## _wp11130

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### Introduction — purpose and main contributions
- Develops a macroeconomic model of risky corporate bank lending that mirrors real-world loan contracts where the lending rate on performing loans is not state-contingent and banks can incur ex-post losses.
- Jointly analyzes the macroeconomic effects of prudential/capital adequacy rules and conventional central bank interest rate rules, using household welfare as the metric and performing grid searches over policy-rule coefficients.
- Main qualitative findings:
  - Prudential rules can have significant positive welfare effects when a significant share of shocks are to borrower creditworthiness.
  - Prudential rules lower minimum capital adequacy requirements in response to contractionary shocks to borrower riskiness and reduce the workload for conventional interest rate policy.
  - Welfare gains from prudential rules are large by the standards of the literature; welfare gains from monetary policy rates are similar to existing findings.
- Distinguishing modeling features:
  - Banks have their own net worth and face endogenously determined, non-diversifiable aggregate risk.
  - Banks are lenders (not holders of risky equity).
  - Lending risk is endogenously risky because debt contracts include non-contingent terms (notably a pre-committed lending rate).
  - Capital regulation is modeled as penalties for falling below the regulatory minimum (leading to endogenous capital buffers), not as continuously binding constraints.
  - Fresh capital acquisition is subject to market imperfections; equity is transferred to households at an exogenous fixed rate (extended-family approach).

### Model overview — economy, agents, and scaling
- Economy: closed; agents include households, capital goods producers, capital investment funds, banks, manufacturers, and the government.
- Aggregate growth: economy grows at constant exogenous growth rate x = T_t / T_{t−1}; real variables are scaled by T_t and denoted ˇz_t; steady state denoted z.
- Full stochastic structure of the financial contracting problem is retained for welfare analysis (no reduced-form external finance premium shortcut).
- Implementation note: functions Γ and G (and their derivatives) involve cumulative distribution functions of the standard normal distribution (suggested implementations: normcdf or erfc in MATLAB/DYNARE/DYNARE++).

### Households — preferences, composition, and budget
- Representative household expected lifetime utility:
  E_0 Σ_{t=0}^∞ β^t [ S_{c,t} (1− v/x) log(c_t(i) − ν c_{t−1}) − ψ h_t(i)^{1+1/η}/(1+1/η) + ζ log( D_t(i)/P_t ) ].
- Preference and parameter notation preserved: β, S_{c,t}, v, x, ν, θ, η, ψ, ζ, (1−v/x).
- Occupational shares and transitions:
  - Workers (1−f), entrepreneurs f(1−b), bankers f b.
  - Entrepreneurs stay in occupation with probability p_e; bankers with probability p_b.
  - Startup fund fractions: ι_c (entrepreneurs), ι_b (bankers).
  - Dividend/ payout aggregation notation: δ and  ̃δ.
- Financial assets and returns:
  - Nominal government debt B_t and nominal bank deposits D_t; real variables b_t = B_t / P_t and d_t = D_t / P_t.
  - Gross nominal interest rates: i_t (government), i_{d,t} (deposits); gross inflation π_t = P_t / P_{t−1}.
  - Gross real interest rates r_t = E_t(i_t / π_{t+1}) and r_{d,t} = E_t(i_{d,t} / π_{t+1}).
- First-order conditions (normalized by T_t):
  - S_{c,t} (1− v/x) ˇc_t − ν/x ˇc_{t−1} = ˇλ_t. (3)
  - ψ h_t^{1/η} = ˇλ_t ˇw_t. (4)
  - ˇλ_t = β/x i_t E_t( ˇλ_{t+1} / π_{t+1} ). (5)
  - ˇλ_t = β/x i_{d,t} E_t( ˇλ_{t+1} / π_{t+1} ) + ζ ˇd_t. (6)
- Household budget constraint (real terms):
  b_t + d_t = r_{t−1} b_{t−1} + r_{d,t−1} d_{t−1} − c_t − τ_t + w_t h_t + ∫_0^1 Π_t(j) dj + Π^k_t + Υ^e_t. (2)

### Capital goods producers and capital accumulation
- Installed capital law of motion:
  ̃k_t(j) = ̃k_{t−1}(j) + I_t(j).
- Investment adjustment costs:
  G_{I,t}(j) = φ_I / 2 * [ I_t ( S_{i,t} (I_t(j)/x) / I_{t−1}(j) − 1 ) ]^2. (7)
- Price notation: Q_t is the nominal price of previously installed capital; price of new capital also Q_t.
- First-order condition / price equation:
  q_t = 1 + φ_I S_{i,t} (ˇI_t / ˇI_{t−1}) [ S_{i,t} ˇI_t / ˇI_{t−1} − 1 ] − E_t β ˇλ_{t+1}/ˇλ_t φ_I S_{i,t+1} (ˇI_{t+1} / ˇI_t)^2 [ S_{i,t+1} ˇI_{t+1}/ˇI_t − 1 ]. (9)
- Physical capital law of motion:
  k_t = (1−∆) k_{t−1} + I_t. (10)
  - ∆ is the depreciation rate.

### Capital investment funds — balance sheets and debt contracts
- Balance sheet (real normalized terms):
  q_t ˇk_t(j) = ˇn_t(j) + ˇℓ_t(j). (11)
- Idiosyncratic productivity shock:
  - After purchase, k_t(j) becomes ω_{t+1} k_t(j), ω_{t+1} unit mean lognormal.
  - Standard deviation of ln(ω_{t+1}) is σ_{t+1} — labeled the borrower riskiness shock.
- Loan contracts:
  - Loan amount L_t(j) and gross nominal retail interest rate i_{r,t} are pre-committed in period t.
  - Borrowers with ω_{t+1} below cutoff ω_{t+1} default; bank recovers fraction (1−ξ); ξ denotes bankruptcy/monitoring cost.
- Banks’ ex-ante zero profit (real) constraint:
  r_{ℓ,t} ˇℓ_t(j) = E_t [ (1−F(ω_{t+1})) r_{r,t} ˇℓ_t(j) + (1−ξ) ∫_0^{ ̄ω_{t+1}} q_t k_t(j) ret_{k,t} ω f(ω) dω ]. (13)
- Definitions:
  - Γ_{t+1} = Γ(ω_{t+1}) ≡ ∫_0^{ ̄ω_{t+1}} ω_{t+1} f(ω_{t+1}) dω_{t+1} + ω_{t+1} ∫_{ ̄ω_{t+1}}^∞ f(ω_{t+1}) dω_{t+1}.
  - ξ G_{t+1} = ξ G(ω_{t+1}) = ξ ∫_0^{ ̄ω_{t+1}} ω_{t+1} f(ω_{t+1}) dω_{t+1}.
  - Capital investment fund’s share: 1 − Γ_{t+1} = ∫_{ ̄ω_{t+1}}^∞ (ω_{t+1} − ω_{t+1}) f(ω_{t+1}) dω_{t+1}.
- Capital investment fund optimization (multiplier notation  ̃λ_t):
  Max over ˇk_t(j),  ̄ω_{t+1} of E_t [ (1−Γ_{t+1}) ret_{k,t}/r_{ℓ,t} q_t ˇk_t(j)/ˇn_t(j) +  ̃λ_t ( (Γ_{t+1} − ξ G_{t+1}) ret_{k,t}/r_{ℓ,t} q_t ˇk_t(j)/ˇn_t(j) − q_t ˇk_t(j)/ˇn_t(j) + 1 ) ]. (14)
- Optimal loan contract condition (Bernanke, Gertler and Gilchrist (1999) structure preserved):
  E_t [ (1−Γ_{t+1}) ret_{k,t}/r_{ℓ,t} + Γ_{ω} ( Γ_{ω}/( Γ_{ω} − ξ G_{ω} ) ) ret_{k,t}/r_{ℓ,t} (Γ_{t+1} − ξ G_{t+1})^{−1} ] = 0. (15)
- Net worth evolution (real terms):
  ˇn_t = r_{ℓ,t−1} ˇn_{t−1} / x + q_{t−1} ˇk_{t−1} / x ( ret_{k,t−1} (1 − ξ G_t) − r_{ℓ,t−1} ) − δ ˇn_t + ˇΛ_{ℓ,t}. (16)
  - Ex-post loan losses by banks:
    ˇΛ_{ℓ,t} / x = r_{ℓ,t−1} ˇℓ_{t−1} − q_{t−1} ˇk_{t−1} ret_{k,t−1} ( Γ_t − ξ G_t ). (17)
  - Interpretation: ˇΛ_{ℓ,t} > 0 when realized defaults exceed expectations so that pre-committed retail lending rates were insufficient.

### Banks — incentives, regulation, and role of capital buffers
- Banks intermediate household deposits to capital investment funds; operate under limited liability.
- Minimum capital adequacy regulation rationale:
  - Moral hazard from limited liability leading to incentive to reduce equity and take on risk.
  - Deposit insurance reduces depositor monitoring but shifts systemic risk to taxpayers; capital adequacy regulation penalizes insufficient equity buffers to protect taxpayers.
- Modeling of regulation:
  - Regulation implemented as penalties imposed on banks if they fall below regulatory minimum capital, generating behavioral incentives for endogenous capital buffers (Milne (2002) idea).
  - Calibration ensures probability of deposit-insurance calls is vanishingly small.
- Capital buffers emerge endogenously from interaction of optimal debt contracts and regulation.
- Acquisition of fresh capital is imperfect; model uses extended-family transfer approach with exogenous transfer rates to households.

### Key mechanisms and channels
- Non-contingent lending rate (pre-committed i_{r,t}) creates possibility of ex-post bank losses when realized defaults exceed expectations; these losses feed back into bank net worth, lending rates, and the real economy.
- Borrower riskiness shock: σ_{t+1} (std. dev. of ln(ω_{t+1})) is stochastic and central to results — prudential policy responds to shifts in borrower riskiness.
- Capital adequacy regulation as penalty-based (not always binding) causes banks to hold endogenous capital buffers that respond to uncertainty and shocks.
- Fresh-equity market imperfections and dividend/transfer rules (δ,  ̃δ, ι_c, ι_b) make bank and entrepreneur net worth salient state variables that affect equilibrium leverage and lending.

### Welfare analysis, numerical implementation, and computational notes
- Welfare measured at household level; analysis uses grid searches over monetary and prudential policy rule coefficients.
- Numerical implementation requires preserving cumulative normal function evaluations for Γ and G — recommended functions: normcdf or erfc in MATLAB/DYNARE/DYNARE++.
- The paper retains the stochastic contracting primitives rather than using reduced-form external finance premia, enabling welfare computations tied to full contract structure.

### Section III.D — unlimited liability alternative (benchmark)
- Banks’ balance sheet under unlimited liability: ˇℓt(j) = ˇdt(j)+ ˇet(j). (18)
- Regulatory discontinuous penalty at t+1 if pre-dividend net worth falls below γt times asset value; penalty is χ ˇℓt(j). (19)
- Heterogeneity via idiosyncratic lognormal shock ̃ωt+1 with E(̃ωt+1)=1 and Var(̃ωt+1)= ̃σ2t+1, density f(̃ωt+1), CDF F(̃ωt+1).
- Ex-ante and ex-post cutoff definitions:
  - Ex-ante cutoff E t(̃ωt+1) ≡ rd,t ˇdt (1−γt) rℓ,t ˇℓt. (20)
  - Ex-post cutoff ̃ωt ≡ rd,t−1 ˇdt−1 + ˇΛℓt (1−γt−1) rℓ,t−1 ˇℓt−1. (21)
- Banks maximize expected pre-dividend net worth including expected penalties; first-order condition links rℓ,t to rd,t and terms involving χ, F(̃ωt+1), f(̃ωt+1), and γt. (23)

### Regulatory rule, monetary policy, and equilibrium structure
- Prudential rule allows γt to vary with gaps. (35)
- Monetary policy rule (forward-looking):
  - Includes smoothing mi, inflation feedback mπ, output feedback my, and loan gaps mℓ and md in deviations; π4,t is four-quarter inflation. (34)
- Goods market clearing:
  ˇyt = ˇct + ˇIt + ˇgt. (36)
- Shock processes:
  - Five stochastic processes Sa t, Sc t, Si t, Sg t, σt follow AR(1) in logs. (37)

### Calibration and key steady-state parameters (quarterly, United States illustrative)
- Data period used for some parameters: 1990Q1 - 2010Q2.
- Selected targets and parameter values:
  - Real growth rate: 2% per annum.
  - Steady state real interest rate: 3% per annum.
  - Steady-state inflation rate: 2% per annum.
  - Habit parameter: 0.7.
  - Labor supply elasticity η = 1.
  - Investment adjustment cost φI = 2.5.
  - Price adjustment cost φp = 200.
  - Gross markup ≥ 1.1 (implies average price contract duration roughly 5 quarters).
  - Capital income share (including markups net of fixed costs): 40%.
  - Private investment to GDP ratio: 19%.
  - Steady state government spending to GDP ratio: 18%.
  - Capital investment funds’ steady state leverage ratio (debt/equity): 100%.
  - Banks’ steady state CAR: 10.5%; γ̄ is fixed so penalties start below an 8% minimum CAR (motivated by Basel III: 8% minimum + 2.5% conservation buffer = 10.5% steady state).
  - Steady state real deposit rate: 2.75%.
  - Steady state wholesale real lending rate: 3.3% (55 basis points margin over deposit rate).
  - Steady state retail real lending rate: 4.3% (100 basis points margin over wholesale rate; model’s external finance premium).
  - Bernanke–Gertler–Gilchrist (1999) external finance premium in steady state: 65 basis points.
  - Share of capital investment funds going bankrupt each quarter: 1%.
  - Share of banks hitting minimum CAR each quarter: 2%.
  - Penalty calibration: regulatory penalties χ calibrated at one third of one percent of the outstanding loan volume (0.33%); sensitivity analysis includes penalty rate 0.5%.
  - In illustrative calibration shocks to borrower riskiness account for a 16% share in the volatility of GDP.
- Monetary-rule calibration reference: mi = 0.7, mπ = 2.0, mygr = 0.25 for SIGMA-style rule used in shock calibration note.

### Numerical results — optimal coefficients and policy design
- Grid-search findings and rule simplifications:
  - Prudential smoothing parameter pγ ≈ 0; set pγ = 0 thereafter.
  - Loan gap / loans-to-output gap in monetary rule (mℓ, md) provides no welfare benefit once other coefficients optimized; set mℓ = md = 0.
  - Search range limit for mπ ∈ [1.5, 3.0].
  - When prudential policy responds to the output gap, welfare gains are smaller than when it responds to the loan gap or loans-to-output gap.
  - Loan gap slightly superior to loans-to-output gap; the focus is on the loan gap case.
  - Overall optimal coefficient combination for loan-gap case: mi = 0, mπ = 3, my = 0.1, pℓ = 6.0 (prudential coefficient restricted to ≤ 6 to avoid unrealistically large CAR volatility).

### Impulse responses to borrower-riskiness shock and key mechanism magnitudes
- Comparison for pℓ ∈ {0, 3, 6} with monetary rule mi = 0, mπ = 3, my = 0.1:
  - Countercyclical capital adequacy (lowering CAR in face of contraction) reduces volatility of output, hours, consumption, investment, and required policy interest rate fluctuations.
  - For a one standard deviation shock to borrower riskiness:
    - Corporate net worth falls more than asset values; corporate leverage (initially 100%) increases by around 2.75 percentage points.
    - Banks, locked into old lending rates on impact, suffer lending losses that reduce net worth by over 3 percent, lowering CAR by around 0.3 percentage points, triggering higher lending rates.
    - Retail rate increases roughly 100 basis points over wholesale rate on impact absent prudential response.
  - Aggressive prudential rule (pℓ = 6) temporarily reduces CAR, lessening banks’ need to rebuild equity quickly, so wholesale rate rises less, loan volumes drop less, and borrower riskiness falls endogenously — generating welfare improvements.
- Nonlinearities: lognormal distributions for idiosyncratic shocks make large shocks disproportionately important for volatility and welfare; damping such shocks via prudential rules yields significant welfare benefits.

### Welfare, policy instrument volatility, and trade-offs
- Welfare metric: compensating consumption variation η in percent (equations (38)–(39)); computed with DYNARE++ over a multidimensional grid.
- Main quantitative welfare findings (relative to best policy in considered grid: pℓ = 6, mπ = 3):
  - Increasing mπ from 1.5 to 3.0 yields welfare gain ≈ 0.04% of steady state consumption (when all shocks present).
  - Raising pℓ from 0 to 6 gives a larger welfare gain ≈ 0.20% of steady state consumption.
  - Overall welfare difference between worst and best rules considered in baseline: around 0.24% (Figure 3 baseline).
- Policy instrument volatility:
  - More aggressive prudential rules reduce nominal interest rate volatility by around 30 basis points for a given inflation coefficient.
  - Aggressive prudential rules increase volatility of minimum CAR; at pℓ = 6 standard error of minimum CAR can exceed 3 percentage points — motivating restriction pℓ ≤ 6.
- Attribution of gains:
  - Welfare gains from countercyclical prudential policy are attributable almost exclusively to borrower-riskiness shocks.
  - Gains from more active monetary policy arise mostly under traditional demand and supply shocks.

### Robustness and sensitivity analysis
- Making banks riskier (steady state ̃σ rising so fraction of banks hitting minimum CAR increases from 2% to 3%) raises welfare difference between worst and best rules from 0.24% to 0.32%.
- Increasing penalty rate from 0.33% to 0.5% produces similar amplification of welfare differences.
- Alternative prudential rule responding directly to borrower riskiness σt:
  - γt = γ̄ − pb ln(σt/̃σ) / 100. (40)
  - This σ-targeting rule yields much smaller welfare gains (maximum ≈ 0.08% around mπ = 3 and pb = 6), and welfare declines beyond pb = 6.
  - Reason: borrower-riskiness shocks die out comparatively quickly, while their balance-sheet effects are persistent; responding to loans (loan gap) targets long-lived balance-sheet impairment better than an immediate σ-based rule.

### Moral hazard, unlimited liability counterfactual, and potential gains
- With limited liability, moral hazard arises unless capital adequacy regulations ensure buffers.
- Under the benchmark calibrated prudential rule, moral hazard effects are effectively eliminated: banks hold buffers well above minimum CAR; depositors perceive returns as safe (residual bankruptcy risk is extremely close to zero).
- Counterfactual experiment: eliminate capital adequacy regulation by setting χ = 0 and impose unlimited liability so households/bankers supply any equity shortfall:
  - Policymaker can choose it and γt freely; minimum CAR becomes irrelevant.
  - Optimal steady state bank equity becomes exactly zero (infinite leverage); bank equity exhibits a unit root.
  - Welfare gain of this unconstrained optimum over the best simple rule in the grid equals 2.52%.
  - Interpretation: eliminating moral hazard effects yields welfare gains an order of magnitude larger than gains from optimizing realistic countercyclical prudential rules.

### Conclusion — policy implications and trade-offs
- Prudential rules that lower CAR in response to contractionary borrower-riskiness shocks can materially reduce macroeconomic volatility (output, hours, consumption, investment) and reduce the workload of conventional monetary policy, improving welfare when borrower-riskiness shocks account for a significant share of GDP volatility.
- Trade-offs:
  - Aggressive countercyclical prudential rules can generate large volatility in minimum CAR (standard errors > 3 percentage points at high pℓ), which may be impractical.
  - Directly targeting borrower riskiness with prudential rules yields smaller welfare gains than targeting loan gaps because balance-sheet impairments are persistent.
- Institutional change removing moral hazard (unlimited liability with contingent household equity provision) would produce far larger welfare gains (2.52%) than achievable through optimizing plausible prudential and monetary rule coefficients alone.

*Source: _wp11130 — References*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   23

### _wp11130 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   23

### Introduction — purpose and main contributions
- Develops a macroeconomic model of risky corporate bank lending that mirrors real-world loan contracts where the lending rate on performing loans is not state-contingent and banks can incur ex-post losses.
- Jointly analyzes the macroeconomic effects of prudential/capital adequacy rules and conventional central bank interest rate rules, using household welfare as the metric and performing grid searches over policy-rule coefficients.
- Main qualitative findings (as stated in the text):
  - Prudential rules can have significant positive welfare effects when a significant share of shocks are to borrower creditworthiness.
  - Prudential rules lower minimum capital adequacy requirements in response to contractionary shocks to borrower riskiness and reduce the workload for conventional interest rate policy.
  - Welfare gains from prudential rules are large by the standards of the literature; welfare gains from monetary policy rates are similar to existing findings.
- Differentiating dimensions of the paper relative to prior literature:
  - Banks have their own net worth and face endogenously determined, non-diversifiable aggregate risk.
  - Banks are lenders (not holders of risky equity) — different macro transmission than equity-holder banking models.
  - Lending risk is endogenously risky because debt contracts include non-contingent terms (notably a pre-committed lending rate).
  - Capital regulation is modeled as penalties for falling below the regulatory minimum (leading to endogenous capital buffers), not as continuously binding constraints.
  - Fresh capital acquisition is subject to market imperfections; equity is transferred to households at an exogenous fixed rate (extended-family approach).

### Model overview — economy, agents, and scaling
- Economy: closed; agents include households, capital goods producers, capital investment funds, banks, manufacturers, and the government.
- Aggregate growth: economy grows at constant exogenous growth rate x = T_t / T_{t−1}; real variables are scaled by T_t and denoted ˇz_t; steady state denoted z.
- Modeling choices emphasized:
  - Full stochastic structure of the financial contracting problem is retained for welfare analysis (no reduced-form external finance premium shortcut).
  - Functions Γ and G (and their derivatives) involve cumulative distribution functions of the standard normal distribution (implementation note: normcdf or erfc in MATLAB/DYNARE).

### Households — preferences, composition, and budget
- Representative household utility (expected lifetime) is:
  E_0 Σ_{t=0}^∞ β^t [ S_{c,t} (1− v/x) log(c_t(i) − ν c_{t−1}) − ψ h_t(i)^{1+1/η}/(1+1/η) + ζ log( D_t(i)/P_t ) ].
- Key preference/parameter notation retained exactly: β, S_{c,t}, v, x, ν, θ (CES elasticity of substitution), η, ψ, ζ, (1−v/x) scale factor.
- Household composition parameters and occupational transitions:
  - Fractional shares: workers (1−f), entrepreneurs f(1−b), bankers f b.
  - Entrepreneurs stay in occupation with probability p_e; bankers with probability p_b.
  - Startup fund fractions: ι_c (entrepreneurs), ι_b (bankers).
  - Dividend/ payout aggregations summarized into δ and  ̃δ (notation preserved).
- Financial assets and returns:
  - Nominal government debt B_t and nominal bank deposits D_t; real variables b_t = B_t / P_t and d_t = D_t / P_t.
  - Gross nominal interest rates: i_t (government), i_{d,t} (deposits); gross inflation π_t = P_t / P_{t−1}.
  - Gross real interest rates r_t = E_t(i_t / π_{t+1}) and r_{d,t} = E_t(i_{d,t} / π_{t+1}).
- First-order conditions (normalized by T_t) preserved exactly:
  - S_{c,t} (1− v/x) ˇc_t − ν/x ˇc_{t−1} = ˇλ_t. (3)
  - ψ h_t^{1/η} = ˇλ_t ˇw_t. (4)
  - ˇλ_t = β/x i_t E_t( ˇλ_{t+1} / π_{t+1} ). (5)
  - ˇλ_t = β/x i_{d,t} E_t( ˇλ_{t+1} / π_{t+1} ) + ζ ˇd_t. (6)
- Household budget constraint (real terms):
  b_t + d_t = r_{t−1} b_{t−1} + r_{d,t−1} d_{t−1} − c_t − τ_t + w_t h_t + ∫_0^1 Π_t(j) dj + Π^k_t + Υ^e_t. (2)

### Capital goods producers and capital accumulation
- Capital producers purchase previously installed capital  ̃k_{t−1}(j) and investment goods I_t(j) to produce new installed capital  ̃k_t(j) =  ̃k_{t−1}(j) + I_t(j), subject to investment adjustment costs:
  G_{I,t}(j) = φ_I / 2 * [ I_t ( S_{i,t} (I_t(j)/x) / I_{t−1}(j) − 1 ) ]^2. (7)
- Price notation: Q_t is the nominal price of previously installed capital; price of new capital also Q_t.
- First-order condition / price equation (preserved exactly):
  q_t = 1 + φ_I S_{i,t} (ˇI_t / ˇI_{t−1}) [ S_{i,t} ˇI_t / ˇI_{t−1} − 1 ] − E_t β ˇλ_{t+1}/ˇλ_t φ_I S_{i,t+1} (ˇI_{t+1} / ˇI_t)^2 [ S_{i,t+1} ˇI_{t+1}/ˇI_t − 1 ]. (9)
- Physical capital law of motion:
  k_t = (1−∆) k_{t−1} + I_t. (10)
  - ∆ is the depreciation rate.

### Capital investment funds — balance sheets and debt contracts
- Balance sheet (real normalized terms):
  q_t ˇk_t(j) = ˇn_t(j) + ˇℓ_t(j). (11)
- Idiosyncratic productivity shock: after purchase, k_t(j) becomes ω_{t+1} k_t(j), where ω_{t+1} is unit mean lognormal; standard deviation of ln(ω_{t+1}) is σ_{t+1} — this stochastic σ_{t+1} is labeled the borrower riskiness shock.
- Capital investment fund nominal loan contract:
  - Loan amount L_t(j) and gross nominal retail interest rate i_{r,t} are pre-committed in period t (key departure from some existing models).
  - Borrowers with ω_{t+1} below cutoff ω_{t+1} default; bank recovers fraction (1−ξ) of value; ξ denotes bankruptcy/monitoring cost.
- Banks’ ex-ante zero profit (real) constraint (preserved exactly):
  r_{ℓ,t} ˇℓ_t(j) = E_t [ (1−F(ω_{t+1})) r_{r,t} ˇℓ_t(j) + (1−ξ) ∫_0^{ ̄ω_{t+1}} q_t k_t(j) ret_{k,t} ω f(ω) dω ]. (13)
- Definitions (preserved exactly):
  - Lender’s gross share in nominal capital earnings:
    Γ_{t+1} = Γ(ω_{t+1}) ≡ ∫_0^{ ̄ω_{t+1}} ω_{t+1} f(ω_{t+1}) dω_{t+1} + ω_{t+1} ∫_{ ̄ω_{t+1}}^∞ f(ω_{t+1}) dω_{t+1}.
  - Lender’s monitoring costs share: ξ G_{t+1} = ξ G(ω_{t+1}) = ξ ∫_0^{ ̄ω_{t+1}} ω_{t+1} f(ω_{t+1}) dω_{t+1}.
  - Capital investment fund’s share: 1 − Γ_{t+1} = ∫_{ ̄ω_{t+1}}^∞ (ω_{t+1} − ω_{t+1}) f(ω_{t+1}) dω_{t+1}.
- Capital investment fund optimization (preserved structure; multiplier notation  ̃λ_t):
  Max over ˇk_t(j),  ̄ω_{t+1} of E_t [ (1−Γ_{t+1}) ret_{k,t}/r_{ℓ,t} q_t ˇk_t(j)/ˇn_t(j) +  ̃λ_t ( (Γ_{t+1} − ξ G_{t+1}) ret_{k,t}/r_{ℓ,t} q_t ˇk_t(j)/ˇn_t(j) − q_t ˇk_t(j)/ˇn_t(j) + 1 ) ]. (14)
- Optimal loan contract condition (identical to Bernanke, Gertler and Gilchrist (1999); preserved exactly):
  E_t [ (1−Γ_{t+1}) ret_{k,t}/r_{ℓ,t} + Γ_{ω} ( Γ_{ω}/( Γ_{ω} − ξ G_{ω} ) ) ret_{k,t}/r_{ℓ,t} (Γ_{t+1} − ξ G_{t+1})^{−1} ] = 0. (15)
- Net worth evolution (real terms) preserved exactly:
  ˇn_t = r_{ℓ,t−1} ˇn_{t−1} / x + q_{t−1} ˇk_{t−1} / x ( ret_{k,t−1} (1 − ξ G_t) − r_{ℓ,t−1} ) − δ ˇn_t + ˇΛ_{ℓ,t}. (16)
  - Ex-post loan losses by banks:
    ˇΛ_{ℓ,t} / x = r_{ℓ,t−1} ˇℓ_{t−1} − q_{t−1} ˇk_{t−1} ret_{k,t−1} ( Γ_t − ξ G_t ). (17)
  - Interpretation: ˇΛ_{ℓ,t} > 0 when realized defaults exceed expectations so that pre-committed retail lending rates were insufficient.

### Banks — incentives, regulation, and role of capital buffers
- Banks intermediate household deposits to capital investment funds; operate under limited liability.
- Minimum capital adequacy regulation rationale:
  - Moral hazard from limited liability leading to incentive to reduce equity and take on risk.
  - Deposit insurance reduces depositor monitoring but shifts systemic risk to taxpayers; capital adequacy regulation penalizes insufficient equity buffers to protect taxpayers.
- Modeling of regulation:
  - Regulation implemented as penalties imposed on banks if they fall below regulatory minimum capital, generating behavioral incentives for endogenous capital buffers (Milne (2002) idea).
  - Calibration ensures probability of deposit-insurance calls is vanishingly small (paper’s calibration choice).
- Capital buffers emerge as optimal equilibrium outcomes from interaction of optimal debt contracts and regulation, rather than as ad-hoc quadratic costs.
- Acquisition of fresh capital is imperfect (recapitalization imperfections necessary for regulation to have effects); model uses extended-family transfer approach with exogenous transfer rates to households.

### Key mechanisms and channels emphasized in the content
- Non-contingent lending rate (pre-committed i_{r,t}) creates possibility of ex-post bank losses when realized defaults exceed expectations; these losses feed back into bank net worth, lending rates, and the real economy.
- Borrower riskiness shock: σ_{t+1} (std. dev. of ln(ω_{t+1})) is stochastic and central to results — prudential policy responds to shifts in borrower riskiness.
- Capital adequacy regulation as penalty-based (not always binding) causes banks to hold endogenous capital buffers that respond to uncertainty and shocks.
- Fresh-equity market imperfections and dividend/transfer rules (δ,  ̃δ, ι_c, ι_b) make bank and entrepreneur net worth salient state variables that affect equilibrium leverage and lending.

### Welfare analysis and computational notes
- Welfare measured at household level; analysis uses grid searches over monetary and prudential policy rule coefficients.
- Numerical implementation requires preserving cumulative normal function evaluations for Γ and G — recommended functions: normcdf or erfc in MATLAB/DYNARE/DYNARE++.
- The paper retains the stochastic contracting primitives rathern than using reduced-form external finance premia, enabling welfare computations tied to full contract structure.

*Source: _wp11130 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   23*

### Section III.D, another alternative to capital adequacy regulations that is generally treated

### Section III.D, another alternative to capital adequacy regulations that is generally treated as a benchmark in the literature, the case of banks operating under unlimited liability

### Model setup: banks, heterogeneity, and the regulatory discontinuity
- Banks’ only asset: loans ˇℓt(j); liabilities: deposits ˇdt(j) and equity ˇet(j), with balance sheet ˇℓt(j) = ˇdt(j)+ ˇet(j) (equation (18)).
- Liquidity management is ignored; no demand for government-provided real cash balances.
- Regulatory framework: a discontinuous penalty at t+1 if pre-dividend net worth falls below γt times asset value (equation (19)); penalty is a real lump-sum χ ˇℓt(j).
- To generate a continuum of outcomes and a time-varying small fraction of penalized banks, banks are made heterogeneous via an idiosyncratic lognormal shock ̃ωt+1 to loan returns with E(̃ωt+1)=1 and Var(̃ωt+1)= ̃σ2t+1, density f(̃ωt+1) and CDF F(̃ωt+1).
- Cut-off shock definitions:
  - Ex-ante cutoff E t(̃ωt+1) ≡ rd,t ˇdt (1−γt) rℓ,t ˇℓt (equation (20)).
  - Ex-post cutoff ̃ωt ≡ rd,t−1 ˇdt−1 + ˇΛℓt (1−γt−1) rℓ,t−1 ˇℓt−1 (equation (21)).
- Banks maximize expected pre-dividend net worth including expected penalties:
  - Optimality condition (first-order) (equation (23)) links wholesale lending rate rℓ,t to deposit rate rd,t and terms involving χ, F(̃ωt+1), f(̃ωt+1), and capital adequacy γt.
- Banks set wholesale lending rate rℓ,t as a premium over rd,t; retail rate rr,t is another premium over rℓ,t to compensate for capital investment funds’ bankruptcy risk.

### Regulatory rule, monetary policy, and equilibrium
- Prudential rule (allows γt to vary with gaps) (equation (35)).
- Monetary policy rule (forward-looking interest rate rule) (equation (34)), with coefficients mi (smoothing), mπ (inflation feedback), my (output feedback), mℓ and md (loan stock / loans-to-output deviations). π4,t is four-quarter inflation (product).
- Goods market clearing: ˇyt = ˇct + ˇIt + ˇgt (equation (36)).
- Five stochastic processes for shocks Sa t, Sc t, Si t, Sg t, σt follow AR(1) in logs (equation (37)).

### Calibration and key steady-state parameters (quarterly, United States illustrative)
- Data period used for some parameters: 1990Q1 - 2010Q2.
- Selected targets and parameter values:
  - Real growth rate: 2% per annum.
  - Steady state real interest rate: 3% per annum.
  - Steady-state inflation rate: 2% per annum.
  - Habit parameter: 0.7.
  - Labor supply elasticity η = 1.
  - Investment adjustment cost φI = 2.5.
  - Price adjustment cost φp = 200.
  - Gross markup ≥ 1.1 (implies average price contract duration roughly 5 quarters).
  - Capital income share (including markups net of fixed costs): 40%.
  - Private investment to GDP ratio: 19%.
  - Steady state government spending to GDP ratio: 18%.
  - Capital investment funds’ steady state leverage ratio (debt/equity): 100%.
  - Banks’ steady state CAR: 10.5%; γ̄ is fixed so penalties start below an 8% minimum CAR (motivated by Basel III: 8% minimum + 2.5% conservation buffer = 10.5% steady state).
  - Steady state real deposit rate: 2.75%.
  - Steady state wholesale real lending rate: 3.3% (55 basis points margin over deposit rate).
  - Steady state retail real lending rate: 4.3% (100 basis points margin over wholesale rate; model’s external finance premium).
  - Bernanke–Gertler–Gilchrist (1999) external finance premium in steady state: 65 basis points.
  - Share of capital investment funds going bankrupt each quarter: 1%.
  - Share of banks hitting minimum CAR each quarter: 2%.
  - Penalty calibration: regulatory penalties χ calibrated at one third of one percent of the outstanding loan volume (0.33%); sensitivity analysis includes penalty rate 0.5%.
  - In illustrative calibration shocks to borrower riskiness account for a 16% share in the volatility of GDP.
- Monetary-rule calibration reference: mi = 0.7, mπ = 2.0, mygr = 0.25 for SIGMA-style rule used in shock calibration note.

### Numerical results — optimal coefficients and policy design
- Overall optimization (grid search) findings:
  - Prudential smoothing parameter pγ ≈ 0; set pγ = 0 thereafter.
  - Loan gap / loans-to-output gap in monetary rule (mℓ, md) provides no welfare benefit once other coefficients optimized; set mℓ = md = 0.
  - Search range limit for mπ ∈ [1.5, 3.0].
  - When prudential policy responds to the output gap, welfare gains are smaller than when it responds to the loan gap or loans-to-output gap.
  - Loan gap slightly superior to loans-to-output gap; focus on loan gap case.
  - Overall optimal coefficient combination for loan-gap case: mi = 0, mπ = 3, my = 0.1, pℓ = 6.0 (prudential coefficient restricted to ≤ 6 to avoid unrealistically large CAR volatility).

### Impulse responses to borrower-riskiness shock and mechanism
- Impulse response comparison for pℓ ∈ {0, 3, 6} with monetary rule mi = 0, mπ = 3, my = 0.1:
  - Countercyclical capital adequacy (lowering CAR in face of contraction) reduces volatility of output, hours, consumption, investment, and required policy interest rate fluctuations.
  - For a one standard deviation shock to borrower riskiness:
    - Corporate net worth falls more than asset values; corporate leverage (initially 100%) increases by around 2.75 percentage points.
    - Banks, locked into old lending rates on impact, suffer lending losses that reduce net worth by over 3 percent, lowering CAR by around 0.3 percentage points, triggering higher lending rates.
    - Retail rate increases roughly 100 basis points over wholesale rate on impact absent prudential response.
  - Aggressive prudential rule (pℓ = 6) temporarily reduces CAR, lessening banks’ need to rebuild equity quickly, so wholesale rate rises less, loan volumes drop less, and borrower riskiness falls endogenously — generating welfare improvements.
- Nonlinearities: lognormal distributions for idiosyncratic shocks make large shocks disproportionately important for volatility and welfare; damping such shocks via prudential rules yields significant welfare benefits.

### Welfare, policy instrument volatility, and trade-offs
- Welfare metric: compensating consumption variation η in percent (equations (38)–(39)); computed with DYNARE++ over a multidimensional grid.
- Main quantitative welfare findings (relative to best policy in considered grid: pℓ = 6, mπ = 3):
  - Increasing mπ from 1.5 to 3.0 yields welfare gain ≈ 0.04% of steady state consumption (when all shocks present).
  - Raising pℓ from 0 to 6 gives a larger welfare gain ≈ 0.20% of steady state consumption.
  - Overall welfare difference between worst and best rules considered in baseline: around 0.24% (Figure 3 baseline).
- Policy instrument volatility implications:
  - More aggressive prudential rules reduce nominal interest rate volatility by around 30 basis points for a given inflation coefficient (policy rate acts less to stabilize when prudential policy adjusts bank balance sheets).
  - Aggressive prudential rules increase volatility of minimum CAR; at pℓ = 6 standard error of minimum CAR can exceed 3 percentage points — potentially impractically large, motivating restriction pℓ ≤ 6.
- Attribution of gains:
  - Welfare gains from countercyclical prudential policy are attributable almost exclusively to borrower-riskiness shocks.
  - Gains from more active monetary policy arise mostly under traditional demand and supply shocks.

### Robustness and sensitivity analysis
- Making banks riskier (steady state ̃σ rising so fraction of banks hitting minimum CAR increases from 2% to 3%) raises welfare difference between worst and best rules from 0.24% to 0.32%.
- Increasing penalty rate from 0.33% to 0.5% produces similar amplification of welfare differences.
- Alternative prudential rule responding directly to borrower riskiness σt:
  - γt = γ̄ − pb ln(σt/̃σ) / 100 (equation (40)).
  - This σ-targeting rule yields much smaller welfare gains (maximum ≈ 0.08% around mπ = 3 and pb = 6), and welfare declines beyond pb = 6.
  - Reason: borrower-riskiness shocks die out comparatively quickly, while their balance-sheet effects are persistent; responding to loans (loan gap) targets long-lived balance-sheet impairment better than an immediate σ-based rule.

### Moral hazard, unlimited liability scenario, and large potential gains
- With limited liability, moral hazard arises unless capital adequacy regulations ensure buffers.
- Under the benchmark calibrated prudential rule, moral hazard effects are effectively eliminated: banks hold buffers well above minimum CAR; depositors perceive returns as safe (residual bankruptcy risk is extremely close to zero).
- Counterfactual policy experiment: eliminate capital adequacy regulation by setting χ = 0 and impose unlimited liability so households/bankers supply any equity shortfall (contingent claim by extended family):
  - Policymaker can choose it and γt freely; minimum CAR becomes irrelevant.
  - Optimal steady state bank equity becomes exactly zero (infinite leverage); bank equity exhibits a unit root.
  - Welfare gain of this unconstrained optimum over the best simple rule in the grid equals 2.52%.
  - Interpretation: eliminating moral hazard effects yields welfare gains an order of magnitude larger than gains from optimizing realistic countercyclical prudential rules.

### Conclusion: policy implications and trade-offs
- Prudential rules that lower CAR in response to contractionary borrower-riskiness shocks can materially reduce macroeconomic volatility (output, hours, consumption, investment) and reduce the workload of conventional monetary policy, improving welfare when borrower-riskiness shocks account for a significant share of GDP volatility.
- Trade-offs:
  - Aggressive countercyclical prudential rules can generate large volatility in minimum CAR (standard errors > 3 percentage points at high pℓ), which may be impractical.
  - Directly targeting borrower riskiness with prudential rules yields smaller welfare gains than targeting loan gaps because balance-sheet impairments are persistent.
- Removing moral hazard via institutional changes (unlimited liability with contingent household equity provision) would produce far larger welfare gains (2.52%) than achievable through optimizing plausible prudential and monetary rule coefficients alone.

*Source: Section III.D of the supplied IMF working paper content unit.*

### References

### _wp11130 - References

### References
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- Bernanke, B., Gertler, M. and Gilchrist, S. (1999), “The Financial Accelerator in a Quantitative Business Cycle Framework”, in: Taylor, J.B., Woodford, M. (eds.), Handbook of Macroeconomics, Volume 1C. Elsevier, Amsterdam, pp. 1341-1393.
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### Figure 1. Firm Riskiness Shock - Impulse Responses
- Title: Firm Riskiness Shock - Impulse Responses
- Axes and series labels appearing in the figure:
  - GDP (% Difference)
  - Consumption (% Difference)
  - Investment (% Difference)
  - Policy Nominal Rate (pp Difference)
  - Inflation (pp Difference)
  - Policy Real Rate (pp Difference)
  - Corporate Asset Value (% Difference)
  - Corporate Net Worth (% Difference)
  - Corporate Leverage (pp Difference)
  - Bank Loans (% Difference)
  - Bank Net Worth (% Difference)
  - Bank Basel Ratio (pp Difference)
  - Retail Real Rate (pp Difference)
  - Wholesale Real Rate (pp Difference)
  - Loan Losses/Bank Net Worth (pp Difference)
  - XFP vs. Wholesale Rate (pp Difference)
  - XFP vs. Policy Rate (pp Difference)
  - Hours (% Difference)
- Legend note: Prudential Rule Feedback Coefficients on Loans: ... = 0, - - = 3, – = 6

### Figure 2. Welfare and Instrument Volatility
- Title: Welfare and Instrument Volatility
- Panels and labels appearing in the figure:
  - Welfare − Baseline Calibration; Welfare Gain; axes labeled m, pie, p, l
  - Welfare − Borrower Riskiness Shocks; Welfare Gain; axes labeled m, pie, p, l
  - Change in Std. Dev. of Nominal Policy Rate; ∆ sd(Nominal Policy Rate); axes labeled m, pie, p, l
  - Change in Std. Dev. of Nominal Policy Rate; ∆ sd(Nominal Policy Rate); axes labeled m, pie, p, l
  - Change in Std. Dev. of Target Basel Ratio; ∆ sd(Target Basel Ratio); axes labeled m, pie, p, l
  - Change in Std. Dev. of Target Basel Ratio; ∆ sd(Target Basel Ratio); axes labeled m, pie, p, l

### Figure 3. Welfare - Different Model Parameterizations
- Title: Welfare - Different Model Parameterizations
- Panels and labels appearing in the figure:
  - Welfare − Baseline Calibration; Welfare Gain; axes labeled m, pie, p, l
  - Welfare − 50% Higher Bank Risk of Reaching Minimum CAR; Welfare Gain; axes labeled m, pie, p, l
  - Welfare − Direct Policy Response to Borrower Riskiness; Welfare Gain; axes labeled m, pie, p, b
  - Welfare − 50% Higher Penalty when Reaching Minimum CAR; Welfare Gain; axes labeled m, pie, p, l

*Content unit: _wp11130 - References*

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