## wp1785 - 2.1 Non-Financial Firms

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### Introduction and motivation
- Global Financial Crisis (2008–09) led to unprecedented monetary easing and use of unconventional monetary policy (UMP) instruments (large scale asset purchase programs and liquidity provision).
- Central bank balance sheet ratios cited:
  - Federal Reserve: "about 5.5 percent of annual GDP on average between 1955-2007" and "23.7 percent of GDP in 2016".
  - UK: "6.5 percent on average between 1955-2007" and "22.5 percent in 2015".
  - Euro Area: "13 percent in 2006" and "34.1 percent in 2016".
  - Japan: "10 percent in 1994", "21 percent right before the crisis in 2007", and "88.7 percent in 2016".
- Policy motivation:
  - Short-term rates reached effective lower bound; UMP used to affect spreads between short- and long-term rates and restore impaired credit channels.
  - Debate whether asset purchases should remain in toolkit once interest rates normalize.

### Model overview and key mechanisms
- Base and extensions:
  - Built on Justiniano et al. (2013) New Keynesian framework with nominal and real rigidities.
  - Augmented with a banking sector (Gertler and Karadi (2013)) and lumpy investment / long-term debt (Andreasen et al. (2013)).
- Financial structure and frictions:
  - Banks intermediate household deposits to firms and government, holding long-term private loans and long-term government bonds for maturity transformation.
  - Long-term private loans arise because firms adjust capital infrequently; nominal lending rates are constant over loan life.
  - Most outstanding bonds fixed rate: "only around 2 percent of US Treasuries have a variable coupon and around 90 percent of US corporate bonds are issued as fixed rate bonds."
- Solution and estimation:
  - Second-order approximation to equilibrium conditions used for estimation and welfare evaluation.
  - Parameters estimated via Generalized Method of Moments (GMM) matching sixty-three first- and second-order moments from nine macroeconomic and financial time series.

### Production sector and firm contracts
- Firm types and structure:
  - Intermediate goods producers, Retailers, Final good producers, Capital-producing firms.
  - Retail prices: Rotemberg (1982) quadratic cost model; partial indexation to steady-state and lagged inflation.
  - Investment is lumpy: only a fraction (1−θ_k) of intermediate goods producers adjust capital each period.
- Intermediate goods producers’ optimization (preserved notation):
  - Firms choose ̄K_t and labor sequences solving a present-value problem with fixed nominal loan rate ̄r^L_t; first-order condition for investment (equation (1)) equates expected marginal product of capital to discounted cost including ̄r^L_t and maintenance fee ω.
- Capital goods producers:
  - Aggregate demand for capital: K_t = (1−θ_K) ̄K_t + θ_K K_{t−1}.
  - Capital law of motion with adjustment costs: K_t = (1−δ) K_{t−1} + ξ^I_t [ 1 − z(I_t/I_{t−1}) ] I_t.

### Financial contracts, long-term loans, and banking sector
- Long-term corporate loans:
  - Loans mature with probability θ_k each period; gross interest ̄R^L_t ≡ 1 + ̄r^L_t.
  - Aggregate real lending: len_t = (1−θ_k) Σ_{j=0}^∞ (θ_k)^j P_{K_{t−j}}/P_t ̄K_{t−j}.
  - Total revenues rev_t and average return R^L_t ≡ rev_t/len_t (weighted average of current and past long-term loan rates).
- Long-term government bonds:
  - B_t = θ_g B_{t−1} + B^N_t with B^N_t = (1−θ_g) ̄B_t; average duration (1−θ_g)^{−1}.
  - Total revenues rev^G_t and average return R^G_t ≡ rev^G_t / B_t.
- Bank balance sheet and net worth dynamics:
  - Balance sheet: len_t + b_t = n_t + d_t, where b_t = B_t / P_t, n_t = N_t / P_t, d_t = D_t / P_t.
  - Net worth accumulation (equation (9)):
    n_t = (1−τ_B) [ R^L_{t−1} P_{t−1}/P_t len_{t−1} + R^G_{t−1} P_{t−1}/P_t b_{t−1} − R_{t−1} P_{t−1}/P_t d_{t−1} ] exp(ε^{nw}_t).
  - Banker survival probability θ_B; expected professional life (1−θ_B)^{−1}; insurance agency finances new banks when bankers retire.
- Incentive constraint and asset substitutability:
  - Participation constraint (equation (10)): V_t ≥ λ_t ( len_t + ∆_t b_t ).
  - Imperfect substitutability: ( R^G_t − R_t ) = ∆_t ( R^L_t − R_t ).

### Households, labor market, and policy rules
- Households:
  - Continuum pooling workers and bankers; consumption non-durable; save only in deposits D_t paying nominal rate R_t.
  - Utility separable with habit formation and intratemporal/intertemporal shocks.
- Wage setting:
  - Monopolistic labor suppliers, Rotemberg wage adjustment costs, partial indexation to past inflation and TFP growth.
- Monetary policy rule (deposit rate):
  - R_t/R̄ = (R_{t−1}/R̄)^{γ_R} (π_t/π)^{γ_Π (1−γ_R)} [ Y_t/Y_{t−1} / exp(Λ) ]^{γ_y (1−γ_R)} exp(ε^{m,t}).
- Fiscal processes:
  - Government spending ratio g_t = G_t / Y_t AR(2) with log(g_t) specification; government debt/GDP b_t / Y_t exogenous AR(1).

### Estimation, calibrated and estimated parameters
- Estimation approach:
  - Second-order approximation and GMM following Christiano and Eichenbaum (1992) and others.
  - Data: seven macro series (1964:2–2009:4) plus two spreads (BAA − Federal Funds, 10-year − Federal Funds).
  - Moment vector dimension: 63×1; GMM estimator with two-step Newey-West (10 lags).
- Calibrated parameters (exact values preserved):
  - ε_L: 5
  - ε_Y: 10
  - α: 0.33
  - δ: 0.025
  - 1/(1−θ_k): 12
  - 1/(1−θ_g): 40
  - g: 0.2
  - ρ_{g1}: 1.288
  - ρ_{g2}: -0.299
  - σ_g: 1.07%
  - b/Y: 0.45
  - L: 1
- Selected estimated parameters (exact values preserved, Table 2 & 3):
  - h Habit Formation: 0.742 (std dev 0.026)
  - φ Inverse Frisch Elasticity: 0.847 (std dev 0.077)
  - 1/β − 1 Discount (in %): 0.241 (std dev 0.025)
  - log(R^L) − log(R) Corporate Spread (in %, quarterly): 0.388 (std dev 0.011)
  - log(R^B) − log(R) Government Spread (in %, quarterly): 0.144 (std dev 0.006)
  - Λ TFP Growth (in %, quarterly): 0.425 (std dev 0.015)
  - η_i Investment Adjustment Costs: 8.43 (std dev 0.85)
  - θ_w Wage Adjustment Cost: 175.33 (std dev 17.78)
  - χ_w Wage Indexation: 0.707 (std dev 0.041)
  - θ_p Price Adjustment Cost: 62.76 (std dev 4.61)
  - χ_p Price Indexation: 0.421 (std dev 0.044)
  - ω Capital Goods Producer Fees: 0.0248 (std dev 0.0009)
  - θ_b Probability of Banker Survival: 0.919 (std dev 0.044)
  - φ Steady-state Leverage Ratio: 15.96 (std dev 1.35)
  - γ_Π Taylor Rule Coefficient: Inflation: 1.255 (std dev 0.071)
  - γ_R Interest Rate Smoothing: 0.606 (std dev 0.036)
  - γ_y Taylor Rule Coefficient: Output Growth: 0.12 (std dev 0.007)
  - π Inflation Target: 0.972 (std dev 0.097)
  - ρ_b AR(1) Government Debt: 0.833 (std dev 0.098)
  - ρ_λ AR(1) Lambda: 0.999 (std dev 0.0003)
  - σ_{nw} SD Net Worth: 0.184 (std dev 0.021)
  - (Full list preserved in source Tables 2 and 3.)

### Model fit, risk corrections, and shock decomposition
- Overidentification and fit:
  - T = 183; objective function value at optimum = 0.108; n_m = 63; n_Θ = 39; p-value = 0.71 (null not rejected).
  - Model matches means and many second-moment properties; important second-order risk corrections.
- Selected mean and second-order risk-corrected statistics:
  - Non-stochastic steady-state spreads:
    - corporate sector: 0.38 percent (quarterly)
    - government: 0.14 percent
  - Second order spreads:
    - corporate: 0.61 percent
    - government: 0.23 percent
  - Inflation mean:
    - steady state: 0.97
    - second order: 0.91
  - Hours mean:
    - steady state: 0
    - second order: 0.16 percent
- Fit tables (selected entries preserved exactly):
  - GDP Growth Mean (Data): 0.40; Mean (Model): 0.42.
  - Spread BAA-FFR Mean (Data): 0.71; Mean (Model): 0.61.
  - Spread 10Y Bond-FFR Mean (Data): 0.22; Mean (Model): 0.23.
- Shock decomposition (percent of variance explained, selected entries preserved):
  - GDP Growth: TFP 40.7, Inv 9.7, Pref 19.0, Fin 15.8, Mark-ups 2.6, Govt 3.8, Mon 8.6.
  - Investment Growth: TFP 21.2, Inv 14.5, Pref 25.2, Fin 28.4, Mon 8.0.
  - Spread BAA-FFR: Fin 85.5, Mon 9.9.
  - Spread 10Y Bond-FFR: Fin 85.6, Mon 9.8.
- Interpretation:
  - Financial shocks explain about 15 percent of GDP fluctuations, about 28 percent of investment fluctuations, and more than 80 percent of spread fluctuations.
  - Preference and TFP shocks explain large shares of other macro variable variances.

### Implementing UMP in the model
- UMP channels:
  1. Direct lending to firms (central bank credit to private sector):
     - len_t = len^p_t + len^{cb}_t; central bank lending reduces banking sector leverage and corporate spreads.
  2. Purchases of government bonds:
     - B_t = B^p_t + B^{cb}_t; central bank purchases raise bond prices, lower yields and government spreads which, via imperfect substitutability, reduce corporate spreads and stimulate investment.
- Asset-purchase processes used in impulse responses:
  - AR(1) specification coefficient: first lag = 0.985.
  - AR(2) specification coefficients: first lag = 1.3 and second lag = -0.31; largest root = 0.985.
- Main transmission distinction:
  - Direct private lending more effective at reducing corporate spreads than government bond purchases because it directly substitutes private intermediation.

### Impulse-response evidence and dynamics
- Under persistent AR(1) shock:
  - Stock of assets increases on impact; central bank starts unwinding at t = 1.
  - Immediate expansionary effect (reduced spreads, increased investment, labor demand, GDP); effect short-lived and turns negative three quarters after initial shock due to worsened balance sheets of intermediaries when UMP is scaled down.
  - Inflation barely increases on impact and then starts declining.
- Under AR(2) shock (with announcement/commitment effects):
  - Stronger, more persistent expansionary effects; increase in inflation; decline in bank net worth stronger because larger fall in spreads tightens participation constraint more.
  - Persistence lowers spreads longer, improving refinancing conditions and inducing persistent, hump-shaped increases in output and employment.
- Comparative effectiveness:
  - Central bank lending directly to firms yields stronger impact than purchasing government bonds because direct lending targets private spreads more directly.
- Shock-by-shock IRF highlights:
  - Net worth shock: with estimated Taylor rule, lending and GDP fall; with UMP (central bank lending directly to firms) shock is fully offset in model responses.
  - Government debt supply shock: initial crowding-out rises spreads; UMP very effective at insulating real economy.
  - Temporary TFP shock: UMP stabilizes spreads and raises investment/GDP somewhat, but welfare gains small.
  - Investment-specific, consumption preference, and government consumption shocks: UMP has limited macro impact; welfare gains small.

### Welfare analysis, optimal UMP rules, and conditional results
- Welfare evaluation: second-order approximation to utility; central bank UMP rules optimized over persistence ρΨ and responsiveness γΨ, subject to constraint that mean of lencbt or Bcbt (% of GDP) stays within [0,50].
- Main unconditional welfare result (using estimated Taylor rule):
  - "Under an estimated Taylor rule, welfare gains from using Unconventional Monetary Policies (UMP) can be up to 1.45 percent of steady-state consumption."
- Table 7 (selected optimal rules and welfare C.E. in % preserved exactly):
  - Corp., RLt − Rt : ρΨ = 0.6363799 / γΨ = 2.7 / Wt = -577.56 / C.E. (in %) = 1.45
  - Gov., RLt − Rt : ρΨ = 0.767659 / γΨ = 34.6 / Wt = -577.56 / C.E. (in %) = 1.45
  - Gov., R̄Bt − Rt : ρΨ = 0 / γΨ = 0 / Wt = -583.6 / C.E. (in %) = 0
- Conditional (shock-group) optimality:
  - Financial shocks: welfare gain from optimal UMP = 1.34 percent of steady-state consumption; optimal UMP often uses government bonds affecting average corporate lending spread with high inertia ρΨ ≈ 0.971 when only financial shocks present.
  - Demand shocks: small welfare gains (0.35 percent) for some UMP rules; many optimal coefficients zero.
  - Supply shocks: very small welfare gains (0.07 percent) or optimal UMP often not used.
- Sensitivity to conventional monetary rule:
  - Under strict inflation targeting: results similar to estimated rule; welfare gains ≈ 1.45 percent; UMP most valuable for financial shocks.
  - Under optimized Taylor rule targeting price and wage inflation (γR = 0.00, γΠ = 23403.33, γW = 7784.26): welfare improvements from UMP become much smaller; when all shocks are present, optimal unconventional policy may be not to intervene.

### Policy trade-offs, costs, and caveats
- Main benefits:
  - UMP provides an additional instrument for macro stabilization, especially effective for financial shocks affecting bank capital and spreads.
- Recognized costs and limits (not quantified in model):
  - Central bank intermediation inefficiency relative to private financial sector (Gertler and Karadi (2011)); if central bank less efficient, this reduces net gains from direct lending.
  - Interactions with fiscal authority if central bank incurs losses (Del Negro and Sims (2015); Hall and Reis (2015)).
  - Political economy and sectoral allocation costs when central bank lends directly or picks sectors.
  - Diminishing returns to unconventional measures (Krishnamurthy and Vissing-Jorgensen (2011)); UMP effective mainly for large financial shocks or politically feasible scales.
- Robust conclusion:
  - If conventional policy follows a standard estimated Taylor rule or strict inflation targeting, UMP (asset purchases of government and corporate bonds) can yield sizable welfare benefits, concentrated on financial shocks.
  - Under a policy that is nearly optimal for price and wage stabilization, the marginal benefits of UMP are much smaller and may not justify intervention.
  - When costs of direct private lending are large, government bond purchases targeting average lending spreads are preferable.

*Source: wp1785 - 2.1 Non-Financial Firms (excerpt).*

### 2.1 Non-Financial Firms  ...............................................................................................

### wp1785 - 2.1 Non-Financial Firms

### Introduction: context and motivation
- The Global Financial Crisis (2008–09) prompted unprecedented monetary easing and the deployment of unconventional monetary policy (UMP) instruments, including large scale asset purchase programs and liquidity provision.
- Central bank balance sheet ratios cited:
  - Federal Reserve: "about 5.5 percent of annual GDP on average between 1955-2007" and "23.7 percent of GDP in 2016".
  - UK: "6.5 percent on average between 1955-2007" and "22.5 percent in 2015".
  - Euro Area: "13 percent in 2006" and "34.1 percent in 2016".
  - Japan: "10 percent in 1994", "21 percent right before the crisis in 2007", and "88.7 percent in 2016".
- Motivation for studying permanent inclusion of UMP in the policy toolkit:
  - Short-term rates reached effective lower bound; UMP used to further affect spreads between short- and long-term rates and restore impaired credit channels.
  - Debate whether asset purchases should remain in the toolkit once interest rates return to normal.

### Model overview and key mechanisms
- Model base and extensions:
  - Built on Justiniano et al. (2013) New Keynesian framework with nominal and real rigidities.
  - Augmented with a banking sector as in Gertler and Karadi (2013) and lumpy investment / long-term debt as in Andreasen et al. (2013).
- Financial structure and frictions:
  - Banks intermediate household deposits to firms and government, facilitate maturity transformation by holding long-term private loans and long-term government bonds.
  - Long-term private loans arise because firms adjust capital infrequently and cannot renegotiate debt every period; nominal lending rates are constant over loan life.
  - Most outstanding bonds are fixed rate: "only around 2 percent of US Treasuries have a variable coupon and around 90 percent of US corporate bonds are issued as fixed rate bonds."
- Modeling choices and solution method:
  - Second-order approximation to equilibrium conditions.
  - Parameters estimated via Generalized Method of Moments (GMM) matching sixty-three first- and second-order moments from nine macroeconomic and financial time series.
  - Second-order solution used for both estimation and welfare evaluation to capture precautionary motives.

### Transmission channels of UMP in the model
- UMP instruments modeled:
  - Asset purchase programs targeted at either private sector debt (direct lending to firms) or government bonds (Quantitative Easing).
- Transmission via direct purchases of private sector debt:
  - Central bank financing crowds out private intermediation by banks.
  - Because banks are leverage constrained (but the central bank is not), such central bank lending is non-neutral and particularly effective when financial shocks hit bank capital.
- Transmission via purchases of government bonds:
  - Increases banking sector liquidity and lowers government bond yields.
  - Induces a portfolio rebalancing by banks into private securities, reducing corporate spreads and stimulating investment.
- Distinction from conventional policy:
  - Conventional monetary policy affects the short-term deposit rate.
  - UMP targets spreads between long- and short-term rates directly and can affect credit costs of borrowers differently than conventional policy.

### Main quantitative findings and welfare implications
- Welfare gains:
  - "Under an estimated Taylor rule, welfare gains from using Unconventional Monetary Policies (UMP) can be up to 1.45 percent of steady-state consumption."
- Shock-specific effectiveness:
  - UMP is "mostly useful to react to financial shocks", which typically affect bank capital, private sector spreads, investment, and employment.
  - UMP "does not help much with normal 'business cycle' supply and demand shocks", including:
    - TFP and investment-specific technology shocks,
    - Mark-up shocks to price and wage setting,
    - Preference shocks to consumption and labor supply,
    - Government spending shocks.
- Modality of UMP:
  - "Providing direct credit to firms or purchasing government bonds delivers a very similar result."
- Policy-rule dependence:
  - Similar welfare gains from UMP arise under a strict inflation targeting regime.
  - Benefits are "much lower when the central bank follows an optimized Taylor rule that targets price and wage inflation."

### Caveats, costs, and limits acknowledged
- The paper does not model or quantify potential costs of UMP, which may include:
  - Interactions with the fiscal authority if the central bank incurs losses.
  - Less efficient credit intermediation if the central bank lends directly or chooses sectors for credit.
  - Diminishing returns to unconventional measures (e.g., Krishnamurthy and Vissing-Jorgensen, 2011) suggesting UMP may be effective only for large financial shocks or at politically feasible scales.
- Uncertainty about long-term implications:
  - Possible welfare costs when affecting yield curve slope away from market-driven equilibrium values.
  - Potential smaller fiscal revenues if central bank losses occur (Hall and Reis (2015), Del Negro and Sims (2015)).

### Relation to existing literature and contribution
- Contrasts with frameworks that place friction at the household level (Chen et al., 2012) or focus on asset illiquidity (Del Negro et al., 2016).
- Aligns with literature emphasizing intermediation frictions (Gertler and Kiyotaki, 2010; Gertler and Karadi, 2011; Cúrdia and Woodford, 2011) where direct central bank lending or targeted purchases can mitigate disruptions.
- Closest recent studies:
  - Ellison and Tischbirek (2014): advocate coordination where conventional policy responds to inflation and UMP offsets output gap fluctuations (their loss function is non-microfounded).
  - Carlstrom et al. (2016): underscore usefulness of UMP to counter bank-intermediation-rooted shocks.
- Paper's distinctive features:
  - Incorporates long-term credit contracts and a maturity-transformation motive for banks.
  - Estimates the model non-linearly (second-order), enabling accounting for precautionary motives and more accurate welfare evaluation.

### Structure of the paper (summary of following sections)
- Section 2: description of structural model (agents: households, bankers, intermediate goods producers, retailers, final goods producers, capital goods producers, financial intermediaries, central bank, fiscal authority).
- Section 3: econometric methodology for parameter estimation.
- Section 4: model fit.
- Section 5: introduction and implementation of UMP and its transmission.
- Section 6: welfare-maximizing policy analysis.
- Section 7: concluding remarks.

*Source: wp1785 - 2.1 Non-Financial Firms (excerpt) from the provided IMF PDF content.*

### 2.1    Non-Financial Firms

### 2.1    Non-Financial Firms

### Firm types and production structure
- Four firm types in the production sector:
  - Intermediate goods producers: hire labor and purchase capital to produce a homogeneous good; face a Calvo (1983)-type restriction when they upgrade capital (investment lumpy).
  - Retailers: purchase homogeneous goods and produce differentiated goods under monopolistic competition; charge a time-varying mark-up over marginal cost; retail price stickiness implemented via Rotemberg (1982) quadratic cost model; retail prices partially indexed to a combination of steady-state and lagged inflation.
  - Final good producers: aggregate differentiated goods into final goods used for consumption, investment, and government spending.
  - Capital-producing firms: create capital goods subject to flow adjustment costs; sell capital goods to intermediate goods producers and provide maintenance services for a fee proportional to the price of capital (ωP_K_t ̄K_t).

### Intermediate goods producers — optimization and financial contract
- Capital adjustment:
  - Only a fraction (1−θ_k) of intermediate goods producers adjust capital each period (Andreasen et al. (2013)); capital adjusted in current period denoted ̄K_t.
  - Adjustment is financed by credit from financial intermediaries at a fixed nominal interest rate ̄r^L_t until next Calvo signal.
  - Contract interpreted as issuance of a perpetual bond with an embedded option allowing redemption when re-optimizing; contract allows selling capital back to capital goods producers at original price and requires payment of a maintenance fee ωP_K_t ̄K_t.
  - Physical capital is valued at the price of capital when the contract is signed (leasing-like relationship).
- Production and shocks:
  - Cobb-Douglas production: Y^M_t = A_t^(1−α) Z_t (K_{t−1})^α (L^D_t)^(1−α).
  - Productivity shocks:
    - Z_t: stationary shock, AR(1) in logs.
    - A_t: non-stationary shock, AR(1) in logs and first differences.
- Optimization problem (notation preserved from source):
  - Firms solve:
    max_{ ̄K_t, L^D_{t+j|t} } E_t Σ_{j=1}^∞ { (θ_k)^{j−1} β^j Ξ_{t+j}/Ξ_t [ P^M_{t+j}/P_{t+j} Y^M_{t+j|t} − ̄r^L_t ( Π_{i=1}^j P_{t+i}/P_{t+i−1} )^{−1} P_{K_t}/P_t ̄K_t − ω ( Π_{i=1}^j P_{t+i}/P_{t+i−1} )^{−1} P_{K_t}/P_t ̄K_t − W_{t+j} L^D_{t+j} ] }
  - First-order condition for investment (equation (1) in source):
    E_t Σ_{j=1}^∞ { (θ_k)^{j−1} β^j Ξ_{t+j}/Ξ_t [ P^M_{t+j}/P_{t+j} α Y^M_{t+j|t}/ ̄K_t − ( Π_{i=1}^j π_{t+i} )^{−1} ( ̄r^L_t + ω ) P_{K_t}/P_t ] } = 0,
    where π_t ≡ P_t/P_{t−1}.
- Labor: firms adjust labor every period and equate real wages with marginal product of labor; capital-labor ratios identical across firms; aggregate production depends on aggregate capital.

### Capital goods producers — contracts and capital law of motion
- Objective (equation (2) in source):
  - max E_t Σ_{j=0}^∞ β^j Ξ_{t+j}/Ξ_t ( ω V_{t+j}/P_{t+j} − I_{t+j} ).
- Value of outstanding contracts (equation (3)):
  - V_t/P_t = (1−θ_K) Σ_{j=0}^∞ (θ_K)^j P_{K_{t−j}}/P_t ̄K_{t−j}.
- Aggregate demand for capital (equation (4)):
  - K_t = (1−θ_K) ̄K_t + θ_K K_{t−1}.
- Law of motion for aggregate capital with investment adjustment costs (equation (5)):
  - K_t = (1−δ) K_{t−1} + ξ^I_t [ 1 − z(I_t/I_{t−1}) ] I_t,
  where ξ^I_t is an investment shock (AR(1) in logs) and z(·) is increasing, convex.

### Financial contracts and long-term corporate loans
- Banks provide financing to intermediate goods producers; corporate loans are long-term and mature with probability θ_k each period.
- Definition of gross interest: ̄R^L_t ≡ 1 + ̄r^L_t.
- Aggregate real lending to private sector (equation (6)):
  - len_t = (1−θ_k) Σ_{j=0}^∞ (θ_k)^j P_{K_{t−j}}/P_t ̄K_{t−j}.
- Total real revenues on the loan portfolio (equation (7)):
  - rev_t = (1−θ_k) Σ_{j=0}^∞ (θ_k)^j ̄R^L_{t−j} P_{K_{t−j}}/P_t ̄K_{t−j}.
- Average return on private loan portfolio:
  - R^L_t ≡ rev_t/len_t (weighted average of current and past long-term loan rates).

### Long-term government bonds and portfolio returns
- Government issues new debt B^N_t with gross interest ̄R^G_t; net interest ̄r^G_t = ̄R^G_t − 1 every period; principal repaid with probability 1−θ_g; average duration (1−θ_g)^{−1}.
- Law of motion for government bonds (equation (8)):
  - B_t = θ_g B_{t−1} + B^N_t.
- With notation B^N_t = (1−θ_g) ̄B_t, government bond law of motion parallels equation (6).
- Total revenues rev^G_t and average return R^G_t ≡ rev^G_t / B_t (weighted average of past long-term government bond rates).

### Banking sector structure, net worth dynamics, and incentive constraint
- Bank balance sheet (real terms):
  - len_t + b_t = n_t + d_t, where b_t = B_t / P_t, n_t = N_t / P_t, d_t = D_t / P_t.
- Net worth accumulation (equation (9)):
  - n_t = (1−τ_B) [ R^L_{t−1} P_{t−1}/P_t len_{t−1} + R^G_{t−1} P_{t−1}/P_t b_{t−1} − R_{t−1} P_{t−1}/P_t d_{t−1} ] exp(ε^{nw}_t),
    where R_t is the short-term nominal deposit rate, τ_B is insurance premium tax on banks’ profit, and ε^{nw}_t is iid shock to banks’ net worth.
- Banker professional life: each period banker stays with probability θ_B; expected professional life (1−θ_B)^{−1}.
- Insurance agency and startup funds: insurance agency financed by τ_B creates new banks when bankers retire to guarantee outstanding contracts (needed because of long-term debt portfolio heterogeneity).
- Incentive (participation) constraint (equation (10)):
  - V_t ≥ λ_t ( len_t + ∆_t b_t ),
    where V_t is expected terminal wealth, λ_t is time-varying fraction of loans that can be diverted, and λ_t ∆_t is time-varying fraction of government bonds that can be embezzled.
- Implications:
  - If ∆_t < 1, easier to divert corporate bonds than government bonds; excess return on government bonds is only fraction ∆_t of excess return on private securities.
  - Optimal portfolio choice condition (from source):
    (1−τ_B) E_t β Ξ_{t+1}/Ξ_t Ω_{t+1} ( R^L_t − R_t ) P_t/P_{t+1} = λ_t Θ_t/(1+Θ_t),
    where Θ_t is Lagrange multiplier on (10) and Ω_t is shadow value of unit net worth; binding participation constraint implies (R^L_t − R_t) > 0.
  - Imperfect substitutability between corporate and government bonds:
    ( R^G_t − R_t ) = ∆_t ( R^L_t − R_t ).

### Households and wage setting
- Continuum of households comprising workers and bankers who perfectly pool consumption risk; consumption C_t is non-durable; households save only in deposits D_t paying nominal deposit rate R_t.
- Utility: separable in consumption and hours worked; internal habit formation in consumption; intertemporal and intratemporal disturbances enter preferences.
- Labor supply: each household is a monopolistic supplier of specialized labor (Erceg et al. (2000)); wage mark-ups are time-varying; wage rigidity implemented via Rotemberg (1982) quadratic wage adjustment costs; wages partially indexed to past inflation and TFP growth.

### Government, monetary policy, and fiscal variables
- Monetary policy rule targeting CPI inflation π_t and real output growth Y_t / Y_{t−1} with notation:
  - π: inflation target of central bank.
  - R̄: steady-state level of nominal interest rate (denoted R̄ earlier as R̄? preserved notation in source).
  - exp(Λ): growth rate of GDP along balanced growth path.
  - ε^{m,t}: i.i.d. monetary policy shock.
- Deposit rate rule (as in source):
  - R_t/R̄ = (R_{t−1}/R̄)^{γ_R} (π_t/π)^{γ_Π (1−γ_R)} [ Y_t/Y_{t−1} / exp(Λ) ]^{γ_y (1−γ_R)} exp(ε^{m,t}).
- Government spending ratio g_t = G_t / Y_t follows AR(2):
  - log(g_t) = (1−ρ_{g1} − ρ_{g2}) log(g) + ρ_{g1} log(g_{t−1}) + ρ_{g2} log(g_{t−2}) + ε_{g,t}, ε_{g,t} ∼ N(0, σ_g).
- Supply of government bonds as percent of GDP exogenous AR(1):
  - b_t / Y_t = (1−ρ_b) b/Y + ρ_b b_{t−1}/Y_{t−1} + ε_{b,t}, ε_{b,t} ∼ N(0, σ_b).
- Government adjusts lump-sum transfers to satisfy budget given paths for spending and debt/GDP.

### Model estimation approach
- Second-order approximation to equilibrium conditions and household utility for welfare evaluation.
- Estimation via Generalized Method of Moments (GMM) following Christiano and Eichenbaum (1992), Ruge-Murcia (2007), Andreasen et al. (2016) because higher-order solution invalidates standard linearized Kalman-filter likelihood approach.
- Data: seven macroeconomic series (real GDP, real consumption, real investment, hours worked, nominal wage growth, GDP deflator inflation, Federal Funds target) between 1964:2 and 2009:4 (from Justiniano et al. (2013)), plus two spreads: BAA corporate yield − Federal Funds rate, and 10-year Treasury − Federal Funds rate.
- Moment vector M_t composed of first moments, contemporaneous second moments, and persistence terms; dimension 63×1.
- GMM estimator:
  - ˆΘ_GMM = arg min ( 1/T Σ_{t=1}^T M_t − E[M(Θ)] )' W ( 1/T Σ_{t=1}^T M_t − E[M(Θ)] ).
- Two-step weighting: identity matrix first, then inverse of variance-covariance of sample moments using Newey-West with 10 lags.

### Calibrated parameters (Table 1 entries, exact values preserved)
- ε_L Elasticity of Substitution between Labor: 5
- ε_Y Elasticity of Substitution between Goods: 10
- α Capital Share of Output: 0.33
- δ Depreciation Rate: 0.025
- 1/(1−θ_k) Average Duration between Capital Stock Changes: 12
- 1/(1−θ_g) Average Duration of Government Debt: 40
- g Government Spending/Output Ratio: 0.2
- ρ_{g1} AR(1) Coefficient for G_t/Y_t Ratio: 1.288
- ρ_{g2} AR(2) Coefficient for G_t/Y_t Ratio: -0.299
- σ_g Standard Deviation Innovation G_t/Y_t Ratio: 1.07%
- b/Y Debt to GDP Ratio: 0.45
- L Steady-State Hours: 1

### Estimated parameters (Table 2 and Table 3 entries, exact values preserved)
- From Table 2 (GMM estimates and standard deviations):
  - h Habit Formation: 0.742 (std dev 0.026)
  - φ Inverse Frisch Elasticity: 0.847 (std dev 0.077)
  - 1/β − 1 Discount (in %): 0.241 (std dev 0.025)
  - log(R^L) − log(R) Corporate Spread (in %, quarterly): 0.388 (std dev 0.011)
  - log(R^B) − log(R) Government Spread (in %, quarterly): 0.144 (std dev 0.006)
  - Λ TFP Growth (in %, quarterly): 0.425 (std dev 0.015)
  - η_i Investment Adjustment Costs: 8.43 (std dev 0.85)
  - θ_w Wage Adjustment Cost: 175.33 (std dev 17.78)
  - χ_w Wage Indexation: 0.707 (std dev 0.041)
  - θ_p Price Adjustment Cost: 62.76 (std dev 4.61)
  - χ_p Price Indexation: 0.421 (std dev 0.044)
  - ω Capital Goods Producer Fees: 0.0248 (std dev 0.0009)
  - θ_b Probability of Banker Survival: 0.919 (std dev 0.044)
  - φ Steady-state Leverage Ratio: 15.96 (std dev 1.35)
  - γ_Π Taylor Rule Coefficient: Inflation: 1.255 (std dev 0.071)
  - γ_R Interest Rate Smoothing: 0.606 (std dev 0.036)
  - γ_y Taylor Rule Coefficient: Output Growth: 0.12 (std dev 0.007)
  - π Inflation Target: 0.972 (std dev 0.097)
- From Table 3 (GMM estimates and standard deviations):
  - ρ_b AR(1) Government Debt: 0.833 (std dev 0.098)
  - ρ_u AR(1) Preference: 0.967 (std dev 0.015)
  - ρ_I AR(1) Investment: 0.558 (std dev 0.067)
  - ρ_λ AR(1) Lambda: 0.999 (std dev 0.0003)
  - ρ_ψ AR(1) Labor supply: 0.623 (std dev 0.053)
  - ρ_Z AR(1) Transitory TFP: 0.947 (std dev 0.033)
  - ρ_A AR(1) Permanent TFP: 0.289 (std dev 0.029)
  - ρ_{ε_Y} AR(1) Goods Elasticity: 0.871 (std dev 0.186)
  - ρ_∆ AR(1) Delta: 0.124 (std dev 0.019)
  - σ_b SD Government Debt: 0.673 (std dev 0.088)
  - σ_u SD Preference: 0.019 (std dev 0.005)
  - σ_I SD Investment: 0.075 (std dev 0.016)
  - σ_λ SD Lambda: 0.046 (std dev 0.009)
  - σ_ψ SD Labor Supply: 0.144 (std dev 0.023)
  - σ_Z SD Transitory TFP: 0.007 (std dev 0.0005)
  - σ_A SD Permanent TFP: 0.005 (std dev 0.0007)
  - σ_{ε_Y} SD Price Markup: 0.034 (std dev 0.009)
  - σ_∆ SD Delta: 0.138 (std dev 0.039)
  - σ_{ε_L} SD Wage Markup: 0.244 (std dev 0.046)
  - σ_m SD Monetary: 0.0033 (std dev 0.0003)
  - σ_{nw} SD Net Worth: 0.184 (std dev 0.021)

### Key estimation remarks and model fit implications
- Estimation matches first moments, contemporaneous second moments, and persistence using GMM on a second-order approximation.
- Several parameters calibrated ex ante (see Calibrated parameters list); calibrations chosen for identification or external data (e.g., steady-state mark-ups 10% in product and 25% in labor market via ε_Y and ε_L; average duration of capital upgrades 12 quarters; average duration of government debt 40 quarters as 10-year bond counterpart).
- Estimated parameters broadly consistent with previous literature using Bayesian estimation on linearized models; habit formation 0.74; inverse Frisch elasticity 0.84; implied β ≈ 0.9975.
- Estimated mean of financial shock affecting tightness of participation constraint for government bonds (∆_t) implies mean of ∆_t is 0.78 (source statement).

*Source: IMF Working Paper — chapter section 2.1 "Non-Financial Firms" (pdf filename: wp1785 - 2.1    Non-Financial Firms).*

### 1.6 percent annual rate, as in Christianoet al.(2014).  The parameters related to the behavior of

### wp1785 - 1.6 percent annual rate, as in Christianoet al.(2014).  The parameters related to the behavior of

### Parameter estimates and calibration
- Steady-state leverage ratio is close to 16.
- Taylor rule estimates:
  - reaction to inflation deviations: 1.25
  - reaction to output growth deviations: 0.12
  - interest rate smoothing coefficient: 0.6
- Only one shock is extremely persistent: the λ_t shock included in the participation constraint (10).
- The growth rate of the permanent TFP shock displays low persistence.
- Investment adjustment costs, price and wage rigidities, and behavior of the banking sector are within ranges of other model-based evidence or empirical studies.

### Model fit and overidentification test
- The model is overidentified; the J-test specification is J = T h(M_t, Θ)' (S_o)^{-1} h(M_t, Θ) d → χ^2_{n_m−n_Θ}.
- J-test results:
  - T = 183
  - value of the objective function at the optimum = 0.108
  - n_m = 63
  - n_Θ = 39
  - p-value for the null hypothesis that the model is valid = 0.71 (null cannot be rejected)
- Estimation matches E(M_t), E(M_t M'_t), and diag[E(M_t M'_{t−1})]; means, standard deviations and correlations presented for assessing goodness of fit.
- Key observations on means and second-order risk corrections:
  - Model mean fit is very good except investment growth, which is higher in the model than in the data.
  - Non-stochastic steady-state spreads over the Federal Funds rate:
    - corporate sector: 0.38 percent (quarterly basis)
    - government: 0.14 percent
  - Second order (risk-corrected) spreads:
    - corporate sector: 0.61 percent
    - government: 0.23 percent
  - Inflation mean:
    - non-stochastic steady state: 0.97
    - up to second order: 0.91
  - Hours mean:
    - steady state: 0
    - second order approximation: 0.16 percent
- Model has important risk corrections; welfare analysis up to second order requires second order model solution to align baseline welfare with data.

### Table 4 — Selected model fit statistics (Data vs Model)
- GDP Growth:
  - Mean (Data): 0.40
  - Std Dev (Data): 0.86
  - Autocorr (Data): 0.32
  - Mean (Model): 0.42
  - Std. Dev. (Model): 0.85
  - Autocorr (Model): 0.38
- Consumption Growth:
  - Mean (Data): 0.50; Std Dev: 0.52; Autocorr: 0.47
  - Mean (Model): 0.42; Std Dev: 0.50; Autocorr: 0.70
- Investment Growth:
  - Mean (Data): 0.26; Std Dev: 3.32; Autocorr: 0.30
  - Mean (Model): 0.42; Std Dev: 3.33; Autocorr: 0.26
- Wage Growth:
  - Mean (Data): 1.35; Std Dev: 0.73; Autocorr: 0.46
  - Mean (Model): 1.33; Std Dev: 0.71; Autocorr: 0.68
- Inflation:
  - Mean (Data): 0.95; Std Dev: 0.60; Autocorr: 0.87
  - Mean (Model): 0.91; Std Dev: 0.62; Autocorr: 0.89
- Federal Funds Rate:
  - Mean (Data): 1.54; Std Dev: 0.84; Autocorr: 0.95
  - Mean (Model): 1.56; Std Dev: 0.76; Autocorr: 0.87
- Hours:
  - Mean (Data): 0.00; Std Dev: 3.74; Autocorr: 0.98
  - Mean (Model): 0.16; Std Dev: 3.75; Autocorr: 0.95
- Spread BAA-FFR:
  - Mean (Data): 0.71; Std Dev: 0.53; Autocorr: 0.90
  - Mean (Model): 0.61; Std Dev: 0.69; Autocorr: 0.85
- Spread 10Y Bond-FFR:
  - Mean (Data): 0.22; Std Dev: 0.43; Autocorr: 0.88
  - Mean (Model): 0.23; Std Dev: 0.26; Autocorr: 0.84

### Table 5 — Contemporaneous correlations (selected pairs, Data vs Model)
- (GDP, C): 0.57 vs 0.62
- (GDP, INV): 0.88 vs 0.85
- (GDP, W): -0.13 vs -0.13
- (GDP, INFL): -0.24 vs -0.38
- (GDP, FFR): -0.14 vs -0.23
- (GDP, H): 0.12 vs 0.10
- (GDP, BAA-FFR): 0.05 vs 0.06
- (GDP, 10Y-FFR): 0.22 vs 0.06
- (C, INV): 0.34 vs 0.28
- (C, W): -0.11 vs -0.06
- (C, INFL): -0.33 vs -0.45
- (C, FFR): -0.10 vs -0.33
- (C, H): 0.20 vs 0.13
- (C, BAA-FFR): 0.00 vs 0.11
- (C, 10Y-FFR): 0.17 vs 0.11
- (INV, W): -0.05 vs -0.13
- (INV, INFL): -0.11 vs -0.24
- (INV, FFR): -0.10 vs -0.11
- (INV, H): 0.03 vs 0.02
- (INV, BAA-FFR): 0.05 vs 0.01
- (INV, 10Y-FFR): 0.21 vs 0.01
- (W, INFL): 0.66 vs 0.65
- (W, FFR): 0.46 vs 0.52
- (W, H): -0.18 vs -0.24
- (W, BAA-FFR): -0.42 vs -0.03
- (W, 10Y-FFR): -0.43 vs -0.03
- (INFL, FFR): 0.65 vs 0.76
- (INFL, H): -0.38 vs -0.31
- (INFL, BAA-FFR): -0.49 vs -0.07
- (INFL, 10YFFR): -0.52 vs -0.07
- (FFR, H): -0.38 vs -0.42
- (FFR, BAA-FFR): -0.49 vs -0.24
- (FFR, 10YFFR): -0.52 vs -0.24
- (H, BAA-FFR): -0.33 vs -0.26
- (H, 10Y-FFR): -0.24 vs -0.26
- (BAA-FFR, 10Y-FFR): 0.94 vs 0.997

- The estimation procedure gets the bilateral contemporaneous correlation sign right in all thirty-six cases.

### Second-moment fit and limitations
- The model explains standard deviations and autocorrelations fairly well overall.
- Difficulty matching volatility of spreads:
  - Both mean and standard deviation of government sector spreads are a fraction Δ_t of the mean and standard deviation of corporate spreads.
  - Estimated model cannot overcome tight relationship because Δ_t is not volatile enough.
- Persistence:
  - Model explains persistence well overall.
  - Model overpredicts persistence of real consumption and nominal wage growth.

### Table 6 — Shock decomposition (percent of variance explained by aggregated shocks)
- Aggregated shocks: TFP (temporary and permanent), Investment-specific, Preference (intertemporal and intratemporal), Financial (λ_t, Δ_t, net worth, debt supply), Market power (prices and wages), Government, Monetary.
- GDP Growth: TFP 40.7, Inv 9.7, Pref 19.0, Fin 15.8, Mark-ups 2.6, Govt 3.8, Mon 8.6
- Consumption Growth: TFP 47.9, Inv 1.6, Pref 39.4, Fin 0.9, Mark-ups 1.6, Govt 4.5, Mon 4.1
- Investment Growth: TFP 21.2, Inv 14.5, Pref 25.2, Fin 28.4, Mark-ups 2.0, Govt 0.6, Mon 8.0
- Wage Growth: TFP 28.7, Inv 6.8, Pref 56.9, Fin 3.2, Mark-ups 2.2, Govt 1.1, Mon 1.2
- Inflation: TFP 39.3, Inv 3.9, Pref 46.1, Fin 3.3, Mark-ups 4.2, Govt 1.9, Mon 1.3
- Federal Funds Rate: TFP 24.1, Inv 4.1, Pref 4.1, Fin 41.4, Mark-ups 3.5, Govt 2.5, Mon 19.2, (note: table lists monetary as 22.6 in another column alignment)
- Hours: TFP 9.1, Inv 8.3, Pref 50.3, Fin 18.2, Mark-ups 2.9, Govt 9.3, Mon 2.0
- Spread BAA-FFR: TFP 2.5, Inv 0.2, Pref 1.5, Fin 85.5, Mark-ups 0.4, Govt 0.1, Mon 9.9
- Spread 10Y Bond-FFR: TFP 2.5, Inv 0.2, Pref 1.4, Fin 85.6, Mark-ups 0.4, Govt 0.1, Mon 9.8

- Interpretation of decomposition:
  - Both TFP shocks explain around 40 percent of fluctuations in output growth, consumption growth, and inflation.
  - Financial shocks explain about 15 percent of GDP fluctuations; they explain about 28 percent of investment fluctuations and about 18 percent of hours fluctuations.
  - Financial shocks explain more than 80 percent of fluctuations in both spreads (government bonds and corporate bonds).
  - Preference shocks explain between one-third and one-half of fluctuations of most macroeconomic variables, and up to 50 percent of the volatility of hours.
  - Mark-up, monetary policy and government spending shocks explain small fractions of economic fluctuations.

### Implementing Unconventional Monetary Policy (UMP) in the model
- UMP implemented via asset purchase programs; central bank purchases either private (corporate) or government sector debt.
- Two implementation channels discussed:
  1. Direct lending to firms (central bank credit to private sector)
     - Aggregate lending law of motion:
       - len_t = (1−θ_k) (P K_t / P_t) ̄K_t + θ_k (P_{t−1}/P_t) len_{t−1}
       - where len_t = len^p_t + len^{cb}_t; len^p_t = commercial bank credit to private sector; len^{cb}_t = central bank credit to private sector.
     - Central bank lending to private sector reduces banking sector leverage, putting downward pressure on corporate sector spreads.
  2. Purchases of government bonds
     - Government bonds law of motion:
       - B_t = (1−θ_g) ̄B_t + θ_g (P_{t−1}/P_t) B_{t−1}
       - where B_t = B^p_t + B^{cb}_t; B^p_t = commercial bank credit to government; B^{cb}_t = central bank purchases of government bonds.
     - Central bank purchases reduce amount of government debt financed by private financial sector, raising bond prices, lowering yields and government bond spreads, which in turn reduce corporate spreads through imperfect asset substitutability and increase investment, employment and GDP.

### Effects of UMP (impulse-response evidence)
- Impulse responses computed with central bank stock of assets following AR(1) or AR(2) and using a second order approximation to equilibrium conditions.
- AR(1) specification: coefficient of first lag = 0.985.
- AR(2) specification: coefficients of first lag = 1.3 and second lag = -0.31; largest root in both processes = 0.985.
- Two main results:
  - Central bank lending directly to firms yields stronger impact than purchasing government bonds because direct lending directly affects private sector spreads which have stronger macroeconomic impact.
  - AR(2) process yields stronger and more persistent expansionary effects than AR(1); announcement effects and commitment not to unwind UMP in near future are key.
- Under a persistent AR(1) shock:
  - Stock of assets increases on impact and central bank starts unwinding UMP at t = 1.
  - Policies are expansionary on impact due to reduced spreads, increased investment, labor demand and GDP.
  - Effect is short-lived and turns negative three quarters after initial shock because central bank worsens balance sheet position of financial intermediaries; when UMP is scaled down, reduced bank net worth constrains credit supply and leads to investment contraction.
  - Inflation barely increases on impact (wages sticky) and starts declining; quantitative inflation effect is small.
- Under AR(2) shock:
  - Expansionary effects are long-lasting and lead to an increase in inflation.
  - Decline in net worth is stronger due to pronounced fall in spreads so incentive constraint (10) tightens more than under AR(1).
  - Persistence lowers spreads for longer, improving refinancing conditions; forward-looking firms increase investment leading to persistent, hump-shaped increases in output and employment.

### Welfare analysis and policy trade-offs
- Main benefit of UMP as an additional instrument: provides an additional tool for macroeconomic stabilization.
- Main cost of implementing UMP via lending to private sector (Gertler and Karadi (2011)):
  - central bank is more inefficient than private financial sector in intermediating credit; otherwise central bank would optimally replace commercial banks.
  - Difficult to measure inefficiency in data; results provide estimates of how large those costs must be for UMP policies not to be worth pursuing.
- When UMP implemented via purchases of government bonds:
  - unclear that central bank incurs additional inefficiency cost relative to banking sector buying them.
  - estimates provide an upper bound on how large the costs of intermediating credit must be for government-bond purchases to be preferable to direct lending.
- Fiscal implications:
  - If central bank incurs losses due to large scale asset purchases, there may be fiscal implications (Del Negro and Sims (2015); Hall and Reis (2015)).
- Next step in analysis:
  - Compare effects of UMP under the estimated Taylor rule and under more optimal conventional monetary policy rules; study optimality of UMP rules that explicitly react to credit spreads.

*Italic: Source — wp1785 (excerpt).*

### 6.1    Using The Estimated Taylor Rule

### 6.1    Using The Estimated Taylor Rule

### Welfare criterion and methodology
- Welfare criterion: representative household utility
  Wt = ξUt [ log(Ct − hCt−1) − ψt L1+φt / (1+φ) ] + βEt Wt+1
- Second order approximation taken to equilibrium conditions and to the welfare function.
- All parameters set at calibrated or estimated values (as in Tables 1 to 3).
- Central bank lending rule:
  lencbt = ρΨ lencbt−1 + γΨ (RLt/Rt − RL/R), with RL/R the non-stochastic steady state spread.
  - Also experimented with rule reacting to spread on new lending rates ( R̄Lt/Rt − RL/R ).
- Central bank government bond purchases rule:
  Bcbt = ρΨ Bcbt−1 + γΨ (RLt/Rt − RL/R).
  - Also experimented with reactions to spreads on new government debt and to spread between government bond rates and short-term rates.
- For each UMP rule, welfare is optimized over coefficients ρΨ and γΨ, holding the model equilibrium conditions fixed.
- Large penalty in optimization when mean of lencbt or Bcbt (as percent of GDP) falls outside [0,50]; prevents extreme short-selling or huge accumulation of securities.

### 6.1.1 Optimal coefficients — main results
- All policies deliver quantitatively very similar welfare results.
- Highest welfare achieved when central bank buys corporate or government bonds to target the average spread on private sector securities; these policies imply very large responses to spreads that effectively flatten spreads.
- Using corporate versus government bonds yields similar results in estimation reflecting normal-times spread behavior; differences would matter if Δt shock were very volatile.
- UMP policies are not desirable when central bank buys government bonds to affect the spread on new government debt.
- If providing credit to private sector entails inefficiency cost, purchases of government debt targeting the average lending spread are preferred.
- When UMP is endogenous, optimal persistence is high but not unit-root; second lag not studied.

- Table 7: Optimal UMP Policy (policy rows show Policy / ρΨ / γΨ / Wt / C.E. (in %))
  - Corp.,  R̄Lt − Rt : 0.972314 / 142.9 / -577.72 / 1.41
  - Corp., RLt − Rt : 0.6363799 / 2.7 / -577.56 / 1.45
  - Gov.,  R̄Lt − Rt : 0.786566 / 88.6 / -577.8 / 1.4
  - Gov., RLt − Rt : 0.767659 / 34.6 / -577.56 / 1.45
  - Gov.,  R̄Bt − Rt : 0 / 0 / -583.6 / 0
  - Gov., RBt − Rt : 0.953379 / 85.4 / -577.66 / 1.43

### Conditional UMP: shocks grouped and optimality by group
- Shock groups defined:
  - Supply shocks: (i) permanent TFP, (ii) transitory TFP, (iii) investment-specific technology, (iv) labor supply, (v) price mark-up, (vi) wage mark-up.
  - Demand shocks: (i) intertemporal preference consumption shock, (ii) government spending shock, (iii) monetary shock.
  - Financial shocks: (i) bank capital (net worth) shock, (ii) fraction of corporate securities that can be diverted by the banker, (iii) fraction of government securities that can be diverted by the banker, (iv) government debt supply shock.

- Main conditional result: UMP policies are most relevant and deliver largest welfare gains when financial shocks hit the economy.
  - Welfare gain from optimally responding to financial shocks: 1.34 percent of steady-state consumption.
  - When only financial shocks are present, optimal UMP: government bonds affecting the average spread on corporate loans with inertia coefficient ρΨ = 0.971; highly inertial policy.
  - Under demand or supply shocks, UMP brings very small welfare gains (0.35 and 0.07 of lifetime consumption respectively).
  - Under conventional business cycle shocks the best policy often uses government bonds to target the spread on new government rates (a policy not chosen when all shocks are considered).

- Table 8: Optimal UMP Policy, Conditional (selected rows)
  - Demand shocks:
    - Corp.,  R̄Lt − Rt : 0.9926 / 352.9 / -575.9 / 6.13
    - Corp., RLt − Rt : 0.581000000 / 0 / -576.1 / 0.07
    - Gov.,  R̄Lt − Rt : 0.997174 / 4.87 / -575.9 / 6.13
    - Gov., RLt − Rt : 0.581000000 / 0 / -576.1 / 0.07
    - Gov.,  R̄Bt − Rt : 0.051406 / 67.2 / -575.0 / 0.35
    - Gov., RBt − Rt : 0.841000000 / 0 / -576.1 / 0.07
  - Supply shocks:
    - Corp.,  R̄Lt − Rt : 0.050 / 0 / -577.4 / 10
    - Corp., RLt − Rt : 0.871000000 / 0 / -577.2 / 0.05
    - Gov.,  R̄Lt − Rt : 0 / 0 / -577.4 / 10
    - Gov., RLt − Rt : 0.871000000 / 0 / -577.2 / 0.05
    - Gov.,  R̄Bt − Rt : 0.111 / 136.9 / -577.1 / 0.07
    - Gov., RBt − Rt : 0.991000000 / 0 / -577.2 / 0.05
  - Financial shocks:
    - Corp.,  R̄Lt − Rt : 0.911923 / 6.6 / -575.76 / 1.33
    - Corp., RLt − Rt : 0.806204 / 17.1 / -575.76 / 1.33
    - Gov.,  R̄Lt − Rt : 0.801543 / 46.5 / -575.76 / 1.33
    - Gov., RLt − Rt : 0.971929 / 2.1 / -575.74 / 1.34
    - Gov.,  R̄Bt − Rt : 0 / 64317.3 / -576.23 / 1.22
    - Gov., RBt − Rt : 0.955368 / 00.4 / -575.81 / 1.32

- Caveat: costs of asset purchases (e.g., central bank inefficiency relative to commercial banks, fiscal/political costs) are not quantified in the model and should be weighed against benefits.

### 6.1.2 Impulse response analysis — overview by shock
- Analysis compares estimated model vs. model with UMP conducted by purchasing corporate bonds reacting to average lending-deposit spread (optimal UMP in Table 7).
- Impulse responses computed after second order approximation.

- Net worth shock (adverse bank capital shock):
  - Estimated Taylor rule: shock reduces banks’ net worth → decline in lending to private sector → increases spreads for corporate and government sector → investment, employment, private consumption, real GDP decrease. Inflation response muted (lending rates up; real wages fall). Monetary policy follows inflation via Taylor rule.
  - With UMP: shock is completely offset. Central bank lends directly to firms → aggregate lending does not fall and spreads do not move.

- Government debt supply shock (increase in government debt financed by banking sector):
  - Initial crowding out: spreads on government and corporate debt increase → lending, investment, labor demand, GDP, consumption fall. Bank balance sheet position improves due to increased lending markings.
  - Shock is not very persistent (AR(1) coefficient estimated to be 0.833) → spreads return rapidly and macro variables rebound.
  - UMP extremely effective at insulating real economy from this shock.

- Temporary TFP shock:
  - Standard effects: GDP, consumption, investment increase; hours worked and inflation decline. Central bank cuts interest rates.
  - Lending increases immediately but banks take time to accumulate net worth; leverage ratio increases and lending-deposit spread increases.
  - UMP removes financial friction, stabilizes spread completely and generates larger effect on investment and GDP; smaller effects on consumption and labor, explaining small welfare gains from UMP under supply shocks.

- Investment-specific technology shock:
  - Standard effects: investment, GDP, consumption, labor increase; inflation increases (capital goods technology improvement raises marginal cost in consumption goods sector).
  - Total lending declines (price of capital goods cheaper → less borrowing in nominal/consumption terms) → lower spreads, but quantitatively small.
  - UMP targeting spreads yields little change in main macro variables.

- Consumption preference shock:
  - Consumption boom, higher inflation, but lower investment and GDP.
  - Financial accelerator effects small → UMP deployment does not substantially affect main variables including consumption and hours.

- Government consumption shock (lump-sum taxes adjust so debt constant):
  - Standard effects: government spending increases GDP and labor demand; consumption and investment crowded out; inflation and short-term interest rates increase.
  - Investment decline reduces lending → lower spreads and reduced bank net worth.
  - UMP stabilizes financial variables (spreads and bank capital) almost fully; macro variables (consumption, labor) remain very similar, explaining small welfare effect under demand shocks.

### 6.2 Using alternative monetary policy rules — robustness
- Two alternative conventional monetary policy rules analyzed:
  - (i) strict inflation targeting rule.
  - (ii) optimized Taylor rule targeting price and wage inflation.

- Table 9: Optimal UMP Policy, Strict Inflation Targeting (selected rows)
  - All shocks / Corp.,  R̄Lt − Rt : 0.149 / 9.62 / -553.83 / 1.45
  - Demand shocks / Gov., RBt − Rt : 0.841000000 / 0 / -576.1 / 0.31
  - Supply shocks / All : 0 / 0 / -553.67 / 0
  - Financial shocks / Gov.,  R̄Lt − Rt : 0.979 / 163.7 / -575.74 / 1.18

- Under strict inflation targeting:
  - Results virtually unchanged relative to estimated rule: UMP matters; welfare gains 1.45 percent of lifetime consumption.
  - Gains largest under financial shocks.
  - Under supply shocks the optimal UMP is typically not to use it (coefficients zero across alternatives).
  - Under demand shocks welfare gains slightly higher than under the estimated rule.

- Optimized Taylor rule targeting both price and wage inflation:
  - Estimated optimized policy form:
    Rt / R = (Rt−1 / R)^{γR} (πt / π)^{γΠ (1−γR)} [ (W̃t / W̃t−1) exp(Λt) / exp(Λ) ]^{γW (1−γR)} exp(εm,t)
    - γR = 0.00, γΠ = 23403.33, γW = 7784.26
  - Welfare improvements from UMP become even smaller.
  - When all shocks considered, optimal unconventional policy is to not intervene.
  - When evaluated under subsets of shocks, UMP can still be optimal under demand or financial shocks, but effects are substantially lower than under estimated Taylor rule or strict inflation targeting.

- Table 10: Optimal UMP Policy, Price and Wage Inflation Targeting (selected rows)
  - All shocks / All : 0 / 0 / -554.49 / 0
  - Demand / Corp.,  R̄Lt − Rt : 0.0319 / 128.8 / -577.02 / 0.09
  - Supply / All : 0 / 0 / -553.67 / 0
  - Financial / Gov.,  R̄Bt − Rt : 0 / 1.39 / -580.25 / 0.17

### Conclusions and policy implications (from section)
- If monetary policy follows a standard, estimated Taylor rule, adding UMP (asset purchases of government and corporate bonds) as a second instrument can have sizable welfare benefits, especially when the economy is hit by financial shocks.
- This main result holds when central bank follows strict inflation targeting.
- Under a rule targeting price and wage inflation, welfare effects of UMP are much smaller and the optimal policy may be not to intervene when all shocks are considered.
- Under traditional supply and demand shocks, benefits of UMP are much smaller; likely UMP should not be used under supply shocks given small benefits.
- Providing credit to private sector versus purchasing government bonds yields very similar macro effects. If government bond purchases entail lower (or no) costs compared to direct lending to private sector, government bond purchases will be preferable.
- Costs of asset purchases (central bank intermediation inefficiency, fiscal interactions, political economy costs) are not quantified in the model and should be weighed against benefits.

*Source: wp1785 - 6.1 Using The Estimated Taylor Rule (chapter excerpt).*

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*Source: wp1785 - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp1785.pdf_