## _wp15142 ## Source details **Canonical URL:** [_wp15142](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15142.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15142.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15142.pdf.json) --- ### 1. Introduction — background, framework, and main findings - Background and motivation: - Pre-GFC mainstream view: using interest rate policy to counteract financial exuberance was costly or ineffective (Bernanke and Gertler [7], Gilchrist and Leahy [22], Greenspan [24]). - GFC raised question of whether monetary policy should explicitly include financial stability as an independent objective and use specific financial variables as intermediate targets (Borio [9]). - Belief: financial architecture is inherently fragile and asset price corrections can have systemic impacts on financial intermediation and the real economy (Adrian et al [2]). - Objective and modeling approach: - Re-assess optimal monetary policy when financial intermediation can be disrupted, triggering adverse real-economy effects. - New-Keynesian model extended for time-varying systemic risk and potentially highly non-linear financial–real interactions. - Two financial frictions (He and Krishnamurthy [28]): 1. Separation between ownership and management inducing excessively pro-cyclical risk-taking. 2. An equity constraint that makes it difficult for intermediaries to raise funds during stress; the constraint is occasionally binding, creating asymmetry between good and bad times. - Crisis propagation mechanism (key channels): - Initial drop in asset prices → fall in return on equity → equity constraint binds. - Difficulty raising equity forces intermediaries to bear more risk → Sharpe ratio rises. - Higher Sharpe ratio → price of capital q_t must be lower → asset prices fall further → investment falls → macro deteriorates → vicious cycle into financial crisis. - Calibration, solution method, and empirical anchors: - Baseline calibration extends He and Krishnamurthy [28] and follows Adrian and Shin [1]; intermediaries expand balance sheets by borrowing, producing procyclical leverage and asset price co-movement. - Empirical evidence: Adrian and Shin [1]; Nuno and Thomas [33]; Kalemli-Ozcan et al. [29]; Greenlaw et al. [23]. - Model replicates cross-correlation: leverage lags output and is more persistent over the business cycle. - Use third-order solution method to capture changes in risk premia and tail events. - Policy experiments and rules: - Laboratory: analyze simple monetary policy rules within a flexible inflation-targeting framework. - Emphasis on observable systemic risk measure: leverage. - Benchmark comparison: optimal interest rate policy vs. targeted macroprudential tax/subsidy on intermediaries according to leverage deviations from unconditional mean. - Main findings (summary): - Surprise monetary tightening: - Does not necessarily reduce systemic risk, particularly when financial sector is fragile. - Negatively impacts output, inflation, asset prices; raises funding costs and reduces profitability without altering risk-taking behavior. - Negative effects mitigated when financial sector is strong and surprise is small. - Systematic monetary policy reaction: - Simple Taylor-type rules that include financial variables (e.g., leverage) can improve welfare by balancing inflation and output stabilization and reducing likelihood of financial stress. - Limitations of leaning-against-leverage: - Requires leverage to be pro-cyclical; empirical evidence is mixed. - If leverage increases because of asset price falls, raising policy rates can exacerbate corrections and induce crises. - Leaning without distinguishing reasons for leverage increases can lead to policy mistakes. - Alternative financial variable: mis-pricing of risk is appealing (risk aversion rises in crises) but is not directly observable and is less affected by monetary policy. - Macroprudential alternative: counter-cyclical tax (subsidy) on intermediaries is substantially better than interest-rate rule at limiting leverage buildup and preventing crises. - Volatility paradox and output stabilization: - Excessive output stabilization compresses risk premia, raises asset prices and investment, increases leverage needed to finance investment, and heightens vulnerability to sharper negative shocks—analog to Brunnermeier and Sannikov [10]. - Monetary policy reacting too strongly to output in presence of financial frictions can lower welfare. ### 2. The Model — sectors, frictions, and policy instruments - Household sector: - Utility and preferences: - U_t = E_0 Σ_{t=0}^{∞} β^t u(C_t, L_t) with β. - u(C_t, L_t) = [C_t − h C_{t−1} − L_t^{1+η_L}/(1+η_L)]^{1−γ} − 1/(1−γ). - Marginal utility u_{c,t} = [C_t − h C_{t−1} − L_t^{1+η_L}/(1+η_L)]^{−γ}. - Portfolio and participation: - Minimum fraction s of wealth in risk-free deposits B_t earning gross real return R^f_{t−1} = (1 + i_{t−1})/π_{t}. - Remaining fraction 1−s invested in risky financial assets V_t or deposits; portfolio allocation constrained by intermediary reputation e_t: - V_t = min{ e_{t−1}, (1−s) W_t / W̄^{1−τ}_t }. - When τ = 1, in good times V_t/W_t = 1−s; in bad times equity share falls with e_t → procyclical leverage. - Wage setting and wage Phillips curve with Rotemberg costs: - w^n_t = μ_{w,t} mrs_t; π_{w,t} = (1−ω_w) E_t π_{w,t+1} + ω_w π_{w,t−1} − ζ_w (μ_{w,t} − μ_{ss}). - Nominal wage adjustment cost term 0.5 ε_{cw} π_{w,t}^2 Y_t. - Real sector: - Production: - Y_t = A_t L_t^{β} K_{t-1}^{1−β} − ϑ_Y. - y^i_t = (p^i_t / P_t)^{−ε} Y_t. - Capital accumulation: - K_t = (1−δ) K_{t−1} + I_t, investment I_t driven by financial intermediaries; q_t = Q_t / P_t. - Investment relation: I_t = δ K_t + (q_t − 1)/φ (per source expression). - Price adjustment (Rotemberg) Phillips curve (non-linear): - ε_{cp} π_t (1 + π_t) Y_t + (ε − 1) Y_t = ε mc_t Y_t + β E_t [ε_{cp} π_{t+1} (1 + π_{t+1}) Y_{t+1}]. - Marginal cost: - mc_t = w_t^{β} r_{k,t}^{1−β} / A_t^{β} * (1−β)^{1−β}. - TFP processes: - A_t = g_t + ζ_A Ā + (1−ζ_A) A_{t−1} + σ_A ε^A_t. - g_t = ζ_g g_{t−1} + σ_g ε^g_t. - Financial sector: - Intermediary structure: - Ownership–control separation; manager runs intermediary and raises equity V_t and debt B_t so W_t = V_t + B_t to purchase capital. - Reputation evolves: e_t = e_{t−1}^m ṼR_t with m > 0. - Returns and leverage: - Real return on equity: ṼR_t = θ_{t−1} R_t − (θ_{t−1} − 1)(1 + φ_t) R^f_{t−1} = R_t + (θ_{t−1} − 1)(R_t − (1 + φ_t) R^f_{t−1}). - θ_t = W_t / V_t; θ_t − 1 = B_t / V_t. - Optimal leverage: θ_t = e * [E_t R_{t+1} − R^f_t] / (m var_t(R_{t+1})), with E_t R_{t+1} = E_t R_{t+1} + ϖ_t and ϖ_t ~ AR(1). - Realized return and dividends: - R_t = (q_t K_t + D_t) / (q_{t−1} K_{t−1}) (equation 2.5). - D_t = Y_t − δ K_t − w_t L_t. - Sharpe ratio and amplification: - S_{a,t} = m θ_t σ_{R,t}, σ_{R,t} = sqrt{var_t(R_{t+1})}; S_{a,t} rises with θ_t and m. - Binding capital constraint: when e_t < (1−s) W_t / W̄^{1−τ}_t, intermediaries can only raise V_t = e_t equity; negative shocks reduce W_t and e_t, amplifying shocks and potentially increasing leverage when constraint binds; higher leverage raises S_{a,t}, implying lower q_t and depressed investment. - Mispricing of risk: - ω_t = β E_t [u_{c,t+1} / u_{c,t} (e R_{t+1} − R^f_t)]. - In absence of financial frictions ω_t = 0. Mispricing is counter-cyclical: underpricing in good times, overpricing in bad times. - Monetary policy: - Taylor-type rule with smoothing and responses to composite inflation, activity, and financial vulnerability: - i_t = ρ_i i_{t-1} + (1 − ρ_i) (ϕ_π π^c_t + ϕ_x x_t + ϕ_θ #_t) + ε^m_t. - Composite inflation: π^c_t = (1 − ω_w) π_t + ω_w π_{w,t}. - Baseline: ϕ_π = 1.5. - Macroprudential policy: - Countercyclical tax rule: - τ_t = τ̄ + τ' (θ_t − E θ_t). - Stationary equilibrium neutral: τ̄ = 0. - Rule raises cost of issuing debt during high-leverage periods to reduce sensitivity to real shocks. - Aggregation and equilibrium: - Goods market clearing: - Y_t = C_t + I_t + 0.5 φ (i_t − δ)^2 K_t + 0.5 (ε_{cp} π_t^2 + ε_{cw} π_{w,t}^2) Y_t. - Financial sector portfolio equality: Q_t K_t = W_t = V_t + B_t. - Aggregate reputation S_t with Poisson death intensity ξ > 0: S_t = S_{t−1} [e^{m ṼR_t} − ξ]/ξ. - Aggregate equity: V_t = min{ S_t, (1−s) W_t / W̄^{1−τ}_t } (equation 2.6). - Quantitative solution and approximation: - Occasionally binding constraint (2.6) replaced with differentiable function: - V_t = (1−s) W_t / W̄^{1−τ}_t * [1 + 1/Θ (W_t / S_{t−1} − S̄ )^{−3}] (functional form per source). - Simulations use third-order perturbation in Dynare 4.1.1. - Baseline calibration (selected parameter values and shocks): - Main parameters (text paragraph choices): - β = 0.96 - h = 0.3 - L̄ = 1 - η_L = 1 - γ = 1 - ω_w = 0.5 - δ = 0.10 - β (production labor share) = 0.6 - φ = 25 - ϑ_Y = 0.2 - ξ = 0.10 - s = 0.5 - m = 3.75 - τ = 1 - ζ_p = 0.10 - ζ_w = 0.02 - Shock and persistence parameters: - ζ_A = 0.92 - ζ_g = 0.65 - σ_A = 0.005 - σ_g = 0.003 - Calibration notes: - 20 main parameters: 17 conventional; 3 model-specific (s, τ, m). - φ = 25 to match consumption/investment volatility. - Fixed cost ϑ_Y = 0.2 (20% of output) to determine average markup. - Monetary policy composite inflation elasticity ϕ_π = 1.5. - Simulation protocol: - Simulate 3,000 periods, discard first 500 as burn-in. - Financial cycle and crisis definitions: - Systemic crisis when equity constraint binds: V_t / S_t = 1. - Using differentiable approximation, recessions defined when elasticity > 0.5 ⇒ reputation threshold S_t < 0.95. - Systemic crisis when S_t < 0.83 (elasticity ≥ 1). - Baseline probability of systemic crisis ≈ 3 percent (≈ every thirty years on average). - Severe systemic crisis when S_t < 0.66 (point where equity becomes negative in third-order approximation). ### 3. Model Analysis — dynamics, shocks, and welfare - Financial cycle vs. business cycle: - Third-order approximation used to compare impulse responses across states (average reputation vs. low reputation). - Under baseline: - Leverage is more persistent than de-trended output. - Leverage lags the business cycle. - Data: broker-dealer leverage is positively and significantly correlated with output gap, highest correlation at a few quarters lag. - Impact of real shocks and amplification: - Negative productivity shock reduces realized returns for financial sector → lowers risk appetite → excess pricing of risk → asset prices fall → investment and output fall. - In bad state (low reputation) the equity constraint amplifies responses: - Equity premium increases more. - Asset prices decrease more. - Leverage increases rather than decreases in baseline state. - Impact of a monetary policy tightening surprise: - Surprise tightening negatively impacts output, inflation, asset prices; raises funding costs and equity premium; reduces financial sector return on investment and reputation at impact. - Generally reduces leverage; however, if surprise occurs during a bad state, persistent falls in asset prices and output can reduce equity sufficiently to subsequently increase leverage. - Monetary policy surprise produces losses without persistently altering risk-taking; systematic policy behavior is more influential on risk-taking. - State dependence: in bad state, after 4 quarters probability of a more fragile financial sector (negative reputation) after a surprise tightening is higher than absent shock. - Welfare analysis: methodology - Recursive household utility: U_t = u(C_t; L_t) + β E_t U_{t+1}; compute unconditional expectation U = E[U_t] via third-order approximation around deterministic steady state. - Welfare metric: steady state consumption equivalent ϕ (fraction of consumption loss equating U_ss(ϕ) to U_policy). - Numerical search for optimal simple monetary and macroprudential rules over grid {ε_i; ε_p; ε_x, ε_κ} with grid values: - ε_i = 0, - ε_p = [1;3], - ε_x = [0;2], - ε_κ = [0;0:5] - Simulation setup: simulate 100 years (400 quarters) after dropping first 500 observations; penalize instability by adjusting welfare with share of stable simulations. - Key results and policy comparisons: - Absence of systemic risk: - Standard New Keynesian prescriptions prevail. - Optimal central bank target includes composite wage inflation with nominal wage rigidities. - Reacting to output beyond composite inflation stabilization is not welfare improving. - Reacting to output — the volatility paradox: - Reacting to output reduces macro volatilities in mild shocks but compresses risk premia, lowers real rates, raises asset prices and investment, increases leverage, and heightens tail vulnerability. - Intensification of tail events reduces welfare; reacting to output beyond usual reaction is not warranted by financial stability concerns. - Reacting to leverage — trade-offs and instability: - Systematic reaction to leverage improves welfare in normal times but the improvement is small: - Modest reaction ε_κ ≈ 0:28 (policy rate change ~3 to 5 bps) improves welfare by about 0:25 percent in steady state consumption equivalent under baseline calibration. - Leaning-against-leverage reduces inflation and output volatility modestly. - Sensitivities: - Requires leverage pro-cyclicality; if leverage rises due to asset price falls, raising policy rates can exacerbate crises. - Higher weight on leverage increases frequency of severe crises; unadjusted welfare may increase but adjusted welfare (penalized for crises) eventually decreases. - Mass of leverage distribution shifts lower but tails enlarge; inflation volatility increases. - Policy implication: interest-rate-based responses to leverage create trade-offs with inflation mandate and can unintentionally destabilize. - Reacting to mis-pricing of risk: - Less destabilizing than reacting to leverage; overall welfare effects smaller. - Optimal coefficient found 0:56. - Increasing reaction does not increase instability but yields vanishing welfare benefits because mis-pricing is less affected by monetary policy. - Macroprudential policy: - Countercyclical tax on intermediaries (akin to countercyclical capital requirement) delivers largest welfare improvement across considered rules. - Leaning-against-leverage via macroprudential policy yields relative welfare benefit up to 1:2% of steady state consumption equivalent. - Macroprudential policy is more targeted and breaks negative feedback linking equity availability and asset prices (low returns → low equity → low asset prices → low returns). - Interest-rate reactions to leverage are blunt: stabilizing leverage tends to destabilize inflation and reduce output. - Macroprudential gains derive from stabilizing good times and mitigating probability/severity of systemic events. ### 4. Quantitative moments, figures, and selected statistics - Table 1. Summary Statistics (in percent) — Baseline Simulation (Std Dev Recession / Std Dev Non‐Recession / Mean Recession − Mean Non‐Recession): - Real GDP Growth Rate: 5.58 2.94 −5.39 - Private Consumption: 4.57 2.34 −3.67 - Private Business Fixed investment: 14.98 7.12 −14.91 - Hours worked: 6.92 2.74 −4.76 - Moments based on NBER Recessions (1960Q1 − 2014Q2): - Real GDP Growth Rate: 6.00 4.07 −4.06 - Private Consumption: 2.87 2.38 −1.62 - Private Business Fixed investment: 17.50 10.48 −12.68 - Hours worked: 4.62 2.93 −2.93 - Figures and diagnostics (captions and notes summary): - Figure 1: Cyclical properties of debt and equity — ΔW versus ΔV and ΔB (model vs. broker-dealer data, 1960–2014). - Figure 2: Correlation between output and leverage — cross-correlations, HP-filtered real GDP and broker-dealer leverage (1980Q1–2014Q2). - Figure 3: Negative TFP shock — conditional impulse responses for bad vs. average states. - Figure 4a/4b: Monetary policy tightening shock — confidence bands combining monetary with demand/supply shocks and initial states. - Figures 5–11: Welfare comparisons across rules (baseline, leverage vs. output, volatility paradox distributions, monetary policy trade-offs, distribution of leverage, mispricing vs. output, macroprudential vs. interest rates). ### 5. Conclusions — policy implications and trade-offs - Surprise monetary tightenings do not necessarily reduce systemic risk; they reduce output, inflation, asset prices and raise funding costs without persistently altering risk-taking. - Simple Taylor-type rules including financial variables can improve welfare by reducing financial stress risk, but they are not robust: - Leaning-against-leverage depends on leverage pro-cyclicality; empirical evidence is mixed. - Reacting to leverage indiscriminately can exacerbate crises when leverage rises due to asset price falls. - Monetary policy should consider non-linear responses that distinguish normal times from crisis build-up. - Mis-pricing of risk is appealing as an intermediate target but is less influenced by monetary policy and yields modest welfare gains. - Macroprudential countercyclical rules are substantially more effective than interest-rate-based rules in limiting leverage buildup and preventing crises; they are more targeted and better at breaking the equity–asset price feedback loop. - Volatility paradox: excessive stabilization of output can increase systemic vulnerability and reduce welfare. *Source: _wp15142 - sections 1, 2, 4, and references as provided in the source PDF.* ### 1. Introduction ### 1. Introduction ### Background and motivation - Prior to the great Önancial crisis the mainstream view among central banks was that using interest rate policy to counteract Önancial exuberance (such as asset price bubbles) was costly or ine§ective (Bernanke and Gertler [7], Gilchrist and Leahy [22], Greenspan [24]).1 - The global Önancial crisis (GFC) challenged this "benign neglect" approach and raised the question whether monetary policy should explicitly include Önancial stability as an independent objective and use speciÖc Önancial variables as intermediate targets (Borio [9]). - There is a widely held belief that the current Önancial architecture is inherently fragile and that widespread externalities stemming from asset price corrections can have a systemic impact on the Önancial sector, disrupting Önancial intermediation and jeopardizing the normal functioning of the real economy (Adrian et al [2]). ### Model framework and key mechanisms - Objective: Re-assess optimal monetary policy when the Önancial intermediation sector can be subject to disruptions that trigger adverse e§ects for the real economy. - Modeling approach: - Use a New-Keynesian model extended to introduce time-varying systemic risk and to accommodate potentially highly non-linear behavior of Önancial variables and their two-way interaction with the real economy. - Include two Önancial frictions following He and Krishnamurthy [28]: 1. Separation between ownership and management of Önancial intermediaries inducing excessively pro-cyclical risk-taking behavior of the Önancial sector. 2. An equity constraint that makes it di¢cult for Önancial intermediaries to raise funds during periods of Önancial stress. - The equity constraint is occasionally binding, creating substantial asymmetry between good and bad times. - Crisis propagation mechanism: - An initial drop in asset prices can induce a su¢cient fall in the return on equity of the Önancial sector to make the equity constraint bind. - Di¢culty in raising equity forces intermediaries to bear more risk, raising the Sharpe ratio (analog to a rise in risk aversion). - A higher Sharpe ratio implies the price of capital must be lower to deliver higher expected returns, lowering asset prices further. - Lower asset prices reduce investment in physical capital, deteriorating the macroeconomic environment and potentially generating a vicious cycle into a Önancial crisis. ### Calibration, empirical anchors, and solution method - Baseline calibration extends He and Krishnamurthy [28] and follows Adrian and Shin [1], assuming Önancial intermediaries expand their balance sheet by borrowing rather than raising equity, resulting in a positive co-movement between leverage and asset prices.2 - Empirical evidence cited: - Adrian and Shin [1]: using pre-crisis aggregate Önancial accounts data, find leverage increases with total assets for broker-dealers. - Nuno and Thomas [33]: extend data up to 2011 and find leverage is procyclical for both broker dealers and commercial banks. - Kalemli-Ozcan et al. [29]: pre-crisis microdata show leverage is procyclical for investment banks and for large commercial banks; similar to Greenlaw et al. [23]. - Model can qualitatively replicate the cross-correlation between leverage and output found in the data: leverage lags output and is more persistent over the business cycle.3 - Use a third order solution method to account for changes in risk premia and tail events, capturing non-linear behavior of macroeconomic variables in periods of Önancial stress—crucial for evaluating welfare costs related to systemic risk. ### Policy experiments and rules analyzed - Laboratory: analyze the e§ect of simple monetary policy rules on stochastic properties of Önancial variables, systemic risk, and welfare. - Focus: simple, predictable, and learnable rules consistent with a áexible ináation targeting framework. - Observable measures of systemic risk emphasized: leverage. - Benchmark comparison: optimal interest rate policy vs. a simple targeted macroprudential rule where a time-varying tax (subsidy) is levied on Önancial intermediaries according to leverage being above (below) its unconditional mean. ### Main findings - Surprise monetary tightening: - A monetary policy tightening surprise does not necessarily reduce systemic risk, particularly when the Önancial sector is fragile. - The surprise tightening negatively impacts output, ináation, asset prices, and raises funding costs for Önancial intermediaries, reducing proÖtability without altering risk taking behavior. - Negative e§ects of a surprise tightening are mitigated when the Önancial sector is strong and the surprise is small. - Systematic monetary policy reaction: - Systematic policy based on a simple (Taylor type) rule that includes Önancial variables such as leverage can improve welfare by balancing ináation and output stabilization and reducing the likelihood of Önancial stress. - Limitations of leaning-against-leverage: - Requires leverage to be pro-cyclical; empirical evidence is mixed across sectors and time. - If leverage increases because of a sharp fall in asset prices, raising policy interest rates can exacerbate the asset price correction and potentially induce a crisis. - Leaning without distinguishing reasons for leverage increases can lead to policy mistakes. - Alternative Önancial variables: - Measures of mis-pricing of risk have appealing properties because risk aversion (asset price undervaluation) always increases in crisis times. - However, such measures are not directly observable and are less a§ected by monetary policy, yielding only modest welfare improvements. - Macroprudential alternative: - A simple macroprudential rule acting like a counter-cyclical capital requirement (making it more costly to raise debt during good times and vice versa) is substantially better than the interest policy rule at limiting leverage buildup and preventing crisis. - Volatility paradox and output stabilization: - Excessive stabilization of output compresses risk premia, raises asset prices and investment levels, and thus increases leverage required to Önance higher investments. - Higher leverage creates vulnerability to sharper negative shocks, producing sharper downturns—analog to the volatility paradox described in Brunnermeier and Sannikov [10].4 - Consequently, monetary policy reacting to output more than warranted in the absence of Önancial frictions can lower welfare. ### Relation to existing literature - Many existing studies find little or no welfare beneÖt from monetary policy targeting or leaning against Önancial variables5; however, they often differ from this paper in important ways: - Several models limit credit frictions to non-Önancial borrowers or treat asset price deviations/ Önancial shocks as exogenous. - Solution techniques in some literature remove non-linear dynamics crucial for describing crisis impacts and welfare implications. - Notable exception: - Brunnermeier and Sannikov [10] place Önancial frictions at the center of monetary transmission, but in their model Önancial frictions are the only source of ine¢ciencies, removing trade-o§s with traditional monetary policy goals by assumption. - Comparisons: - Svensson [39] and Ajello et al [3] focus on monetary policy surprises and find little welfare gains; the present paper broadly conÖrms those results regarding surprise tightenings but emphasizes the welfare e§ects of systematic policy reactions to Önancial variables that are internalized by private agents.6 ### Paper organization - Section II: present the model. - Section III: calibrate the model and describe how the model matches data for key macro and Önancial variables. - Section IV: discuss model properties and perform welfare analysis for alternative policy rules. - Section V: concluding summary of results. *Source: _wp15142 - 1. Introduction.* ### 2. The Model ### 2. The Model ### Household Sector - Representative household maximizes expected utility flow: - U_t = E_0 Σ_{t=0}^{∞} β^t u(C_t, L_t) with discount factor β. - Instantaneous utility: u(C_t, L_t) = [C_t − h C_{t−1} − L_t^{1+η_L}/(1+η_L)]^{1−γ} − 1/(1−γ), where γ is the inverse of inter-temporal elasticity of substitution, 1/η_L is the Frisch elasticity of labor supply, h captures external habit formation, and parameter >0 accounts for steady state of L_t. - Intertemporal budget constraint (in nominal terms): - W_t = w^n_t L_t − P_t C_t + ṼR_{t−1} + R^f_{t−1} B_{t−1} + D^k_t − 0.5 ε_{cw} π_{w,t}^2 Y_t. - Real wage w_t = w^n_t / P_t; P_t is price of final consumption bundle. - Nominal wage adjustment cost term equals 0.5 ε_{cw} π_{w,t}^2 Y_t. - Asset allocation and portfolio: - Minimum fraction s of household wealth is channeled into risk-free deposits B_t earning gross real return R^f_{t−1} = (1 + i_{t−1})/π_{t}. - Real Euler equation: β E_t [u_{c,t+1} / u_{c,t} * (1 + i_{t−1})/(1 + π_{t+1})] = 1. - Marginal utility u_{c,t} = [C_t − h C_{t−1} − L_t^{1+η_L}/(1+η_L)]^{−γ}. - Remaining fraction 1−s is invested in risky financial assets V_t (stochastic return ṼR_t) or in deposits. - Portfolio allocation to risky assets constrained by intermediary reputation e_t: - V_t = min{ e_{t−1}, (1−s) W_t / W̄^{1−τ}_t } (equation notation per source). - When τ = 1, in good times V_t/W_t = 1−s; in bad times equity share falls with e_t, generating procyclical leverage. - Wage setting and labor supply: - Marginal rate of substitution mrs_t = L_t^{η_L}. - Households have market power in setting nominal wages: w^n_t = μ_{w,t} mrs_t. - Wage Phillips curve governing wage inflation π_{w,t}: - π_{w,t} = (1−ω_w) E_t π_{w,t+1} + ω_w π_{w,t−1} − ζ_w (μ_{w,t} − μ_{ss}). - Wage inflation cost to households: 0.5 ε_{cw} π_{w,t}^2 Y_t. - Parameter ε_{cw} is a function of ζ_w; Rotemberg and Calvo adjustment cost interpretations noted. ### Real Sector (Production) - Firms and technology: - Continuum of monopolistic firms produce differentiated goods: Y_t = A_t L_t^{β} K_{t−1}^{1−β} − ϑ_Y. - Demand for individual firm output: y^i_t = (p^i_t / P_t)^{−ε} Y_t. - Capital accumulation: - K_t = (1−δ) K_{t−1} + I_t, but investment decision I_t is driven by financial intermediaries (firms are owned by intermediaries). - Labor demand: - w_t = mc_t β A_t L_t^{β−1} K_{t}^{1−β}. - Price adjustment and Phillips curve: - Firms face Rotemberg price adjustment costs parameterized by ε_{cp}, yielding a non-linear Phillips curve: - ε_{cp} π_t (1 + π_t) Y_t + (ε − 1) Y_t = ε mc_t Y_t + β E_t [ε_{cp} π_{t+1} (1 + π_{t+1}) Y_{t+1}]. - ε_{cp} is functionally related to Calvo parameter λ_p; first-order equivalence to Calvo noted. - Marginal cost: - mc_t = w_t^{β} r_{k,t}^{1−β} / A_t^{β} * (1−β)^{1−β} (source expression in text). - Total factor productivity (TFP) processes: - A_t = g_t + ζ_A Ā + (1−ζ_A) A_{t−1} + σ_A ε^A_t. - g_t = ζ_g g_{t−1} + σ_g ε^g_t. - Capital goods producers (owned by households) buy I_t to produce investment sold to intermediaries at price Q_t: - Real value of capital q_t = Q_t / P_t; investment optimality condition implies: - I_t = δ K_t + (q_t − 1)/φ (exact relation: I_t = K_t = δ + (q_t − 1)/φ per source). - Capital producers’ profits rebated to households: D^k_t = (q_t − 1)(δ + (q_t − 1)/2φ) K_t; zero only in deterministic steady state. ### Financial Sector - Intermediary structure: - Separation of ownership and control; manager runs intermediary and raises funds from households as equity V_t and debt B_t so that W_t = V_t + B_t used to purchase capital. - Manager maximizes reputation e_t determined by history of realized returns on intermediary equity: e_t = e_{t−1}^m ṼR_t, with m > 0 (manager risk aversion parameter). - Intermediary returns and leverage: - Real return on equity ṼR_t = θ_{t−1} R_t − (θ_{t−1} − 1)(1 + φ_t) R^f_{t−1} = R_t + (θ_{t−1} − 1)(R_t − (1 + φ_t) R^f_{t−1}), where θ_t > 1 is intermediary leverage (assets / equity). - In equilibrium θ_t = W_t / V_t and θ_t − 1 = B_t / V_t. - Optimal leverage from manager maximization (mean–variance type): - θ_t = e * [E_t R_{t+1} − R^f_t] / (m var_t(R_{t+1})), where E_t R_{t+1} = E_t R_{t+1} + ϖ_t and ϖ_t follows AR(1). - Realized return on investment: - R_t = (q_t K_t + D_t) / (q_{t−1} K_{t−1}) (equation 2.5). - Dividends D_t = Y_t − δ K_t − w_t L_t. - Sharpe ratio and risk amplification: - Sharpe ratio S_{a,t} = m θ_t σ_{R,t}, where σ_{R,t} = sqrt{var_t(R_{t+1})}; Sharpe ratio rises with intermediary leverage θ_t and risk aversion m. - Binding capital constraint amplifying shocks: - If e_t < (1−s) W_t / W̄^{1−τ}_t, intermediaries can only raise V_t = e_t equity; negative shocks reduce W_t = q_t K_t and reduce e_t more than one-for-one due to leverage and m > 1, amplifying shocks and potentially increasing leverage when constraint binds. - Higher leverage raises S_{a,t}, which implies lower q_t to deliver higher expected returns (via equation 2.5), depressing investment. - Mispricing of risk: - Defined as ω_t = β E_t [u_{c,t+1} / u_{c,t} (e R_{t+1} − R^f_t)]. - In absence of financial frictions ω_t = 0 (see Appendix referenced). Mispricing is counter-cyclical: underpricing during good times and overpricing during bad times; important mechanism for risk build-up. ### Monetary Policy - Nominal interest rate follows Taylor-type rule with interest rate smoothing and responses to composite inflation, activity, and financial vulnerability: - i_t = ρ_i i_{t−1} + (1 − ρ_i) (ϕ_π π^c_t + ϕ_x x_t + ϕ_θ #_t) + ε^m_t. - Composite inflation: π^c_t = (1 − ω_w) π_t + ω_w π_{w,t}. - Monetary policy shock ε^m_t appended and possibly autocorrelated. - Baseline: monetary policy responds to composite inflation deviations from target with elasticity 1.5 (ϕ_π = 1.5 in baseline calibration). ### Macroprudential Policy - Two motives for macroprudential policy highlighted: - Households do not internalize systemic effects of portfolio allocation; household investment in equity is price insensitive. - Intermediary investment decisions maximize returns without fully capturing household risk preferences or externalities on asset prices → risk mispricing and asset price distortions. - Macroprudential tax rule: - τ_t = τ̄ + τ' (θ_t − E θ_t), where tax increases cost of issuing debt during high-leverage periods, reducing sensitivity of financial sector to real shocks. - In stationary equilibrium τ set to make policy neutral (τ̄ = 0 and E(θ_t − E θ_t) = 0) so net impact on intermediary revenues is zero while raising attractiveness of issuing outside equity. - Rule applies to all intermediaries; practical implementation issues (risk shifting across financial system) acknowledged but are beyond scope. ### Equilibrium Conditions and Aggregation - Goods market clearing: - Y_t = C_t + I_t + 0.5 φ (i_t − δ)^2 K_t + 0.5 (ε_{cp} π_t^2 + ε_{cw} π_{w,t}^2) Y_t. - Financial sector portfolio value equality: - Q_t K_t = W_t = V_t + B_t. - Aggregate reputation and equity: - Aggregating intermediary reputation S_t with Poisson death intensity ξ > 0: S_t = S_{t−1} [e^{m ṼR_t} − ξ]/ξ (source expression per text). - In equilibrium aggregate equity V_t = min{ S_t, (1−s) W_t / W̄^{1−τ}_t } (equation 2.6). ### Quantitative Analysis — Solution Method and Approximation - Non-linear model normally requires global methods; due to dimensionality, occasionally binding constraint (2.6) replaced with differentiable function: - V_t = (1−s) W_t / W̄^{1−τ}_t * [1 + 1/Θ (W_t / S_{t−1} − S̄ )^{−3}] (functional form per source) capturing higher cost of raising equity during bad times. - Simulations use third-order perturbation procedures in Dynare 4.1.1. ### Calibration (baseline parameter choices and data matching) - Main parameter choices (as listed): - β = 0.96 (annual discount rate in text there is inconsistency: in calibration paragraph "The annual discount rate, β, is set at 0.96" while table lists 0.6 — follow paragraph: 0.96). [Note: source lists both; numerical fidelity preserved as presented in text] - h = 0.3 (habit) - L̄ = 1 (steady state labor) - η_L = 1 (inverse Frisch labor elasticity) - γ = 1 (intertemporal elasticity of substitution) - ω_w = 0.5 (wage indexation) - δ = 0.10 (depreciation of capital a.r.) - β (production labor share) = 0.6 (CD-labor share) - φ (capital adjustment cost) = 25 - ϑ_Y = 0.2 (fixed cost in production; equals 20% of output) - ξ = 0.10 (financial intermediary exit rate) - s = 0.5 (liquidity service share) - m = 3.75 (manager risk aversion) - τ (leverage cyclicality parameter) = 1 - ζ_p = 0.10 (price stickiness) - ζ_w = 0.02 (wage stickiness) - Shock and persistence parameters (Table): - ζ_A = 0.92 (supply process persistence) - ζ_g = 0.65 (demand process persistence) - σ_A = 0.005 (supply shock standard dev.) - σ_g = 0.003 (demand shock standard dev.) - Calibration notes and data matching: - 20 main parameters: 17 conventional; 3 specific to this model (s, τ, m). - Depreciation rate 0.10 (10%). - φ = 25 produces realistic relative volatility of consumption and investment. - Wage stickiness ζ_w = 0.02 (wages more sticky than prices ζ_p = 0.10); wage indexation ω_w = 0.5 gives good welfare performance. - Fixed cost chosen equal to 20% of output determines average markup and elasticity of substitution. - Baseline monetary policy response to composite inflation elasticity = 1.5. - Demand and supply processes calibrated to postwar US data from 1960Q1 to 2014Q2 for PCE inflation, real GDP, private consumption, private business fixed investment to match growth-rate volatilities. - Simulation protocol: simulate 3,000 periods, discard first 500 as burn-in. - Financial cycle calibration and crisis definitions: - τ chosen so correlation between leverage and intermediary portfolio value W_t matches data in normal and crisis periods (broker-dealer evidence from Adrian and Shin). - Systemic crisis defined where equity constraint binds: elasticity of equity to reputation = 1 (V_t / S_t = 1). - Using differentiable approximation, recessions defined when elasticity > 0.5, implying reputation threshold S_t < 0.95. - Systemic crisis when reputation S_t < 0.83 (elasticity ≥ 1). - Under baseline calibration probability of a systemic crisis ≈ 3 percent (i.e., systemic crises occur approximately every thirty years on average); chosen to reflect three major financial crises in US over last 100 years. - Severe systemic crisis defined when S_t < 0.66 (technical point where equity becomes negative in third-order approximation — point of no return). *Source: _wp15142 - 2. The Model (chapter/section content from provided PDF).* ### 4. Model Analysis ### 4. Model Analysis ### Financial cycle vs. business cycle - Model uses a third-order approximation and compares impulse responses across states: average reputation vs. low reputation ("bad" state). - Empirical observation: the business cycle and the financial cycle are not perfectly aligned. - Under the baseline calibration: - Leverage is more persistent than de-trended output. - Leverage lags the business cycle. - In the data, broker-dealer leverage is positively and significantly correlated with the output gap, with the highest correlation at a few quarters lag. ### Impact of a real shock to the financial sector and the amplification mechanism - A negative productivity shock in the real sector: - Reduces realized returns in the financial sector and its perception of health. - Reduces risk appetite leading to excess pricing of risk. - Lowers asset price valuations, which in a vicious feedback loop implies lower investment and output. - Amplification in a bad state (financial sector reputation already low): - Equity constraint amplifies the mechanism: sudden drop in capacity of the financial sector to bear risk exacerbates reductions in investment and output. - Equity premium increases more substantially. - Asset prices decrease more substantially. - Leverage increases rather than decreasing in the baseline state. ### Impact of a monetary policy shock on the financial sector - A surprise monetary policy tightening: - Negatively impacts output, inflation, and asset prices. - Raises funding costs and the equity premium, reducing financial sector return on investment and reducing reputation at impact. - Generally leads to a reduction in leverage; however: - If the surprise occurs during a bad state, the more persistent fall in asset prices and deeper fall in output and inflation triggers a sharper reduction in financial equity which can actually lead to a subsequent increase in leverage. - Monetary policy surprise leads to losses without persistently altering risk taking behavior in the financial sector; risk taking is more affected by systematic monetary policy behavior. - Impact on systemic risk is state dependent. - In the bad state, after4quarters the probability of a more fragile financial sector (with negative reputation) after a surprise monetary policy tightening is higher than in absence of the shock. ### Welfare analysis: methodology and computation - Household utility expressed recursively: U_t = u(C_t; L_t) + β E_t U_{t+1}, with U_t = E_t Σ_{j=0}^1 β^j u(C_{t+j}; L_{t+j}). - Third-order approximation of U_t around the deterministic steady state; compute unconditional expectation U = E[U_t] for policy comparisons. - Welfare is ranked by steady state consumption equivalent, ϕ, defined as the fraction of consumption loss required to equate U_ss(ϕ) to U_policy. - Numerical search for optimal simple monetary and macroprudential policy rules over grid {ε_i; ε_p; ε_x, ε_κ} with grid values: - ε_i = 0, - ε_p = [1;3], - ε_x = [0;2], - ε_κ = [0;0:5] - Simulation setup: - Simulate model for 100 years (400 quarters) after dropping the first 500 observations and compute the average value of U_t. - If during simulation reputation drops below a point of no return (about S_t <0:66) record as unstable (severe crisis) and draw another seed; repeat until N_stable simulations obtained. - Penalize instability by adjusting welfare: adjusted (weighted) welfare = average computable welfare times frequency of stable simulations. ### Results in absence of systemic risk - Standard New Keynesian results prevail without financial frictions. - With nominal wage rigidities, optimal central bank target is a composite index including wage inflation. - Once composite inflation is stabilized sufficiently, reacting to the level of output is not welfare improving. - Conclusion: time-varying risk premia and higher order non-linearities do not alter traditional policy prescription in absence of financial frictions. ### Reacting to output: the volatility paradox - Reacting to output (in addition to the baseline prescription) reduces macroeconomic volatilities during mild shocks, at the cost of higher inflation volatility. - Compressing macroeconomic volatility reduces risk premia, lowers real rates, increases asset prices, investment, and capital stock above efficient levels; average output is higher. - Financial sector finances a larger investment portfolio through borrowing and higher leverage, making it more vulnerable to boom-bust cycles when benign shocks are followed by negative shocks. - Intensification of tail events can negatively affect welfare; the number of simulations where reputation drops below its lower bound threshold increases. - Policy implication: reacting to output beyond the usual reaction is not warranted by financial stability concerns. ### Reacting to leverage: risk of financial dominance and unintended consequences - Systematic reaction to leverage improves welfare in normal times but the improvement is small: - A modest reaction to leverage, with ε_κ '0:28, which would typically induce a change in the policy rate that is 3to5bps larger than otherwise, improves welfare by about0:25percent in terms of steady state consumption equivalent under baseline calibration. - Modest leaning-against-the-wind reduces both inflation and output volatility. - Sensitivities and trade-offs: - Result depends on leverage being procyclical. - A higher weight on leverage increases frequency of severe crises; unadjusted welfare may increase but adjusted welfare (penalized for crises) eventually decreases as weight on leverage increases. - Crises cause sharp drops in asset prices, reducing equity more than debt and putting upward pressure on leverage; reacting to leverage in these times can exacerbate crisis. - While mass of leverage distribution concentrates at a lower value, tails become larger. - Higher weight on leverage increases inflation volatility. - Policy implication: reacting to systemic risks via interest rate rules creates a trade-off with the inflation mandate and can produce unintended destabilization. ### Reacting to mis-pricing of risk: a modest effect - Reaction to mis-pricing of risk, ϑ_t = ψ_t: - Entails less destabilization than reacting to leverage. - Overall welfare effects are smaller. - Optimal coefficient found is 0:56. - Increasing the reaction does not lead to increased instability but welfare benefits vanish even with relatively mild shocks. - Mis-pricing of risk is not highly affected by monetary policy, making it a less appealing intermediate target despite more favorable stabilization properties. ### Macroprudential policy - A macroprudential rule (countercyclical tax on financial intermediaries akin to countercyclical capital requirement) that increases cost of funding when funding costs are low and leverage is high delivers the largest welfare improvement across considered rules. - Leaning-against-leverage via macroprudential policy relative to interest rate policy yields a relative welfare benefit up to1:2%of steady state consumption equivalent. - Rationale: macroprudential policy is more targeted and breaks the negative feedback loop linking equity availability and asset prices (low returns → low equity → low asset prices → low returns). - Interest rate reactions to leverage are blunt: in stabilizing leverage they tend to destabilize inflation and reduce output. - Macroprudential gains derive both from stabilizing in good times and mitigating probability and severity of systemic events. ### Conclusions (key takeaways) - Monetary policy surprises (tightenings) do not necessarily reduce systemic risk, especially when the financial sector is fragile. Surprise tightenings reduce output, inflation, and asset prices and raise funding costs, lowering financial profitability without altering risk-taking; systematic monetary policy affects risk-taking more. - Simple Taylor-type rules that include financial variables such as leverage can improve welfare by balancing inflation/output stabilization and reducing likelihood of financial stress, but: - Simple reactions to leverage are not robust or sufficiently nuanced. - Leaning against leverage requires leverage to be pro-cyclical; empirical evidence is mixed. - When leverage increases due to sharp asset price falls, raising policy rates can exacerbate corrections and induce crises. - Monetary policy should go beyond simple linear rules; consider non-linear responses that differentiate normal times from crisis buildup. - Alternative financial variables, like measures of mis-pricing of risk, have appealing properties but are less affected by monetary policy and yield modest welfare improvements. - A simple macroprudential countercyclical rule is substantially better than an interest-rate-based rule in limiting leverage buildup and preventing crisis. *Source: _wp15142 - 4. Model Analysis* ### References ### _wp15142 - References ### References (bibliographic list) - [1] Adrian, T., and Shin, H., (2013), "Procyclical Leverage and Value-at-Risk", National Bureau of Economic Research, Working Paper Series 18943. - [2] Adrian, T., D. Covitz, N. Liang, (2014), "Financial Stability Monitoring", Federal Reserve Bank of New York Sta§ Reports, N 601. - [3] Ajello, A., Laubach, T., Lopez-Salido, D., and Nakata., T., (2015), "Financial Stability and Optimal Interest-Rate Policy", working paper, Federal Reserve Board. - [4] Angeloni, I., and Faia, E., (2013), "Capital regulation and monetary policy with fragile banks," Journal of Monetary Economics, Elsevier, vol. 60(3), pages 311-324. - [5] Bagehot, W., (1873), "Lombard Street, a description of the money market", Scriber, Armstrong, and Co, New York. - [6] BIS 2011. ìThe Transmission Channels Between the Financial and Real Sectors: A Critical Survey of the Literature.î BIS Working Paper 18. - [7] Bernanke, B., and Gertler, M., (2001), "Should Central Banks Respond to Movements in Asset Prices?" American Economic Review, 91(2): 253-257 - [8] Bernanke, B.S., Gertler, M., Gilchrist, S., (1999). "The Önancial accelerator in a quantitative business cycle framework". In: Taylor, J.B., Woodford, M.(Eds.), Handbook of Macroeconomics, vol.1., Elsevier, Amsterdam, TheNetherlands, pp. 1341ñ1393. (Chapter21). - [9] Borio, C., (2014), "The Önancial cycle and macroeconomics: What have we learnt?", Journal of Banking & Finance, Elsevier, vol. 45(C), pages 182-198. - [10] Brunnermeier, M., and Sannikov, Y., (2014), ìA Macroeconomic Model With A Financial Sectorî. American Economic Review 104.2: 379-421. - [11] Carlstrom, C. T., and Fuerst, T. S. (1997), "Agency costs, net worth, and business áuctuations: A computable general equilibrium analysis", American Economic Review, 893-910. - [12] Carlstrom, C., T., Fuerst, T., S., and Paustian, M., (2010), ìOptimal Monetary Policy in a Model with Agency Costsî, Journal of Money, Credit and Banking 42 (s1): 7ñ70. - [13] Clouse, J.,.(2013), "Monetary Policy and Financial Stability Risks," Finance and Economics Discussion Series 2013-41. Board of Governors of the Federal Reserve System (U.S.). - [14] Christiano, L., Eichenbaum, M., and Evans, C., (2005), "Nominal Rigidities and the Dynamics E§ects of a Shock to Monetary Policy", Journal of Political Economy 113 (1), 1-45. - [15] Cociuba, S., Shukayev, M., and Ueberfeldt, A., (2012). "Collateralized Borrowing and Risk Taking at Low Interest Rates?," University of Western Ontario, Economic Policy Research Institute Working Papers 20121. - [16] De Groot, O., (2014). "The Risk Channel of Monetary Policy," Finance and Economics Discussion Series 2014-31. Board of Governors of the Federal Reserve System (U.S.). - [17] Dewachter, H., Wouters, R., (2014), "Endogenous risk in a DSGE model with capital-constrained Önancial intermediaries", Journal of Economic Dynamics and Control, 43: 241-268. - [18] Erceg, C., Henderson, D., and Levin, A., (2000), "Optimal monetary policy with staggered wage and price contracts", Journal of Monetary Economics, Elsevier, vol. 46(2), pages 281-313, October. - [19] Faia, E., and Monacelli, T., (2007), "Optimal Interest Rate Rules, Asset Prices and Credit Frictions", Journal of Economic Dynamics and Control, vol. 31(10), 3228-3254. - [20] Galati, G., and Moessner, R., (2013), "Macroprudential Policy ñA Literature Review," Journal of Economic Surveys, Wiley Blackwell, vol. 27(5), pages 846-878, December. - [21] Gertler, M., and Karadi, P., (2011), "A model of unconventional monetary policy", Journal of Monetary Economics, Elsevier, vol. 58(1), pages 17-34, January. - [22] Gilchrist, S., and Leahy, J., (2002), "Monetary policy and asset prices," Journal of Monetary Economics, Elsevier, vol. 49(1), pages 75-97, January. - [23] Greenlaw, D., Hatzius, J., Kashyap, A., and Shin, H. (2008): "Leveraged Losses: lessons from the Mortgage market Meltdown", Proceedings of the U.S. Monetary Policy Forum 2008. - [24] Greenspan, A., (2002). ìOpening Remarks,î Federal Reserve Bank of Kansas City Economic Symposium Rethinking Stabilization Policy: 1-10. - [25] Greenwood, J., Hercowitz, Z., and Hu§man, G., (1988), "Investment, capacity utilization, and the real business cycle", American Economic Review 78 (3): 402-17. - [26] Guerrieri, L., and Iacoviello, M., (2015), "Occbin: A Toolkit to Solve Models with Occasionally Binding Constraints Easily", Journal of Monetary Economics, March, vol. 70, pages 22-38. - [27] Hayashi, F., (1982), "Tobinís marginal Q and average Q: A neoclassical interpretation", Econometrica, 50, 215-224. - [28] He, Z., and Krishnamurthy, A., (2014), "A Macroeconomic Framework for Quantifying Systemic Risk", NBER Working Papers 19885, National Bureau of Economic Research, Inc. - [29] Kalemli-Ozcan, S., Sorensen, B., and Yesiltas, S. (2012): "Leverage Across Firms, Banks and Countries", Journal of International Economics 88(2). - [30] Kim, S., Kollmann, R., and Kim, J., (2010) "Solving the incomplete market model with aggregate uncertainty using a perturbation method", Journal of Economic Dynamics and Control 34(1), 50-58. - [31] Kiyotaki, N., and Moore, J., (1997) "Credit cycles" Journal of Political Economy 105(2), 211ñ48 - [32] Lombardo, G., & Vestin, D., (2008). "Welfare implications of Calvo vs. Rotemberg-pricing assumptions", Economics Letters, 100(2), 275-279. - [33] Nuno, G., and Thomas, D. (2013): "Bank Leverage Cycles", ECB working paper 1524. - [34] Quint, D., and Rabanal, P., (2014), "Monetary and Macroprudential Policy in an Estimated DSGE Model of the Euro Area", International Journal of Central Banking, Vol. 10, No. 2, pp. 169-236. - [35] Rotemberg, J., and Woodford, M., (1999), "Interest rate rules in an estimated sticky price model" In ëMonetary Policy Rulesí NBER Chapters (National Bureau of Economic Research) pp. 57ñ126. - [36] Schmitt-Grohe, S., and Uribe, M., (2007), "Optimal, Simple, and Implementable Monetary and Fiscal Rules", Journal of Monetary Economics 54, September, 1702-1725. - [37] Smets, F., and Wouters, R., (2007), "Shocks and Frictions in US Business Cycles: A Bayesian DGSE Approach", îAmerican Economic Review 97(3), 586-606. - [38] Svensson, L., E., O., (2010), "Ináation Targeting", in Friedman, Benjamin M., and Michael Woodford, eds., Handbook of Monetary Economics, Volume 3B, chapter 22, Elsevier 2010. - [39] Svensson, L., E., O., (2014), Ináation Targeting and "Leaning against the Wind", International Journal of Central Banking, 10(2), 103-114. - [40] Taylor, J., B., (1979), ìEstimation and Control of a Macroeconomic Model with Rational Expectations,î Econometrica 47(5), 1267ó 1286. - [41] Taylor, J., B., (1993), "Discretion versus policy rules in practice", Carnegie-Rochester Conference Series on Public Policy 39, 195ñ214. - [42] Woodford, M., (2012), ìInáation Targeting and Financial Stability,î Sveriges Riksbank Economic Review, 2012:1, p. 7-32. ### Appendix: The E¢cient Allocation (summary of model structure and key equations) - Household sector (no financial sector): - Objective: maximize expected utility flow U_t = E_0 Σ_{t=0}^1 β^t u(C_t; L_t) with discount factor β (denoted as in text). - Intertemporal budget constraint: C_t + q_t I_t = w_t L_t + r^k_t K_{t-1} + Div_cp_t + Div_t. - Law of motion of physical capital: K_t = (1 - ϵ) K_{t-1} + I_t (.1). - Capital Euler: q_t = β E_t (u_{c,t+1}/u_{c,t}) [(1 - ϵ) q_{t+1} + r^k_{t+1}] (.2). - Labor supply: w_t = -u_{l,t}/u_{c,t} = L_t^ϕ (equation (.3) denotes w_t = L^_t form in source). - Household with explicit financial sector: - Modified budget constraint: C_t + p^s_t x_t + B_t = w_t L_t + (p^s_t + d_t) x_{t-1} + (1 + r_{t-1}) B_{t-1} + Div_cp_t. - Asset pricing conditions: - p^s_t = β E_t (u_{c,t+1}/u_{c,t}) [p^s_{t+1} + d_{t+1}] (.4). - 1/(1 + r_t) = β E_t (u_{c,t+1}/u_{c,t}) (.5). - Definitions: V_t = p^s_t x_t, eR_{t+1} = (p^s_{t+1} + d_{t+1}) / p^s_t, W_t = V_t + B_t. - Consolidated household constraint: C_t + W_t = w_t L_t + eR_t V_{t-1} + (1 + r_{t-1}) B_{t-1} + Div_cp_t. - Asset pricing with adjusted discount factor ~m_{t,t+1} = m_{t,t+1} x_{t-1}/x_t: - p^s_t = E_t ~m_{t,t+1} [p^s_{t+1} + d_{t+1}] (.7). - q_t = E_t ~m_{t,t+1} [(1 - ϵ) q_{t+1} + r^k_{t+1}] (.8). - 1 = E_t ~m_{t,t+1} [1 + r_t] (.9). - Veil condition: if m_{t,t+1} = β u_{c,t+1}/u_{c,t} and equilibrium implies E_t ~m_{t,t+1} = E_t m_{t,t+1}, then x_t = x_{t-1} and banking sector is a veil. ### Appendix: Real Sector (production, costs, and investment) - Production technology and demand: - Y_t = F(L_t, K_{d,t}) - ̄Y = A_t L_t^α K_{t-1}^{1-α} - ̄Y (.10). - Demand for individual firm output: y^i_t = (p^i_t / P_t)^{-ε} Y_t. - Marginal cost and markup: - Marginal cost mc_t = w_t^α r_{k,t}^{1-α} A_t^{-α} (1 - α)^{1-α} = 1/μ_p with μ_p = ε/(ε - 1). - Labor demand condition: w_t = α A_t (K_{t-1} / L_t)^{1-α} = μ_p (.11). - Rental rate: r^k_t = (1 - α) A_t (L_t / K_{t-1})^α = μ_p. - Capital producers: - Investment cost function: ̄(I_t/K_{t-1}; K_{t-1}) = 0.5 χ (i_t - ϵ)^2 / K_t (text notation). - Optimal investment relation: I_t / K_t = ϵ + (q_t - 1)/χ (.12). - Capital producer profits: Div_cp_t = (q_t - 1)(ϵ + (q_t - 1)/(2χ)) K_t. - Resource constraint: - Aggregate output: Y_t = C_t + I_t + ̄(I_t/K_{t-1}; K_{t-1}). ### Tables and key statistics (as reported) - Table 1. Summary Statistics (in percent) — Baseline Simulation (Std Dev Recession / Std Dev Non‐Recession / Mean Resession - Mean Non‐Recession) - Real GDP Growth Rate: 5.58 2.94 -5.39 - Private Consumption: 4.57 2.34 -3.67 - Private Business Fixed investment: 14.98 7.12 -14.91 - Hours worked: 6.92 2.74 -4.76 - Moments based on NBER Recessions (1960Q1 - 2014Q2) - Real GDP Growth Rate: 6.00 4.07 -4.06 - Private Consumption: 2.87 2.38 -1.62 - Private Business Fixed investment: 17.50 10.48 -12.68 - Hours worked: 4.62 2.93 -2.93 ### Figures: captions and notes (selected) - Figure 1. Cyclical Properties of Debt and Equity - Scatter plots of changes in total financial assets (ΔW) versus changes in equity (ΔV) and debt (ΔB) in the model and in the data for broker-dealers from financial accounts. Data sample is 1960-2014. - Figure 2. Correlation between Output and Leverage - Charts represent cross-correlation of de-trended output (denoted by 1) and leverage (denoted by 2) in the model and the data. C11 is the autocorrelation of output; C22 is the autocorrelation of leverage; C12 is the correlation between output and leverage. Data is HP-filtered (lambda=1600) real GDP and broker-dealer leverage 1980Q1-2014Q2. - Figure 3. Negative Total Factor Productivity Shock - Note: A bad state refers to a state with low reputation. A bad state impulse response function is defined as the mean reaction conditional to the 4-quarter average of reputation being below its 2.5th percentile. The average state impulse response function is defined as the unconditional mean reaction. - Figure 4a / 4b. Monetary Policy Tightening Shock - Confidence bands show the uncertainty related to the combination of the monetary policy shock with demand and supply shocks and the initial state of the economy. - Figure 5. Welfare: Baseline without Systemic Risk - Welfare is measured in terms of weighted steady state consumption equivalent. Higher (less negative) values in red indicate higher welfare. - Figure 6. Welfare: Leverage vs. Output stabilization - Welfare is measured in terms of weighted steady state consumption equivalent. Higher (less negative) values in red indicate higher welfare. - Figure 7. Volatility Paradox: Distribution of Output, Inflation and Leverage - Red circled distributions are under the baseline Taylor-type monetary rule where there is no reaction to output gap (߶_π = 0). The black solid line is when the central bank reacts to output gap with a coefficient of 2 (߶_π = 2). - Figure 8. Monetary Policy Trade-Offs: Leaning against Leverage - Top left: composite inflation and output volatility as the weight on leverage goes from 0 (indicated with a star) to 0.5. - Top right: adjusted welfare as a function of the weight on leverage. - Bottom left: non-adjusted welfare as a function of the weight on leverage. - Bottom right: share of stable simulations as the weight on leverage increases. - Numeric axis labels excerpted in figure: 0.5 1 1.5 2; other plotted axes include 0 0.1 0.2 0.3 0.4 and welfare values -2.8 to -2.55, consumption equivalent welfare values -2.2 to -1.6, share of stable simulations 0.55 to 0.8. - Figure 9. Distribution of leverage - Histograms of leverage for a path of the simulated economy with the baseline monetary policy rule (߶_R = 0) and with a monetary policy rule with a higher weight on leverage (߶_R = 0.25). - Figure 10. Welfare: Risk Mispricing vs. Output stabilization - Welfare is measured in terms of weighted steady state consumption equivalent. Higher (less negative) values in red indicate higher welfare. - Figure 11. Welfare: Macroprudential vs. Interest Rates - X-axis is the weight on leverage in both the macroprudential and interest rate rule. Welfare is expressed in terms of weighted steady state consumption equivalent. *Source: _wp15142 - References (source PDF content provided).* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp15142.pdf_