## wp17179 — Selected chapters: Households; Entrepreneurs; Estimation results; Robustness; Welfare

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### 2.1 Households — preferences, habits, labor and budget
- Households indexed j∈(0,1), infinitely-lived.
- Inter-temporal utility:
  - U_{j,t} = E_t ∑_{s=0}^∞ e^{B_{t+s}} β_{t+s} [ (X_{j,t+s})^φ (1−H_{j,t+s})^{1−φ} ]^{1−σ_c} / (1−σ_c)
  - Parameters/shocks:
    - β∈(0,1); e^{B}_t preference shock; σ_c relative risk aversion; φ preference parameter.
- Habit-adjusted consumption and habit stock:
  - X_{t}^j = C_{t}^j − θ S_{c,t−1}, with θ∈(0,1).
  - S_{c,t} = ρ S_{c,t−1} + (1−ρ) C_t, with ρ∈(0,1).
- Labor supply and wage setting:
  - Each household is monopolistic provider of a differentiated labor service.
  - H_{j,t} = (w_{j,t}/w_t)^{−e_W} H_t, where w_{j,t} real wage, w_t average real wage, η_W intra-temporal elasticity (exponent notation e_W), e^{W}_t wage mark-up shock, H_t average demand for labor services.
  - Households choose w_{j,t}.
- Wage adjustment costs (Rotemberg):
  - ξ_W/2 (W_{j,t}/(γ W_{j,t−1}) − Π)^2 w_t H_t = ξ_W/2 (w_{j,t}/(γ w_{j,t−1} Π_t) − Π)^2 w_t H_t
  - ξ_W wage adjustment cost parameter; Π_t = P_t / P_{t−1}; Π steady-state inflation; γ deterministic growth rate.
- Budget constraint and savings:
  - Households enter with D_{j,t−1} real deposits, choose C_{j,t}, D_{j,t}, w_{j,t}, supply H_{j,t}, receive w_{j,t} H_{j,t}, profits P_t, pay T_t, earn R^D_t on deposits.
  - Constraint:
    - C_{j,t} + D_{j,t} + ξ_W/2 (w_{j,t}/(γ w_{j,t−1} Π_t) − Π)^2 w_t H_t ≤ w_{j,t} H_{j,t} + (1 + R^D_{t−1})/Π_t D_{j,t−1} + P_t − T_t
- Household optimization:
  - Maximize U_{j,t} w.r.t. C_{j,t}, D_{j,t}, w_{j,t} subject to habit-adjusted consumption, habit stock dynamics, labor supply, and budget constraint above.

### 2.2 Entrepreneurs — firm decisions, deep lending habits, financing constraint
- Entrepreneurs e∈(0,1); firms owned by households; stochastic discount factor Λ_{t,t+1} from household marginal rate of substitution.
- Decision variables: employment H^e_t, capital K^e_{t+1}, investment I^e_t, utilization U^e_t, firm-specific price P^e_t (inter-temporal profit maximization, equation (9)).
- Two-tier optimization:
  - Intra-temporal (cost minimization / lending composition):
    - Firms choose loan demand L^{e}_{b,t} across banks to minimize borrowing costs subject to a deep-habit Dixit–Stiglitz-type constraint and SL_{b,t} persistence.
    - Key objects: R^{L}_{b,t} net lending rate by bank b; θL degree of habit in lending; SL_{b,t} stock of external habit; ηL elasticity of substitution across loan varieties; %L persistence parameter.
    - Intra-temporal solution yields downward-sloping demand for loans; enters banks’ profit-maximization (equation (15)).
  - Inter-temporal (real decisions / profit maximization):
    - Flow terms: wage bill w_t H_t; investment I^e_t; utilization cost Ψ(U^e_t) K^e_t with Ψ(U)=γ1(U−1)+γ2/2(U−1)^2; Rotemberg price-adjustment cost ξ/2 (P^e_{t}/P^e_{t−1} − Π̄)^2 Y_t; deep-habit loan aggregate (X^{L}_t)^e; lump-sum Φ_t; repayment integral ∫ (1+R^{L}_{b,t+s−1}) P_{t+s−1}/P_{t+s} L^{e}_{b,t+s−1} db.
    - Capital accumulation:
      - K^e_{t+1} = I^e_t [1 − S(I^e_t / I^e_{t−1})] e^{I_t} + (1 − δ) e^{K_t} K^e_t
      - S(·) investment adjustment costs; quadratic form S(I_t / I_{t−1}) = ψ/2 (I_t / I_{t−1} − γ)^2, ψ > 0.
    - Financing constraint (equation (11)):
      - L^e_t ≥ ζ I^e_t, ζ∈(0,1). Binds in equilibrium and ensures external credit role; ζ calibrated/estimated.
    - Production and pricing:
      - Y^e_t = (P^e_t / P_t)^{-η} Y_t = F(e^{A}_t, e^{K}_t, U^e_t, K^e_t, H^e_t)
      - Technology: F(e^{A}_t, H_t, K_t, U_t) = e^{A}_t H_t^{α} (e^{K}_t U_t K_t)^{1−α}; Dixit–Stiglitz demand with elasticity η and price mark-up shock e^{P}_t.
- Deep-habit lending formal elements:
  - Aggregator adjusted for past lending: equation (7) with θL SL_{b,t−1}; SL_{b,t} = %L SL_{b,t−1} + (1−%L) L_{b,t}.
  - Φ_t ensures equilibrium equality between effective borrowing and sum of loans:
    - (X^{L}_t)^e + Ξ_t − Φ_t = ∫ L^{e}_{b,t} db
    - Ξ_t ≡ θL ∫ (1+R^{L}_{b,t})/(1+R^{L}_t) SL_{b,t−1} db
    - Φ_t = ∫ (R^{L}_{b,t} − R^{L}_t /(1+R^{L}_t)) [ ((1+R^{L}_{b,t})/(1+R^{L}_t))^{−ηL} (X^{L}_t)^e + θL SL_{b,t−1} ] db
  - Properties of Φ_t:
    - (i) increases with RL_{b,t} − RL_t;
    - (ii) decreases if market power decreases (ηL increases);
    - (iii) increases if lending relationships stronger (θL increases);
    - (iv) Φ_t = 0 if ηL → ∞ or θL = 0.
- Implications for banking–firm interactions:
  - Intra-temporal loan demand imposes bank-specific demand schedule entering banks’ problem.
  - Deep habits and lending relationships generate persistence/"lock-in", supporting bank market power and differentiated loan pricing.
  - Financing constraint links real decisions (investment, utilization, labor) to financial frictions.

### 3.2 Estimation results — posterior estimates, model comparison, dynamics
- Key posterior means and 90% credible intervals:
  - Consumption habits θ: 0.5732 [0.4781;0.6671].
  - Habit persistence ρ: 0.6563 [0.5155;0.8031].
  - Deep banking habits θL: 0.5769 [0.4336;0.7188].
  - Habit persistence in banking %L: 0.7173 [0.5273;0.8930].
  - Price stickiness ξ: 45.4310 [38.4095;52.2765].
  - Wage stickiness ξ_W: 93.8691 [75.1532;112.6440].
  - Interest rate smoothing ρ_r: 0.8390 [0.8132;0.8666].
  - Inflation response ρ_π: 1.9983 [1.7193;2.2547].
  - Output gap response ρ_y: 0.0271 [0.0059;0.0466].
  - Output gap growth response ρ_∆y: 0.1574 [0.0405;0.2674].
  - Credit growth response ρ_s: 0.2522 [0.1626;0.3426].
  - Marginal log-likelihood (baseline): -739.661.
- Model comparison baseline vs restricted (ρ_s = 0):
  - Restricted marginal log-likelihood: -751.132.
  - Bayes factor: 9.59×10^4.
  - Kass–Raftery statistic: 22.94.
  - Interpretation: “decisive evidence” in favor of credit-growth-augmented Taylor rule.
- Additional parameter estimates (selected):
  - σ_c: 1.5505 [1.4095;1.7305]; ψ: 2.5747 [0.8315;3.9961]; η_u: 0.8588 [0.7776;0.9500]; ̄γ: 0.3917 [0.3266;0.4580]; ̄π: 0.6546 [0.5826;0.7343]; ̄rn: 1.2738 [1.1714;1.3792]; ̄rL: 1.9357 [1.8532;2.0242]; ρA: 0.9630 [0.9378;0.9890]; ρG: 0.9576 [0.9378;0.9785]; ρI: 0.3647 [0.0585;0.6484]; ρK: 0.9311 [0.8992;0.9660]; ρP: 0.8017 [0.7279;0.8817]; ρW: 0.7775 [0.6971;0.8589].
- Dynamic properties and impulse-response implications:
  - Eight structural shocks: five demand-type (preference, investment-specific, capital quality, government, monetary) move output and inflation together; three supply-type (technology, wage mark-up, price mark-up) move output and inflation opposite.
  - Lending generally comoves positively with real output except for preference and government spending shocks; bank spread is counter-cyclical.
  - Credit-augmented Taylor rule partially counteracts lending fluctuations: stabilizes output when lending pro-cyclical; slightly increases output contraction when lending counter-cyclical.
- Variance decomposition highlights:
  - Output: three supply shocks ~30% on impact, 62% at one-year horizon, ~70% at two-year horizon.
  - Lending: investment-specific and capital quality shocks explain 57% on impact; three supply shocks >60% of long-term lending variance.
  - Bank spread: dominated on impact by investment-specific and capital quality shocks; spread movements imply counter-cyclical spread response.
- Model fit to credit spreads:
  - Model spread aligns closest with Bank Prime (BP) − FFR proxy, capturing rapid surges during recessions and low lending rates in late 1990s and dot-com burst.

### 3.5 Robustness to alternative Taylor-type rules — six specifications and evidence
- Six Taylor-rule formulations (I–VI) considered: implementable; implementable + output growth; baseline + stock-price gap; AR(2) smoothing; AR(2) + stock prices; baseline with potential output w/o stickiness/FF.
- Across all six specifications:
  - Point estimate of ρ_s near 0.25; credible sets well above zero.
  - Examples of posterior ρ_s:
    - (I) 0.2861 [0.1963;0.3726]; (II) 0.2512 [0.1599;0.3369]; (III) 0.2393 [0.1500;0.3215]; (IV) 0.1956 [0.1103;0.2845]; (V) 0.1931 [0.1087;0.2808]; (VI) 0.2494 [0.1609;0.3403].
- Model comparison (unconstrained vs constrained ρ_s = 0) — selected Bayes factors and KR statistics:
  - (I) BF 1.03×10^6; KR 27.69.
  - (II) BF 6.20×10^4; KR 22.07.
  - (III) BF 3.58×10^4; KR 20.97.
  - (IV) BF 3.96×10^2; KR 11.96.
  - (V) BF 2.16×10^2; KR 10.75.
  - (VI) BF 9.64×10^4; KR 22.95.
  - Interpretation: decisive evidence that US monetary policy leaned against loan-market developments during the Great Moderation.
- Implementable and AR(2) variants:
  - Implementable (I) ρ_r: 0.8450 [0.8187;0.8689]; ρ_π: 2.0208 [1.7482;2.2845]; ρ_y: 0.0213 [0.0056;0.0371]; ρ_s: 0.2861 [0.1963;0.3726].
  - AR(2) (IV) ρ_1: 1.3270 [1.2089;1.4512]; ρ_2: -0.4623 [-0.5823;-0.3492]; ρ_s: 0.1956 [0.1103;0.2845].

### Optimal policy exercises — design, welfare, and counterfactuals
- Re-parametrized Taylor-type rule for optimization (log form, equation (23)):
  - log(R^n_t / R̄^n) = ρ_r log(R^n_{t−1} / R̄^n) + α_π log(Π_t / Π) + α_y log(Y_t / Y^f_t) + α_∆y [log(Y_t / Y^f_t) − log(Y_{t−1} / Y^f_{t−1})] + α_s log(L_t / (γ L_{t−1}) Π_t / Π)
  - α_π ≡ (1−ρ_r) ρ_π; α_y ≡ (1−ρ_r) ρ_y; α_∆y ≡ (1−ρ_r) ρ_∆y; α_s ≡ (1−ρ_r) ρ_s.
- Welfare objective: present value of lifetime utility Ω_t with second-order approximation; ZLB asymmetry approximated via Θ(R^n_t) with weight w_r calibrated s.t. Pr(R^n_t < 1) = 0.01 for tightest calibration.
- Ramsey policy results (selected w_r and outcomes):
  - w_r = 5.00 → Pr(ZLB) = 0.067 ; Ω = -246.6634.
  - w_r = 18.00 → Pr(ZLB) = 0.017 ; Ω = -246.6750 ; 100×ω = 0.04.
  - w_r = 22.00 → Pr(ZLB) = 0.010 ; Ω = -246.6777 ; 100×ω = 0.05.
  - Increasing w_r reduces Pr(ZLB) and yields small welfare losses in consumption-equivalent terms.
- Optimized Taylor-type rules (matching Ramsey Pr(ZLB) cases) — selected rows:
  - w_r = 3.85 → Pr(ZLB) = 0.067 ; ρ_r = 0.875 ; α_π = 2.101 ; α_y = 0.016 ; α_∆y = 0.000 ; α_s = 0.000 ; Ω = -246.8130 ; 100×ω = 0.56.
  - w_r = 11.50 → Pr(ZLB) = 0.017 ; ρ_r = 1.000 ; α_π = 1.122 ; α_y = 0.006 ; α_∆y = 0.000 ; α_s = 0.000 ; Ω = -246.8268 ; 100×ω = 0.61.
  - w_r = 22.00 → Pr(ZLB) = 0.010 ; ρ_r = 1.000 ; α_π = 0.885 ; α_y = 0.000 ; α_∆y = 0.000 ; α_s = 0.000 ; Ω = -246.8329 ; 100×ω = 0.63.
- Estimated Taylor-type rule (data):
  - ρ_r = 0.839 ; α_π = 0.322 ; α_y = 0.004 ; α_∆y = 0.025 ; α_s = 0.041 ; Ω = -246.9724 ; 100×ω = 1.15.
  - Text: estimated rule implies Pr(ZLB) = 0.017 (hit ZLB once every 15 years); welfare loss relative to Ramsey (same Pr(ZLB)) = 1.15% consumption-equivalent.
- Main normative result on α_s (responsiveness to nominal credit growth):
  - In optimized Taylor-type rules for all w_r considered, optimal α_s = 0.
  - Optimal α_∆y = 0 (no response to changes in output gap).
  - Higher w_r → lower optimal α_π.
  - With w_r = 22 the welfare loss relative to Ramsey with w_r = 5 equals 0.63% − 0.05% = 0.58% in consumption-equivalent terms.

### 4.3 Welfare implications and counterfactual experiments — trade-offs and robustness
- Effect of responsiveness to nominal credit growth (α_s):
  - Holding ρ_r, α_π, α_y at optimal values (w_r = 22), varying α_s ∈ [0,0.5]:
    - Estimated α_s = 0.041 implies consumption-equivalent loss ≈ 0.06%.
    - Doubling α_s approximately doubles welfare loss.
    - Conclusion: monetary policy responding to credit growth is not optimal; estimated responsiveness implies small welfare loss (< 0.1%).
- Role of supply vs demand shocks in trade-off:
  - Supply shocks (technology, price mark-up, wage mark-up) jointly explain ~63% of unconditional variance of lending and ~72% of inflation variance.
  - Contractionary supply shock → lending decreases, inflation rises → leaning against lending requires accommodative policy that worsens inflation–output trade-off.
  - Single instrument (nominal interest rate) cannot offset both financial frictions and price stickiness.
- Counterfactuals:
  - Fix α_r = 1, α_y = 0, vary α_s ∈ [0,0.1], optimize α_π (w_r = 22):
    - Optimal α_π increases with α_s; welfare virtually same along combinations — if forced to react more to lending, optimal to react more to inflation.
  - Increasing volatility of capital quality shocks (double/triple STD of output growth):
    - For capital quality shock, inflation, output gap and credit growth can be stabilized simultaneously ("trinity").
    - Under higher volatility driven by capital quality shocks, optimal α_s ≈ 0.1 (leaning against lending becomes optimal).
    - If all shocks' volatilities proportionally increase, optimal α_s remains ≈ 0.
  - Suppressing wage mark-up shocks:
    - Optimal α_s ≈ 0.2 (i.e., lean against lending becomes optimal absent wage mark-up shocks).
    - Wage mark-up shocks are important for inflation and lending variance; their inclusion explains difference with literature finding leaning optimal.
- Robustness checks (appendix H):
  - Superinertial rules and price-level rules: negligible welfare effect; optimal α_s remains zero in baseline with estimated shocks.
  - Alternative financial variables in Taylor rule and implementable specifications yield similar optimal α_s ≈ 0 and similar welfare conclusions.
- Policy implications:
  - Given supply shocks dominate medium-to-long run drivers of output, lending and inflation in the estimated model, optimal monetary policy responds to inflation rather than credit growth.
  - Optimality of leaning-against-the-wind is sensitive to shock composition; when shocks that make lending and inflation move together dominate, leaning can be optimal.
  - Results support using macroprudential instruments to address financial stability rather than tilting the policy interest rate toward nominal credit growth.

*Source: wp17179 — IMF Working Paper (selected excerpts).*

### 2.1  Households  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

### 2.1  Households

### Preferences and utility
- Households are infinitely-lived and indexed by j∈(0,1).
- Inter-temporal utility:
  - U_j_t = E_t ∑_{s=0}^∞ e^{B_{t+s}} β_{t+s} [ (X_{j,t+s})^φ (1−H_{j,t+s})^{1−φ} ]^{1−σ_c} / (1−σ_c)
  - Parameters and shocks explicitly stated:
    - β∈(0,1) is the discount factor.
    - e^{B}_t is a preference shock.
    - σ_c is the relative risk aversion parameter.
    - φ is a preference parameter affecting labor supply.
- Total time normalized to unity so 1−H_{j,t} represents leisure.

### Habit-adjusted consumption and habit stock
- Habit-adjusted consumption:
  - X_t^j = C_t^j − θ S_{c,t−1}
  - θ∈(0,1) is the degree of habit formation.
- Habit stock dynamics:
  - S_{c,t} = ρ S_{c,t−1} + (1−ρ) C_t
  - ρ∈(0,1) is the persistence of the stock of habit.

### Labor supply and wage setting
- Each household j is a monopolistic provider of a differentiated labor service.
- Labor supply schedule:
  - H_{j,t} = (w_{j,t}/w_t)^{−e_W} H_t
  - where w_{j,t} is household j’s real wage, w_t is the average real wage, η_W is the intra-temporal elasticity of substitution across labor services, e_W denotes the exponent notation in the source, and e^{W}_t is a wage mark-up shock.
- H_t denotes average demand for labor services by firms.
- Households choose w_{j,t} (real wage) as part of their optimization.

### Wage adjustment costs (Rotemberg)
- The household budget constraint includes a Rotemberg quadratic cost of adjusting the nominal wage W_{j,t}, zero at steady state.
- Cost specification proportional to average real value of labor services:
  - ξ_W/2 (W_{j,t}/(γ W_{j,t−1}) − Π)^2 w_t H_t = ξ_W/2 (w_{j,t}/(γ w_{j,t−1} Π_t) − Π)^2 w_t H_t
  - ξ_W is the wage adjustment cost parameter.
  - Π_t = P_t / P_{t−1} is the gross inflation rate of price index P_t.
  - Π is the steady state value of inflation.
  - γ is deterministic growth rate parameter (model-wide specification referenced elsewhere).

### Budget constraint and savings
- Households enter period t with D_{j,t−1} units of real deposits in the bank.
- During period t they choose consumption C_{j,t}, supply H_{j,t}, receive real wage w_t, receive profits of financial and non-financial firms P_t, incur wage adjustment cost, pay lump-sum taxes T_t, and allocate savings in deposits D_{j,t} that pay net nominal interest rate R^D_t one period ahead.
- Budget constraint:
  - C_{j,t} + D_{j,t} + ξ_W/2 (w_{j,t}/(γ w_{j,t−1} Π_t) − Π)^2 w_t H_t ≤ w_{j,t} H_{j,t} + (1 + R^D_{t−1})/Π_t D_{j,t−1} + P_t − T_t

### Household optimization
- Each household maximizes the inter-temporal utility function with respect to C_{j,t}, D_{j,t}, and w_{j,t} subject to:
  - Habit-adjusted consumption: X_t^j = C_t^j − θ S_{c,t−1}
  - Habit stock dynamics: S_{c,t} = ρ S_{c,t−1} + (1−ρ) C_t
  - Labor supply: H_{j,t} = (w_{j,t}/w_t)^{−e_W} H_t
  - Budget constraint above

*Source: wp17179 - 2.1  Households (excerpt) — IMF Working Paper*

### 2.2  Entrepreneurs

### 2.2 Entrepreneurs

### Household ownership and objective
- Entrepreneurs are distributed over a unit interval and indexed by e∈(0,1).
- Firms are owned by households; the stochastic discount factor for firms is Λt,t+1, given by the inter-temporal marginal rate of substitution of households.
- Entrepreneurs maximize the expected discounted value of lifetime profits by choosing employment He t, capital Ke t+1, investment Ie t, capital utilization Ue t, and the firm-specific price Pe t (inter-temporal problem, equation (9)).

### Two-tier optimization structure
- Intra-temporal problem (cost minimization / lending composition):
  - Entrepreneurs choose demand for each variety of loans Le bt to minimize overall borrowing costs (equation (6)) subject to a deep-habit Dixit–Stiglitz-type constraint (equation (7)) and persistence in the stock of habit (equation (8)).
  - Key objects and interpretation:
    - RL bt: net lending rate charged by bank b.
    - Le bt: demand by firm e for loans issued by bank b.
    - θL: degree of habit in lending (deep lending habits).
    - SL bt: stock of (external) habit in lending.
    - ηL: elasticity of substitution across varieties of loans.
    - (XL t) e: demand for loans by firm e augmented by lending relationships.
    - %L: persistence of lending relationships.
  - The deep-habit-adjusted Dixit–Stiglitz aggregator (equation (7)) captures:
    - (i) banks differentiating loan products, and
    - (ii) firms holding multiple bank relationships.
  - Solving the intra-temporal cost minimization yields a downward-sloping demand for loans; this demand is a constraint in banks’ profit-maximization (enters as equation (15) in the banks’ problem).

- Inter-temporal problem (profit maximization / real decisions):
  - Entrepreneurs maximize the flow of discounted profits given in equation (9), where terms include:
    - Wage bill wtHt; investment expenditure Ie t; utilization cost Ψ(Ue t)Ke t with Ψ(Ut)=γ1(Ut−1)+γ2/2(Ut−1)2.
    - Rotemberg convex price-adjustment cost ξ/2 (Pe t+s/Pe t+s−1 − Π̄)2 Yt+s.
    - Deep-habit-adjusted aggregate of loans (XL t) e, and a deep-habit-specific term Ξt ≡ θL ∫ 1 0 (1+RL bt)/(1+RL t) SL bt−1 db.
    - Lump-sum term Φt needed to ensure consistency between effective borrowing and actual sum of loans.
    - Repayment term ∫ 1 0 (1+RL bt+s−1) Pt+s−1/Pt+s Le bt+s−1 db.
  - Capital accumulation law of motion (equation (10)):
    - Ke t+1 = Ie t [1 − S(Ie t / Ie t−1)] eIt + (1 − δ)eK t K e t, with investment adjustment costs S(·); S(γ) = S′(γ) = 0 and S′′(γ) > 0.
    - Following Smets and Wouters (2007), investment adjustment costs are quadratic: S(I t / I t−1) = ψ/2 (I t / I t−1 − γ)2, ψ > 0.
  - Financing constraint (equation (11)):
    - Le t ≥ ζ Ie t, where ζ∈(0,1) is the fraction of investment that must be financed externally by banks.
    - Constraint (11) is a financing constraint that binds in equilibrium (holds with equality) and ensures external credit plays a role in the model; it defines total loans demanded by firm e.
  - Production and pricing (equation (12)):
    - Ye t = (Pe t / Pt)−e P t η Y t = F(eA t, eK t, Ue t, Ke t, He t).
    - Firms face a Dixit–Stiglitz demand with intra-temporal elasticity of substitution η and price mark-up shock eP t.
    - Technology: F(eA t,U t, K t, H t) = eA t (H t)α (eK t U t K t)1−α, a Cobb–Douglas technology with α as the labor share and eA t a TFP shock.

### Deep-habit and lending relationships: formal elements and implications
- Deep habits in lending:
  - Equation (7) imposes deep habits via a Dixit–Stiglitz aggregator adjusted for past lending habits (θL SL bt−1).
  - Equation (8): SL bt = %L SL bt−1 + (1 − %L) L bt imposes persistence in the stock of habit.
  - Deep-habit framework is not a formal asymmetric-information setup but produces similar effects in symmetric equilibrium (see note referencing Aksoy et al., 2013).
- Interpretation of Φt (switching cost):
  - At equilibrium (XL t) e + Ξt = ∫ 1 0 (1+RL bt)/(1+RL t) Le bt db, and Le bt = ( (1+RL bt)/(1+RL t) )−ηL (XL t) e + θL SL bt−1.
  - Φt = ∫ 1 0 (RL bt − RL t /(1+RL t)) [ ( (1+RL bt)/(1+RL t) )−ηL (XL t) e + θL SL bt−1 ] db.
  - Properties of Φt:
    - (i) Φt is increasing in the difference between bank b’s rate and the market rate (RL bt − RL t).
    - (ii) Φt decreases if market power decreases (ηL increases).
    - (iii) Φt increases if lending relationships are stronger (θL increases).
    - (iv) Φt = 0 under perfect competition (ηL → ∞) or no lending relationships (θL = 0).

### Additional modeling details and normalizations
- Utilization normalization:
  - Steady-state utilization rate u = 1. It follows that Ψ(u) = 0, Ψ′(u) = γ1, Ψ′′(u) = γ2.
  - The elasticity of the utilization rate to changes in the marginal utilization cost is Ψ′(u)/Ψ′′(u)u = γ1/γ2 ≡ σu ≡ 1 − ηu ηu. Following Smets and Wouters (2007) the parameter ηu∈[0,1].
- Shocks and additional terms:
  - eI t is an investment-specific technology shock; eK t is a capital quality shock entering as in Gertler and Karadi (2011).
  - Ξt ≡ θL ∫ 1 0 (1+RL bt)/(1+RL t) SL bt−1 db is a deep-habit-specific term analogous to Ravn et al. (2006).
  - Φt is a lump-sum cost ensuring equilibrium equality between effective and actual borrowing: (XL t) e + Ξt − Φt = ∫ 1 0 Le bt db.
- Financing constraint economic role:
  - Constraint (11) is necessary because, with market power in banking, absent a financing constraint firms would always finance via internal funds; ζ is calibrated/estimated to match data rather than imposed at ζ = 1.

### Implications for banking–firm interactions
- The intra-temporal demand for loans derived from the entrepreneurs’ cost minimization imposes a bank-specific demand schedule (equation (15) in the banks’ problem) that feeds into bank optimization.
- Deep habits and lending relationships generate persistence and "lock-in" effects in firm–bank linkages, supporting a role for bank market power and differentiated loan products in equilibrium demand and pricing of credit.
- The model links real decisions (investment, utilization, labor) to financial frictions via the financing constraint, deep-habit loan aggregator, and the associated switching cost Φt.

*Source: wp17179 - 2.2 Entrepreneurs, IMF Working Paper*

### 3.2  Estimation results

### 3.2 Estimation results

### Key estimated parameters and identification
- Habits in consumption θ: posterior mean 0.5732 [0.4781;0.6671].
- Habit persistence in consumption ρ: posterior mean 0.6563 [0.5155;0.8031].
- Deep habits in banking θL: posterior mean 0.5769 [0.4336;0.7188].
- Habit persistence in banking %L: posterior mean 0.7173 [0.5273;0.8930].
- Price stickiness ξ: posterior mean 45.4310 [38.4095;52.2765] (implies firms adjust prices about every three quarters and a half).
- Wage stickiness ξW (Rotemberg): posterior mean 93.8691 [75.1532;112.6440] (higher than price stickiness).
- Interest rate smoothing ρr: posterior mean 0.8390 [0.8132;0.8666].
- Inflation response in Taylor rule ρπ: posterior mean 1.9983 [1.7193;2.2547] (monetary policy was aggressive on inflation).
- Output gap response ρy: posterior mean 0.0271 [0.0059;0.0466] (weak response).
- Output gap growth response ρ∆y: posterior mean 0.1574 [0.0405;0.2674] (stronger response to changes in output gap).
- Credit growth response in Taylor rule ρs: posterior mean 0.2522 [0.1626;0.3426] (novel result: positive response to nominal credit growth).
- Marginal log-likelihood (baseline model): -739.661.

Notes on identification:
- Figure 2 (prior and posterior densities) shows Taylor rule parameters well identified by the data; posterior density for ρs is entirely positive and tightly gathered around the posterior mean despite a loose prior.

### Model comparison: baseline vs. standard Taylor rule
- Baseline (credit-augmented Taylor rule) marginal log-likelihood: -739.661.
- Restricted model (ρs = 0) marginal log-likelihood: -751.132.
- Bayes factor (BF): 9.59×10^4.
- Kass-Raftery (KR) statistic: 22.94.
- Interpretation: With a BF well above 100 the authors conclude “decisive evidence” in favor of the credit-growth-augmented Taylor rule; KR = 22.94 points to “very strong” evidence in favor of the unconstrained baseline model versus the restricted standard Taylor rule.

### Additional notable parameter estimates (selected from Table 3)
- Relative risk aversion σc: posterior mean 1.5505 [1.4095;1.7305].
- Investment adjustment costs ψ: posterior mean 2.5747 [0.8315;3.9961].
- Capital utilization ηu: posterior mean 0.8588 [0.7776;0.9500].
- Trend growth rate ̄γ: posterior mean 0.3917 [0.3266;0.4580].
- Inflation rate ̄π: posterior mean 0.6546 [0.5826;0.7343].
- Interest rate ̄rn: posterior mean 1.2738 [1.1714;1.3792].
- Lending rate ̄rL: posterior mean 1.9357 [1.8532;2.0242].
- Technology persistence ρA: posterior mean 0.9630 [0.9378;0.9890].
- Government spending persistence ρG: posterior mean 0.9576 [0.9378;0.9785].
- Investment-specific persistence ρI: posterior mean 0.3647 [0.0585;0.6484].
- Capital quality persistence ρK: posterior mean 0.9311 [0.8992;0.9660].
- Price mark-up persistence ρP: posterior mean 0.8017 [0.7279;0.8817].
- Wage mark-up persistence ρW: posterior mean 0.7775 [0.6971;0.8589].

### Interpretation and substantive findings
- Evidence of both consumption habits and habit persistence (means 0.57 and 0.66 respectively).
- Deep banking habits and persistence are sizable (means ~0.58 and 0.72).
- Technology and government spending shocks are more persistent than other shocks.
- Monetary policy during the Great Moderation shows inertia and a strong response to inflation (ρπ ≈ 1.9983).
- There is a weak direct response to the output gap (ρy = 0.0271) but a stronger response to output gap growth (ρ∆y = 0.1574).
- Novel empirical finding: monetary authority responds to nominal credit growth (ρs = 0.2522), with posterior distribution strictly positive.
- Model comparison provides decisive evidence that augmenting the Taylor rule with credit growth improves fit (BF = 9.59×10^4, KR = 22.94).

---

### Dynamic properties and impulse-response implications
- The model features eight structural shocks: five aggregate demand shocks (preference, investment-specific, capital quality, government spending, monetary policy) where output and inflation move together, and three aggregate supply shocks (technology, wage mark-up, price mark-up) where output and inflation move in opposite directions.
- Impulse responses (Figure 3) compare responses under the baseline credit-augmented Taylor rule (solid line) and a counterfactual standard Taylor rule ρs = 0 (dashed line).
- General effects observed:
  - Sign of impulse responses preserved across Taylor rule specifications, but the severity of downturns differs.
  - Except for preference and government spending shocks, lending positively comoves with real output and the bank spread is counter-cyclical.
  - Preference shock: shorter-lived output contraction and overshooting—fall in current consumption increases savings, raises loan supply, lowers spreads, and boosts future activity.
  - Government spending shock: similar crowding-in of private investment and counter-cyclical lending response.
- Role of credit-augmented Taylor rule:
  - The central bank partially counteracts lending fluctuations.
  - For shocks where lending is pro-cyclical, leaning against the wind stabilizes output (contraction of output is more severe without the augmented rule).
  - For shocks that generate counter-cyclical lending (preference and government spending), the augmented rule slightly increases output contraction.
- General lesson: leaning against the wind stabilizes output for shocks that imply pro-cyclical lending and counter-cyclical bank spreads.

### Variance decomposition (summary of major drivers)
- Output:
  - On impact, demand shocks dominate output dynamics; medium-to-long term, technology plus wage and price mark-up shocks become main drivers.
  - These three supply shocks: about 30% of output fluctuations on impact, 62% at a one-year horizon, and almost 70% at a two-year horizon.
  - Government and monetary policy shocks decay in importance over time.
- Inflation:
  - Price and wage mark-up shocks are dominant for short-run and medium-run inflation movements (consistent with Smets and Wouters (2007)).
- Lending:
  - Three supply shocks account for more than 60% of long-term variation in lending.
  - Investment-specific technology and capital quality shocks dominate lending on impact (explaining 57% of lending fluctuations on impact) and account for around 30% in the longer term.
  - Dominant shocks for lending are those that foster a pro-cyclical lending response.
- Bank spread:
  - On impact, investment-specific technology shock and, to a lesser extent, capital quality shocks are main exogenous sources of variation in the spread; price and wage mark-up shocks also matter.
  - Shocks dominant for spread movements imply a counter-cyclical spread response (consistent with the negative relationship between bank spread and lending in the model).

### Model fit to credit spreads
- Figure 4 compares the model-implied demeaned spread (one-step-ahead forecast) with three empirical proxies:
  - Bank Prime (BP) rate minus FFR (used as observable in estimation).
  - Moody’s Baa minus long-term Treasury constant maturity rate (TCMR).
  - Gilchrist and Zakrajšek (2012) corporate bond spread.
- The model generates a spread closest in magnitude and behavior to BP-FFR, captures rapid surges during recessions, low lending rates of the late 1990s, the dot-com bubble burst, and the increase at end-of-sample crisis; other proxies are more volatile.

*Source: wp17179 - 3.2 Estimation results (IMF working paper PDF).*

### 3.5  Robustness to alternative Taylor-type interest-rate rules

### 3.5  Robustness to alternative Taylor-type interest-rate rules

### Purpose and alternative formulations considered
- Investigates whether evidence in favor of a leaning-against-the-wind policy is robust to different specifications of the Taylor rule.
- Six formulations are considered (Table 6):
  - (I) Implementable: responds to output in deviation from steady state (Y) instead of a latent output gap.
  - (II) Implementable augmented: adds a response to output growth.
  - (III) Baseline augmented with stock-price gap response: stock-price gap defined as Q_t − Q^f_t (gap between stock price in sticky-price model and in model without nominal stickiness).
  - (IV) AR(2) interest-rate smoothing: second-order autoregressive interest-rate smoothing (ρ_1, ρ_2) à la Coibion and Gorodnichenko (2012).
  - (V) AR(2) smoothing plus stock-price response.
  - (VI) Baseline but potential output Y^f_t computed as the level that would prevail in the absence of price and wage stickiness and financial frictions.
- When stock prices are included (formulations III and V) an additional measurement equation is estimated:
  - ∆Q^o_t = Q̄ + Q̂_t − Q̂_{t−1} + ε^{me}_{Qt}  (equation (22))
  - Q̄ is the average quarterly growth rate of the stock price index; ε^{me}_{Qt} is a measurement error.

### Parameter estimation results (summary of Table 7)
- Prior and posterior estimates reported for Taylor-rule parameters in all six formulations.
- Point estimates and 90% credible intervals (square brackets) for key parameters (selected highlights):
  - Implementable (I):
    - Interest rate smoothing ρ_r: posterior mean 0.8450 [0.8187;0.8689]
    - Inflation ρ_π: posterior mean 2.0208 [1.7482;2.2845]
    - Output ρ_y: posterior mean 0.0213 [0.0056;0.0371]
    - Credit growth ρ_s: posterior mean 0.2861 [0.1963;0.3726]
  - Output growth (II):
    - ρ_r: 0.8393 [0.8129;0.8670]
    - ρ_π: 2.0052 [1.7295;2.2784]
    - ρ_∆y: 0.1250 [0.0344;0.2140]
    - ρ_s: 0.2512 [0.1599;0.3369]
  - Stock prices (III):
    - ρ_r: 0.8430 [0.8170;0.8688]
    - ρ_π: 2.0114 [1.7422;2.2659]
    - ρ_q (stock prices): 0.1320 [0.0407;0.2196]
    - ρ_s: 0.2393 [0.1500;0.3215]
  - AR(2) smoothing (IV):
    - ρ_1: 1.3270 [1.2089;1.4512]
    - ρ_2: -0.4623 [-0.5823;-0.3492]
    - ρ_π: 2.0225 [1.7660;2.2883]
    - ρ_s: 0.1956 [0.1103;0.2845]
  - Stock prices & AR(2) (V):
    - ρ_1: 1.3113 [1.1947;1.4312]
    - ρ_2: -0.4416 [-0.5598;-0.3330]
    - ρ_q: 0.1020 [0.0279;0.1695]
    - ρ_s: 0.1931 [0.1087;0.2808]
  - Potential output w/o financial frictions (VI):
    - ρ_r: 0.8381 [0.8117;0.8664]
    - ρ_π: 2.0048 [1.7383;2.2799]
    - ρ_y: 0.0273 [0.0064;0.0477]
    - ρ_∆y: 0.1544 [0.0370;0.2618]
    - ρ_s: 0.2494 [0.1609;0.3403]

- Cross-model summary statement:
  - In all six cases the point estimate of the interest-rate response to credit growth (ρ_s) is in the vicinity of 0.25.
  - Credible sets of ρ_s are well above zero across models; ρ_s is well identified in the data.
  - Estimates of AR(2) smoothing parameters (ρ_1, ρ_2) show same signs and comparable magnitudes as Ascari et al. (2011).
  - Responsiveness to stock prices (ρ_q) is close to Castelnuovo and Nisticò (2010).

### Model comparison and evidence for leaning against the wind (Table 8)
- Models estimated with Taylor rules (I)-(VI) are compared to constrained versions with ρ_s = 0 via marginal log-likelihood, Bayes factor, and Kass–Raftery statistics.
- Selected reported values:
  - Implementable (I):
    - Marginal log-likelihood (unconstrained vs constrained): −742.302 vs −756.149
    - Bayes factor: 1.03×10^6
    - Kass–Raftery statistic: 27.69
  - Output growth (II):
    - Marginal log-likelihood: −741.275 vs −752.310
    - Bayes factor: 6.20×10^4
    - Kass–Raftery statistic: 22.07
  - Stock prices (III):
    - Marginal log-likelihood: −1086.730 vs −1097.215
    - Bayes factor: 3.58×10^4
    - Kass–Raftery statistic: 20.97
  - AR(2) (IV):
    - Marginal log-likelihood: −726.302 vs −732.284
    - Bayes factor: 3.96×10^2
    - Kass–Raftery statistic: 11.96
  - Stock prices & AR(2) (V):
    - Marginal log-likelihood: −1074.571 vs −1079.948
    - Bayes factor: 2.16×10^2
    - Kass–Raftery statistic: 10.75
  - Potential output w/o FF (VI):
    - Marginal log-likelihood: −739.639 vs −751.115
    - Bayes factor: 9.64×10^4
    - Kass–Raftery statistic: 22.95
- Interpretation:
  - For all specifications the Bayes factor is always well above 100 and the Kass–Raftery statistic is always greater than 10, pointing to decisive evidence that during the Great Moderation US monetary policy leaned against the wind from the loan market.

### Optimized simple monetary policy rules: design and welfare objective
- Re-parametrized Taylor-type rule used for optimization (equation (23)):
  - log(R^n_t / R̄^n) = ρ_r log(R^n_{t−1} / R^n) + α_π log(Π_t / Π) + α_y log(Y_t / Y^f_t) + α_∆y [log(Y_t / Y^f_t) − log(Y_{t−1} / Y^f_{t−1})] + α_s log(L_t / (γ L_{t−1}) Π_t / Π)
  - α_π ≡ (1−ρ_r) ρ_π ; α_y ≡ (1−ρ_r) ρ_y ; α_∆y ≡ (1−ρ_r) ρ_∆y ; α_s ≡ (1−ρ_r) ρ_s
- Welfare objective maximized is the present value of lifetime utility (equation (24)):
  - Ω_t = E_t[Σ_{s=0}^∞ β^s U( X̃_{t+s}, 1−H_{t+s} )]
  - Second-order approximation around the Ramsey steady state used because of frictions.
- Zero lower bound (ZLB) incorporated via modified welfare with penalty Θ(R^n_t) to approximate ZLB asymmetry:
  - Ω^*_t = E_t[Σ_{s=0}^∞ β^s [ U( X̃_{t+s}, 1−H_{t+s} ) − Θ(R^n_{t+s}) ]]
  - Θ(R^n_t) ≡ w_r (R^n_t − R^n)^2 exp[ − w_r (R^n_t − 1) ]
  - Weight w_r chosen so that Pr(ZLB) ≡ Pr(R^n_t < 1) = 0.01 (once every 25 years) for the tightest calibration.
  - Probability of hitting ZLB computed by simulation: 50 random draws of shock vectors × 500 quarters each → 25,000 realizations; frequency R^n_t < 1 is Pr(ZLB).

### Optimal monetary policy results (Table 9) — Ramsey and optimized rules
- Ramsey policy (selected rows):
  - w_r = 5.00 → Pr(ZLB) = 0.067; Ω = -246.6634
  - w_r = 18.00 → Pr(ZLB) = 0.017; Ω = -246.6750 ; 100×ω = 0.04
  - w_r = 22.00 → Pr(ZLB) = 0.010; Ω = -246.6777 ; 100×ω = 0.05
  - Increasing w_r reduces Pr(ZLB) and yields small welfare losses (less than 0.1% in consumption-equivalent terms).
  - w_r = 5 implies probability of hitting ZLB once every ~15 years? (table shows Pr(ZLB)=0.067 → once every ~15 quarters? Note: table shows interpretation in text: w_r = 5 gives probability once every four years; w_r = 18 gives once every 15 years; w_r = 22 gives once every 25 years.)
- Optimized Taylor-type rules (for same Pr(ZLB) as Ramsey policy):
  - Optimized coefficients and welfare (selected rows from Table 9):
    - w_r = 3.85 → Pr(ZLB) = 0.067 ; ρ_r = 0.875 ; α_π = 2.101 ; α_y = 0.016 ; α_∆y = 0.000 ; α_s = 0.000 ; Ω = -246.8130 ; 100×ω = 0.56
    - w_r = 11.50 → Pr(ZLB) = 0.017 ; ρ_r = 1.000 ; α_π = 1.122 ; α_y = 0.006 ; α_∆y = 0.000 ; α_s = 0.000 ; Ω = -246.8268 ; 100×ω = 0.61
    - w_r = 22.00 → Pr(ZLB) = 0.010 ; ρ_r = 1.000 ; α_π = 0.885 ; α_y = 0.000 ; α_∆y = 0.000 ; α_s = 0.000 ; Ω = -246.8329 ; 100×ω = 0.63
- Estimated Taylor-type rule (estimated from data, shown for comparison):
  - Pr(ZLB) not reported in table header for estimated rule row, but coefficients given:
    - ρ_r = 0.839 ; α_π = 0.322 ; α_y = 0.004 ; α_∆y = 0.025 ; α_s = 0.041 ; Ω = -246.9724 ; 100×ω = 1.15
  - Text interpretation:
    - Estimated monetary policy rule implies Pr(ZLB) = 0.017 (nominal interest rate hits ZLB once every 15 years).
    - Consumption-equivalent welfare loss under the estimated rule is 1.15% relative to the Ramsey policy associated to the same probability of hitting the ZLB.
- Main normative result regarding response to loan market:
  - In the optimized Taylor-type rules (across values of w_r), the optimal responsiveness to nominal credit growth (α_s) is zero for any value of w_r.
  - As w_r increases, optimal response to the output gap approache s zero; it is always optimal not to respond to changes in the output gap (α_∆y = 0).
  - A higher penalty w_r delivers a lower optimal α_π.
  - With w_r = 22 the welfare loss relative to Ramsey with w_r = 5 is 0.63% − 0.05% = 0.58% in consumption-equivalent terms.

### Additional robustness checks
- Appendices H.1–H.3 check robustness to:
  - (i) Taylor rules with potential output defined as level that would prevail absent price stickiness, wage stickiness and financial frictions;
  - (ii) interest-rate rules reacting to alternative financial variables (e.g., lending and the bank spread);
  - (iii) the implementable specification of the Taylor rule allowing a response to output growth.

*Source: IMF Working Paper — "3.5  Robustness to alternative Taylor-type interest-rate rules" (wp17179).*

### 4.3  Welfare implications and counterfactual experiments

### 4.3  Welfare implications and counterfactual experiments

### Effect of responsiveness to nominal credit growth
- Isolating the effect of responsiveness to nominal credit growth by keeping ρ_r, α_π and α_y fixed at their optimal values (obtained with w_r = 22) and varying α_s in the interval [0, 0.5]:
  - Estimated value α_s = 0.041 implies a consumption-equivalent loss of around 0.06%.
  - A doubling of α_s yields an approximately double welfare loss.
- Summary conclusion:
  - It is not optimal for monetary policy to respond to credit growth.
  - The estimated responsiveness implies a small welfare loss (equivalent to a permanent loss in consumption of less than 0.1%).
  - Bolder responses to financial conditions can represent a substantial detriment to welfare.

### Role of supply and demand shocks in generating the trade-off
- Supply shocks (technology, price mark-up and wage mark-up) explain the largest share of business-cycle fluctuations of lending and inflation:
  - The three supply shocks together explain around 63% of the unconditional variance of lending and 72% of the unconditional variance of inflation.
- Mechanism:
  - A contractionary supply-type shock causes lending to decrease and inflation to rise.
  - A monetary policy that leans against windy bank lending becomes more accommodative toward inflation to boost lending, which is not optimal because of the inflation–output trade-off.
  - With only one policy instrument (the nominal interest rate), the instrument cannot neutralize both financial frictions and price stickiness; a strong anti-inflationary stance leads to the highest level of welfare.

### Counterfactual experiments and scenarios
- Experiment fixing α_r = 1, α_y = 0 (their optimal values), letting α_s vary in [0, 0.1], and optimizing over α_π (for w_r = 22):
  - The optimal α_π monotonically increases when α_s increases.
  - Welfare attains virtually the same level for any combination of α_s and α_π in this exercise.
  - Interpretation: if policy is forced to react more to lending, it is optimal to react more to inflation as well, highlighting the conflicting objectives and absence of a "divine coincidence."
- Counterfactual changing volatility of capital quality shocks:
  - Baseline vs artificially increased volatilities (double and triple standard deviation of output growth) while keeping ρ_r, α_π, α_y and α_∆y at their optimal values (w_r = 22) and varying α_s in [0, 0.5]:
    - For the capital quality shock, inflation, the output gap and credit growth can be stabilized simultaneously ("trinity").
    - Capital quality shock accounts for 38% of the unconditional variance of output and 30% of the unconditional variance of lending.
    - Under higher volatility driven by capital quality shocks, it becomes optimal to lean against windy bank lending, with an optimal α_s around 0.1.
  - If the increase in volatility is due to a proportional increase in the standard deviations of all shocks, it remains optimal not to lean against windy bank lending.
- Suppressing wage mark-up shocks:
  - When wage mark-up shocks are removed (keeping ρ_r, α_π, α_y and α_∆y at their optimal values for w_r = 22):
    - Optimal α_s would be around 0.2, i.e., it would be optimal to lean against windy bank lending in the absence of wage mark-up shocks.
    - Wage mark-up shocks are prominent in the unconditional variance decomposition of inflation and important for output and lending fluctuations.
  - This helps reconcile with other literature (models without wage mark-up shocks tend to find leaning-against-the-wind optimal).

### Robustness and policy implications
- Robustness:
  - Appendix H.5: superinertial rules for the interest rate have negligible effects on welfare relative to price level rules.
  - Even with superinertial interest-rate rules, it is optimal not to lean against windy bank lending (in the baseline specification with estimated shocks).
- Policy implications:
  - Given supply shocks are the main medium-to-long run drivers of output, lending and inflation in the estimated model, and given the procyclical behavior of lending, it is optimal for monetary policy to respond entirely to inflation rather than to credit growth.
  - The optimality of a leaning-against-the-wind policy is sensitive to the sources of business-cycle fluctuations.
  - The findings support the recent tendency in central banking to move toward macroprudential instruments as tools to promote financial stability; further research is needed to identify effective instruments and design rules that reduce financial instability without conflicting with inflation stabilization.

*Source: 4.3 Welfare implications and counterfactual experiments (wp17179).*

### References

### wp17179 - References

### Bibliographic corpus
- Comprehensive list of cited works covering topics in monetary policy, banking, financial frictions, DSGE modeling, Bayesian estimation, and macro-financial interactions. Representative authors and works include:
  - Airaudo, M. and Olivero, M. P. (2014). Optimal monetary policy with counter-cyclical credit spreads.
  - Ajello, A., Laubach, T., Lopez-Salido, J. D., and Nakata, T. (2016). Financial stability and optimal interest-rate policy.
  - Bernanke, B., Gertler, M., and Gilchrist, S. (1999). The financial accelerator in a quantitative business cycle model.
  - Blanchard, O. and Galí, J. (2007). Real wage rigidities and the New Keynesian model.
  - Cúrdia, V. and Woodford, M. (2009, 2010). Credit frictions and credit spreads and monetary policy.
  - Smets, F. and Wouters, R. (2007). Shocks and frictions in US business cycles: A Bayesian DSGE approach.
  - Woodford, M. (2003). Interest and prices: Foundations of a theory of monetary policy.
- The references list spans theoretical and empirical contributions relevant to the model and estimation exercises reported in the paper.

### Appendix A — Narrative analysis of FOMC meetings on concerns about credit
- Contains a chronology of FOMC meeting dates, intended FFR change, and quotes about credit conditions.
- Selected entries (date / Intended FFR change / Quote):
  - 01/31/84 — 9.3750 — "Growth in total credit at U.S. commercial banks remained strong in December"
  - 03/27/84 — 10.1250.375 — "rate of credit growth, which had accelerated considerably in early 1984 and which appeared to be an important factor in recent interest rate increases"
  - 05/22/84 — 10.50 — "credit growth had shown no sign of slowing so far"
  - 12/18/84 — 8.75-0.625 — "strong growth in private credit"
  - 11/14/89 — 8.50 — "substantially stronger expansion of bank credit"
  - 07/03/90 — 8.25-0.25 — "some businesses were finding it more difficult to obtain credit from banks"
  - 01/29/08 — 3-1.25 — "tightening in credit availability to businesses"
  - 06/24/08 — 2 0 — "tighter standards and terms on business credit"
- Table A.1: Intended changes in the federal funds rate (FFR) around FOMC meetings and quotes on credit conditions found in FOMC minutes.

### Appendix B — Symmetric equilibrium (model equations)
- Production function and marginal products:
  - Y_t = e^{A_t} H_t^{α} (U_t e^{K_t} K_t)^{1−α} (B.1)
  - F_{K,t} = (1−α) μ_t Y_t / (U_t e^{K_t} K_t) (B.2)
  - F_{H,t} = α μ_t Y_t / H_t (B.3)
- Marginal utilities, Euler equation, and wage setting (examples):
  - U_{X,t} = φ(1−H_t)^{1−φ} [X_t^{φ} (1−H_t)^{1−φ}]^{−σ_c} X_t^{φ−1} (B.4)
  - e^{B_t} U_{X,t} = β E_t [ e^{B_{t+1}} U_{X,t+1} (1 + R^D_t)^{-1} Π_{t+1}^{-1} ] (B.8)
  - Wage setting: (1−e^{W_t} η_W) + e^{W_t} η_W μ_{W,t} − ξ_W ( Π_{W,t}^{γ} − Π ) / Π_{W,t}^{γ} + ξ_W E_t[ ... ] = 0 (B.9)
- Loan demand/supply, spreads, and Taylor rule:
  - L_t = ζ I_t (B.23)
  - spread_t = (1 + R^L_t) / (1 + R^D_t) (B.29)
  - 1 + R^D_t = R^n_t ; log( R^n_t / R^n ) specified with smoothing ρ_r and responses ρ_π, ρ_y, ρ_∆y, ρ_s (B.30–B.31)

### Appendix C — Detrending
- Follows Smets and Wouters (2007): all real variables grow at rate γ; defines detrended variables with tildes, e.g.  ̃Y_t = Y_t / γ_t, ̃K_t = K_t / γ_t, etc.
- Presents detrended counterparts of equilibrium conditions (C.1)–(C.33), including:
  - γ ̃K_{t+1} = (1−δ) e^{K_t} ̃K_t + ̃I_t [1 − ̃S(γ ̃I_t / ̃I_{t−1})] e^{I_t} (C.12)
  - Λ_{t,t+1} = γ β E_t[ ̃U_{X,t+1} / ̃U_{X,t} ] (C.20)
- Resource constraint in detrended form (C.32).

### Appendix D — Steady state
- Steady-state relations include:
  - Π_W = γ Π (D.1)
  - Λ = γ β (D.2)
  - U = 1 (D.3)
  - μ_W = ̃η / (̃η − 1) (D.4)
  - MC = (η − 1) / η (D.5)
  - R^D = Π / (γ β) − 1 (D.7)
  - I = (γ − 1 + δ) K (D.8)
  - Y = H^{α} K^{1−α} (D.9)
  - C = Y − I − G (D.11)
  - S_c = (1 − ρ) / (1 − ρ / γ) C (D.12)
- Steady-state solution algorithm: K and H solve equations (B.26) and (B.3) evaluated at steady state; remaining variables computed recursively using Section D relationships.

### Appendix E — Ramsey problem
- Lagrangian L_t constructed (E.1) with a comprehensive set of constraints and 32 Lagrange multipliers.
- First-order conditions: system of 55 dynamic equations in 55 unknowns (33 endogenous variables and 32 Lagrange multipliers).
- Solution method: Dynare solver with second-order Taylor expansion around Ramsey-optimal steady state; steady state computed numerically as described in Section E.3.

### Appendix F — Data sources and transformations
- Nine observables used in estimation; sources:
  - GDP, GDP deflator inflation, commercial and industrial loans, Federal funds rate, three-month bank prime loan rate, civilian population (CNP160V), civilian employment (CE160V) — FRED.
  - Private consumption expenditures and fixed private investment — NIPA Table 1.1.5, BEA.
  - Average weekly hours (PRS85006023) and compensation per hour (PRS85006103) — BLS.
- Transformations:
  - GDP, consumption, investment, and loans → real per-capita: divide nominal values by GDP deflator and civilian population.
  - Real wages = compensation per hour / GDP deflator.
  - Observables of GDP, consumption, investment, wages, loans expressed in first differences; hours worked multiplied by civilian employment (per-capita) and demeaned.
  - Inflation = quarter-on-quarter difference of log GDP deflator.
  - Federal funds rate and lending rate expressed in quarterly terms; remaining variables = 100 × log.
  - All series seasonally adjusted by sources.

### Appendix G — Robustness of results to alternative Taylor rules
- Figure G.1: Prior and posterior densities show ρ_s identified and positive across alternative Taylor rules.
- Marginal log-likelihoods and selected posterior estimates (prior → posterior mean):
  - Table G.1 (Implementable (I)): Marginal log-likelihood = -742.302; Credit growth-Taylor rule ρ_s = 0.2861 [0.1963;0.3726]
  - Table G.2 (Implementable with output growth (II)): Marginal log-likelihood = -741.275; ρ_s = 0.2512 [0.1599;0.3369]
  - Table G.3 (Stock prices (III)): Marginal log-likelihood = -1086.730; ρ_s = 0.2393 [0.1500;0.3215]
  - Table G.4 (AR(2) smoothing (IV)): Marginal log-likelihood = -726.302; ρ_s = 0.1956 [0.1103;0.2845]
  - Table G.5 (Stock prices and AR(2) smoothing (V)): Marginal log-likelihood = -1074.571; ρ_s = 0.1931 [0.1087;0.2808]
  - Table G.6 (Potential output with no financial frictions (VI)): Marginal log-likelihood = -739.639; ρ_s = 0.2494 [0.1609;0.3403]
- Posterior posterior means for ρ_r (interest rate smoothing) and other parameters vary by specification; across tables interest rate smoothing ρ_r posterior means reported (examples):
  - Table G.1: ρ_r = 0.8450 0.8187[0.8689;]
  - Table G.2: ρ_r = 0.8393 [0.8129;0.8670]
  - Table G.3: ρ_r = 0.8430 [0.8170;0.8688]
  - Table G.4: ρ_1 = 1.3270 [1.2089;1.4512] (AR(2) specification)
- Table G.1–G.6 include posterior means and 90% confidence intervals for a wide set of structural and exogenous process parameters (σ_c, θ, ρ, ξ, ψ, η_u, ρ_π, ρ_y, etc.).

### Appendix H — Robustness exercises for optimal policy
- Scope: modifications to DSGE model to test robustness of optimal policy results:
  - (i) Alternative potential output (no price/wage stickiness, no financial frictions)
  - (ii) Alternative financial variables in Taylor rule
  - (iii) Implementable Taylor-type rules
  - (iv) Higher volatility of structural shocks
  - (v) Superinertial rules
- Key quantitative findings:
  - Table H.1 (Alternative output gap): Estimated Taylor-type rule welfare loss (consumption-equivalent) = 1.48% (reported as "the estimated Taylor-type rule features a considerable welfare loss, equivalent to a permanent loss in consumption of1.48%").
  - Optimal responsiveness to nominal credit growth under alternative output-gap specification: 0.003 (negligible).
  - Table H.2 (Alternative financial variables): Optimized rules with S_t as L_t/(γ L_{t−1}), L_t/L, or spread_t/spread yield similar optimal ρ_r, α_π, α_y, α_∆y for Pr(ZLB)=0.010 and similar welfare loss ω.
  - Table H.3 (Implementable Taylor-type rules): Estimated implementable Taylor-type rule implies consumption-equivalent welfare loss = 1.12%.
  - Table H.3 also reports optimized implementable rules (examples):
    - w_r = 13.000.0 ; Pr(ZLB) = 0.017 ; ρ_r = 1.066 ; α_π = 0.011 ; α_y = 0.000 ; α_∆y = 0.000 ; Ω100×ω = -246.82650.61
    - estimated implementable rule row: –0.0170.839 0.322 0.003 0.020 0.040 -246.87021.12
  - H.4 (Highly volatile economy): Figure H.1 shows welfare cost (CE loss) associated with increasing α_s in economies with Baseline STD, Double STD, Triple STD; higher STD increases trade-off and welfare loss for larger α_s.
  - H.5 (Superinertial rules): Figure H.2 shows:
    - Raising ρ_r from 0.5 to 1 (price-level rule case) delivers a welfare gain of around 0.15% in consumption-equivalent terms (conditional on α_s = 0).
    - Further increases in ρ_r (ρ_r > 1) produce negligible additional welfare changes.
    - Optimal value of α_s is zero for every level of ρ_r; higher α_s is welfare-detrimental.
- Selected numeric optimization outputs (Table H.1 / H.3 excerpts):
  - Optimized implementable Taylor-type rules with alternative financial variables (examples):
    - S_t = L_t/L_{t−1} with w_r = 300.0 Pr(ZLB)=0.010 ρ_r=1.000 α_π=0.825 α_y=0.008 α_∆y=0.000 α_s=0.000 Ω100×ω = -246.83260.63
    - S_t = L_t/L with w_r = 400.0 Pr(ZLB)=0.010 ρ_r=1.000 α_π=0.806 α_y=0.000 α_∆y=0.000 α_s=0.007 Ω100×ω = -246.82930.62
  - Table H.1 optimized rows:
    - w_r = 10.000.0 Pr(ZLB)=0.017 ρ_r=1.000 α_π=0.834 α_y=0.003 α_∆y=0.000 α_s=0.003 Ω100×ω = -246.74650.31
    - w_r = 15.000.0 Pr(ZLB)=0.010 ρ_r=1.000 α_π=0.695 α_y=0.000 α_∆y=0.000 α_s=0.003 Ω100×ω = -246.92140.96
- Overall robustness conclusion: across multiple alternative Taylor-rule specifications, alternative financial variables, implementability constraints, and higher shock volatility scenarios, the optimal policy results are broadly robust; in particular the model finds negligible optimal responsiveness of the policy interest rate to nominal credit growth (α_s ≈ 0) and higher α_s tends to be welfare-detrimental.

*Italic: Source — wp17179 - References (appendices and reference list as provided in the PDF chapter).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17179.pdf_