## wpiea2019043 - 2.1 Real Economy and 3.1 Monetary shock

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### Households: preferences, budget, and first-order conditions
- Preferences:
  - max_{C_s,N_s,D_s} E_t (sum_{s=t}^{∞} β^{s−t} u(C_s, N_s))
  - u(C_t, N_t) = (C_t)^{1−σ}/(1−σ) − η N_t^{1+φ}/(1+φ)
- Definitions and parameters:
  - C_t: consumption; N_t: hours worked.
  - σ: household risk aversion; φ: inverse of the Frisch elasticity of labor supply; η: disutility weight for work.
- Budget constraint:
  - C_t + D_t + T_t ≤ R^D_{t−1} D_{t−1} / π_t + W_t N_t + Π^R_t + Π^K_t + (1−α_W) Π^B_t
  - π_t = P_t / P_{t−1} is inflation.
- First-order conditions:
  - C_t^{−σ} = λ_t
  - η N_t^{φ} = λ_t W_t
  - λ_t = β E_t[(λ_{t+1} / π_{t+1}) R^D_t]

### Wholesale producers: production, factor returns, and marginal cost
- Production function:
  - Y^W_t = e^{a_t} K_t^{1−α−ρ} N_t^{α} (N^E_t)^{ρ}
  - a_t follows an AR(1) process.
- Entrepreneurial labor:
  - Share very small; N^E_t normalized to one in later assumptions.
- Factor returns (zero profit FOCs):
  - R^W_t = (1−α−ρ) P^W_t Y^W_t / K_t
  - W_t = α P^W_t Y^W_t / N_t
  - W^E_t = ρ P^W_t Y^W_t / N^E_t
- Marginal cost:
  - P^W_t = W_t^{α} (W^E_t)^{ρ} (R^W_t)^{1−α−ρ} exp(a_t) α^{α} ρ^{ρ} (1−α−ρ)^{1−α−ρ}

### Capital producers: accumulation, adjustment costs, and Tobin’s Q
- Capital accumulation:
  - K_{t+1} = (1−δ) K_t + Φ(X_t / X_{t−1}) X_t
- Adjustment cost (Christiano et al. (2010) nonlinear form; Φ′ specified):
  - Φ′(x_t) = χ (x_t − 1)/x_t [ (x_t − 1)/(2 x_t) − x_t ] with x_t = X_t / X_{t−1}
- Tobin’s Q condition:
  - Q_t [ (X_t / X_{t−1}) Φ′_t(x_t) + Φ(x_t) ] = 1 + (1 / R^D_t) E_t[ Q_{t+1} Φ′(x_{t+1}) (X_{t+1}/X_t)^2 ]
- Realized profit:
  - Π^K_t = Q_t K_{t+1} − (1−δ) Q_t K_t − X_t
  - Profit zero in steady state; nonzero during transitions due to pre-determined X_t.

### Retailers: pricing, Calvo rigidity, and market clearing
- CES aggregation and price index:
  - P_t = [ ∫_0^1 P_t(z)^{1−θ} dz ]^{1/(1−θ)}
  - Y_t = [ ∫_0^1 Y_t(z)^{(θ−1)/θ} dz ]^{θ/(θ−1)}
  - Y_t(z) = (P_t(z)/P_t)^{−θ} Y_t
- Calvo price stickiness:
  - Firms can change price with probability 1−α_p each period; price remains fixed with prob. α_p.
  - Optimal reset price P^*_t(z) satisfies:
    - E_t ∑_{s=t}^{∞} M_{s,t} α_p^{s−t} Y^*_{s,t}(z) [ P^*_t(z) − θ/(θ−1) P^W_s ] = 0
- Market clearing and retailer profits:
  - ∫_0^1 Y_t(z) dz = Y^W_t
  - Π^R_t = ∫_0^1 [ P_t(z) − P^W_t ] Y_t(z) dz = Y_t − P^W_t Y^W_t

### Entrepreneurs: financing, default, and expected profits
- Financing identity:
  - Q_t K_{t+1} = L_t + NW_t
- Ex-post and aggregate returns:
  - Ex-post return subject to idiosyncratic ω: ω R^E_t
  - R^E_t = r^W_t + (1−δ) Q_t / Q_{t−1}
- Default cutoff:
  - ω_{t+1} defined by ω_{t+1} R^E_{t+1} Q_t K_{t+1} = R^L_t L_t
- Expected profit V_t:
  - V_t = E_t [ ∫_{ω_{t+1}}^{∞} ω R^E_{t+1} Q_t K_{t+1} dF(ω) − (1−F(ω_{t+1})) R^L_t L_t ]
  - Simplified: V_t = E_t [ ( ∫_{ω_{t+1}}^{∞} ω dF(ω) − ω_{t+1} ∫_{ω_{t+1}}^{∞} dF(ω) ) R^E_{t+1} Q_t K_{t+1} ]

### Banking sector — Lending banks: zero-profit pricing and leverage dynamics
- Zero-profit condition (wholesale loan funding at R^I_t):
  - ∫_0^{∞} ω_{t+1} R^L_t L_t dF(ω) + (1−μ) ∫_0^{ω_{t+1}} ω R^E_{t+1} Q_t K_{t+1} dF(ω) = R^I_t L_t
- Key aggregates and functions:
  - Γ_t(ω_{t+1}) = ω_{t+1} ∫_{ω_{t+1}}^{∞} dF(ω) + ∫_0^{ω_{t+1}} ω dF(ω)
  - G_t(ω_{t+1}) = ∫_0^{ω_{t+1}} ω dF(ω)
  - Leverage ̺_t = Q_t K_{t+1} / NW_t
- Relationships:
  - [ Γ(ω_{t+1}) − μ G(ω_{t+1}) ] ̺_t R^E_{t+1} / R^I_t = ̺_{t−1}
  - R^E_{t+1} / R^I_t = Λ(ω_{t+1}) with Λ′(.) > 0
  - Q_t K_{t+1} = Ψ( R^E_{t+1} / R^I_t ) NW_t where Ψ(1) = 1 and Ψ′(.) > 0
- Net worth dynamics:
  - NW_t = γ V_t + W^E_t where C^E_t = (1−γ) V_t consumed.
  - V_t = R^E_t Q_{t−1} K_t − [1−d^L_{t−1}] R^I_{t−1} L_{t−1} − μ ∫_0^{ω_t} ω R^E_t Q_{t−1} K_t dF(ω)
- External finance premium:
  - Expressed via μ ∫_0^{ω_t} ω R^E_t Q_{t−1} K_t dF(ω) / (Q_{t−1} K_t − NW_t)
- Lending banks’ leverage choice constraint:
  - L_t = ρ_l L_{t−1} + (1−ρ_l) e K_t

### Banking sector — Deposit banks: deposit pricing and adjustment costs
- Deposit demand for bank j:
  - D_{j,t} = (R^D_{j,t} / R^D_t)^{ε} D_t
- Symmetric equilibrium (R^D_{j,t} = R^D_t) FOC:
  - (1 + ε)/ε R^D_t = R^{IB}_t − κ_D ε [ (R^D_t / R^D_{t−1} − 1)^2 ] R^D_t / R^D_{t−1} + M κ_D ε [ (R^D_{t+1} / R^D_t − 1)^2 ] R^D_{t+1} / R^D_t
- Interpretation:
  - Deposit interest rate is a markup (markdown) relative to interbank rate; spread time-varying and increases with net marginal adjustment cost.

### Banking sector — Wholesale banks: capital requirements and balance-sheet effects
- Balance sheet identity:
  - D_t + K^{WB}_t = B_t + L_t
- Capital requirement:
  - K^{WB}_t = S_t + λ_W ( δ^b_t B_t + δ^l_t L_t )
  - If S_t < 0 pay cost Ω(S_t) with Ω(S_t) > 0 for S_t < 0 and Ω(0) = 0; Ω convex.
- Risk-weighted coefficients (time-varying):
  - δ^b_t = δ^b (1 + d^B_t)^{η_b}
  - δ^l_t = δ^l (1 + d^L_t)^{η_l}
  - d^L_t and d^B_t are external default probabilities in t+1 for lending banks and government bonds.
- Bank capital accumulation:
  - K^W_t = (1−ρ_W) K^W_{t−1} + α_W Π^B_t
  - Π^B_t = Π^S_t + Π^{WB}_t
- Wholesale bank cash flow:
  - Π^{WB}_t = (1−d^L_t) R^I_{t−1} L_{t−1} + (1−d^B_t) R^B_{t−1} B_{t−1} / π_t − R^{IB}_{t−1} D_{t−1} / π_t − K^{WB}_t − Ω(S_t)
- First-order conditions for B_t and L_t link expected returns, default probabilities, and Ω′(S_t).
- Assumption:
  - Interbank rate equals central bank policy rate: R^{IB}_t = R_t.
- Under-capitalization (S_t < 0) increases spreads due to Ω′(S_t) < 0 and convex Ω.

### Central bank, government, and market clearing
- Central bank (Taylor rule):
  - log(R_t / R) = ρ_π log(π_t / π) + ρ_y log(Y_t / Y) + ε^R_t
- Government budget:
  - G_t + R^B_{t−1} (1−δ^b_t) B_{t−1} / π_t = B_t + T_t
- Tax process:
  - T_t = (1−ρ_T) τ Y_t + ρ_T T_{t−1}
- Debt issuance limit:
  - B_t = [ (1−d^B_t) B_{t−1} ]^{ρ_b} B^{1−ρ_b}
  - B steady-state debt chosen as 60 percent of GDP.
- Goods market clearing:
  - Y_t = C_t + C^E_t + G_t + X_t + μ ∫_0^{ω_t} ω dF(ω) R^E_t Q_{t−1} K_t + (κ_D / 2) (R^D_t / R^D_{t−1} − 1)^2 D_t + Ω(S_t) + K^{WB}_t − (1−ρ_W) K^{WB}_{t−1}

### Results and Calibration (euro zone quarterly calibration)
- Calibration highlights (selected parameters and targets):
  - β Discount factor: 0.99 (implies annual deposit rate of 0.041 percent).
  - Frisch elasticity of labor: 0.1 (high elasticity interpretation).
  - σ Elasticity of consumption: 0.5.
  - η calibrated so steady state C/Y = 0.6076.
  - (C + C^E)/Y = 0.7298.
  - α = 0.7; ρ = 0.01.
  - δ = 0.025 (quarterly).
  - χ = 20 (capital-producer adjustment cost parameter).
  - α_p = 0.75 (one price change per year).
  - θ = 6 (markup = 1.2 at steady state).
  - Steady-state entrepreneurial leverage ratio = 2.
  - External finance premium at steady state = 114 b.p. quarterly.
  - Deposit banks’ monopolistic power ε = 50.
  - κ_D = 2.4.
  - λ_W = 0.08.
  - Risk-weight parameters: δ^b = 0.20; δ^l = 0.50.
  - η̄_b = 46.27; η̄_l = 17.5.
  - Taylor rule: ρ_π = 1.5; ρ_y = 0.05.
  - ρ_G = 0.8.
  - Investment to output = 0.0524.
  - Public spending to GDP = 0.1728.
- Additional calibrated entries (from section 3.1 parameter table):
  - χ Coefficient of adjustment cost for investment: 600
  - αp Nominal rigidities: 0.85
  - τ Tax rate: 0.215
  - μ Entrepreneurs’ monitoring cost: 0.21
  - σω Standard deviation for log-normal distribution of ω: 0.082
  - ηb Power in risk weighing asset framework for bonds: 45.27
  - ηl Power in risk weighing asset framework for loans: 17.5
- Steady-state statistics (selected):
  - π Inflation: 1
  - Rf Policy rate: 1.0303
  - RD Deposit rate: 1.0101
  - ̺ Entrepreneurs’ leverage ratio: 2.0282
  - R_E/R_I External risk premium: 1.0202
  - K/Y Capital to GDP ratio: 2.09
  - X/Y Investment to GDP ratio: 0.0524
  - (C + C^E)/Y Consumption to GDP: 0.7298
  - G/Y Public Spending to GDP: 0.1728
  - K_WB/Y Bank Capital to GDP: 0.1330
  - L/Y Loan to GDP ratio: 1.1480
  - D/Y Deposit to GDP ratio: 1.6153
  - B/Y Debt to GDP ratio: 0.4432

### 3.1 Monetary shock: transmission, roles of frictions, and numerical experiments
- Monetary shock scenario:
  - Unanticipated 25 basis point decline in the policy interest rate (annual basis).
- Core transmission (full model):
  - Output, consumption, investment, and capital stock increase.
  - Higher capital demand → higher demand for loans.
  - Entrepreneurs’ net worth rises as loan rates decline and default rate declines → reduced external premium.
  - Financial accelerator amplifies real-side responses via bank balance-sheet channels.
- Role of deposit-rate adjustment costs (Rotemberg cost):
  - With Rotemberg cost (full model):
    - Deposit rate does not fully follow the policy interest rate.
    - Deposits increase at the expense of households’ consumption; consumption increases only moderately.
    - Banks become momentarily over-capitalized as more valuable loan assets appear; leverage ratios fall initially then rise as loan demand builds.
    - Financial accelerator is dampened relative to the no-Rotemberg case.
  - Without Rotemberg cost:
    - Deposit rate follows the policy rate by a mark-down.
    - Initial decrease in deposit rate lowers deposit volume, later offset by higher household income → deposit accumulation.
    - Financial accelerator reinforces the shock more aggressively: investment and output rise faster; financial variables show higher sensitivities.
    - Banks choose to hold higher capital levels because deposits drop more, forcing banks to raise capital to meet both higher loan demand and reduced deposit funding.
- Role of bank capital cost and capital requirements:
  - Wholesale banks’ capital holding requirements show little impact under monetary expansion.
  - Wholesale banks with better capital positions lower interest-rate spreads → better transmission of monetary policy.
  - Overall impact of bank capital cost on most variables is negligible for a monetary expansion.
- Basel treatment:
  - Models for Basel I and for Basel II & III produce the same impulse-response functions for the monetary shock because risk weights do not change following a monetary shock.
- Reporting conventions:
  - Charts except interest rates report percentage deviations from steady state.
  - Interest rate graphs shown as absolute deviation from steady state in percentage points.

### Sovereign risk shock: scenario, channels, and amplification mechanisms
- Shock scenario:
  - Unexpected sovereign default risk shock of 2 percent, implemented as a series of unexpected shocks lasting around 2 years.
- Direct financial-market responses:
  - Interest rate on bonds rises sharply.
  - Demand for bonds falls to close to 3.5 percent of its steady state level.
  - Following the 2 percent increase in sovereign default risk:
    - GDP can shrink by more than 3 percent.
    - Interest rate on bonds rises by 5 percent.
    - Banks’ leverage ratio grows higher to around 30 percent of its steady state.
- Banking-channel mechanisms:
  - Funding cost rises as banks’ asset values diminish and leverage ratios increase.
  - In risk-weighted asset framework, bond risk weights increase with rating deterioration → higher capital charges.
  - Both effects reduce bank profitability and capital positions, widening interest-rate spreads.
  - Outcomes: lower loan supply, lower bond demand, asset price declines, higher market interest rates → amplified spillovers to real economy.
  - Sovereign default risk modeled as exogenous (not endogenously affected by higher interest rates).
- Real-economy dynamics:
  - Lower loan supply and bond demand reduce asset prices and raise interest rates.
  - Retailers and capital producers suffer profit contractions → labor declines → households’ incomes fall.
  - Monetary policy reacts with a contraction in the policy rate; deposit rates follow, allowing households to smooth consumption by dissaving initially.
  - Investment declines due to interest-rate and external-premium channels; consumption initially increases (lower deposit rates and dissaving) but later declines as activity weakens and deposits recover → output shrinks driven by falling investment, public spending, and private consumption in later stages.
- Role of the financial accelerator:
  - Fixing the external premium to steady state removes accelerator and reduces persistence.
  - With the accelerator (full model), responses show higher persistence; accelerator works through monetary policy channel affecting interest rates, inflation, and investment.
- Role of bank capital cost (α_S) and capital adequacy:
  - Without bank capital cost (α_S = 0):
    - Wholesale banks’ profits remain higher, banks better capitalized, leverage ratio remains lower when default risk rises.
  - In the full model, spread between bond/loan rates and policy rate is higher; R_Et does not rise enough to compensate R_It → external premium R_Et/R_It lower in full model.
  - Capital adequacy condition amplifies sovereign shocks via:
    - Balance-sheet channel: wholesale banks reshuffle portfolios and cut balance sheets to recover capital/leverage.
    - Interest-rate channel: bond and loan rates increase, depressing investment and output.
  - Monetary authority reacts more strongly in full model because economy plunges deeper; monetary policy partly offsets market-rate hikes.
- Rotemberg cost in sovereign shock:
  - Full model and no-Rotemberg model show similar behavior except deposit rate responses are more pronounced without adjustment cost.
- Basel I vs Basel II/III:
  - Time-varying risk-weight mechanisms (Basel II & III) matter for amplifying sovereign-default shocks; Basel I (no time-varying risk weights) shows weaker amplification.

### Model-wide conclusions and policy-relevant implications
- Bank balance sheets are central in amplifying sovereign-risk shocks to the real economy.
- Capital adequacy:
  - Attenuates effects of technology and monetary shocks.
  - Reinforces impacts of sovereign default and interbank risk shocks that directly impair bank balance sheets.
- Risk-weight dynamics:
  - Time-varying risk weights (Basel II & III style) interact with sovereign risk to increase capital charges and amplify bank distress following sovereign shocks.
- Policy implications:
  - Monetary and macroprudential policies can attenuate negative spillovers from sovereign default shocks by addressing balance-sheet and interest-rate channels.
  - Rotemberg adjustment cost is more important for monetary shocks; capital requirement costs and time-varying risk weights are more important for sovereign-risk shocks.

*Source: wpiea2019043 - 2.1 Real Economy; 3.1 Monetary shock*

### 2.1    Real Economy

### wpiea2019043 - 2.1 Real Economy

### Households
- Households maximize lifetime utility:
  - max_{C_s,N_s,D_s} E_t (sum_{s=t}^{∞} β^{s−t} u(C_s, N_s))
  - u(C_t, N_t) = (C_t)^{1−σ}/(1−σ) − η N_t^{1+φ}/(1+φ)
- Definitions and roles:
  - C_t: consumption; N_t: hours worked.
  - σ: household risk aversion; φ: inverse of the Frisch elasticity of labor supply; η: disutility weight for work.
- Budget constraint:
  - C_t + D_t + T_t ≤ R^D_{t−1} D_{t−1} / π_t + W_t N_t + Π^R_t + Π^K_t + (1−α_W) Π^B_t
  - Π^R_t, Π^K_t and (1−α_W)Π^B_t are profits from good retailers, capital producers, and dividends from the banking sector respectively.
  - π_t = P_t / P_{t−1} is inflation.
- First-order conditions:
  - C_t^{−σ} = λ_t
  - η N_t^{φ} = λ_t W_t
  - λ_t = β E_t[(λ_{t+1} / π_{t+1}) R^D_t]

### Wholesale Producers
- Production function (Cobb-Douglas with entrepreneurial labor N^E_t):
  - Y^W_t = e^{a_t} K_t^{1−α−ρ} N_t^{α} (N^E_t)^{ρ}
  - a_t follows an AR(1) process.
- Entrepreneurial labor share is very small; entrepreneurial labor supply normalized to one in later assumptions.
- Wholesale producers rent capital r^W_t and pay wages W_t and W^E_t.
- Zero profit, first-order conditions:
  - R^W_t = (1−α−ρ) P^W_t Y^W_t / K_t
  - W_t = α P^W_t Y^W_t / N_t
  - W^E_t = ρ P^W_t Y^W_t / N^E_t
- Marginal cost:
  - P^W_t = W_t^{α} (W^E_t)^{ρ} (R^W_t)^{1−α−ρ} exp(a_t) α^{α} ρ^{ρ} (1−α−ρ)^{1−α−ρ}

### Capital Producers
- Capital accumulation:
  - K_{t+1} = (1−δ) K_t + Φ(X_t / X_{t−1}) X_t
  - Φ(x) = 1 − (χ/2) (x − 1)^2 x/(x−1)? (Φ definition provided via Christiano et al. (2010) nonlinear adjustment cost; Φ′ specified below)
  - Parameter χ measures concavity of adjustment costs.
- Investment adjustment and Tobin’s Q:
  - Capital producers maximize discounted Q_s K_{s+1} − (1−δ) Q_s K_s − X_s subject to capital dynamics.
  - Stochastic discount factor:
    - M_{s,t} = β^{s−t} λ_s / λ_t 1/π_{s,t} (with special case 1 if s=t, and product form if s>t).
  - First-order (Tobin’s Q) condition:
    - Q_t [ (X_t / X_{t−1}) Φ′_t(x_t) + Φ(x_t) ] = 1 + (1 / R^D_t) E_t[ Q_{t+1} Φ′(x_{t+1}) (X_{t+1}/X_t)^2 ]
  - Φ′(x_t) expression:
    - Φ′(x_t) = χ (x_t − 1)/x_t [ (x_t − 1)/(2 x_t) − x_t ] with x_t = X_t / X_{t−1}
- Realized profit:
  - Π^K_t = Q_t K_{t+1} − (1−δ) Q_t K_t − X_t
  - Profit is zero in steady state; nonzero during transitions due to pre-determined X_t.

### Retailers
- Continuum of retailers indexed by z; buy Y^W_t from wholesale producers and sell differentiated goods.
- Price index:
  - P_t = [ ∫_0^1 P_t(z)^{1−θ} dz ]^{1/(1−θ)}
- Representative firm maximizes Y_t − ∫_0^1 P_t(z) Y_t(z) dz subject to CES aggregation:
  - Y_t = [ ∫_0^1 Y_t(z)^{(θ−1)/θ} dz ]^{θ/(θ−1)}
- CES demand:
  - Y_t(z) = (P_t(z)/P_t)^{−θ} Y_t
- Price rigidity via Calvo (1983):
  - Firms can change price with probability 1−α_p each period; price remains fixed with prob. α_p.
  - Optimal price P^*_t(z) solves:
    - E_t ∑_{s=t}^{∞} M_{s,t} α_p^{s−t} Y^*_{s,t}(z) [ P^*_t(z) − θ/(θ−1) P^W_s ] = 0
- Market clearing:
  - ∫_0^1 Y_t(z) dz = Y^W_t
- Aggregate nominal profit:
  - Π^R_t = ∫_0^1 [ P_t(z) − P^W_t ] Y_t(z) dz = Y_t − P^W_t Y^W_t

### Entrepreneurs
- Financing and returns:
  - Q_t K_{t+1} = L_t + NW_t (loans plus net worth).
  - Ex-post return subject to aggregate and idiosyncratic risk: ω R^E_t where ω is idiosyncratic.
  - Aggregate ex-post return:
    - R^E_t = r^W_t + (1−δ) Q_t / Q_{t−1}
- Default and monitoring:
  - Entrepreneurs default if ω < ω_{t+1}; in default banks pay monitoring cost and seize (1−μ) ω R^E_t Q_{t−1} K_t.
  - Cut-off ω_{t+1} defined by:
    - ω_{t+1} R^E_{t+1} Q_t K_{t+1} = R^L_t L_t
- Expected profit:
  - V_t = E_t [ ∫_{ω_{t+1}}^{∞} ω R^E_{t+1} Q_t K_{t+1} dF(ω) − (1−F(ω_{t+1})) R^L_t L_t ]
  - Simplified to:
    - V_t = E_t [ ( ∫_{ω_{t+1}}^{∞} ω dF(ω) − ω_{t+1} ∫_{ω_{t+1}}^{∞} dF(ω) ) R^E_{t+1} Q_t K_{t+1} ]

### Banking Sector

#### Lending banks
- Lending banks obtain wholesale loan L_t at R^I_t; perfect competition implies zero profit condition:
  - ∫_0^{∞} ω_{t+1} R^L_t L_t dF(ω) + (1−μ) ∫_0^{ω_{t+1}} ω R^E_{t+1} Q_t K_{t+1} dF(ω) = R^I_t L_t
- Define:
  - Γ_t(ω_{t+1}) = ω_{t+1} ∫_{ω_{t+1}}^{∞} dF(ω) + ∫_0^{ω_{t+1}} ω dF(ω)
  - G_t(ω_{t+1}) = ∫_0^{ω_{t+1}} ω dF(ω)
- Relationship (using ̺_t = Q_t K_{t+1} / NW_t leverage):
  - [ Γ(ω_{t+1}) − μ G(ω_{t+1}) ] ̺_t R^E_{t+1} / R^I_t = ̺_{t−1}
- Mapping between external fund premium and cutoff:
  - R^E_{t+1} / R^I_t = Λ(ω_{t+1}) with Λ′(.) > 0
- Entrepreneurs’ demand for new capital:
  - Q_t K_{t+1} = Ψ( R^E_{t+1} / R^I_t ) NW_t where Ψ(1) = 1 and Ψ′(.) > 0
- Net worth dynamics:
  - NW_t = γ V_t + W^E_t where γ V_t is equity held from prior profits and C^E_t = (1−γ) V_t consumed.
  - V_t = R^E_t Q_{t−1} K_t − [1−d^L_{t−1}] R^I_{t−1} L_{t−1} − μ ∫_0^{ω_t} ω R^E_t Q_{t−1} K_t dF(ω)
- Reorganized external finance premium expression:
  - NW_t = γ [ R^E_t Q_{t−1} K_t − (1−d^L_{t−1}) R^I_{t−1} L_{t−1} − μ ∫_0^{ω_t} ω R^E_t Q_{t−1} K_t dF(ω) ] + W^E_t
  - External finance premium term shown as μ ∫_0^{ω_t} ω R^E_t Q_{t−1} K_t dF(ω) / (Q_{t−1} K_t − NW_t)
- Lending banks restrict leverage:
  - L_t = ρ_l L_{t−1} + (1−ρ_l) e K_t

#### Deposit banks
- Deposit demand for bank j:
  - D_{j,t} = (R^D_{j,t} / R^D_t)^{ε} D_t
- Deposit banks set R^D_{j,t} and face quadratic Rotemberg adjustment cost.
- Collected deposits transferred to wholesale bank at rate R^{IB}_t.
- Deposit bank maximization (symmetric equilibrium R^D_{j,t} = R^D_t) yields:
  - (1 + ε)/ε R^D_t = R^{IB}_t − κ_D ε [ (R^D_t / R^D_{t−1} − 1)^2 ] R^D_t / R^D_{t−1} + M κ_D ε [ (R^D_{t+1} / R^D_t − 1)^2 ] R^D_{t+1} / R^D_t
- Interpretation:
  - Deposit interest rate is a markup (markdown) relative to interbank rate; spread time-varying and increases with net marginal adjustment cost.

#### Wholesale banks
- Wholesale banks are quantity takers and use bank capital K^{WB}, receive funds from deposit banks, and access central bank liquidity; all funding sources are perfect substitutes.
- Capital requirement:
  - K^{WB}_t = S_t + λ_W ( δ^b_t B_t + δ^l_t L_t )
  - If S_t < 0 pay cost Ω(S_t) with Ω(S_t) > 0 for S_t < 0 and Ω(0) = 0; Ω convex.
- Risk-weighted coefficients (risk-sensitive):
  - δ^b_t = δ^b (1 + d^B_t)^{η_b}
  - δ^l_t = δ^l (1 + d^L_t)^{η_l}
  - d^L_t and d^B_t are external default probabilities in t+1 for lending banks and government bonds.
- Balance sheet identity:
  - D_t + K^{WB}_t = B_t + L_t
- Bank capital accumulation:
  - K^W_t = (1−ρ_W) K^W_{t−1} + α_W Π^B_t
  - Π^B_t = Π^S_t + Π^{WB}_t (profits of deposit banks and wholesale banks)
- Wholesale bank cash flow (simplified using balance sheet):
  - Π^{WB}_t = (1−d^L_t) R^I_{t−1} L_{t−1} + (1−d^B_t) R^B_{t−1} B_{t−1} / π_t − R^{IB}_{t−1} D_{t−1} / π_t − K^{WB}_t − Ω(S_t)
- First-order conditions (for B_t and L_t):
  - M_{t+1,t} E_t [ (1−d^L_t) R^I_t − R^{IB}_t / π_{t+1} + ρ_l ζ_{t+1} ] = ζ_t − λ_W δ^l Ω′(S_t)
  - M_{t+1,t} E_t [ (1−d^B_t) R^B_t − R^{IB}_t / π_{t+1} ] = − λ_W δ^b Ω′(S_t)
- Assumption:
  - Interbank rate equals central bank policy rate: R^{IB}_t = R_t.
- Under-capitalization (S_t < 0) increases spreads on bond assets and loans due to Ω′(S_t) < 0 and convex Ω.

### Central Bank, Government, and Market Clearing

#### Central bank
- Taylor rule:
  - log(R_t / R) = ρ_π log(π_t / π) + ρ_y log(Y_t / Y) + ε^R_t
  - R, π, Y are steady-state policy rate, inflation, output.

#### Government
- Government budget identity:
  - G_t + R^B_{t−1} (1−δ^b_t) B_{t−1} / π_t = B_t + T_t
- Tax process:
  - T_t = (1−ρ_T) τ Y_t + ρ_T T_{t−1}
  - τ is tax rate; ρ_T is autocorrelation for taxes.
- Debt issuance limit tied to repayment history:
  - B_t = [ (1−d^B_t) B_{t−1} ]^{ρ_b} B^{1−ρ_b}
  - B steady-state debt chosen as 60 percent of GDP.

#### Market clearing for goods
- Final goods equality:
  - Y_t = C_t + C^E_t + G_t + X_t + μ ∫_0^{ω_t} ω dF(ω) R^E_t Q_{t−1} K_t + (κ_D / 2) (R^D_t / R^D_{t−1} − 1)^2 D_t + Ω(S_t) + K^{WB}_t − (1−ρ_W) K^{WB}_{t−1}

### Results and Calibration (euro zone quarterly calibration)
- Discount factor: 0.99 (implies annual deposit rate of 0.041 percent).
- Frisch elasticity of labor: 0.1 (high elasticity interpretation).
- Elasticity of consumption (σ): 0.5.
- η calibrated so steady state C/Y = 0.6076.
- Total consumption (households + entrepreneurs) to GDP = 0.7298.
- Labor share in production function (α) = 0.7; entrepreneurial labor share 0.01.
- Quarterly capital depreciation δ = 0.025 (implies almost a 10 percent annual rate).
- Capital producer adjustment cost parameter χ = 20.
- Calvo parameter α_p = 0.75 (corresponds to one price change per year).
- Monopolistic power of retailers θ calibrated to 6 (markup = 1.2 at steady state).
- Steady-state entrepreneurial leverage ratio = 2.
- External finance premium at steady state = 114 b.p. quarterly.
- Deposit banks’ monopolistic power = 50.
- Interest rate adjustment parameter for deposits = 2.4 (to match average historical R^D_t).
- Minimum capital to risk-weighted assets λ_W = 0.08 (Basel II/III).
- Assumed ratings:
  - Wholesale bank holds "A" rated government bonds; lending banks rated "A".
- Risk-weight parameters:
  - δ^b = 0.20 for "A" rated sovereign bonds.
  - δ^l = 0.50 for "A" rated banks.
- Calibration of η̄_b and η̄_l using implied default probabilities and rating transitions:
  - Italy average default probability before downgrade ~ 6.5 percent; after downgrade to BBB+ average default probability ~ 8.5 percent.
  - η̄_b = 46.27 (from Basel II/III risk-weight change from 0.2 to 0.5 and default probs).
  - η̄_l = 17.5 (from bank risk-weight change 0.50 to 1.00 and default prob differences).
- Taylor rule parameters:
  - ρ_π = 1.5; ρ_y = 0.05 (ρ_π > 1 for determinacy).
- Government spending autocorrelation ρ_G = 0.8.
- Additional steady-state targets:
  - Investment to output = 0.0524.
  - Public spending to GDP = 0.1728.
- Tables 1 and 2 summarize calibrated parameters and steady-state values (as referenced).

*Source: wpiea2019043 - 2.1 Real Economy*

### 3.1    Monetary shock

### wpiea2019043 - 3.1 Monetary shock

### Transmission of a monetary shock (unanticipated 25 basis point annual decline in policy rate)
- Scenario: unanticipated 25 basis point decline in the policy interest rate (annual basis).
- Key dynamic responses:
  - Output, consumption, investment, and capital stock increase.
  - Higher capital demand → higher demand for loans.
  - Entrepreneurs’ net worth rises as loan rates decline and default rate declines → reduced external premium.
  - Financial accelerator operates by amplifying real-side responses via bank balance-sheet channels.
- Role of deposit-rate adjustment costs (Rotemberg cost):
  - With Rotemberg cost (full model):
    - Deposit rate does not fully follow the policy interest rate.
    - Deposits increase at the expense of households’ consumption; consumption increases only moderately.
    - Banks become momentarily over-capitalized as more valuable loan assets appear on balance sheets; leverage ratios fall initially then rise as loan demand builds.
    - Financial accelerator is dampened relative to the no-Rotemberg case.
  - Without Rotemberg cost:
    - Deposit rate follows the policy rate by a mark-down.
    - Initial decrease in deposit rate lowers deposit volume, later offset by higher household income → deposit accumulation.
    - Financial accelerator reinforces the shock more aggressively: investment and output rise faster; financial variables show higher sensitivities.
    - Banks choose to hold higher capital levels because deposits drop more, forcing banks to raise capital to meet both higher loan demand and reduced deposit funding.
- Role of bank capital cost and capital requirements in monetary shock:
  - Wholesale banks’ capital holding requirements show little impact under monetary expansion.
  - Wholesale banks with better capital positions lower interest-rate spreads (between bonds/loans and policy rates) → better transmission of monetary policy.
  - Overall impact of bank capital cost on most variables is negligible for a monetary expansion.
- Basel treatment comparison:
  - Models for Basel I and for Basel II & III produce the same impulse response functions for the monetary shock because risk weights do not change following a monetary shock.
- Interest rates reporting conventions in figures:
  - All charts except those for interest rates report percentage deviations from steady state.
  - Interest rate graphs are shown as absolute deviation from steady state in percentage points.
- Explicit references to figures:
  - Figure 3: full model (black line) vs model without Rotemberg adjustment cost (blue dashed line).
  - Figure 4: full model vs model without wholesale banks’ under-capitalization cost function (blue dashed line).

### Sovereign risk shock (2 percent increase in sovereign default risk)
- Scenario: unexpected sovereign default risk shock of 2 percent, implemented as a series of unexpected shocks lasting around 2 years in the model.
- Direct financial-market responses:
  - Interest rate on bonds rises sharply.
  - Demand for bonds falls to close to 3.5 percent of its steady state level.
  - Following the 2 percent increase in sovereign default risk:
    - GDP can shrink by more than 3 percent.
    - Interest rate on bonds rises by 5 percent.
    - Banks’ leverage ratio grows higher to around 30 percent of its steady state.
- Mechanisms operating through banks’ balance sheets:
  - Two simultaneous effects on banks:
    1. Funding cost rises as banks’ asset values diminish and leverage ratios increase, compelling them to pay extra cost.
    2. In the risk-weighted asset framework, the weight associated to bonds increases with asset rating deterioration → higher capital management costs.
  - Both effects reduce bank profitability and deteriorate capital positions, widening interest-rate spreads (equations (41) and (42)).
  - Outcomes: lower loan supply, lower bond demand, asset price declines, higher market interest rates → amplifies transmission of sovereign risk to the real economy.
  - Note: in the model the sovereign default risk is exogenous and is not endogenously affected by the rise in interest rates.
- Real-economy consequences:
  - Lower loan supply and lower bond demand reduce asset prices and raise interest rates.
  - Retailers and capital producers experience profit contractions → labor declines → households’ incomes fall.
  - Monetary policy reacts with a contraction in the policy rate; deposit rates follow, allowing households to smooth consumption by dissaving initially.
  - Investment declines due to interest-rate and external-premium channels; consumption initially increases (due to lower deposit rates and dissaving) but later declines as economic activity weakens and deposits recover → output shrinks driven by falling investment, public spending, and private consumption in later stages.
- Role of the financial accelerator:
  - Removing the financial accelerator by fixing the external premium to its steady state reduces persistence.
  - With the financial accelerator (full model, solid black line in figures), responses show higher persistence; the accelerator works through the monetary policy channel affecting interest rates, inflation, and investment.
- Role of bank capital cost (α_S) and capital adequacy:
  - Without bank capital cost (α_S = 0):
    - Wholesale banks’ profits remain higher, banks are better capitalized, and leverage ratio remains lower when default risk rises.
  - In the full model, equation (42) implies the spread between bond/loan rates and the policy rate is higher; R_Et does not increase sufficiently to compensate rise in R_It → external premium R_Et/R_It is lower in the full model.
  - The capital adequacy condition amplifies sovereign bond-market shocks via:
    - Balance-sheet channel: wholesale banks reshuffle portfolios and cut balance sheets to recover capital and leverage positions.
    - Interest-rate channel: interest rates on bonds and loans increase, further depressing investment and output.
  - Monetary authority reacts more strongly in the full model because the economy plunges deeper; monetary policy reaction partly offsets market-rate hikes.
- Rotemberg cost in sovereign shock:
  - Comparing full model and model without Rotemberg cost shows similar behavior for a default-risk shock except for deposit rate responses, which are more pronounced without adjustment cost.
- Basel I vs Basel II/III in sovereign shock:
  - Figure 7 compares full model (Basel II and III, black line) and model without time-varying risk weights (Basel I framework, blue dashed line). The capital adequacy/time-varying risk-weight mechanisms matter for amplifying sovereign-default shocks.

### Model-wide conclusions relevant for policy and macroprudential design
- Bank balance sheets are central in amplifying sovereign-risk shocks to the real economy.
- Capital adequacy:
  - Attenuates effects of technology and monetary shocks.
  - Reinforces impacts of sovereign default and interbank risk shocks that directly impair bank balance sheets.
- Risk-weight dynamics:
  - Time-varying risk weights (Basel II & III style) interact with sovereign risk to increase capital charges and amplify bank distress following sovereign shocks.
- Monetary vs macroprudential policy implications:
  - The model enables analysis of how monetary and macroprudential policies can attenuate negative spillovers from sovereign default shocks by addressing the balance-sheet and interest-rate channels.
- Specific model-element importance by shock type:
  - Rotemberg adjustment cost is more important for monetary shocks.
  - Capital requirement costs and time-varying risk weights are more important for sovereign-risk shocks.

### Key calibrated parameters and steady-state statistics (selected)
- Calibrated parameters (selected):
  - β Discount factor: 0.99
  - σ Households’ risk aversion parameter: 0.5
  - φ Household utility elasticity for work: 0.1
  - α Share of households’ labour: 0.7
  - ρ Share of entrepreneur’s labour: 0.01
  - δ Capital depreciation rate: 0.025
  - χ Coefficient of adjustment cost for investment: 600
  - αp Nominal rigidities: 0.85
  - θ Elasticity in the CES production function: 6
  - εD Deposit banks monopoly power: 50
  - κD Coefficient of adjustment cost for deposit interest rate: 2.4
  - τ Tax rate: 0.215
  - μ Entrepreneurs’ monitoring cost: 0.21
  - σω Standard deviation for log-normal distribution of ω: 0.082
  - λW Capital-to-Asset ratio requirement: 0.08
  - δ̄b Risk weight for government bonds: 0.2
  - δ̄l Risk weight for interbank loans: 0.5
  - ηb Power in risk weighing asset framework for bonds: 45.27
  - ηl Power in risk weighing asset framework for loans: 17.5
- Steady-state values (selected):
  - π Inflation: 1
  - Rf Policy rate: 1.0303
  - RD Deposit rate: 1.0101
  - ̺ Entrepreneurs’ leverage ratio: 2.0282
  - R_E/R_I External risk premium: 1.0202
- Steady-state ratios (selected):
  - K/Y Capital to GDP ratio: 2.09
  - X/Y Investment to GDP ratio: 0.0524
  - (C + C^E)/Y Consumption to GDP: 0.7298
  - G/Y Public Spending to GDP: 0.1728
  - K_WB/Y Bank Capital to GDP: 0.1330
  - L/Y Loan to GDP ratio: 1.1480
  - D/Y Deposit to GDP ratio: 1.6153
  - B/Y Debt to GDP ratio: 0.4432

*Source: wpiea2019043 - 3.1 Monetary shock (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019043.pdf_