρ}.
  - Model linearized around the steady state associated with the long-run mean civilian unemployment rate 5.7%.
  - Solution uses Sims (2001) methods; likelihood via Kalman filter (diffuse Kalman filter for unit eigenvalue); MCMC draws: 100,000, keep 50,000.

### Key parameter choices and priors (as used in estimation)
- Fixed / calibration:
  - β = 0.99 (fixed).
  - u^* (optimal unemployment rate) = 0.045 (fixed) — used to pick Γ so that l^* matches u^*.
- Prior means and distributions (Table 2 excerpts):
  - a (≈ Capital share if ρ = 0): beta prior, prior mean 0.33, Std. Dev. 0.15.
  - ε_{k,l} (Elasticity between capital and labor): beta prior, prior mean 0.50, Std. Dev. 0.25.
  - δ (Capital depreciation): beta prior, prior mean 0.03, Std. Dev. 0.015.
  - λ (Labor productivity persistence): beta prior, prior mean 0.90, Std. Dev. 0.05.
  - χ (Expectations persistence): beta prior, prior mean 0.90, Std. Dev. 0.05.
  - θ (Elasticity of matching function): beta prior, prior mean 0.5, Std. Dev. 0.25.
  - σ_p (St.dev. of productivity innovation ε_p): Inv. Gamma prior, prior mean 0.02, Std. Dev. 0.01.
  - σ_b (St.dev. of belief shock ε_b): Inv. Gamma prior, prior mean 0.02, Std. Dev. 0.01.
- Calibration notes:
  - Prior mean for ε_{k,l} = 0.5 informed by literature (Klump, McAdam and Willman (2007)).
  - Quarterly depreciation prior mean 0.03 corresponds to annual depreciation 12 percent.
  - Prior mean for expectations speed 0.8 similar to equilibrium persistence in Evans and Ramey (2006).
  - Γ chosen so socially optimal unemployment u^* = 4.5 percent (Hall (2011)); Γ re-chosen for each θ during likelihood evaluation to satisfy l^* condition.

### Empirical and theoretical conclusions
- Wealth declines, including belief-driven permanent-income declines, can explain the persistence of unemployment and “jobless” recoveries observed since 1990 and during the Great Recession.
- A model closed with adaptive expectations about permanent income generates:
  - Hysteresis and path dependence in unemployment.
  - Persistence in unemployment from belief shocks even if those shocks are i.i.d.
  - Coexistence of high persistent unemployment with positive real growth.
- Both demand (belief/permanent-income) and supply (TFP) shocks are required to match postwar macro dynamics; combining them in appropriate proportions can replicate both typical postwar recessions and post-1990 jobless recoveries.

*Source: _wp1477 - 2008. The X-axis corresponds to the number of months from peak employment in the corresponding (PDF).*

### 0.28 in Shimer (2005) to 0.54 in Mortensen and Nagypal (2007). Farmer (2012b) fixesθat 0.5.

### _wp1477 - 0.28 in Shimer (2005) to 0.54 in Mortensen and Nagypal (2007). Farmer (2012b) fixesθat 0.5.

### Priors and calibration
- Prior mean for θ set at 0.5 with a standard deviation of 0.25.
- Prior mean for the standard deviation of the productivity innovations set to 2 percent (steady state TFP calibration).
- Same prior mean chosen for the standard deviation of belief shocks, σ_b, and the labor demand shock, σ_LD.
- Prior standard deviations chosen "as large as possible" given means and natural parameter limits (rule of thumb: distance between prior mean and closest natural limit < two standard deviations).

### Posterior estimates (MCMC outcome, Table 3)
- MCMC details: log L = 2101; MCMC accept. rate 32.84%; 100000 draws; 50000 kept.
- For every parameter the prior mean, posterior mean and the 90 percent posterior confidence interval centered around the posterior mode are reported; posterior estimates are stable to priors.
- Posterior means (parameter — posterior mean [90% CI]):
  - a ≈ Capital share (= if ρ = 0): 0.4585 [0.3929, 0.5212]
  - ε_{k,l} Elasticity b/w capital and labor: 0.9209 [0.8804, 0.9611]
  - δ Capital depreciation: 0.0082 [0.0079, 0.0086]
  - β Discount factor: ––
  - u^* Optimal unemployment rate: ––
  - λ Labor productivity persistence: 0.9175 [0.8784, 0.9531]
  - χ Expectations persistence: 0.9513 [0.9223, 0.9820]
  - θ Elasticity of the matching function: 0.6284 [0.3680, 0.8920]
  - σ_p St.dev. of ε_p: 0.0156 [0.0141, 0.0172]
  - σ_b St.dev. of ε_b: 0.0082 [0.0076, 0.0089]
  - σ_LD St.dev. of ε_LD: 0.0245 [0.0227, 0.0263]
- Interpretations and derived quantities:
  - a = 0.4585 and ε_{k,l} = 0.9209 imply ρ = -0.089; together a and ρ imply the U.S. economy can be well approximated by a Cobb-Douglas function.
  - Estimated capital depreciation rate corresponds to approximately 3.3 percent per year (0.0082 quarterly).
  - The three parameters a, ρ, δ together with β imply households consume 84 percent of their permanent income (reference: Equation (25)); compares with Friedman (1957) estimate of 90 percent.
  - Persistence parameters: λ = 0.92 and χ = 0.95 (posterior means), implying sufficient persistence but both far from unity.
  - θ (matching elasticity) is not well identified; reported CI contains values from 0.4 to 0.9.
  - Posterior standard deviations of realized shocks are relatively small:
    - σ_p = 0.0156 (technology shock standard error ≈ 1.6 percent, below standard calibration 2 percent)
    - σ_b = 0.0082 (belief shocks standard deviation < 1 percent)
    - σ_LD = 0.0245

### Identification and posterior vs prior
- Posterior densities for all estimated parameters, except θ, differ significantly from priors, indicating identification; posterior for θ does not differ much from prior, indicating little information in data on θ.
- Low σ_b implies non-fundamental (belief) disturbances need not be highly volatile to explain the data.
- Low σ_p suggests belief shocks explain a share of observed volatility usually attributed to TFP shocks.

### Model performance (Monte-Carlo exercise, R = 10000, T = 256)
- Simulation design: R = 10000 Monte-Carlo samples of length T = 256; report Monte-Carlo average and 90 percent Monte-Carlo confidence interval.
- Series analyzed (log-deviations from statistical means): investment in wage units (I_t^w), consumption in wage units (c_t^w), output in wage units (y_t^w), employment rate (l_t).
- Standard deviations (Simulations MC Avg. [90% CI] — Data):
  - I_t^w: 14.468 [10.883,18.017] — Data: 12.363
  - c_t^w: 5.589 [2.319,8.849] — Data: 5.648
  - y_t^w: 6.007 [2.831,9.173] — Data: 3.730
  - l_t: 5.972 [2.784,9.149] — Data: 1.767
- Persistence (Corr(X_t, X_{t−1}), Simulations MC Avg. [90% CI] — Data):
  - I_t^w: 0.904 [0.858,0.950] — Data: 0.904
  - c_t^w: 0.981 [0.963,0.999] — Data: 0.988
  - y_t^w: 0.968 [0.942,0.997] — Data: 0.966
  - l_t: 0.971 [0.942,0.997] — Data: 0.970
- Unit root tests (Simulations: share of simulated series where unit root not rejected at 5% — Data: p-value for unit root null):
  - I_t^w: Simulations share 0.0022 — Data p-value 0.0001 (investment stationary in >99% of simulations)
  - c_t^w: Simulations share 0.7358 — Data p-value 0.3407 (consumption behaves as random walk in 73% of simulations)
  - y_t^w: Simulations share 0.5167 — Data p-value 0.0307 (output borderline)
  - l_t: Simulations share 0.5518 — Data p-value 0.0758 (employment borderline)
- Key performance findings:
  - Model generates standard deviations similar to data; Monte-Carlo means close to actual standard deviations for all series except employment rate.
  - Investment is most volatile; consumption less volatile and smooth; GDP in model more volatile than measured GDP in data partly because data GDP includes government expenditures (excluding government purchases raises data GDP std from 3.730 to 5.204).
  - Model matches persistence of all series almost exactly despite belief shocks being i.i.d. and productivity persistence being only 0.9175.
  - Simulations reproduce unit-root behavior similar to data: investment mostly stationary, consumption often random walk, output and employment borderline.

### Mechanism of the model
- Structural features:
  - Estimated policy rule always has an eigenvalue equal to unity, making simulated series nonstationary; economy shifts between steady states (hysteresis).
  - Consumption and output nonstationarity driven by belief shocks; investment appears more stationary because its steady-state shifts from belief shocks are small relative to productivity effects.
- Decomposition exercises (figures summarized):
  - Both shocks (productivity + belief): consumption smooth and persistent; investment stationary and less persistent; output volatile and persistent.
  - Belief shocks only: all series highly persistent and belief shocks drive persistence but produce very small investment volatility in levels because consumption share ≈ five times investment share.
  - Productivity shocks only: largest effects on investment; consumption almost flat due to adaptive expectations smoothing; investment fluctuations translate almost one-to-one into output fluctuations.
- Interaction result:
  - Belief shocks dominate consumption dynamics (persistence), productivity shocks dominate investment volatility; combined produce persistent and volatile output as in data.

### Case study: The Great Recession (2007:4–2011:4)
- Experiment setup:
  - Standard RBC model (separable linear disutility from labor) calibrated to same parameter values as benchmark model; ψ calibrated via ψ·c_t = w_t.
  - Raw quarterly TFP series from San Francisco FRB used to back out implied TFP innovations assuming log(s_t) = λ̂·log(s_{t−1}) + ε_p_t with λ̂ = 0.9175 and s_{2007:3} = 1.
  - Economy assumed at steady state prior to 2007:3; linearized policy functions used to generate series.
- Findings comparing models:
  - RBC model with TFP shocks fails to explain persistent decline in employment and real output after 2007; predicts recovery by mid-2010 which did not occur.
  - Benchmark multiple-equilibrium model with same TFP shocks produces more persistent effects: recovery only at end of sample, employment rate more persistent, matches recovery in real output almost exactly.
- Identification of belief shocks from data:
  - Use the benchmark model’s result that a 1 percent negative belief shock permanently increases real wage by 1 percent and unemployment by 1 percent, affecting real variables negligibly.
  - Obtain ε̂_b_t from differences between actual log real wage and log real wage implied by TFP-only model:
    - ε̂_b_t = (log(w_data_t) − log(w_TFP_t)) − (log(w_data_{t−1}) − log(w_TFP_{t−1}))
- Result of including implied belief shocks with TFP shocks:
  - Model with both shock series can explain almost the entire decline in employment since 2007; real output largely unaffected by belief shocks.

### Impulse response functions (IRFs) — qualitative summary
- IRF to a negative 1 percent TFP shock (≈ 60 percent of σ_p):
  - Productivity process autocorrelated with λ = 0.9175, initial effects disappear after ≈ 45 quarters.
  - Left panel (output and components): investment most volatile; consumption changes by less than 0.25 percent in absolute value (consumption drops less than in RBC model due to adaptive expectations), leading to larger initial fall in investment and slower recovery of output.
  - Long run: convergence to same real output level as before shock but with slightly higher investment and slightly lower consumption.
  - Right panel (unemployment and real wage): unemployment rate effect is persistent — only 2/3 of initial impact disappears after 50 quarters; after 75 periods 80 percent absorbed and 10 percent permanent; real wage exhibits small overall change (< 0.2 percent) — endogenous real rigidity.
- Mechanism for apparent real-wage rigidity:
  - Real wage w_t = MPL_t · Θ_t^ρ = b s_t^ρ (y_t / l_t)^{1−ρ} · (1 − l_t^{1−θ}/θ · Γ)^{ρ}.
  - MPL_t falls after negative TFP shock but externality term Θ_t^ρ increases as unemployment rises; these effects nearly cancel, yielding small net real-wage movement.
  - Rigidity is endogenous, not imposed.
- Belief shock IRF:
  - One-time negative 1 percent belief shock: unemployment rate and real wage increase by 1 percent permanently; real output, investment, consumption change by magnitudes ~20 times less than initial shock (practically unaffected).
  - Belief shocks act as drops in consumption demand expressed relative to real wages → price drop → increase in real wage while real quantities unchanged.
  - Hysteresis: negative belief shock moves economy to new steady state with higher unemployment while leaving real output similar, explaining "jobless recoveries."

### Key implications and conclusions
- Constructed and estimated a general equilibrium rational expectations model displaying hysteresis in unemployment (Blanchard and Summers (1987) type).
- Model explains both "jobless" post-1990 recoveries and postwar recoveries prior to 1990.
- Model reproduces stylized business cycle dynamics in output, consumption, investment and real wages; in particular, real wages are slightly countercyclical as in the data.
- Advances literature by:
  - Expanding second-generation endogenous business cycle models to include investment, consumption and real wages.
  - Providing micro-foundations for endogenous real-wage rigidity with respect to TFP shocks (without imposing wage rigidity).
- Two driving shocks: TFP shocks (business-cycle frequencies) and belief shocks (permanent shifts in unemployment).
  - Productivity shocks have transitory effects and also long-lasting effects on unemployment via endogenous propagation.
  - Belief shocks explain permanent increases in unemployment and jobless recoveries; they need not be highly volatile to matter.
- Neither demand (belief) nor supply (TFP) shocks alone can explain postwar employment and GDP dynamics: demand-only generates unemployment volatility puzzle; supply-only cannot explain investment volatility.
- Interpretation of belief shocks: akin to Keynesian "animal spirits" (Farmer (2012b)); stock market and housing wealth falls can be self-fulfilling belief shocks affecting permanent income expectations and explaining aspects of the Great Recession.

*Italic: Source — content unit from _wp1477 (text provided).*

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### _wp1477 - REFERENCES

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*Source: _wp1477 - REFERENCES.*

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