## wp17211

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### Introduction, purpose, and headline results
- Aim:
  - Investigate transmission of micro and macro uncertainty shocks to the real economy and quantify their role in business cycle fluctuations.
  - Examine amplification via credit frictions, price stickiness, monetary policy response, and household preferences.
- Approach:
  - General equilibrium model with sticky prices and credit frictions (BGG framework variant).
  - Macro uncertainty: time-varying variance of aggregate TFP; micro uncertainty: time-varying cross-sectional dispersion of establishment-level TFP.
  - Compute responses to a mean-preserving shock to variance of aggregate productivity and to variance of idiosyncratic productivity.
- Key headline results:
  - Both macro and micro uncertainty shocks are contractionary; they propagate via sticky prices and the financial accelerator.
  - Monetary policy response and households’ preference specification materially affect output impacts.
  - Quantitatively:
    - A one standard deviation shock to micro uncertainty leads to a 0.8 percent fall in total output, over 30 times larger than a one standard deviation shock to macro uncertainty.
  - Unconditional simulations:
    - Business cycle statistics from a model without uncertainty shocks are almost identical to baseline, implying uncertainty shocks unlikely to be a major driver of business cycle fluctuations.
    - When introduced alone, micro uncertainty shocks can explain around 20 percent of the total volatility of output.

### Model structure and agents
- Agents and sectors:
  - Optimizing households; intermediate-goods producing firms; competitive final-good assemblers; capital producers; entrepreneurs subject to credit friction; financial intermediaries; policy maker setting interest rates.
- Sources of uncertainty:
  - Macro uncertainty: stochastic volatility in aggregate TFP (W_t affecting σ_A).
  - Micro uncertainty: stochastic volatility in dispersion of entrepreneurial idiosyncratic productivity (S_t related to σ^2_ω).
- Key preference specification:
  - Baseline: GHH preferences U(C_t, N_t) = 1/(1−ρ) [ C_t − τ N^{1+υ}_t ]^{1−ρ} with U_{c,t} = (C_t − τ N^{1+υ}_t)^{−ρ}.
  - Rationale: GHH removes wealth effects on labor supply, mitigating labor supply shifts from consumption movements.

### Optimal loan contract, external finance premium, and capital demand
- Entrepreneur financing and default:
  - Borrowing: B_j_{t+1} = Q_t K_j_{t+1} − NW_j_{t+1}. (10)
  - Expected nominal income per unit finished capital Y^k_{t+1} defined in (11); return 1+R^k_{t+1} = Y^k_{t+1} / Q_t. (12)
  - Default cutoff: ω_j_{t+1} < ω̄_j_{t+1} = B_j_{t+1} (1+R^L_{t+1}) / (Y^k_{t+1} K_j_{t+1}). (13)
  - Banks audit at monitoring cost μ ∈ [0,1] of realized gross return when default occurs.
- Bank participation (zero profit) condition:
  - Y^k_{t+1} K_j_{t+1} ( Γ(ω̄_j_{t+1}) − μ G(ω̄_j_{t+1}) ) ≥ (1 + R^n_t) B_j_{t+1}. (14)
  - G(ω̄) ≡ ∫_0^{ω̄} ω dF(ω); Γ(ω̄) ≡ [1 − ∫_0^{ω̄} dF(ω)] ω̄ + G(ω̄).
- External finance premium and lending rate:
  - Equilibrium relation: (1 + R^k_{t+1}) / (1 + R^n_t) = ψ_t. (16)
  - ψ_t defined by (17); f′(ω̄) > 0 so ψ_t = f(ω̄_j_{t+1}).
  - Lending rate premium: (1 + R^L_{t+1}) / (1 + R^n_t) = ψ_t ω̄_j_{t+1} ( 1 − NW_j_{t+1} / (Q_t K_j_{t+1}) ). (18)
  - Implication: higher leverage (lower NW_j_{t+1} / (Q_t K_j_{t+1})) raises the premium on entrepreneurial loans.
- Capital demand:
  - K_j_{t+1} = ( 1 / ( 1 − ψ_t ( Γ(ω̄_j_{t+1}) − μ G(ω̄_j_{t+1}) ) ) ) NW_j_{t+1} / Q_t. (19)

### Evolution of entrepreneurial net worth and final-good sector
- Net worth dynamics:
  - Entrepreneurs survive with probability γ; surviving entrepreneurs accumulate NW^j_{t+1} = γ Y^k_{t+1} K^j_{t+1} (1 − Γ(ω̄^j_{t+1})).
  - Entrepreneurial consumption: C^e_t = (1−γ) Y^k_{t+1} K^j_{t+1} (1 − Γ(ω̄^j_{t+1})).
  - Net worth positively related to Q_t and K; aggregate R^k_{t+1} has ambiguous effect on NW because it raises both returns and the external finance premium.
- Final good aggregator and intermediate firms:
  - CES aggregator Y_t = ( ∫_0^1 Y_t(i)^{(ε−1)/ε} di )^{ε/(ε−1)}. (22)
  - Demand Y_t(i) = ( P_t(i) / P_t )^{−ε} Y_t. (23)
  - Intermediate production Y_t(i) uses labor N_t(i) and finished entrepreneurial capital K_t(i); aggregate TFP A_t is central to macro uncertainty. (24)
  - Price adjustment costs ω_p/2 (P_t(i)/P_{t−1}(i) − π)^2; ω_p = 0 implies flexible prices. (25)
  - NK Phillips curve in symmetric equilibrium given by (28).

### Monetary policy, market clearing, and exogenous processes
- Monetary policy rule (log-linear):
  - (1+R^n_t)/(1+R^n) = ((1+R^n_{t−1})/(1+R^n))^{φ_r} ((1+π^n_t)/(1+π))^{(1−φ_r)φ_π} ((1+Y_t)/(1+Y_{t−1}))^{(1−φ_r)φ_y}. (29)
  - Parameters: φ_r ∈ [0,1), φ_π > 0, φ_y ≥ 0.
- Aggregate resource constraint includes monitoring costs:
  - Y_t = C_t + C^e_t + I_t + ω_p/2 (π_t − π)^2 + μ G(ω̄) Y^k_t P_t K_t. (30)
- Exogenous stochastic processes:
  - A_t = ρ_A A_{t−1} + e^W_t σ_A ε^A_t, ε^A_t ~ N(0,1). (31)
  - W_t = ρ_W W_{t−1} + σ_W ε^W_t, ε^W_t ~ N(0,1). (32) — macro uncertainty driven by W_t.
  - log( S_t / ̄S ) = ρ_S log( S_{t−1} / ̄S ) + σ_S ε^S_t, ε^S_t ~ N(0,1). (33) — micro uncertainty via S_t.

### Calibration, empirical inputs, and solution method
- Time unit: quarter.
- Fixed parameters and calibration targets:
  - π = 2 percent (annual steady-state inflation).
  - β = 0.994 (targets annualized average real risk–free rate of 2.4 percent).
  - γ = 0.985; μ = 0.25; ̄S = 0.225.
  - Quarterly steady-state default probability ≈ 1 percent (close to 0.974 percent in Carlstrom and Fuerst (1997)).
  - Implied steady-state external finance premium ≈ 188 basis points.
  - Steady-state leverage ratio ≈ 2.
  - τ set so N = 1/3; ρ = 2; ν = 1; δ = 0.025; α = 0.3; ε = 10 (markup ≈ 11 percent).
  - ω_p chosen so prices fixed 4 quarters on average in Calvo equivalence.
  - φ_π = 1.8; φ_y = 0.25; φ_r = 0.8.
- Table 1 highlights (as presented):
  - μ = 0.25; γ = 0.985; α = 0.3; δ = 0.025; β = 0.994; ρ = 2; ν = 1; τ = 2.5; ε = 10; θ = 105; φ_k = 30; π = 2%; ̄S = 0.225; ρ_r = 0.8; ρ_y = 0.3; ρ_π = 1.8.
- Exogenous-process estimates (Table 2):
  - ρ_S = 0.86; σ_S = 0.023
  - ρ_A = 0.98; σ_A = 0.007
  - ρ_W = 0.88; σ_W = 0.140
- Data and mapping:
  - Micro uncertainty proxy σ_micro_t from Bloom and others (2012) establishment-level TFP innovations (1972–2009).
  - Quarterly conversion: annual ρ_S = 0.56 → quarterly ρ_S = 0.56^{1/4} = 0.86; σ_S set to 0.023.
  - Macro volatility σ_macro_t computed with rolling eight-quarter window; cyclical component yields ρ_W = 0.88 and σ_W = 0.140.
- Solution method:
  - Third-order Taylor expansion around non-stochastic steady state with Dynare 4.3; impulse responses computed as deviations from ergodic mean.

### Impulse-response findings (baseline)
- Shock definitions:
  - Macro uncertainty IRF: 1 standard deviation increase in ε^W_t raises σ_A from 0.007 to about 0.008 (≈ 15 percent increase).
  - Micro uncertainty IRF: 1 standard deviation increase in ε^S_t corresponds to a 2.5 percent increase in micro uncertainty.
- Macro uncertainty transmission (baseline):
  - Operates via precautionary saving → reduced consumption; sticky prices amplify via markups and lower labor demand.
  - Quantitative impacts:
    - Output falls by less than 0.02 percent on impact.
    - Risk premium increases by less than 1.5 annualized basis point.
- Micro uncertainty transmission (baseline):
  - Increased dispersion raises bankruptcy probability and expected monitoring costs → higher lending rates → lower capital demand and investment; sticky prices align consumption and investment declines.
  - Quantitative impacts:
    - A 1 standard deviation micro uncertainty shock leads to a fall in total output of about 0.8 percent on impact.
    - Risk premium increases by more than 60 basis points.
    - Net worth falls by 1 percent.
- Comparative insight:
  - Micro uncertainty shocks have much larger macroeconomic and financial effects than macro uncertainty shocks under the baseline calibration; macro shocks act mainly through demand/precautionary channels.

### Price stickiness, credit frictions, and other ingredients
- Price stickiness:
  - Essential for comovement of consumption and investment and for amplifying effects of uncertainty shocks.
  - Flexible prices (ω_p = 0) with GHH imply unchanged hours and wages for macro shocks and negligible output response to micro uncertainty (Only Micro Uncertainty output volatility = 0.01).
  - Greater price stickiness increases effect of both macro and micro uncertainty on output.
- Credit frictions (monitoring cost μ):
  - 'High credit friction' exercise sets μ = 0.5.
  - This raises steady-state spread by 25 basis points relative to baseline and increases monitoring output loss.
  - Amplification:
    - Macro uncertainty: larger impacts on investment, net worth, price of capital, risk premium, and total output.
    - Micro uncertainty: investment falls about 0.15 percent more than baseline; risk premium response 20 basis points higher; total output falls by more than 0.9 percent (baseline 0.8 percent).
- Households’ preferences:
  - KPR (log-separable) vs GHH:
    - KPR reduces the output impact of both macro and micro uncertainty relative to GHH because labor supply exhibits wealth effects under KPR.
    - Example: Only Micro Uncertainty output volatility falls to 0.12 under KPR (baseline 0.34).
- Risk aversion:
  - Higher ρ = 5 increases precautionary motives; macro uncertainty shocks produce larger impacts; micro shocks amplification increases (Only Micro Uncertainty = 0.44 for high ρ).
- Monetary policy:
  - More aggressive inflation response (φ_π = 3 “Active MP”) dampens output declines and credit spreads for both shock types.
  - Only Micro Uncertainty volatility falls to 0.19 under Active MP (baseline 0.34).

### Unconditional business cycle statistics and numerical experiments
- Simulation method: 2000 periods; last 164 periods used to compute moments (matching 1972:Q1–2012:Q4); HP-filter smoothing parameter 1600 except EFP.
- Selected moments (Table 3 highlights; volatilities and correlations preserved as presented):
  - Output volatility:
    - Data: 1.55
    - Baseline model: 1.56
    - Only TFP: 1.53
    - Only Micro Uncertainty: 0.34
  - Consumption volatility relative to output:
    - Data: 0.82; Baseline: 0.81; Only Micro Uncertainty: 1.08
  - Investment relative to output:
    - Data: 2.92; Baseline: 1.54; Only Micro Uncertainty: 1.37
  - Hours relative to output:
    - Data: 0.91; Baseline: 0.56; Only Micro Uncertainty: 1.44
  - EFP relative to output:
    - Data: 0.12; Baseline: 0.17; Only Micro Uncertainty: 0.13
  - First-order autocorrelations (output):
    - Data: 0.88; Baseline: 0.76; Only Micro Uncertainty: 0.20
  - Contemporaneous correlations with output (investment):
    - Data: 0.95; Baseline: 0.98; Only Micro Uncertainty: 0.84
- Interpretation:
  - Baseline model reproduces several data features.
  - Only Micro Uncertainty output volatility of 0.34 versus Data 1.55 implies micro uncertainty alone accounts for around 20 percent of observed output volatility under baseline calibration.
- Sensitivity experiments (Table 4 summary: standard deviation of output under variants):
  - Baseline All shocks / Only TFP / Only Micro Uncertainty: 1.56 / 1.53 / 0.34
  - Flex. Price: 1.62 / 1.58 / 0.01
  - High Credit Frictions: 1.61 / 1.56 / 0.47
  - KPR Preferences: 1.12 / 1.11 / 0.12
  - Active Monetary Policy: 1.55 / 1.53 / 0.19
  - High Risk Aversion: 1.68 / 1.59 / 0.44
  - CMR Micro Shock (σ_S = 0.07 as in Christiano, Motto, and Rostagno (2014)): 1.94 / 1.53 / 1.17
  - Note: increasing micro shock standard deviation increases its importance roughly proportionally.

### Main conclusions and policy-relevant implications
- Main conclusions:
  - Mean-preserving variance shocks to aggregate TFP (macro uncertainty) and to dispersion of entrepreneurial productivity (micro uncertainty) are both contractionary but differ greatly in quantitative impact under baseline calibration.
  - Micro uncertainty shocks have larger effects on output, investment, net worth, and risk premia primarily via credit-market frictions and are amplified by price stickiness.
  - Uncertainty shocks added to aggregate TFP shocks do not dominate business cycle fluctuations in baseline calibration, though micro uncertainty can account for a non-trivial share (≈ 20 percent) of output volatility.
- Determinants of quantitative importance of micro uncertainty:
  - Presence and degree of sticky prices (ω_p).
  - Severity of credit frictions (monitoring cost μ).
  - Household preference specification (GHH versus KPR).
  - Monetary policy aggressiveness (φ_π).
  - Degree of risk aversion (ρ).
- Suggested research avenue:
  - Investigate reasons for gap between micro-data-based and macro-data-based estimates of micro uncertainty, given sensitivity of results to calibration of σ_S and model ingredients.

*Source: IMF Working Paper — wp17211 (sections "The Optimal Loan Contract", "Evolution of Net Worth", "Price stickiness", "Credit frictions", and "Other Ingredients")*

### 1.    The Optimal Loan Contract  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    11

### 1.    The Optimal Loan Contract

### Introduction: purpose, scope, and key contributions
- Aim:
  - Investigate the transmission of both micro and macro uncertainty shocks to the real economy.
  - Explore mechanisms that amplify uncertainty shocks: severity of credit frictions, price stickiness, monetary policy response, and consumer preferences.
  - Quantify the role of uncertainty shocks in driving business cycle fluctuations.
- Approach:
  - General equilibrium model with sticky prices and credit frictions in the spirit of Bernanke, Gertler, and Gilchrist (1999) (BGG).
  - Cyclical fluctuations in macro uncertainty characterized using aggregate total factor productivity (TFP) data for the U.S. business sector.
  - Cyclical fluctuations in micro uncertainty characterized using cross-sectional dispersion of establishment-level TFP from the Census panel of manufacturing establishments.
  - Compute model responses to a mean-preserving shock to the variance of aggregate productivity (“macro uncertainty shock”) and to the variance of idiosyncratic productivity (“micro uncertainty shock”).
- Main contributions:
  - First paper (to the authors’ knowledge) to study both macro and micro uncertainty in an environment with credit frictions and sticky prices.
  - Investigates key ingredients in the transmission of both micro and macro uncertainty shocks and reconciles contrasting results in the literature by showing sensitivity to physical environment and frictions.
  - Provides estimates of cyclical fluctuations in micro uncertainty using disaggregated establishment-level data.
- Key headline results:
  - Impulse responses show both macro and micro uncertainty shocks have a contractionary impact on the economy.
  - Both shocks propagate via sticky prices and the financial accelerator mechanism; monetary policy response and households’ preference specification are important determinants of output impacts.
  - Quantitatively different impacts:
    - A one standard deviation shock to micro uncertainty leads to a 0.8 percent fall in total output, over 30 times larger than a one standard deviation shock to macro uncertainty.
  - Unconditional (simulated) results:
    - Business cycle statistics from a model without uncertainty shocks are almost identical to the baseline model, suggesting uncertainty shocks are unlikely to be a major driver of business cycle fluctuations.
    - When introduced alone, micro uncertainty shocks can explain around 20 percent of the total volatility of output.

### Model: structure and agents
- Model overview:
  - Variant of the BGG model (Faia and Monacelli (2007)) with optimizing households; intermediate-goods producing firms; perfectly competitive final-good assemblers; capital producers; entrepreneurs subject to a credit friction; financial intermediaries; and a policy maker setting interest rates.
  - Figure 1 provides a simplified structure linking Households, Financial Intermediaries, Capital Producers, Entrepreneurs, Firms, and the Central Bank (policy rate).
- Sources of uncertainty in the model:
  - Macro uncertainty: time-varying variance of aggregate total factor productivity (TFP).
  - Micro uncertainty: time-varying variance (cross-sectional dispersion) of idiosyncratic establishment-level productivity.

### A. Households: preferences, budget, and first-order conditions
- Households:
  - Continuum indexed by i ∈ (0,1).
  - Choose processes {C_t, N_t}∞_{t=0} and one-period nominal deposits {D_t}∞_{t=0}, taking as given {P_t, W_t, (1+R^n_t)}∞_{t=0} and initial D_0.
- Objective:
  - max_{ {C_t, N_t, D_t} } E_t ∑_{t=0}^∞ β^t U(C_t, N_t).  (equation (1) as presented)
- Budget constraint:
  - P_t C_t + D_{t+1} ≤ (1+R^n_t) D_t + W_t N_t + Π_t.  (equation (2))
- First-order conditions:
  - U_{c,t} = β(1+R^n_t) E_t[ U_{c,t+1} P_t / P_{t+1} ].  (equation (3))
  - W_t / P_t = − U_{n,t} / U_{c,t}.
- Baseline preferences:
  - Greenwood, Hercowitz, and Huffman (1988) (GHH preferences):
    - U(C_t, N_t) = 1/(1−ρ) [ C_t − τ N^{1+υ}_t ]^{1−ρ}.  (equation (4))
    - U_{c,t} = (C_t − τ N^{1+υ}_t)^{−ρ}.  (equation (5))
    - U_{n,t} = − τ (1+υ) N^{υ}_t (C_t − τ N^{1+υ}_t)^{−ρ}.
  - Rationale:
    - GHH preferences remove wealth effects from labor supply, preventing labor supply shifts in response to consumption movements, which mitigates the impact of uncertainty shocks.

### B. Unfinished Capital Producers: production, adjustment costs, and optimality
- Production technology:
  - Constant returns to scale production φ(I_t / K_t) K_t with physical adjustment costs.
  - Specific functional form assumed:
    - φ(I_t / K_t) K_t = [ I_t / K_t − φ_k / 2 ( I_t / K_t − δ )^2 ] K_t.  (equation (6))
- Capital accumulation:
  - K_{t+1} = (1−δ) K_t + I_t − φ_k / 2 ( I_t / K_t − δ )^2 K_t.  (equation (7))
- Pricing and optimality:
  - Define Q_t as the re-sell price of the capital good.
  - Capital producers maximize profits:
    - max_{I_t} Q_t [ I_t − φ_k / 2 ( I_t / K_t − δ )^2 K_t ] − P_t I_t.  (equation (8))
  - First-order condition:
    - Q_t [ 1 − φ_k ( I_t / K_t − δ ) ] = P_t.  (equation (9))

### C. Entrepreneurs: technology, idiosyncratic shocks, financing, and optimal loan contract
- Entrepreneurs:
  - Risk-neutral agents who purchase unfinished capital at price Q_t and transform it into finished capital for production in t+1.
- Idiosyncratic productivity shocks:
  - Productivity shocks ω^j_{t+1} are i.i.d. across entrepreneurs and time.
  - Assumed distribution: ω ∼ logN(1, σ^2_ω), with cumulative distribution function F(ω).
  - In the entrepreneurs’ problem, the variance of ω is taken as a given parameter; time variation in σ^2_ω constitutes “micro uncertainty.”
- Financing and credit friction:
  - Entrepreneurs use internal funds and external loans from financial intermediaries.
  - Asymmetric information between entrepreneurs and banks and costly state verification (Townsend (1979); Gale and Hellwig (1985)).
  - To observe the shock the lender must pay an auditing cost equal to a fixed proportion μ ∈ [0,1] of the realized gross return to capital.
- Optimal loan contract:
  - Optimal contract is a standard debt contract with costly bankruptcy:
    - If entrepreneur does not default, lender receives a fixed payment independent of the idiosyncratic shock realization.
    - If entrepreneur defaults, lender audits and seizes whatever is left.
  - Result: interest rate on entrepreneurial loans includes a spread over the risk-free rate.
  - The optimal contract induces truthful reporting and minimizes expected auditing costs; the derivation is provided in the text that follows.

### Transmission mechanisms and sensitivity
- Amplifying mechanisms identified:
  - Sticky prices: necessary for comovement between consumption and investment and for amplifying output impacts.
  - Financial accelerator (credit frictions): amplifies impacts through higher external finance costs and constrained entrepreneurs.
  - Monetary policy response: crucial; constrained monetary policy (e.g., zero lower bound) can increase conditional impact.
  - Households’ preferences: specification (e.g., GHH) materially affects propagation and magnitude of uncertainty shocks.
- Quantitative sensitivity:
  - The amount of output volatility generated by uncertainty shocks is particularly sensitive to:
    - Degree of price stickiness.
    - Severity of the credit friction.
    - Specification of households’ preferences.
    - Reaction function of monetary policy.
- Comparison with literature:
  - The estimated volatility of micro uncertainty shocks is smaller, but comparable, to Christiano, Motto, and Rostagno (2014)’s estimate.
  - Using establishment-level data (Bloom and others (2012)) produces micro uncertainty estimates close to other “micro” estimates (Chugh (2016); Gilchrist, Sim, and Zakrajsek (2014); Bachmann and Bayer (2013)).

### Unconditional business cycle implications
- Simulated business cycle statistics:
  - A model without uncertainty shocks (aggregate TFP only) yields business cycle statistics almost identical to the baseline model.
  - Implication: uncertainty shocks are unlikely to be a major driver of business cycle fluctuations.
- Role of micro uncertainty:
  - When introduced alone, micro uncertainty shocks explain around 20 percent of the total volatility of output.
- Reconciling contrasting findings:
  - Differences across studies can be explained by variation in model environments and frictions (financial frictions, nominal rigidities, zero lower bound, etc.).

*Source: IMF Working Paper — wp17211 (section 1, “The Optimal Loan Contract”) — text as provided.*

### 1.   The Optimal Loan Contract

### 1.   The Optimal Loan Contract

### Model setup and definitions
- Two agents: entrepreneurs and banks. At the end of period t, an entrepreneur j holds nominal net worth NW_j_t+1 and acquires credit B_j_t+1 to finance capital purchases:
  - B_j_t+1 = Q_t K_j_t+1 − NW_j_t+1. (10)
- Expected nominal income from holding one unit of finished capital, Y_k_t+1, is:
  - Y_k_t+1 ≡ Z_t+1 + Q_t+1 [ (1−δ) − φ_k/2 (I_t+1/K_t+1 − δ)^2 + φ_k (I_t+1/K_t+1 − δ) I_t+1/K_t+1 ]. (11)
- Return to entrepreneurs from holding a unit of capital:
  - 1 + R_k_t+1 ≡ Y_k_t+1 / Q_t. (12)

### Default cutoff and bank participation condition
- An entrepreneur repays only if ω_j_t+1 Y_k_t+1 K_j_t+1 ≥ B_j_t+1 (1 + R_L_t+1), where R_L_t+1 is the lending rate.
- Cutoff idiosyncratic shock separating bankrupt and non-bankrupt entrepreneurs:
  - ω_j_t+1 < ω̄_j_t+1 = B_j_t+1 (1+R_L_t+1) / (Y_k_t+1 K_j_t+1). (13)
  - Entrepreneurs with ω_j_t+1 < ω̄_j_t+1 default; the bank seizes remaining assets after paying monitoring cost.
- Banks operate only if:
  - Y_k_t+1 K_j_t+1 ( Γ(ω̄_j_t+1) − μ G(ω̄_j_t+1) ) ≥ (1 + R_n_t) B_j_t+1. (14)
  - Definitions:
    - G(ω̄_j_t+1) ≡ ∫_0^{ω̄_j_t+1} ω_j_t+1 dF(ω)
    - Γ(ω̄_j_t+1) ≡ [1 − ∫_0^{ω̄_j_t+1} dF(ω)] ω̄_j_t+1 + G(ω̄_j_t+1)
  - Interpretation: Γ(ω̄) is the share of finished capital going to banks; 1 − Γ(ω̄) is the share going to entrepreneurs; G(ω̄) is average idiosyncratic shock among bankrupt entrepreneurs.

### Optimal contract and external finance premium
- Entrepreneurs maximize profits with respect to {ω̄_j_t+1, B_j_t+1}:
  - max_{ω̄_j_t+1, B_j_t+1} Y_k_t+1 K_j_t+1 (1 − Γ(ω̄_j_t+1)), subject to (10) and (14) holding with equality. (15)
- First-order conditions imply equality between returns and define the external finance premium ψ_t:
  - (1 + R_k_t+1) / (1 + R_n_t) = ψ_t. (16)
  - ψ_t = ( (1 − Γ(ω̄_j_t+1)) ( Γ′(ω̄_j_t+1) − μ G′(ω̄_j_t+1) ) / Γ′(ω̄_j_t+1) + ( Γ(ω̄_j_t+1) − μ G(ω̄_j_t+1) ) )^{−1}. (17)
  - As in BGG, ψ_t = f(ω̄_j_t+1) with f′(ω̄_j_t+1) > 0.

### Risk premium, leverage, and lending rate
- The ratio between the lending rate and the risk free rate (risk premium) from the zero profit condition:
  - (1 + R_L_t+1) / (1 + R_n_t) = ψ_t ω̄_j_t+1 ( 1 − NW_j_t+1 / (Q_t K_j_t+1) ). (18)
  - NW_j_t+1 / (Q_t K_j_t+1) is the inverse of the leverage ratio.
  - Implication: in presence of credit market imperfections, the premium on the risk free interest rate for a loan depends on the entrepreneur’s balance-sheet condition; higher leverage implies a higher premium on entrepreneurial risky loans.

### Capital demand
- From the zero-profit condition, a demand function for capital that is increasing in net worth and decreasing in price:
  - K_j_t+1 = ( 1 / ( 1 − ψ_t ( Γ(ω̄_j_t+1) − μ G(ω̄_j_t+1) ) ) ) NW_j_t+1 / Q_t. (19)

*Source: wp17211 - 1.   The Optimal Loan Contract*

### 2.   Evolution of Net Worth

### 2.   Evolution of Net Worth

### Entrepreneurial net worth dynamics
- Entrepreneurs survive to the next period with probability γ. Those who die consume accumulated resources and exit.
- Entrepreneurial consumption in each period:
  - Ce_t = (1−γ) Y^k_{t+1} K^j_{t+1} (1−Γ( ̄ω^j_{t+1} ))
- Surviving entrepreneurs accumulate net worth as:
  - NW^j_{t+1} = γ Y^k_{t+1} K^j_{t+1} (1−Γ( ̄ω^j_{t+1} ))
- Since Y^k_{t+1} = Q_t (1+R^k_{t+1}), net worth is positively related to the price Q_t and the stock of capital K.
- Aggregate return on finished capital R^k_{t+1} has an ambiguous effect on net worth:
  - Higher R^k_{t+1} raises returns per unit of finished capital owned by entrepreneurs.
  - Higher R^k_{t+1} also raises the external finance premium (see equation (16)), contributing to the risk premium and reducing net worth.

### Final good sector (production and demand)
- Aggregate final good uses a CES aggregator over intermediate varieties:
  - Y_t = ( ∫_0^1 Y_t(i)^{(ε−1)/ε} di )^{ε/(ε−1)}  (equation (22))
- Demand for each intermediate variety:
  - Y_t(i) = ( P_t(i) / P_t )^{−ε} Y_t  (equation (23))
- Price index:
  - P_t = ( ∫_0^1 P_t(i)^{−ε} di )^{1/(1−ε)}
- Final good producers earn zero profits in equilibrium.

### Intermediate firms (production, pricing, and optimality)
- Each intermediate firm produces with CRS technology using labor N_t(i) and finished entrepreneurial capital K_t(i):
  - Y_t = Φ( exp(A_t), N_t(i), K_t(i) )  (equation (24))
- A_t is aggregate TFP and is central to the macro uncertainty shock.
- Firms face quadratic price adjustment costs:
  - ω_p/2 ( P_t(i)/P_{t−1}(i) − π )^2  (equation (25)), where π is steady-state inflation and ω_p measures nominal price rigidity.
  - ω_p = 0 implies flexible prices.
- Firms maximize expected discounted real profits (equation (26)) subject to (24) and (23).
- First-order conditions yield:
  - W_t / P_t = mc_t Y_{n,t}
  - Z_t / P_t = mc_t Y_{k,t}
  - A forward-looking Phillips curve in symmetric equilibrium ( ̃p_t = 1):
    - (π_t − π) π_t = β E_t { (U_{c,t+1} / U_{c,t}) (π_{t+1} − π) π_{t+1} } + Y_t (ε / ω_p) (mc_t − (ε−1)/ε)  (equation (28))
- mc_t interpreted as real marginal cost.

### Monetary policy and market clearing
- Monetary policy rule (log-linear form):
  - (1+R^n_t)/(1+R^n) = ((1+R^n_{t−1})/(1+R^n))^{φ_r} ((1+π^n_t)/(1+π))^{(1−φ_r)φ_π} ((1+Y_t)/(1+Y_{t−1}))^{(1−φ_r)φ_y}  (equation (29))
  - Parameters: φ_r ∈ [0,1), φ_π > 0, φ_y ≥ 0.
  - Steady-state nominal interest rate R^n determined by equilibrium given target π.
- Final goods market clearing:
  - Y_t = C_t + C^e_t + I_t + ω_p/2 (π_t − π)^2 + μ G( ̄ω ) Y^k_t P_t K_t  (equation (30))
  - Includes private consumption (households and entrepreneurs), investment, price adjustment costs, and banks’ monitoring costs.

### Sources of uncertainty in the model
- Three exogenous processes:
  1. Aggregate TFP level A_t follows AR(1):
     - A_t = ρ_A A_{t−1} + e^W_t σ_A ε^A_t  (equation (31))
     - ε^A_t ~ N(0,1); σ_A is std. dev. of TFP innovations.
     - e^W_t is a stochastic-volatility shifter of σ_A; W_t is the stochastic volatility (macro uncertainty).
     - Macro uncertainty shocks: movements in W_t that change TFP variance without affecting its level.
  2. Stochastic volatility of TFP W_t follows AR(1):
     - W_t = ρ_W W_{t−1} + σ_W ε^W_t  (equation (32)); ε^W_t ~ N(0,1).
  3. Dispersion of idiosyncratic entrepreneurial productivity (micro uncertainty):
     - If ω ∼ logN(1, σ^2_ω), then log(ω) ∼ N(M, S^2) with S^2 = log(1+σ^2_ω).
     - Log-deviation of S_t from steady state follows:
       - log( S_t / ̄S ) = ρ_S log( S_{t−1} / ̄S ) + σ_S ε^S_t  (equation (33)); ε^S_t ~ N(0,1).
     - Micro uncertainty shocks: exogenous movements in S_t increase dispersion of entrepreneurial outcomes and affect entrepreneurial loans via higher bankruptcy probability, raising lending rates under costly state verification.

### Calibration and solution methodology — key parameter values and targets
- Time unit: quarter.
- Fixed parameters used to pin down entrepreneurial problem:
  - π (annual steady-state inflation) = 2 percent.
  - β = 0.994 (targets annualized average real risk–free rate of 2.4 percent).
- Credit-friction related calibration:
  - γ (survival probability) = 0.985.
  - μ (monitoring cost) = 0.25.
  - ̄S (steady-state std. dev. of idiosyncratic productivity) = 0.225.
  - Calibration targets and implied steady-state results:
    - Quarterly steady-state probability of default ≈ 1 percent (close to 0.974 percent used in Carlstrom and Fuerst (1997)).
    - Implied steady-state external finance premium ≈ 188 basis points.
    - Steady-state leverage ratio ≈ 2.
- Household and technology parameters:
  - τ (scaling factor) set so steady-state hours N = 1/3.
  - ρ (risk aversion) = 2.
  - ν (inverse Frisch elasticity) = 1.
  - δ (quarterly depreciation) = 0.025.
  - α (capital share) = 0.3.
  - ε (elasticity of substitution across varieties) = 10 (markup ≈ 11 percent).
- Price rigidity and monetary rule:
  - ω_p chosen so prices fixed 4 quarters on average in Calvo equivalence.
  - φ_π = 1.8, φ_y = 0.25, φ_r = 0.8 (interest-rate smoothing).
- Table 1 parameter highlights (as presented):
  - μ = 0.25; γ = 0.985; α = 0.3; δ = 0.025; β = 0.994; ρ = 2; ν = 1; τ = 2.5; ε = 10; θ = 105; φ_k = 30; π = 2%; ̄S = 0.225; ρ_r = 0.8; ρ_y = 0.3; ρ_π = 1.8

- Exogenous-process parameter estimates (Table 2):
  - ρ_S = 0.86; σ_S = 0.023
  - ρ_A = 0.98; σ_A = 0.007
  - ρ_W = 0.88; σ_W = 0.140

- Data and estimation notes:
  - Micro uncertainty proxy σ_micro_t comes from Bloom and others (2012) using establishment-level TFP innovations (1972–2009).
  - Quarterly conversion: estimated annual AR(1) persistence ρ_S = 0.56 → quarterly ρ_S = 0.56^{1/4} = 0.86; σ_S adjusted to match annual standard deviation producing σ_S = 0.023.
  - TFP AR(1) fit yields ρ_A = 0.98 and σ_A = 0.007 using U.S. business-sector TFP (1972:Q1–2012:Q4).
  - Macro volatility series σ_macro_t computed with rolling eight-quarter window; cyclical component used to estimate ρ_W = 0.88 and σ_W = 0.140.
  - Estimates may be upper bounds because observed uncertainty movements could partly reflect endogenous responses to first-moment shocks.

- Solution method:
  - Third-order Taylor series approximation around non-stochastic steady state computed with Dynare 4.3 to allow second moments to affect policy functions.
  - Impulse responses computed as deviations from the ergodic mean of the model simulation (following Fernandez-Villaverde and others (2011)).

### Impulse-response analysis — baseline findings
- Shock definitions:
  - Macro uncertainty IRFs: 1 standard deviation increase in ε^W_t; under baseline this raises σ_A from 0.007 to about 0.008 (≈ 15 percent increase).
  - Micro uncertainty IRFs: 1 standard deviation increase in ε^S_t corresponds to an increase in micro uncertainty of 2.5 percent.
- Macro uncertainty (panel A) transmission and quantitative effects:
  - Propagation via precautionary saving → reduced consumption.
  - Sticky prices cause downward pressure on firms’ marginal costs → higher markups → reduced labor demand.
  - With GHH preferences (labor supply fixed in the household schedule), lower labor demand reduces hours and real wage.
  - Demand-driven output decline: reduced consumption -> reduced aggregate demand -> lower labor and capital demand -> lower investment.
  - Quantitative impacts (baseline):
    - Output falls by less than 0.02 percent on impact.
    - Risk premium increases by less than 1.5 annualized basis point.
- Micro uncertainty (panel B) transmission and quantitative effects:
  - Transmission via increased dispersion of idiosyncratic returns → higher bankruptcy probability → higher expected monitoring costs under costly state verification → higher lending rates → reduced capital demand and investment.
  - Sticky prices align consumption and investment declines through demand channel.
  - Quantitative impacts (baseline):
    - A 1 standard deviation micro uncertainty shock leads to a fall in total output of about 0.8 percent on impact (≈ 40 times larger than macro uncertainty shock).
    - Risk premium increases by more than 60 basis points.
    - Net worth falls by 1 percent.
- Comparative insight:
  - Macro uncertainty shocks operate mainly through precautionary saving and demand channels with muted financial amplification under baseline calibration.
  - Micro uncertainty shocks operate mainly through credit-market frictions and entrepreneurial capital demand with large financial amplification.

### Transmission and amplification channels examined
- Key mechanisms that affect transmission:
  - Household preferences (e.g., GHH vs. separable preferences) shape the role of precautionary labor supply and the demand channel.
  - Severity of credit frictions (monitoring cost μ, dispersion S_t) governs amplification of micro uncertainty via lending rates, bankruptcy, and net worth.
  - Degree of price stickiness (ω_p) affects comovement of consumption and investment and demand-determined output responses.
  - Monetary-policy rule parameters (φ_π, φ_y, φ_r) determine interest-rate responses to shocks and thus influence propagation.
- Baseline results indicate:
  - Micro uncertainty shocks have much larger macroeconomic and financial effects than macro uncertainty shocks in this model and calibration.
  - Sticky prices and credit frictions are central to the magnitude and propagation of uncertainty shocks.

*Source: IMF Working Paper — "2. Evolution of Net Worth" (content unit wp17211).*

### 1.   Price stickiness

### 1. Price stickiness

### Role of price stickiness in propagation and amplification
- Price stickiness is crucial for the propagation of uncertainty shocks: it generates comovement between consumption and investment and amplifies the impact of the shocks (Basu and Bundick (2017), Fernandez-Villaverde and others (2011), Born and Pfeifer (2014)).
- Comparison of baseline calibration (dark circles) with a flexible price version obtained by setting the price stickiness parameter asωp'0 shows the importance of sticky prices for macro dynamics.
- With flexible prices and constant mark-ups:
  - Labor demand schedule is unchanged in response to the uncertainty shock.
  - With GHH preferences the labor supply schedule is also fixed, resulting in unchanged hours and wages.
  - Consequently output is unchanged in response to a macro uncertainty shock, implying lower consumption is channelled towards higher investment and hence no comovement between consumption and investment—contrary to business cycle evidence.
- Sticky prices act as a powerful amplifying mechanism: the higher the degree of price stickiness, the larger the effect of both macro and micro uncertainty shocks on output.
- Footnote evidence: In a previous version of the paper the effect of uncertainty shocks on output almost doubles when calibrated to obtain an average probability of changing prices of 5 quarters, instead of 4 quarters as in the baseline.

### Macro uncertainty versus micro uncertainty (mechanisms and outcomes)
- Macro uncertainty shock:
  - With capital predetermined, output can only change via labor movements.
  - Under flexible prices and GHH preferences, hours and wages remain unchanged and output is unchanged.
  - Under sticky prices, the macro shock can reduce both consumption and investment and generate comovement.
- Micro uncertainty shock:
  - Depresses investment through higher expected cost associated with bankruptcies, increasing the cost of external finance.
  - Higher lending rates imply lower demand for capital, generating a sharp fall in investment and in the price of capital.
  - The shock resembles an increase in the tax rate on the return on investment which discourages saving and hence investment, and can boost consumption or leisure (Christiano, Motto, and Rostagno (2014)).
  - Under flexible prices lower investment leads to higher consumption as mark-ups, labor demand and labor supply (under GHH preferences) remain unchanged.

### Quantitative setup and IRF notes
- IRFs reported are responses to a 1 standard deviation increase in macro uncertainty (panel A) and in micro uncertainty (panel B).
- IRFs are computed with respect to the ergodic mean of the variables of interest.
- All responses are in percent, except for the risk premium which is in basis points.
- The unit of the x-axis is quarters.

---

### 2. Credit frictions

### Financial accelerator and role of monitoring costs
- The financial accelerator amplifies shocks that affect entrepreneurs’ net worth in general equilibrium; more severe financial market distortions can increase the impact of uncertainty shocks on economic activity, risk premia, and asset prices.
- The paper examines cases with more pronounced credit frictions by increasing the monitoring cost parameter (μ).
- In the ‘high credit friction’ exercise μ=0.5.
  - This implies that the steady-state level of the spread between the lending rate and the risk free rate is now 25 basis points higher than in the baseline.
  - A larger fraction of total output is “lost” in monitoring activities.

### Amplification effects of stronger credit frictions
- Macro uncertainty (higher μ):
  - Larger impact on investment and financial variables (net worth, the price of capital and the risk premium).
  - Resulting effect on total output is also larger.
- Micro uncertainty (higher μ):
  - The impact on the risk premium and on investment increases substantially.
  - Investment falls by about 0.15 percent more than in the baseline.
  - The response of the risk premium is 20 basis points higher than in the baseline.
  - Total output—which falls by about 0.8 percent in the baseline—falls by more than 0.9 percent when the severity of the credit friction is increased.
- Mechanism: A large monitoring cost introduces a larger wedge in the banks’ zero profit condition, inducing banks to raise the spread they charge on lending interest rates; when credit frictions are more severe (monitoring cost larger) the effect of both macro and micro uncertainty shocks on total output is larger.

*Source: IMF Working Paper (wp17211), section 1. Price stickiness and section 2. Credit Frictions.*

### 3.   Other Ingredients

### 3.   Other Ingredients

### (a) Households’ Preferences
- Experimented with log-separable (KPR) preferences versus baseline GHH preferences.
- Key mechanism:
  - With GHH preferences: macro uncertainty → precautionary savings ↓ consumption → lower labor demand under sticky prices → declines in hours and output.
  - With KPR preferences: decline in consumption produces a “precautionary” outward shift in labor supply that mitigates the decline in hours and output.
- Implications:
  - Impact of a macro uncertainty shock on output is smaller with KPR preferences than with GHH preferences.
  - With micro uncertainty shocks:
    - KPR preferences again reduce the effect on output relative to GHH.
    - With GHH preferences consumption and hours are complements; with separable (KPR) preferences this complementarity is absent, so the micro shock’s amplification via consumption-hours complementarity is weaker under KPR.
- Quantitative references from IRFs (note: IRFs computed to a 1 standard deviation increase in shocks; responses in percent; x-axis in quarters):
  - Panel A (Macro Uncertainty): figures illustrate smaller output responses under KPR than baseline.
  - Panel B (Micro Uncertainty): figures illustrate smaller output responses under KPR than baseline.

### (b) Risk aversion
- Experimented with higher coefficient of relative risk aversion ρ = 5 (noted as at the high end of plausible values).
- Mechanisms:
  - Larger ρ increases prudence (Kimball (1990) sense) and strengthens precautionary motive.
  - Macro uncertainty shocks produce larger impacts on hours, output and investment with higher ρ.
  - For micro uncertainty shocks amplification is smaller but present: main channel is increased consumption-hours complementarity at higher ρ, leading to a larger drop in consumption following a drop in hours.
- Quantitative illustration:
  - Panel A (Macro Uncertainty) shows larger negative output responses with “High Risk Aversion” than baseline.
  - Panel B (Micro Uncertainty) shows larger negative output responses with “High Risk Aversion” than baseline.

### (c) Monetary Policy
- Baseline Taylor-rule coefficient on inflation ρπ = 1.8; alternative “Active MP” uses ρπ = 3.
- Mechanism:
  - More aggressive response to inflation → nominal rate falls by more in response to contractionary shock → smaller fall in inflation → lower real interest rate relative to baseline.
  - Lower real rate dampens effects on consumption, investment, output and the credit spread for both macro and micro uncertainty shocks.
- Quantitative illustration:
  - Panel A and Panel B IRFs show smaller output declines under “Active MP” relative to baseline.

### V. Unconditional business cycle properties (numerical experiments)
- Method: compare baseline model (micro + macro uncertainty shocks + aggregate TFP shocks) to variants with shocks introduced one at a time; simulations use 2000 periods and last 164 periods to compute moments (matching 1972:Q1–2012:Q4).
- Data sample for comparison: US data over 1972:Q1–2012:Q4.
- Key empirical variable: External Finance Premium (EFP) proxied by BAA to AAA spread.
- Table 3 (selected reported moments; all series logged and HP-filtered with smoothing parameter 1600, except EFP):
  - Volatility (standard deviation, relative to output volatility where indicated):
    - Output (Data): 1.55
    - Output (Baseline model): 1.56
    - Output (Only TFP): 1.53
    - Output (Only Micro Uncertainty): 0.34
    - Consumption volatility relative to output (Data): 0.82; Baseline: 0.81; Only TFP: 0.80; Only Micro Uncertainty: 1.08
    - Investment relative to output (Data): 2.92; Baseline: 1.54; Only TFP: 1.58; Only Micro Uncertainty: 1.37
    - Hours relative to output (Data): 0.91; Baseline: 0.56; Only TFP: 0.49; Only Micro Uncertainty: 1.44
    - EFP relative to output (Data): 0.12; Baseline: 0.17; Only TFP: 0.14; Only Micro Uncertainty: 0.13
  - First-order autocorrelations:
    - Output (Data): 0.88; Baseline: 0.76; Only TFP: 0.81; Only Micro Uncertainty: 0.20
    - Consumption (Data): 0.88; Baseline: 0.76; Only TFP: 0.83; Only Micro Uncertainty: 0.17
    - Investment (Data): 0.91; Baseline: 0.73; Only TFP: 0.77; Only Micro Uncertainty: 0.49
    - Hours (Data): 0.95; Baseline: 0.60; Only TFP: 0.85; Only Micro Uncertainty: 0.21
    - EFP (Data): 0.87; Baseline: 0.83; Only TFP: 0.89; Only Micro Uncertainty: 0.76
  - Contemporaneous correlations with aggregate output:
    - Consumption (Data): 0.88; Baseline: 0.99; Only TFP: 1.00; Only Micro Uncertainty: 0.99
    - Investment (Data): 0.95; Baseline: 0.98; Only TFP: 0.99; Only Micro Uncertainty: 0.84
    - Hours (Data): 0.82; Baseline: 0.89; Only TFP: 0.98; Only Micro Uncertainty: 1.00
    - EFP (Data): -0.46; Baseline: -0.58; Only TFP: -0.73; Only Micro Uncertainty: -0.60
- Interpretation from Table 3:
  - Baseline model matches several key features of the data reasonably well despite model simplicity.
  - Micro uncertainty shocks alone drive a small but non-trivial share of output volatility: Only Micro Uncertainty output volatility = 0.34 versus Data output volatility = 1.55 (implying micro uncertainty alone accounts for around 20 percent of observed output volatility in the baseline calibration).
  - Comparison to literature:
    - Chugh (2016): micro uncertainty shocks drive about 5 percent of GDP volatility.
    - Christiano, Motto, and Rostagno (2014): estimate about 20 percent of GDP volatility driven by micro uncertainty shocks.

### Role of model ingredients (Table 4 summary: standard deviation of output under variants; all series logged and HP-filtered with smoothing parameter 1600)
- Baseline (All shocks / Only TFP / Only Micro Uncertainty): 1.56 / 1.53 / 0.34
- Flex. Price: 1.62 / 1.58 / 0.01
  - Flexible prices: micro uncertainty shocks account for less than 1 percent of total output volatility when they are the sole source of variation (Only Micro Uncertainty = 0.01).
- High Credit Frictions: 1.61 / 1.56 / 0.47
  - Increasing severity of credit frictions raises the share of output volatility due to micro uncertainty to about 30 percent (Only Micro Uncertainty = 0.47 vs baseline 0.34).
- KPR Preferences: 1.12 / 1.11 / 0.12
  - KPR preferences dampen both uncertainty and TFP shocks; share of output volatility from micro uncertainty falls to about 12 percent (Only Micro Uncertainty = 0.12).
- Active Monetary Policy: 1.55 / 1.53 / 0.19
  - More aggressive monetary response lowers the importance of uncertainty shocks (Only Micro Uncertainty = 0.19).
- High Risk Aversion: 1.68 / 1.59 / 0.44
  - Higher risk aversion increases the share of output volatility associated with micro uncertainty (Only Micro Uncertainty = 0.44).
- CMR Micro Shock (micro shock standard deviation set to σS = 0.07 as in Christiano, Motto, and Rostagno (2014)): 1.94 / 1.53 / 1.17
  - Increasing the standard deviation of the micro uncertainty shock increases its importance roughly proportionally (Only Micro Uncertainty = 1.17).
- Additional note:
  - Investment adjustment costs set to zero raise investment volatility but leave unconditional business cycle statistics similar to baseline.

### Main conclusions (section VI)
- Study focus: mean-preserving variance shocks to aggregate TFP (macro uncertainty) and to dispersion of entrepreneurs’ idiosyncratic productivity (micro uncertainty) in a financial accelerator DSGE model with sticky prices.
- Calibration/data:
  - Model disciplined by aggregate TFP data for U.S. business sector and establishment-level TFP disaggregated data.
  - Estimated micro uncertainty time series aligns with micro-data-based estimates but is smaller than some macro-based estimates.
- Key findings:
  - Uncertainty shocks, when added to aggregate TFP shocks, are not major drivers of business cycle fluctuations in the baseline calibration.
  - Micro uncertainty shocks have larger impact on total output than macro uncertainty shocks and account for a small but non-trivial share of output volatility in baseline.
  - Determinants of the quantitative importance of micro uncertainty shocks include:
    - Presence of sticky prices.
    - Severity of credit frictions.
    - Specification of households’ preferences (GHH versus KPR).
    - Monetary policy response (aggressiveness).
    - Degree of risk aversion.
- Suggested avenue: understanding the gap between micro-data-based and macro-data-based estimates of micro uncertainty could be a focus for further research.

*Source: IMF Working Paper — "3.   Other Ingredients" (wp17211).*

### REFERENCES

### REFERENCES

### Bibliographic citations (selection)
- Extensive list of works on investment under uncertainty, uncertainty shocks, financial frictions, credit channels, precautionary saving, time-varying volatility, and macroeconomic modeling. Notable entries include:
  - Abel, Andrew B, 1983, “Optimal Investment under Uncertainty,” American Economic Review, Vol. 73, No. 1, pp. 228–33.
  - Bernanke, Ben S., 1983, “Irreversibility, Uncertainty, and Cyclical Investment,” The Quarterly Journal of Economics, Vol. 98, No. 1, pp. 85–106.
  - Bloom, Nicholas, 2009, “The Impact of Uncertainty Shocks,” Econometrica, Vol. 77, No. 3, pp. 623–685.
  - Christiano, Lawrence, Roberto Motto, and Massimo Rostagno, 2014, “Risk Shocks,” American Economic Review, Vol. 104(1), pp. 27–65.
  - Fernandez-Villaverde, Jesus, Pablo Guerron-Quintana, Juan F. Rubio-Ramirez, and Martin Uribe, 2011, “Risk Matters: The Real Effects of Volatility Shocks,” American Economic Review, Vol. 101, No. 6, pp. 2530–61.
  - Dixit, Avinash K., and Robert S. Pindyck, 1994, Investment under Uncertainty (Princeton University Press).
  - Gilchrist, Simon, Jae W. Sim, and Egon Zakrajsek, 2014, “Uncertainty, Financial Frictions, and Investment Dynamics,” NBER Working Papers 20038.
  - Kimball, Miles S, 1990, “Precautionary Saving in the Small and in the Large,” Econometrica, Vol. 58, No. 1, pp. 53–73.
- (Full list of cited works appears in the source document.)

### Appendix A — Equilibrium (model system)
- Definitions:
  - Defineq t ≡ Q t / P t, nw t ≡ NW t / P t, z t ≡ Z t / P t.
- System of equilibrium conditions (for a given path for exogenous processes):
  - Euler equation of households:
    - U c,t = β(1+R n t ) E t [ U c,t+1 / π t+1 ]. A.1
  - Labor supply:
    - mc t Y n,t = − U n,t / U c,t . A.2
  - Marginal product of capital:
    - mc t Y k,t = z t . A.3
  - Price of capital:
    - q t = [ 1 − φ k ( I t / K t − δ ) ] −1 . A.4
  - Zero profit condition:
    - y k t+1 K t+1 ( Γ( ̄ω t+1 ) − μ G( ̄ω t+1 ) ) = (1+R n t )( q t K t+1 − nw t+1 ). A.5
  - NK Phillips curve:
    - ( π t − π ) π t = β E t { U c,t+1 / U c,t ( π t+1 − π ) π t+1 } + Y t ε ω p ( mc t − (ε−1)/ε ). A.6
  - Net worth law of motion:
    - nw t+1 = γ y k t+1 K t+1 ( 1 − Γ( ̄ω t+1 ) ). A.7
  - Entrepreneurs real consumption:
    - C e t = (1−γ)(1−Γ( ̄ω t+1 )) y k t K t . A.8
  - Aggregate resource constraint:
    - A t F(K t , N t ) = C t + C e t + I t + ω p / 2 ( π t − π ) 2 + μ G( ̄ω ) y k t K t . A.9
  - Accumulation of aggregate capital:
    - K t+1 = (1−δ) K t + I t − φ k / 2 ( I t / K t − δ ) 2 K t . A.10
  - Monetary policy rule:
    - 1+R n t / 1+R n = ( 1+R n t−1 / 1+R n ) φ r ( 1+π n t / 1+π ) (1−φ r ) φ π ( 1+Y t / 1+Y t−1 ) (1−φ r ) φ y . A.11
  - Real income from holding one unit of finished capital:
    - y k t = z t + q t [ 1 − δ − φ k / 2 ( I t / K t − δ ) 2 + φ k ( I t / K t − δ ) I t / K t ]. A.12
    - y k t+1 = (1+R k t+1 ) q t / π t+1 . A.13
  - Optimal contract:
    - 1+R k t+1 / 1+R n t = ψ t . A.14
    - ψ t = ( (1−Γ( ̄ω j t+1 ))( Γ′( ̄ω j t+1 ) − μ G′( ̄ω j t+1 ) ) / Γ′( ̄ω j t+1 ) + ( Γ( ̄ω j t+1 ) − μ G( ̄ω j t+1 ) ) ) −1 . A.15

### Appendix A.2 — Households’ preferences (KPR vs GHH)
- Two functional forms compared: KPP (King, Plosser, and Rebelo (1988)) log-separable preferences and GHH (Greenwood, Hercowitz, and Huffman (1988)) preferences.
- KPR preferences (log-separable):
  - Utility: ( C t (1−N t ) τ KPR ) (1−ρ) / (1−ρ) . A.16
  - Marginal utilities:
    - U c,t = C −ρ t (1−N t ) τ KPR (1−ρ) . A.17
    - U n,t = − τ KPR C 1−ρ t (1−N t ) τ KPR (1−ρ)−1 .
  - Implications:
    - Euler equation: C −ρ t (1−N t ) τ KPR (1−ρ) = β(1+R n t ) E t [ C −ρ t+1 (1−N t+1 ) τ KPR (1−ρ) P t / P t+1 ]. A.18
    - Labor supply: W t / P t = τ KPR C t (1−N t ).
    - Interpretation: Expected consumption growth depends on the real interest rate and expected labor growth; consumption appears in the labor supply equation, so labor supply shifts with consumption (wealth effects on labor supply).
- GHH preferences (non-separable; cross-derivative nonzero):
  - Utility: 1/(1−ρ) ( C t − τ GHH N 1+υ t ) 1−ρ . A.19
  - Marginal utilities:
    - U c,t = ( C t − τ GHH N 1+υ t ) −ρ . A.20
    - U n,t = − τ GHH (1+υ) N υ t ( C t − τ GHH N 1+υ t ) −ρ .
  - Implications:
    - Euler equation: ( C t − τ GHH N 1+υ t ) −ρ = β(1+R n t ) E t [ ( C t+1 − τ GHH N 1+υ t+1 ) −ρ P t / P t+1 ]. A.21
    - Labor supply: W t / P t = τ GHH (1+υ) N υ t .
    - Interpretation: Expected consumption growth depends on the real interest rate and expected labor growth; unlike KPR, labor supply depends only on the real wage (no wealth effect on labor supply because the marginal rate of substitution is independent of consumption).

*Source: wp17211 - REFERENCES (excerpt).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17211.pdf_