## 27. Empirical impulse response functions to 1-sd uncertainty shock in the monthly specification of SVAR, period ranges from 1979M1 to 2007M12.

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---

### Introduction and key empirical findings
- Research question: Effects of aggregate macroeconomic uncertainty on employment and wages of skilled (college educated) and unskilled (less-than-college) workers; role of labor market segmentation and capital-skill complementarity in transmission.
- Preliminary correlations (annual averages, cyclical components, 1979–2018):
  - Correlation between uncertainty measure (Jurado, Ludvigson, and Ng (2015), h=1) and relative employment rate: 0.38.
  - Correlation between uncertainty measure and skill premium: -0.1.
- Main empirical result (SVAR 1979Q1–2018Q4):
  - Macroeconomic uncertainty shocks increase the employment-rate gap between skilled and unskilled workers (relative employment).
  - Response of the skilled-to-unskilled wage ratio (skill premium) is negligible.

### Empirical data and construction
- Uncertainty measure:
  - Jurado, Ludvigson, and Ng (2015) macroeconomic uncertainty index (JLN), monthly series, quarterly average of monthly values for h=1.
  - JLN index is taken in logs to interpret IRFs in percentage terms; JLN macroeconomic uncertainty index has a mean of 0.65.
- Labor market microdata:
  - NBER extracts of the Current Population Survey (CPS) Merged Outgoing Rotation Groups (MORG), monthly data 1979M1 to 2018M12.
  - Sample restrictions: individuals age 16 to 64, exclude self-employed, armed forces, observations with missing/negative person-level earnings weights or zero earnings, exclude observations with missing labor force status.
  - Skilled = college and above; unskilled = lower than college degree.
  - Hourly wages: weekly earnings / usual weekly hours for weekly workers; hourly earnings (main job) for hourly workers. Real hourly wages in 2012 dollars using CPI research series.
  - Weighted averages use CPS MORG earnings sampling weight earnwt.
  - Employment rates and relative employment rate ratio constructed from weighted counts; monthly series aggregated to quarterly by three-month averages; seasonally adjusted using X-13-ARIMA.
- Other macro series from FRED: GDPC1 (real GDP), PCEC96 (real personal consumption expenditures), GPDIC1 (real gross private domestic investment), HCOMPBS/GDPDEF for economy-wide hourly real wage, GDPDEF for inflation.

### SVAR methodology and identification
- SVAR-(p) specification with Yt vector:
  - Yt = [σz_t, y_t, i_t, c_t, n_s_t, (n_s/n_u)_t, w_s_t, (w_s/w_u)_t, π_t]′ where:
    - σz_t: JLN macroeconomic uncertainty measure (log).
    - y_t: real GDP (log level).
    - i_t: real gross private domestic investment (log level).
    - c_t: real personal consumption expenditures (log level).
    - n_s_t: skilled employment rate.
    - (n_s/n_u)_t: relative employment rate ratio.
    - w_s_t: weighted average real hourly wage of skilled (log level).
    - (w_s/w_u)_t: wage ratio (skill premium).
    - π_t: quarterly growth rate of GDP implicit price deflator.
- Lag order:
  - Akaike Information Criterion (AIC) indicates p = 2.
- Identification:
  - Uncertainty shock defined as a one standard deviation increase in the JLN index.
  - Structural identification via recursive ordering (Cholesky), with uncertainty ordered first (assumed exogenous contemporaneously).

### Impulse response results (IRFs) to a 1-sd uncertainty shock
- Aggregate responses (median IRFs; quarters refer to quarters after shock):
  - Output:
    - By the 4th quarter, output falls by 0.37%.
  - Consumption:
    - Consumption drops by 0.25%.
  - Investment:
    - Investment drops by 1.9%.
  - Inflation:
    - Contemporaneous fall in inflation (not significant) — suggests uncertainty shock acts like a demand shock.
- Labor market responses:
  - Skilled employment rate:
    - Hump-shaped response; stays down for about 3 years.
    - Falls by 0.14% to the lowest level in the 5th quarter.
  - Relative employment rate ratio (skilled/unskilled):
    - Increases in the 5th quarter by 0.2% — firms adjust unskilled employment more than skilled employment.
  - Wages and skill premium:
    - Movement in skilled wage and wage ratio negligible.
    - Decline in the wage ratio is very insignificant at 0.07% (implies wage rigidities).

### Forecast error variance decomposition (FEVD) — role of macroeconomic uncertainty
- Medium-run contributions (about 4 years):
  - Macroeconomic uncertainty shocks account for around 20% of fluctuations of real GDP.
  - Account for 12% of fluctuations of real consumption.
- Employment variance:
  - Macroeconomic uncertainty accounts for around 15% of fluctuations in skilled employment over the horizon of 12 month.
  - Accounts for 35% of fluctuations in skilled employment and employment rate ratio at longer horizons.
- Inflation:
  - Macroeconomic uncertainty has a negligible impact on forecast variance of inflation at all horizons.
- Wages:
  - Macroeconomic uncertainty is negligible for fluctuations in wages.

### Stylized facts distilled from SVAR (empirical stylized facts to be matched by theory)
- Macroeconomic uncertainty shock is recessionary: lowers aggregate output, consumption, investment, and both skilled and unskilled employment.
- Unexpected rise in macroeconomic uncertainty leads to a significant increase in the relative employment rate of skilled labor.
- The skill premium and wages do not respond significantly to the uncertainty shock.

### Theoretical mechanism and model overview
- Model features:
  - New Keynesian DSGE with stochastic volatility (uncertainty shocks as time-varying volatility of technology following Fernandez-Villaverde and others (2011)).
  - Production exhibits capital-skill complementarity via a CES production function.
  - Households: three types — skilled workers, unskilled workers, entrepreneurs.
    - Population shares πi, i ∈ {s, u, e}, sum to 1; shares are constant (no transitions across types).
    - Entrepreneurs do not supply labor, invest in capital, own firms and receive dividends — isolates labor income and avoids income/labor supply effects from ownership.
  - Firms: wholesalers (intermediate goods producers using skilled and unskilled labor and capital; face capital adjustment costs) and a representative retailer with Calvo price-setting.
  - Wage rigidity: asymmetric wage rigidity — skilled wages more rigid than unskilled wages (motivated by monitoring, bargaining power, higher hiring/training costs).
  - Solution method: calibrated to US data, solved using third-order perturbation and pruning.
- Transmission mechanism intuition:
  - Uncertainty rise lowers investment and employment by reducing relative prices of capital equipment and labor.
  - Firms adjust capital slower than labor → capital-to-skilled-labor ratio increases.
  - Capital-skill complementarity: skilled labor is more complementary with capital → higher capital-to-skilled-labor ratio dampens decline in marginal product of skilled labor → skilled labor demand declines less than unskilled → relative employment rises.
  - Asymmetric wage rigidity (skilled wages stickier) implies negligible change in skill premium; smaller decline in skilled wage implies smaller increase in skilled wage markup vis-à-vis unskilled wage, further amplifying relative employment increase.
  - Precautionary labor supply: skilled households may increase labor supply more to insure against future adverse shocks, reinforcing results.

### Policy implications and recommendations discussed
- General implication: uncertainty shocks have distributional effects across skill groups — negative impact larger on employment of less skilled and educated workers; more skilled/educated workers weather shocks better.
- Policy instruments that could mitigate adverse effects:
  - Stimulus monetary policy can cushion impact of higher uncertainty on aggregate employment.
  - Targeted vacancy subsidies to help preserve employment of unskilled workers.
  - Government spending on education, training, and on-the-job learning to improve productivity of unskilled workers and increase their complementarity with technical capital.
  - Tax policy: reduction in capital income taxes coupled with an increase in labor income taxes could raise capital stock and, via capital-skill complementarity, raise relative marginal product of skilled labor — may encourage skill accumulation and, depending on wage stickiness, lower the skill premium. Net effects depend on country-specific circumstances.
  - Investments in digital skills and subsidizing internet access or low-cost computers for vulnerable groups (highlighted by Covid pandemic experience).
- Caveat: potential distributional consequences of high uncertainty at business-cycle frequencies; further analyses and different models needed for a complete picture.

### Model calibration, solution, and organization of paper
- Model calibration is to US data; solved using third-order perturbation and pruning.
- Selected calibration values (quarterly US data):
  - π^e = 0.10, π^s = 0.21, π^u = 0.69.
  - β = 0.99.
  - σ^s_u = 1, σ^u_u = 1.
  - b_c = 0.5.
  - h^i = 1/3 in steady state.
  - φ^s = 2, φ^u = 2.
  - η^u_w = 0.8, η^s_w = 0.8, θ^u_w = 0.75, θ^s_w = 0.75.
  - κ_p = 0.75, θ_p = 11, χ = 0, χ_w = 0.
  - δ = 0.025, φ_i = 5, σ = 0.401, ρ = −0.495.
  - Resulting elasticities: 1/(1−ρ) = 0.67, 1/(1−σ) = 1.67.
  - μ = 0.62, λ = 0.8.
  - ρ_R = 0.7, ρ_π = 1.5, ρ_y = 0.3, φ_D = 0.3, φ_Y = 0.34, g/y = 0.2. Public debt calibrated to 67% of annual output.
  - ρ_Z = 0.8, σ_Z = 0.01, ρ_{σZ} = 0.65, η_{σZ} = 0.04. Calibrated standard deviation of the volatility shock set to 0.03 to match empirical uncertainty measure.

### Key theoretical and sensitivity results
- Aggregate impact:
  - One standard deviation shock to volatility of productivity causes a persistent downturn; example quantitative response in model with capital-skill complementarity: output decreases by 0.1% rapidly and returns after 10 quarters in one reported variant.
  - Price stickiness is necessary for recessionary response; without nominal price and wage rigidities the shock can raise output and investment.
- Relative labor outcomes:
  - Capital-skill complementarity critical: increases capital-to-skilled-labor ratio, dampens decline in skilled marginal product, raises skilled-to-unskilled employment ratio.
  - Wage stickiness (especially asymmetric with skilled more rigid) produces negligible change in skill premium while amplifying rise in relative skilled employment.
- Sensitivity:
  - Frisch elasticities, strength of capital-skill complementarity (e.g., baseline 1/(1−ρ)=0.67 vs alternatives), price markups, and presence of nominal rigidities materially affect magnitudes and sometimes signs of IRFs.
  - Removing nominal rigidities, habits, and adjustment costs can reverse responses.

*Source: IMF Working Paper content unit "27. Empirical impulse response functions to 1-sd uncertainty shock in the monthly specification of SVAR, period ranges from 1979M1 to 2007M12."*

### 1.    Skilled and Unskilled Worker Households.  .  .  .  .  .  .  .  .  .  .  .  .  .  .    18

### 1.    Skilled and Unskilled Worker Households.

### Table of contents context (neighboring sections and components)
- 2.    Entrepreneurs.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    22
- B.Wholesale Firms.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    24
- C.Retailers.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    26
- D.    Exogenous Processes.  .  .  .  .  .  .  .  .  .  .  .  .  .    27
- E.Monetary Policy.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    28
- F.Fiscal Policy.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    28
- G.    Closing the Model.  .  .  .  .  .  .  .  .  .  .  .  .    29
- IV.Impact of Uncertainty Shocks: Dissecting the Mechanism.  .    29
- V.Solution and Calibration.  .  .  .  .  .  .  .  .  .  .  .    31
  - A.    Solution Method.  .  .  .  .  .  .  .  .  .  .  .    31
  - B.Calibrated Parameters.  .  .  .  .  .  .  .  .  .  .    32
- VI.    Theoretical Results: Impulse Response Analysis and Inspecting the Transmission Channels of Uncertainty.  .    35
  - A.    IRF Analysis with Flexible Wages.  .  .  .  .  .    35
    - 1.    Aggregate Economy.  .  .  .  .  .  .  .  .  .    35
    - 2.    Relative Variables.  .  .  .  .  .  .  .  .  .    37
  - B.Extension: Sticky Wage Model.  .  .  .  .  .  .  .    39
    - 1.    Asymmetric Sticky Wage.  .  .  .  .  .  .  .    40
    - 2.    Comparison with Data.  .  .  .  .  .  .  .    42
  - C.Sensitivity Analysis.  .  .  .  .  .  .  .  .  .  .    43
- VII.  Conclusion.  .  .  .  .  .  .  .  .  .  .  .  .  .  .    51

### Figures and visual components referenced in the document
- 1.Macro Uncertainty and Skill Premium   .  .  .  .  .  .  .  .  .  .    6
- 2.Impulse responses to 1-sd uncertainty shock. .  .  .  .  .  .    15
- 3.Impulse responses to 1-sd uncertainty shock in manufacturing sector.  .    16
- 4.Model intuition. Capital-skill complementarity & flexible wages case.   .    31
- 5.Impulse response functions to TFP uncertainty shock in the model with flexible wages.  .    37
- 6.Impulse response functions to TFP uncertainty shock in the model with flexible wages.  .    38
- 7.Impulse response functions to TFP uncertainty shock in the model with Calvo wages.  .    40
- 8.Impulse response functions to TFP uncertainty shock in the model with Calvo wages.  .    41
- 9.Impulse response functions to TFP uncertainty shock – income shares.  .    42
- 10.    Impulse response functions in the model and in the data.    .  .    43
- 11.    Impulse response functions to TFP uncertainty shock – Frisch elasticity.   .    44
- 12.    Impulse response functions to TFP uncertainty shock – elasticity of substitution btw capital and skilled labor.  .    46
- 13.    Impulse response functions to TFP uncertainty shock – elasticity of substitution btw capital and skilled labor composite and unskilled labor.    .    47
- 14.    Impulse response functions to TFP uncertainty shock – elasticity of substitution btw goods.   .    48
- 15.    Impulse response functions to TFP uncertainty shock in baseline model and in the model without nominal price and wage rigidity.   .    49
- 16.    Impulse response functions to TFP uncertainty shock in baseline model and in the model without nominal price and wage rigidity, habit formation in consumption and investment adjustment costs.  .    50
- 17–26.    SVAR and robustness impulse response figures covering detrending methods, uncertainty orderings, lag specifications, alternative uncertainty measures, inclusion of stock prices, monthly frequency, and pre-2007 period analyses.  .    63–71

### Appendices and robustness checks listed
- A.Additional Details Concerning Model Derivations  .  .  .  .  .  .    57
  - A.    Production Function without Capital-Skill Complementarity  .    57
  - A.1.Wage Stickiness  .  .  .  .  .  .  .  .  .  .  .  .    58
  - A.2.Wholesale Firms .  .  .  .  .  .  .  .  .  .  .  .    61
- B.Empirical Robustness  .  .  .  .  .  .  .  .  .  .  .  .    62
  - A.    Detrending Methods  .  .  .  .  .  .  .  .  .  .    62
  - B.Uncertainty Ordered Last   .  .  .  .  .  .  .  .    62
  - C.Increase the Number of Lags    .  .  .  .  .  .    62
  - D.    Alternative Measure of Uncertainty   .  .    65
  - E.Alternative Measure of Skill Premium .  .  .  .    65
  - F.Control for the Stock Market    .  .  .  .  .    68
  - G.    Monthly Frequency    .  .  .  .  .  .  .    68
  - H.    Period Prior to 2007   .  .  .  .  .  .    70

*Source: wpiea2023155-print-pdf - 1.    Skilled and Unskilled Worker Households.  (https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023155-print-pdf.pdf)*

### 27.    Empirical impulse response functions to 1-sd uncertainty shock in the monthly

### 27. Empirical impulse response functions to 1-sd uncertainty shock in the monthly specification of SVAR, period ranges from 1979M1 to 2007M12.

### Introduction and key empirical findings
- Research question: Effects of aggregate macroeconomic uncertainty on employment and wages of skilled (college educated) and unskilled (less-than-college) workers; role of labor market segmentation and capital-skill complementarity in transmission.
- Preliminary correlations (annual averages, cyclical components, 1979–2018):
  - Correlation between uncertainty measure (Jurado, Ludvigson, and Ng (2015), h=1) and relative employment rate: 0.38.
  - Correlation between uncertainty measure and skill premium: -0.1.
- Main empirical result (SVAR 1979Q1–2018Q4):
  - Macroeconomic uncertainty shocks increase the employment-rate gap between skilled and unskilled workers (relative employment).
  - Response of the skilled-to-unskilled wage ratio (skill premium) is negligible.

### Empirical data and construction
- Uncertainty measure:
  - Jurado, Ludvigson, and Ng (2015) macroeconomic uncertainty index (JLN), monthly series, quarterly average of monthly values for h=1.
  - JLN index is taken in logs to interpret IRFs in percentage terms; JLN macroeconomic uncertainty index has a mean of 0.65.
- Labor market microdata:
  - NBER extracts of the Current Population Survey (CPS) Merged Outgoing Rotation Groups (MORG), monthly data 1979M1 to 2018M12.
  - Sample restrictions: individuals age 16 to 64, exclude self-employed, armed forces, observations with missing/negative person-level earnings weights or zero earnings, exclude observations with missing labor force status.
  - Skilled = college and above; unskilled = lower than college degree.
  - Hourly wages: weekly earnings / usual weekly hours for weekly workers; hourly earnings (main job) for hourly workers. Real hourly wages in 2012 dollars using CPI research series.
  - Weighted averages use CPS MORG earnings sampling weight earnwt.
  - Employment rates and relative employment rate ratio constructed from weighted counts; monthly series aggregated to quarterly by three-month averages; seasonally adjusted using X-13-ARIMA.
- Other macro series from FRED: GDPC1 (real GDP), PCEC96 (real personal consumption expenditures), GPDIC1 (real gross private domestic investment), HCOMPBS/GDPDEF for economy-wide hourly real wage, GDPDEF for inflation.

### SVAR methodology and identification
- SVAR-(p) specification with Yt vector:
  - Yt = [σz_t, y_t, i_t, c_t, n_s_t, (n_s/n_u)_t, w_s_t, (w_s/w_u)_t, π_t]′ where:
    - σz_t: JLN macroeconomic uncertainty measure (log).
    - y_t: real GDP (log level).
    - i_t: real gross private domestic investment (log level).
    - c_t: real personal consumption expenditures (log level).
    - n_s_t: skilled employment rate.
    - (n_s/n_u)_t: relative employment rate ratio.
    - w_s_t: weighted average real hourly wage of skilled (log level).
    - (w_s/w_u)_t: wage ratio (skill premium).
    - π_t: quarterly growth rate of GDP implicit price deflator.
- Lag order:
  - Akaike Information Criterion (AIC) indicates p = 2.
- Identification:
  - Uncertainty shock defined as a one standard deviation increase in the JLN index.
  - Structural identification via recursive ordering (Cholesky), with uncertainty ordered first (assumed exogenous contemporaneously).

### Impulse response results (IRFs) to a 1-sd uncertainty shock
- Aggregate responses (median IRFs; quarters refer to quarters after shock):
  - Output:
    - By the 4th quarter, output falls by 0.37%.
  - Consumption:
    - Consumption drops by 0.25% (at relevant horizon reported alongside output).
  - Investment:
    - Investment drops by 1.9%.
  - Inflation:
    - Contemporaneous fall in inflation (not significant) — suggests uncertainty shock acts like a demand shock.
- Labor market responses:
  - Skilled employment rate:
    - Hump-shaped response; stays down for about 3 years.
    - Falls by 0.14% to the lowest level in the 5th quarter.
  - Relative employment rate ratio (skilled/unskilled):
    - Increases in the 5th quarter by 0.2% — firms adjust unskilled employment more than skilled employment.
  - Wages and skill premium:
    - Movement in skilled wage and wage ratio negligible.
    - Decline in the wage ratio is very insignificant at 0.07% (implies wage rigidities).

### Forecast error variance decomposition (FEVD) — role of macroeconomic uncertainty
- Medium-run contributions (about 4 years):
  - Macroeconomic uncertainty shocks account for around 20% of fluctuations of real GDP.
  - Account for 12% of fluctuations of real consumption.
- Employment variance:
  - Macroeconomic uncertainty accounts for around 15% of fluctuations in skilled employment over the horizon of 12 month.
  - Accounts for 35% of fluctuations in skilled employment and employment rate ratio at longer horizons.
- Inflation:
  - Macroeconomic uncertainty has a negligible impact on forecast variance of inflation at all horizons.
- Wages:
  - Macroeconomic uncertainty is negligible for fluctuations in wages.

### Stylized facts distilled from SVAR (empirical stylized facts to be matched by theory)
- Macroeconomic uncertainty shock is recessionary: lowers aggregate output, consumption, investment, and both skilled and unskilled employment.
- Unexpected rise in macroeconomic uncertainty leads to a significant increase in the relative employment rate of skilled labor.
- The skill premium and wages do not respond significantly to the uncertainty shock.

### Theoretical mechanism and model overview
- Model features:
  - New Keynesian DSGE with stochastic volatility (uncertainty shocks as time-varying volatility of technology following Fernandez-Villaverde and others (2011)).
  - Production exhibits capital-skill complementarity via a CES production function.
  - Households: three types — skilled workers, unskilled workers, entrepreneurs.
    - Population shares πi, i ∈ {s, u, e}, sum to 1; shares are constant (no transitions across types).
    - Entrepreneurs do not supply labor, invest in capital, own firms and receive dividends — isolates labor income and avoids income/labor supply effects from ownership.
  - Firms: wholesalers (intermediate goods producers using skilled and unskilled labor and capital; face capital adjustment costs) and a representative retailer with Calvo price-setting.
  - Wage rigidity: asymmetric wage rigidity — skilled wages more rigid than unskilled wages (motivated by monitoring, bargaining power, higher hiring/training costs).
  - Solution method: calibrated to US data, solved using third-order perturbation and pruning.
- Transmission mechanism intuition:
  - Uncertainty rise lowers investment and employment by reducing relative prices of capital equipment and labor.
  - Firms adjust capital slower than labor → capital-to-skilled-labor ratio increases.
  - Capital-skill complementarity: skilled labor is more complementary with capital → higher capital-to-skilled-labor ratio dampens decline in marginal product of skilled labor → skilled labor demand declines less than unskilled → relative employment rises.
  - Asymmetric wage rigidity (skilled wages stickier) implies negligible change in skill premium; smaller decline in skilled wage implies smaller increase in skilled wage markup vis-à-vis unskilled wage, further amplifying relative employment increase.
  - Precautionary labor supply: skilled households may increase labor supply more to insure against future adverse shocks, reinforcing results.

### Policy implications and recommendations discussed
- General implication: uncertainty shocks have distributional effects across skill groups — negative impact larger on employment of less skilled and educated workers; more skilled/educated workers weather shocks better.
- Policy instruments that could mitigate adverse effects:
  - Stimulus monetary policy can cushion impact of higher uncertainty on aggregate employment.
  - Targeted vacancy subsidies to help preserve employment of unskilled workers.
  - Government spending on education, training, and on-the-job learning to improve productivity of unskilled workers and increase their complementarity with technical capital.
  - Tax policy: reduction in capital income taxes coupled with an increase in labor income taxes could raise capital stock and, via capital-skill complementarity, raise relative marginal product of skilled labor — may encourage skill accumulation and, depending on wage stickiness, lower the skill premium. Net effects depend on country-specific circumstances.
  - Investments in digital skills and subsidizing internet access or low-cost computers for vulnerable groups (highlighted by Covid pandemic experience).
- Caveat: potential distributional consequences of high uncertainty at business-cycle frequencies; further analyses and different models needed for a complete picture.

### Model calibration, solution, and organization of paper
- Model calibration is to US data; solved using third-order perturbation and pruning.
- Paper organization:
  - Section II: SVAR empirical analysis.
  - Section III: theoretical model setup.
  - Section IV: intuitions of transmission mechanisms.
  - Section V: parametrization and solution method.
  - Section VI: results and sensitivity analysis.
  - Final section: concluding remarks.

*Source: IMF Working Paper content unit "27. Empirical impulse response functions to 1-sd uncertainty shock in the monthly specification of SVAR, period ranges from 1979M1 to 2007M12."*

### 1.   Skilled and Unskilled Worker Households

### 1.   Skilled and Unskilled Worker Households

### Household preferences and time constraint
- Two household types indexed i∈(s,u) (skilled and unskilled). Households are heterogeneous indexed by j∈(0; 1) for differentiated labor input.
- Time constraint normalized to 1: ai-type household h_i^t + l_i^t = 1, where h_i^t is hours worked and l_i^t is leisure.
- Utility (separable in consumption and labor) for each i:
  - U_i^t = E_t ∞∑_{t=0} β [ (c_i^t − b_c c_i^{t−1})^{1−σ_i^u} / (1−σ_i^u) − κ_i,h (h_i,j^t)^{1+φ_i} / (1+φ_i) ]
  - Parameters: β∈(0,1) (subjective discount factor), φ_i (inverse of Frisch elasticity), κ_h (scale parameter), σ_i^u (intertemporal elasticity of substitution), b_c (habit in consumption).

### Budget constraint and optimization
- Budget constraint (real terms):
  - c_i^t + t_i^t + B_{t+1} / (π_{t+1} R_t) = w_{i,j}^t h_{i,j}^t + B_t
- Definitions:
  - B_t: zero-coupon nominal non-state contingent risk-free government bond holdings
  - R_t: gross nominal return on bonds
  - t_i^t: tax levied
  - w_{i,j}^t: real wage for household j of type i
  - π_t = p_t / p_{t−1}: inflation rate
- Lagrangian (real terms) with multiplier λ_i^t (marginal utility of wealth):
  - L_i = (c_i^t)^{1−σ_i^u} / (1−σ_i^u) − κ_i h (n_i,j^t)^{1+φ_i} / (1+φ_i) − λ_i^t [ c_i^t + t_i^t + B_{t+1} / (π_{t+1} R_t) − w_{i,j}^t h_{i,j}^t − B_t ]

### Euler equation and consumption
- First order conditions:
  - For B_{t+1} (Euler): β E_t { λ_i^{t+1} R_t / π_{t+1} } = λ_i^t  (Equation 3)
  - For consumption: λ_i^t = (c_i^t)^{−σ_i^u}  (Equation 4)
- Interpretation:
  - Equation 3 determines intertemporal dynamics of marginal utility of consumption as a function of real return on bonds.
  - Equation 4 links marginal utility to current consumption.

### Labor aggregation and labor packer
- Aggregate labor for type i:
  - h_i^t = [ ∫_0^1 (h_{i,j}^t)^{(η_w^i −1)/η_w^i} dj ]^{η_w^i/(η_w^i −1)}  (Equation 5)
  - η_w^i: elasticity of substitution among different types of skilled or unskilled labor (0 ≤ η_w^i ≥ ∞)
- Labor packer input demand (from FOCs):
  - h_{i,j}^t = ( w_{i,j}^t / w_i^t )^{−η_w^i} h_i^t  (Equation 6)
- Aggregate wage:
  - w_i^t = [ ∫_0^1 (w_{i,j}^t)^{1−η_w^i} dj ]^{1/(1−η_w^i)}  (Equation 7)

### Nominal wage stickiness and Calvo setting
- Calvo wage setting:
  - Each period, a fraction (1−θ_s^w) of skilled and (1−θ_u^w) of unskilled households can change their wages. Remaining fraction θ_i^w cannot, or partially index by past inflation.
  - Indexation parameter χ_w ∈ [0,1].
- Relevant part of wage Lagrangian for household i∈(s,u):
  - L_{i,w} = E_t ∞∑_{s=0} (θ_i^w β)^s [ −κ_i h 1+φ_i ( ∏_{k=1}^s (π_{t+k} / π_{t+k−1}^{χ_w})^{−1} w_{i,j}^t / w_{i,t+s} )^{−η_w^i} (h_i^{t+s})^{1+φ_i} / (1+φ_i) + λ_{i,j}^{t+s} ( ∏_{k=1}^s (π_{t+k} / π_{t+k−1}^{χ_w})^{−1} w_{i,j}^t / w_{i,t+s} )^{1−η_w^i} w_{i,t} h_i^{t+s} ]
- Complete markets imply identical wage choices across j; drop j from wages and λ.

### Wage first-order condition and recursive representation
- First order condition with respect to the reset wage w_{i,∗}^t:
  - E_t ∑_{s=0}^∞ (θ_i^w β)^s [ η_w^i w_{i,∗}^t κ_i h ( ∏_{s k=1} (π_{t+k} / π_{t+k−1}^{χ_w})^{−1} w_{i,∗}^t / w_{i+s}^t )^{−η_w^i} (1+φ_i) (h_i^{t+s})^{1+φ_i} + (1−η_w^i) λ_i^{t+s} ( ∏_{s k=1} (π_{t+k} / π_{t+k−1}^{χ_w})^{−1} w_{i,∗}^t / w_{i}^t )^{−η_w^i} h_i^{t+s} ] = 0  (Equation 9)
- Define auxiliary recursive objects:
  - f_{i,1}^t = ( (η_w^i −1)/η_w^i ) w_{i,∗}^t E_t ∞∑_{s=0} (β θ_i^w)^s λ_i^{t+s} ( w_{i,t+s} / w_{i,∗}^t )^{η_w^i} h_i^{t+s} ∏_{k=1}^s (π_{t+k} / π_{t+k−1}^{χ_w})^{η_w^i −1}
  - f_{i,2}^t = E_t ∞∑_{s=0} (β θ_i^w)^s κ_i h s ∏_{k=1}^s (π_{t+k} / π_{t+k−1}^{χ_w})^{η_w^i (1+φ_i)} ( w_{i,∗}^{t+s} / w_{i,∗}^t )^{η_w^i (1+φ_i)} (h_i^{t+s})^{1+φ_i}
  - Equality f_{i,1}^t = f_{i,2}^t is the FOC.

### Recursive laws of motion for wage-setting aggregates
- Define f_i^t = f_{i,1}^t = f_{i,2}^t. Laws of motion:
  - f_i^t = ( (η_w^i −1)/η_w^i ) (w_{i,∗}^t)^{1−η_w^i} λ_i^t (w_i^t)^{η_w^i} h_i^t + (β θ_i^w) E_t ( π_t / π_{t−1}^{χ_w} )^{η_w^i −1} ( w_{i,∗}^{t+1} / w_{i,∗}^t )^{η_w^i −1} f_i^{t+1}  (Equation 10)
  - f_i^t = κ_i h (h_i^t)^{1+φ_i} ( w_i^t / w_{i,∗}^t )^{η_w^i (1+φ_i)} + (β θ_i^w) E_t ( π_t / π_{t−1}^{χ_w} )^{η_w^i (1+φ_i)} ( w_{i,∗}^{t+1} / w_{i,∗}^t )^{η_w^i (1+φ_i)} f_i^{t+1}  (Equation 11)

### Real wage index evolution
- In symmetric equilibrium w_{i,j,∗}^t = w_{i,∗}^t. The real wage index evolves:
  - (w_i^t)^{1−η_w^i} = θ_i^w ( π_t / π_{t−1}^{χ_w} )^{η_w^i −1} (w_i^{t−1})^{1−η_w^i} + (1−θ_i^w) (w_{i,∗}^t)^{1−η_w^i}  (Equation 12)

*Source: 1.   Skilled and Unskilled Worker Households (wpiea2023155-print-pdf)*

### 2.   Entrepreneurs

### 2.   Entrepreneurs

### Household preferences and decisions
- Entrepreneur households:
  - Own firms, invest in physical capital, do not participate in the labor market and enjoy leisure equal to 1.
  - Preferences given by:
    - U^e_t = E_t sum_{t=0}^∞ β [ (c^e_t − b_c c^e_{t−1})^{1−σ^e_u} / (1−σ^e_u) ] (equation labeling preserved from source).
- Entrepreneur budget constraint (real terms):
  - c^e_t + t^e_t + B_{t+1}/π_{t+1} R_t + i^e_t = div_t + B_t + R^k_t k^e_{t−1} (Equation 14).
  - div_t is household’s share of firms’ dividends, net of a government lump-sum tax.
  - Wholesalers’ profits are redistributed to entrepreneur households in the form of dividends (see original source).
- Dividend accounting:
  - π^e div_t = x_t y_t − ( w^s_t n^s_t + w^u_t n^u_t + R^k_t k_t ) (Equation 15).

### Capital accumulation and adjustment costs
- Law of motion for capital:
  - i^e_t = k^e_{t+1} − (1−δ_i) k^e_t + D(k^e_{t+1}, k^e_t) (Equation 16).
- Capital adjustment cost specification:
  - D(k^e_{t+1}, k^e_t) = φ_i/2 ( (k^e_{t+1}/k^e_t − 1)^2 ) k^e_t, with φ_i < 0.
  - δ satisfies 0 < δ < 1 and D(1) = D′(1) = 0.
  - Parameter φ_i governs magnitude of adjustment costs; as φ_i → ∞, investment and stock of capital become constant.

### Optimization and first-order conditions
- Lagrangian in real terms:
  - L^e = (c^e_t − b_c c^e_{t−1})^{1−σ_u} / (1−σ_u) − λ^e_t[ budget constraint ] − Q_t[ capital law of motion ].
  - λ^e_t is multiplier associated with budget constraint (marginal utility of wealth); q^i_t = Q_t / λ^e_t is Tobin’s Q marginal ratio.
- Key first-order conditions preserved exactly:
  - Consumption:
    - λ^e_t = (c^e_t − b_c c^e_{t−1})^{−σ^e_u} − β b_c (c^e_{t+1} − b_c c^e_t)^{−σ^e_u} (Equation 17).
  - Bonds:
    - β E_t λ^e_{t+1} R_t / π_{t+1} = λ^e_t (Equation 18).
  - Capital:
    - λ^e_t(1 + φ_i[ k^e_{t+1}/k^e_t − 1 ]) = β λ^e_{t+1}(1 + R^k_{t+1} − δ + φ_i/2( (k^e_{t+2}/k^e_{t+1})^2 − 1 )) (Equation 19).

### Complete markets and perfect risk-sharing
- Assumption: complete markets, perfect risk-sharing and full insurance between households (following Dolado, Motyovszki, and Pappa (2021)).
- Perfect risk sharing condition:
  - λ^i_{t+1} / λ^e_{t+1} = λ^i_t / λ^e_t = \bar λ^i / \bar λ^e for i ∈ (s,u) (Equation 20).
  - This keeps ratio of different agents’ marginal utilities constant at its steady-state value.

---

### B.   Wholesale Firms (aggregate production and factor demand)
- Wholesale sector:
  - Continuum of perfectly competitive wholesalers produce homogeneous wholesale good y_t and sell to retailers at relative price x_t.
  - Aggregate production: y_t ≡ Z_t f(k_t, n^s_t, n^u_t) with Z_t aggregate TFP and k_t = π^e k^e_t aggregate capital.
  - Labor supplies: n^s_t = π^s h^s_t and n^u_t = π^u h^u_t with population shares π^s and π^u.
- Production function form (three-factor nested CES capturing capital-skill complementarity):
  - y_t = Z_t ( [ μ (n^u_t)^σ + (1−μ)( λ k_t^ρ + (1−λ)(n^s_t)^ρ )^{σ/ρ} ]^{1/σ} ) (Equation 21).
  - Parameters satisfy σ, ρ ∈ (−∞, 1).
  - λ governs capital intensity; μ governs skill-intensity.
- Elasticities:
  - Elasticity of substitution between capital and skilled labor: ε_{k,n^s} = 1/(1−ρ).
  - Elasticity of substitution between capital and unskilled labor (and between skilled and unskilled): ε_{k,n^u} ≡ ε_{n^s,n^u} = 1/(1−σ).
  - Capital-skill complementarity condition: σ > ρ (i.e., 1/(1−ρ) < 1/(1−σ)).
- Relative marginal products and labor demand:
  - Relative marginal product of skilled to unskilled labor:
    - m p l^s_t / m p l^u_t = (1−μ)(1−λ)/μ [ λ (k_t / n^s_t)^ρ + (1−λ) ]^{σ/ρ − 1} (n^u_t / n^s_t)^{(1−σ)} (Equation 22).
  - First-order conditions for factor payments (Equations 23–25):
    - R^k_t x_t = (1−μ) λ ( λ k_t^ρ + (1−λ)(n^s_t)^ρ )^{σ/ρ − 1} k_t^{ρ−1} y_t^{1−σ} (23).
    - w^s_t x_t = (1−μ)(1−λ) ( λ k_t^ρ + (1−λ)(n^s_t)^ρ )^{σ/ρ − 1} (n^s_t)^{ρ−1} y_t^{1−σ} (24).
    - w^u_t x_t = μ (n^u_t)^{σ−1} y_t^{1−σ} (25).

---

### C.   Retailers and price setting
- Retail sector:
  - Unit measure of retailers j ∈ [0,1] purchase wholesale good at price x_t, produce differentiated final goods y_{j,t} and set nominal prices p_{j,t}.
  - Final output aggregator:
    - y_t = ( ∫_0^1 y_{j,t}^{(ε−1)/ε} dj )^{ε/(ε−1)} (Equation 26).
  - Demand for each retailer:
    - y_{j,t} = (p_{j,t} / p_t)^{−ε} y_t (Equation 27) with p_t the aggregate price index (Equation 28).
- Price rigidity:
  - Calvo pricing with fraction (1−κ_p) able to reoptimize each period; indexation parameter χ ∈ [0,1].
  - Firms that update set common price p^*_t; average price duration 1/(1−κ_p).
  - Price index evolution:
    - 1 = κ_p (π_{t−1}^χ π_t)^{1−θ_p} + (1−κ_p) (π^*_t)^{1−θ_p} (Equation 32).
  - Price dispersion v^p_t = ∫_0^1 (p_{j,t} / p_t)^{−θ_p} dj; law of motion:
    - v^p_t = κ_p (π_{t−1}^χ π_t)^{−θ_p} v^p_{t−1} + (1−κ_p) (π^*_t)^{−θ_p} (Equation 33).
  - Aggregate production with price dispersion:
    - y_t = Z_t f(k_t, n^s_t, n^u_t) v^p_t (Equation 34). Note v^p_t ≥ 1 implies output loss due to price dispersion.

---

### D.   Exogenous processes
- Technology level and stochastic volatility:
  - Z_t = ρ_Z Z_{t−1} + σ_{Z,t} ε_{Z,t}, with ε_{Z,t} ∼ N(0,1) (Equation 35).
  - σ_{Z,t} = (1−ρ_{σZ}) σ_Z + ρ_{σZ} σ_{Z,t−1} + η_{σZ} ε_{σZ,t}, with ε_{σZ,t} ∼ N(0,1) (Equation 36).
- Interpretation:
  - ε_{Z,t} is a first-moment (level) shock; ε_{σZ,t} is a second-moment (uncertainty) shock that widens tails while keeping mean unchanged.
  - ρ_Z and ρ_{σZ} drive persistence; η_{σZ} drives magnitude of productivity uncertainty shock.
  - The model uses stochastic volatility approach as in Fernandez-Villaverde and others (2011).

---

### E.   Monetary policy
- Taylor rule with smoothing:
  - R_t / \bar R = (R_{t−1}/\bar R)^{ρ_R} ( (π_t/π)^{ρ_π} (y_t / y_{t−1})^{ρ_y} )^{(1−ρ_R)} (Equation 37).
  - ρ_R ∈ [0,1] smoothing parameter; ρ_π elasticity to inflation deviations; ρ_y elasticity to output gap; R steady-state gross nominal interest rate; y steady-state output.

### F.   Fiscal policy
- Government budget constraint:
  - t_t + B_t = g_t + R_{t−1} B_{t−1} / π_t (Equation 38).
  - Lump-sum taxes distributed equally across households: t_t = ∑_i π_i t^i_t.
- Tax feedback rule (real lump-sum taxes adjusted):
  - t_t / t = (B_{t−1} / B)^{φ_D} (y_t / y)^{φ_Y} (Equation 39).
- Government spending AR(1):
  - log(g_t / \bar g) = ρ_g log(g_{t−1} / \bar g) + ε_{g,t} (Equation 40).

### G.   Closing the model
- Final good market clearing:
  - y_t = c_t + i_t + g_t (Equation 41).
  - Aggregate consumption c_t = ∑_i π_i c^i_t for i ∈ (s,u,e); aggregate investment i_t = π^e i^e_t.

---

### IV. Impact of uncertainty shocks: mechanism overview
- Focus: transmission of uncertainty shocks onto skilled and unskilled labor markets.
- Key stylized facts (from Section II referenced in source):
  - (i) Relative employment of skilled labor increases in response to a rise in macroeconomic uncertainty.
  - (ii) The skill premium (skilled to unskilled wage ratio) does not react.
- Mechanism dissection (flexible wages example):
  - Consider flexible wages (θ^s_w = θ^u_w = 0) so real wages are markups over marginal rate of substitution.
  - Precautionary behavior by households under higher uncertainty:
    - Households reduce consumption, increase savings and labor supply (precautionary labor supply shift right).
    - Increased labor supply lowers firms’ marginal costs; with sticky prices output is demand-determined so lower consumption reduces aggregate demand.
    - Higher markups (price and wage) and fall in labor demand follow.
  - Capital-skill complementarity channel:
    - Capital adjusts slower than labor ⇒ capital-to-skilled labor ratio increases after uncertainty shock.
    - Complementarity implies marginal product of skilled labor declines less than that of unskilled labor.
    - Result: relative labor demand for skilled workers increases (fall in skilled labor demand is smaller than for unskilled).
  - Wealth effect on labor supply:
    - Lower labor income induces households to increase hours worked; if skilled households exhibit a larger wealth effect, fall in equilibrium skilled labor is dampened further.
  - Net result (illustrated in Figure 4 of source):
    - Relative employment of skilled labor increases despite overall demand contraction; skill premium may remain unchanged.

---

### V. Solution and calibration
- Solution method:
  - Model solved and simulated by a third-order perturbation method using the pruning algorithm (Andreasen, Fernandez-Villaverde, and Rubio-Ramírez (2018)).
  - Third-order approximation necessary to analyze uncertainty shocks independently of first-moment shocks: volatility shock enters as independent argument only in third-order approximation.
  - Impulse response functions (IRFs) computed as percentage deviations from the ergodic mean of simulated data in the absence of shocks.
- Calibration highlights (quarterly US data; preserve exact calibrated values):
  - Population and preferences:
    - π^e = 0.10 (share of entrepreneurs).
    - π^s = 0.21 (share of skilled workers), π^u = 0.69.
    - β = 0.99.
    - σ^s_u = 1, σ^u_u = 1.
    - b_c = 0.5 (habit persistence).
    - Households work h^i = 1/3 in steady state.
    - Inverse Frisch elasticities: φ^s = 2, φ^u = 2.
    - Wage rigidity parameters: η^u_w = 0.8, η^s_w = 0.8, θ^u_w = 0.75, θ^s_w = 0.75.
  - Price setting and markups:
    - κ_p = 0.75 (implied average price duration of 4 quarters).
    - θ_p = 11 (elasticity of substitution between goods; steady-state markup of 10%).
    - χ = 0 (price indexation).
    - χ_w = 0 (wage indexation).
  - Capital and production parameters:
    - δ = 0.025 (capital depreciation rate; 10% depreciation rate p.a. noted).
    - φ_i = 5 (investment adjustment cost).
    - σ = 0.401 (substitutability between skilled (or capital) and unskilled labor).
    - ρ = −0.495 (capital-skill complementarity).
    - Resulting elasticities:
      - 1/(1−ρ) = 0.67 (elasticity of capital to skilled labor).
      - 1/(1−σ) = 1.67 (elasticity of capital to unskilled labor).
    - Composite production parameters calibrated:
      - μ = 0.62 to obtain labor share in income of 69%.
      - λ = 0.8 to target skill premium of 1.67.
  - Monetary and fiscal policy parameters:
    - ρ_R = 0.7 (interest rate smoothing).
    - ρ_π = 1.5 (Taylor-coefficient on inflation).
    - ρ_y = 0.3 (Taylor-coefficient on output).
    - φ_D = 0.3 (tax feedback to debt).
    - φ_Y = 0.34 (tax feedback to output).
    - g/y = 0.2 (steady-state government spending to GDP).
    - Public debt calibrated to 67% of annual output (stated in text).
  - Technology and shocks:
    - ρ_Z = 0.8 (technology autoregressive parameter).
    - σ_Z = 0.01 (steady-state TFP volatility).
    - ρ_{σZ} = 0.65 (persistence of volatility of TFP shocks).
    - η_{σZ} = 0.04 (magnitude of the productivity uncertainty shock).
    - Calibrated standard deviation of the volatility shock set to 0.03 in text to match empirical uncertainty measure in SVAR (paragraph discussion).
- Table 2 (benchmark parameter calibration) preserved in content above with exact numeric values provided.

---

### VI. Theoretical results: impulse response analysis and transmission channels
- Aim: assess importance of capital-skill complementarity and other model ingredients for responses to uncertainty shocks.
- Approach:
  - Analyze IRFs isolating pure uncertainty effect (positive shock to σ_{Z,t} in Equation 36 while level shock is zero).
  - Consider alternative model specifications (e.g., flexible vs. sticky wages) to identify roles of channels.
- First step reported in source:
  - IRF analysis begins with model variant featuring flexible wages to isolate role of capital-skill complementarity.

*Italic source attribution: Chapter/section "2. Entrepreneurs" from wpiea2023155-print-pdf*

### 1.   Aggregate Economy

### 1.   Aggregate Economy

### Aggregate responses to a TFP uncertainty shock
- A one standard deviation shock to the volatility of productivity causes a persistent downturn in aggregate economic activity in the model with capital-skill complementarity.
- Quantitative responses (model with capital-skill complementarity):
  - Rapid decrease in output of 0.1%, before output returns to its initial level after 10 quarters.
  - Investment and employment fall, together with wages and capital rents.
  - Inflation rises as the cumulative effect of the aggregate demand channel and the upward nominal pricing bias channel dominates precautionary savings effects.
- Transmission channels emphasized:
  - Precautionary household behavior: risk-averse households reduce consumption and increase savings; labor supply rises as marginal utility of wealth increases.
  - Nominal price rigidities: firms face higher markups when prices cannot adjust instantly, generating a downward shift in labor demand and amplifying falls in investment, labor hours, and output (aggregate demand channel).
  - Upward nominal pricing bias: firms set prices higher due to asymmetry in the profit function under price rigidities, contributing to higher inflation.
  - Capital-skill complementarity: amplifies aggregate responses via a feedback loop—lower investment reduces productivity of complementary skilled workers, which feeds back into further reductions in investment and output.

### Differences with model without capital-skill complementarity
- Without capital-skill complementarity (skilled and unskilled perfect substitutes), the amplification via the capital–skill feedback loop is absent, producing milder aggregate declines.
- Capital-skill complementarity materially increases the responses of aggregate variables to uncertainty shocks.

### Key inference
- Price stickiness is necessary for the recessionary response to uncertainty; in models without nominal price and wage rigidities the same shock can raise output and investment due to precautionary labor supply effects interacting with flexible prices.

---

### Relative Variables: skilled vs unskilled labor responses
- In absence of capital-skill complementarity: relative ratio of skilled labor and the skill premium do not react.
- With capital-skill complementarity:
  - Capital-to-skilled labor ratio increases following the uncertainty shock because capital adjusts slower than labor.
  - Higher capital-to-skilled labor ratio dampens decline in skilled labor demand and marginal product of skilled labor.
  - Model generates an increase in the skilled-to-unskilled labor ratio and an increase in the skill premium in the flexible-wage variant; however, flexible-wage model can generate a reduction in the skill premium in some setups (contradicting empirical non-responsiveness).

---

### Extension: Sticky Wage Model (Calvo wages)
- Wage stickiness parameters used:
  - Symmetric wage rigidity: θs_w =0.75 and θu_w =0.75.
  - Asymmetric wage rigidity (skilled more rigid): θs_w =0.75 and θu_w =0.65.
- Main findings:
  - Wage stickiness does not significantly affect aggregate variables; output, consumption and investment still contract.
  - With both price and wage rigidities, uncertainty increases both price markup and wage markup, somewhat amplifying output decline.
  - Higher wage rigidity attenuates wage declines; symmetric wage rigidity amplifies the rise in relative skilled employment and generates a fall in the skill premium relative to the flexible-wage case.
- Mechanism:
  - Households set wages as a markup over marginal rate of substitution; time-variation in markups is central to transmission.
  - Capital-skill complementarity dampens fall in skilled labor demand, increasing relative labor demand for skilled workers; unskilled labor falls more relative to skilled.
  - Both skilled and unskilled wage markups increase, but unskilled wage markup increases more, reducing the skill premium.

---

### Asymmetric Sticky Wage
- Rationale: higher wage rigidities for high-skilled workers due to monitoring, bargaining power, hiring/training costs, etc.
- Parameterization: θs_w =0.75 and θu_w =0.65.
- Effects:
  - Increase in relative labor ratio (skilled-to-unskilled) is magnified.
  - Fall in relative wage ratio is dampened; skilled wage decreases less, generating an almost flat response of the skill premium—consistent with empirical results.
  - Relatively smaller decline in skilled wage implies smaller increase in skilled wage markup relative to unskilled markup, further increasing relative labor ratio.

---

### Effects on labor income shares
- Both income shares of skilled and unskilled labor fall in response to an increase in uncertainty.
- The relative skilled-to-unskilled labor income share (ws_t n_s_t / (wu_t n_u_t) notation in source) increases, indicating uncertainty shocks are more harmful for lower-skilled workers in relative terms.
- Drivers:
  - Increase in relative employment of skilled workers (capital-skill complementarity raises relative demand for skilled labor).
  - Asymmetric wage rigidity amplifies the rise in relative skilled employment, further increasing the relative skilled-to-unskilled income share.

---

### Comparison with Data
- The model matches the empirical increase in relative employment of skilled labor, though the model’s response is less persistent than in the data.

---

### Sensitivity Analysis
- Exercises conducted in a model with capital-skill complementarity and asymmetric wage rigidity.

- Frisch elasticity of labor supply:
  - Decreasing φs and φu to 1 (increasing Frisch elasticity) lowers the fall in output via a smaller decline in hours worked.
  - Higher Frisch elasticity increases labor supply sensitivity to wages and dampens aggregate employment decline; lower Frisch elasticity produces a stronger recession.

- Capital-skill complementarity: elasticity of substitution between capital and skilled labor:
  - Baseline elasticity between capital and skilled labor: 0.67 (implying 1/(1−ρ)=0.67 in source notation).
  - Elasticity between capital and low-skilled labor in benchmark: 1/(1−σ)=1.67.
  - Strong complementarity alternative: 1/(1−ρ)=0.37, ρ=−1.7 produces larger responses—stronger complementarity amplifies skilled wage responsiveness and output decline.
  - Changing elasticity between capital-skill composite and unskilled labor:
    - Benchmark σ =0.401.
    - Alternative from literature: σ =0.7899 implying 1/(1−σ)=4.76 (higher substitutability).
    - As σ becomes smaller (weaker capital-skill complementarity), effects of uncertainty are more muted; higher elasticity of substitution decreases real rigidity and dampens recessionary effects.

- Nominal rigidities and markups:
  - Price elasticity set to imply steady-state markups of 5% in one specification; baseline calibration of steady-state price markups of 10% is more consistent with micro pricing studies.
  - Larger demand elasticity (lower steady-state markup) leads to larger output effects by increasing convexity of marginal profit and the precautionary pricing effect.

- Simplified model checks:
  - Removing nominal price and wage rigidities can reverse the sign of the response: without nominal rigidities the uncertainty shock can raise output and investment because increased labor supply lowers marginal costs and firms increase production given flexible prices.
  - Removing nominal rigidities, habits in consumption, and investment adjustment costs likewise alters IRFs and demonstrates the necessity of nominal rigidities to generate co-movement between consumption and output and a recessionary response.

---

### Conclusion (key takeaways)
- Aggregate uncertainty contracts economic activity while increasing relative skilled employment and leaving the skill premium approximately unchanged in line with empirical evidence.
- The interaction of capital-skill complementarity, precautionary labor supply, and wage stickiness is crucial to generate these patterns.
- Capital-skill complementarity amplifies responses of both relative labor demand and relative labor supply, increasing employment disparities between skilled and unskilled workers.
- Findings improve understanding of transmission channels of uncertainty and can inform policymaking during periods of elevated uncertainty.

*Source: wpiea2023155-print-pdf - 1.   Aggregate Economy*

### REFERENCES

### REFERENCES

### Bibliographic scope and themes
- The references compile literature on: uncertainty shocks, capital-skill complementarity, wage and price rigidities, labor market frictions, on-the-job search and wage bargaining, macro-financial interactions, identification and measurement of uncertainty, and empirical SVAR strategies.
- Representative cited works and authors (sample from the list): Acemoglu and Autor (2011); Akerlof and Yellen (1990); Altig et al. (2011); Angelini and Fanelli (2019); Bachmann and Bayer (2013); Basu and Bundick (2017a, 2017b); Bloom (2009); Caldara et al. (2016); Calvo (1983); Christiano, Eichenbaum, and Rebelo (2011); Fernandez‑Villaverde et al. (2011, 2015); Jurado, Ludvigson, and Ng (2015); Ludvigson, Ma S., and Ng (2021); Schaal (2017); Shapiro and Stiglitz (1984); Stock and Watson (1999, 2012).
- Methodological topics in the cited literature include: DSGE solution methods (second-order approximations, pruned state-space), SVAR identification, measures of uncertainty (JLN, EPU, SPF dispersion, financial uncertainty indices), and empirical techniques for detrending and seasonal adjustment.

### Appendix A — Additional details concerning model derivations (key elements and parameter values)
- Production function without capital-skill complementarity:
  - Alternative Cobb-Douglas specification: y = Z_t k_t^ι n_t^{1−ι} with n_t a CES composite of skilled and unskilled hours n_t = (ω(n^s_t)^ν + (1−ω)(n^u_t)^ν)^{1/ν}.
  - Skilled and unskilled hours set as perfect substitutes by setting ω = 0.5, ν = 1.
  - Income share of capital ι is calibrated to obtain a labor income share of 69%.
  - Production function given as Equation (42): y = Z_t k_t^ι (ω(n^s_t)^ν + (1−ω)(n^u_t)^ν)^{(1−ι)/ν}.
  - First-order conditions for inputs and expressions for marginal products and labor share are provided as Equations (43)–(47).

- Wage stickiness (Calvo-type) and wage-setting:
  - Households supply differentiated labor; a labor packer aggregates differentiated labor with an aggregator h^i_t defined in Equation (48) with elasticity parameter η^i_w (0 ≤ η^i_w ≥ ∞).
  - Labor packer demand and aggregate wage relations given in Equations (49)–(50).
  - In each period a fraction (1−θ^i_w) of skilled and unskilled households can reset wages; remaining fraction θ^i_w partially index wages by past inflation. Indexation parameter χ_w ∈ [0,1].
  - Household Lagrangian for wage choice and demand constraints are developed (Equations (51)–(54)), leading to first-order condition and recursive definitions f_{i,1,t}, f_{i,2,t} (Equations (56)–(59)), and the evolution of the real wage index (Equation (60)).
  - Symmetric equilibrium assumption: w_{i,j,∗,t} = w_{i,∗,t} for all households.

- Wholesale firms:
  - FOCs for wholesale firms given the benchmark nested CES production function (Equation (21)) yield marginal products with capital and labor interactions; expressions provided in Equations (61)–(63).
  - Labor share in income expression presented as Equation (64).

### Appendix B — Empirical robustness (tests, specifications, and key numeric details)
- Baseline SVAR stylized facts: a 1-sd uncertainty shock raises the employment rate ratio and does not significantly impact the skill premium.
- Robustness dimensions examined:
  - Detrending methods:
    - One-sided HP filter (Stock and Watson (1999)) and linear detrending. Results with one-sided HP and linear detrending qualitatively and quantitatively resemble the benchmark.
  - Alternative Cholesky ordering:
    - Uncertainty ordered last (uncertainty responds on impact to other variables). Baseline results hold for aggregate economy and Manufacturing sector. Number of lags following AIC set to 2 in this specification.
  - Increasing number of lags:
    - Number of lags increased from 2 to 6; baseline results persist.
  - Alternative measures of uncertainty:
    - SPF forecast dispersion (cross-sectional dispersion of quarterly forecasts for nominal GDP growth) — IRFs to a 1-sd shock are qualitatively in line with baseline.
    - EPU (Baker et al. (2016)) — sample using EPU starts in 1985; responses similar to baseline. EPU index measured in arbitrary units with mean of 100.
    - Financial uncertainty index (Ludvigson and Ng (2021)) — index measured in arbitrary units with mean of 0.91; responses similar to baseline.
  - Alternative measure of skill premium:
    - Skill premium from Balleer and van Rens (2013) (1979Q1 to 2005Q4) — baseline IRFs robust to this alternative.
  - Controlling for the stock market:
    - Inclusion of S&P500 ordered first in SVAR; ordering implies uncertainty can be contemporaneously affected by S&P500 shocks. Results similar to baseline.
  - Monthly frequency estimation:
    - Monthly SVAR estimated for period 1979M1 to 2018M12 using monthly variables: σ^z_t (JLN uncertainty), y_t (Industrial Production Index), i_t (real gross private domestic investment — temporally disaggregated where needed), c_t (personal consumption expenditures), n^s_t (skilled employment rate), n^s_t / n^u_t (employment rate ratio), w^s_t (weighted average real hourly wage in skilled category), w^s_t / w^u_t (wage ratio / skill premium), π_t (chain-type price index for personal consumption expenditures).
    - Monthly labor market series seasonally adjusted with X-13-ARIMA; some quarterly series temporally disaggregated using Chow-Lin method via JDemetra+ version 2.2.1.
    - Monthly SVAR results (and versions controlling for S&P500 ordered first) align with quarterly-baseline responses.
  - Pre-2007 sample:
    - Monthly sample truncated to end in 2007M12 to exclude the global financial crisis; results hold when excluding post-2007 years.
- Figures referenced document IRFs across these robustness checks (Figures 17–27). Typical SVAR settings in notes: median IRFs (solid lines), 10th and 90th percentiles (dashed lines), horizontal axes in quarters, logs taken for uncertainty measure to interpret IRFs in percentage terms. Number of lags varies by specification (AIC-guided), examples include 2, 4, and 6 lags.

*Macroeconomic Uncertainty and Capital-Skill Complementarity — Working Paper No. WP/2023/155*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023155-print-pdf.pdf_