A Macroeconomic Model with Financial Panics

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1 A Macroeconomic Model with Financial Panics Mark Gertler, Nobuhiro Kiyotaki, Andrea Prestipino NYU, Princeton, Federal Reserve Board 1 March The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Board or the Federal Reserve System

2 What we do Incorporate banks and banking panics in simple macro model Broad goal: Develop framework to understand dynamics of recent financial crisis Specific goal: Characterize sudden/discrete nature of financial collapse in fall 28 No observable large exogenous shock Gorton (21), Bernanke (21): Bank runs at heart of collapse Explore qualitatively and quantitatively: Spillover of crisis to real activity Role of monetary policy and macro-prudential policy

3 Motivation GDP Growth, Credit Spreads, and Broker Liabilities during the Financial Crisis 1. GDP Growth and Credit Spreads 2. Broker Liabilities Lehman failure 2.5 Lehman failure Nominal GDP Growth -1.5 BAA-1 Year Treasury Spread

4 Model Overview Simple New Keynesian model with investment Banks intermediate funds between households and productive capital Hold imperfectly liquid long term assets and issue short term debt Vulnerable to panic failure of depositors to roll over short term debt Based on GK (215) and GKP (216) In turn based on Cole/Kehoe(21) self-fulfilling sovereign debt Households may directly finance capital, but less efficient at margin than banks

5 Evolution and Financing of Capital End of period capital S t vs. beginning K t S t = Γ( It K t )K t + (1 δ)k t Γ >, Γ < S t K t+1 : K t+1 = ξ t+1 S t ξ t+1 capital quality shock S b t intermediated by banks; S h t directly held by households S t = S b t + S h t

6 Household and Bank intermediation If S h t /S t > γ, (utility) cost to household of direct finance ς(s h t, S t ) = χ 2 ( Sh t S t γ) 2 S t Marginal rate of return on intermediated capital R b t+1 = ξ t+1 Z t+1+(1 δ)q t+1 Q t Marginal rate of return on directly held capital with ς( ) S h t R h t+1 = 1 = max 1+ ζ( ) S h t R 1 t+1 b Qt λt { } χ( Sh t S t γ), For S h t /S t > γ, increasing marginal cost of direct finance

7 Household and Bank Intermediation NO BANK RUN EQUILIBRIUM Q t S b t D t CAPITAL S t N t! " #$ % h Q S t t HOUSEHOLDS CAPITAL S t BANK RUN EQUILIBRIUM Q * t S t HOUSEHOLDS

8 Bankers Bankers exit with exogenous probability 1 σ Objective V t = E t Λ t,t+1 [(1 σ)n t+1 + σv t+1 ] Net worth n t accumulated via retained earnings - no new equity issues n t+1 = R b t+1q t s b t R t+1 d t if no run = if run Balance sheet Q t s b t = d t + n t

9 Deposit Contract R t+1 deposit rate; R t+1 return on deposits p t run probability; x t+1 < 1 recovery rate Deposit contract: (One period) { Rt+1 with prob. 1 p R t+1 = t x t+1 R t+1 with prob. p t

10 Limits to Bank Arbitrage Moral Hazard Problem: After banker borrows funds at t, it may divert fraction θ of assets for personal use. If bank diverts, creditors can recover the residual funds and shut the bank down. Incentive constraint (IC) θq t s b t V t

11 Solution Endogenous leverage constraint: Q t s b t φ t n t φ t depends on aggregate state only Note: n t bank cannot operate (key for run equilbria)

12 Bank Runs Self-fulfilling bank run equilibrium (i.e. rollover crisis) possible iff: A depositor believes that if other households do not roll over their deposits, the depositor will lose money by rolling over. Condition met iff banks net worth nt goes to zero during a run n t = bank would divert any new deposit

13 Existence of Bank Run Equilibrium Forced liquidation Q t < Q t Q t = E t {(Λ t,t+1 ξ t+1 (Z t+1 + (1 δ)q t+1 )} χ( S h t S t γ) 1 λ t evaluated at Sh t S t = 1. Run equilibrium exists if x t = ξ t(z t + (1 δ)q t )S b t 1 R t D t 1 < 1 or equivalently if ξ t < ξ R t x t (ξ R t ) = ξr t (Z t + (1 δ)q t )S b t 1 R t D t 1 = 1

14 Run Equilibrium Run at t + 1 if : (i) A run equilibrium exists (ii) A sunspot occurs Assume sunspot occurs with probability κ. The time t probability of a run at t + 1 is p t = Pr t {ξ t+1 < ξ R t+1} κ

15 Production, Pricing and Monetary Policy (Standard) Production, resource constraint and Q relation for investment Y t = AK α t L 1 α t Y t = C t + I t + G Q t = Φ( It K t ) Monopolistically comp. producers with quadratic costs of nominal price adjustment (Rotemberg) Monetary policy: simple Taylor rule R n t = 1 β ( P t P t 1 ) κπ (Θ t ) κy

16 Level Annual Basis Points Level Annual Basis Points Level Annual Basis Points Level Response to a Capital Quality Shock: No Run Case Response to a Capital Quality Shock (1 std): No Run Case Baseline No Financial Fricitons Capital Quality.1 Run Probability 1 Bank Net Worth Capital Quality Run Threshold Leverage Multiple:? 2 Investment.5 Output Excess Return: ER b -R free 45 Policy Rate 2 Inflation Quarters Quarters Quarters

17 Level Annual Basis Points Level Annual Basis Points Level Annual Basis Points Level Response to a Sequence of Shocks: Run VS No Run Response to a Sequence of Shocks: Run VS No Run RUN (Run Threshold Shock and Sunspot) NO RUN (Run Threshold Shock and No Sunspot) 1 Capital Quality Capital Quality Run Threshold Initial Threshold.15.1 Run Probability -5 Bank Net Worth Leverage Multiple:? 2 Investment Output Excess Return: ER b -R free 6 Policy Rate 5 Inflation Quarters Quarters Quarters

18 Financial Crisis: Model vs Data SHOCKS: -.3% -.6% -.5% -.8% -.7% 27Q4 28Q1 28Q2 28Q3 28Q4 1. Investment 2. XLF index and Net Worth 3. Spreads (AAA-Risk Free) 1 25 Lehman Brothers 4-1 Bear Sterns -2 Data -3 Model -4 Model No Run q3 28q q3 28q q3 28q GDP 5. Labor (Hours) 6. Consumption q3 28q q3 28q q3 28q4 216

19 Conclusion Incorporated banking sector within conventional macro model Banks occasionally exposed to self-fulfilling rollover crises Crises lead to significant contractions in real economic activity Model captures qualitatively and quantitatively Nonlinear dimension of financial crises The broad features of the recent recent collapse Next steps: Macroprudential policy (Run Externality) Lender-of-last resort policies

20 Conditions for Bank Run Equilibrium with We can simplify existence condition for BRE: x t = Rb t φ t 1 R t φ t 1 1 < 1 R b t = ξt[zt+(1 δ)q t ] Q t 1 ; φ t 1 = Q t 1S b t 1 N t 1 Likelihood BRE exists decreasing in Q ( ) and increasing in φ t 1 φ t 1 countercyclical likelihood BRE exists is countercyclical.

21 Run Equilibrium Threshold RUN THRESHOLD No Run-Equilibrium A Possible.8 Negative Capital Quality shock.6.4 B Run-Equilibrium Possible

22 %" Level Level (Annual %) %" %" Net Level (Annual %) Non-Linearities (or Lack Thereof) due to Occasionally Binding Constraints Fig. 5. Non-Linearities due to Occasionally Binding Constraints Constraint Binds Constraint Slack 1 Investment 2 Price of Capital 7 Real Interest Rate: R free % + 1% Capital Quality Shock -3-1% + 1% Capital Quality Shock 2-1% + 1% Capital Quality Shock 4 Bank Net Worth 1 Leverage multiple:? 2.5 Excess Returns: ER b -R % + 1% Capital Quality Shock 6-1% + 1% Capital Quality Shock - 1% + 1% Capital Quality Shock

23 %" Level Level (Annual %) %" %" Net Level (Annual %) Non-Linearities From Runs Fig. 5. Non-linearities from Runs No Sunspot Sunspot Run Threshold: r = -.9% 1 Investment 5 Price of Capital 5 Real Interest Rate: R free r 1% Capital Quality Shock Net Worth r 1% Capital Quality Shock Leverage r 1% Capital Quality Shock Excess Returns: ER b -R r 1% Capital Quality Shock r 1% Capital Quality Shock r 1% Capital Quality Shock

24 Calibration Parameter Description Value Target Standard Parameters β Impatience.99 Risk Free Rate γ h Risk Aversion 2 Literature ϕ Frish Elasticity 2 Literature ɛ Elasticity of subst across varieties 11 Markup 1% α Capital Share.33 Capital Share δ Depreciation.25 I K =.25 η Elasticity of q to i.25 Literature a Investment Technology Parameter.53 Q = 1 b Investment Technology Parameter -.83% I K =.25 G G Government Expenditure.45 Y =.2 ρ jr Price adj costs 1 Slope of Phillips curve.1 κ π Policy Response to Inflation 1.5 Literature κ y Policy Response to Output.5 Literature Financial Intermediation Parameters σ Banker Survival rate.93 Leverage QSb N = 1 ζ New Bankers Endowments as a share of Capital.1% % I in crisis 35% θ Share of assets divertible.23 Spread Increase in Crisis = 1.5% γ Threshold for S.432 b HH Intermediation Costs S =.5 χ HH Intermediation Costs.65 ER b R = 2% Annual κ Sunspot Probability.15 Run Probability 4% Annual σ(ɛ ξ ) std of innovation to capital quality.75% std Output ρ ξ serial correlation of capital quality.7 std Investment

25 Households Within each household, 1 f workers and f bankers Workers earn wages Bankers manage financial intermediaries and pay dividends Perfect consumption insurance within the family Bankers have finite expected horizons With i.i.d. prob. 1 σ, a banker exits next period. expected horizon = 1 1 σ (Run leads to earlier exit) Replaced by new bankers who receive start-up transfer from the family

26 Household Optimization Choose {C h t, L h t, D t, S h t } to maximize U t = E t i= β i [ln C h t+i 1 1+ϕ (Lh t+i )1+ϕ χ 2 ( Sh t+i S t+i γ) 2 S t+i ] s.t. Ct h + D t + Q t St h = w t L h t + R t D t 1 + ξ t [Z t + (1 δ)q t ]St 1 h + Π t T t

27 Optimal Household Asset Demands Λ t,t+1 β i C h t /C h t+1 ; conditional on run; conditional on no run Deposits: {(1 p t )E t (Λ t,t+1 ) + p t E t (Λ t,t+1 x t+1 )} R t+1 = 1 Capital: E t {Λ t,t ζ( ) S h t 1 Qt λt R b t+1 } = 1

28 Run Probability p t Run at t + 1 if : (i) A run equilibrium exists (ii) A sunspot occurs Condition (i) satisfied if x t+1 = ξ t+1(z t+1 + (1 δ)q t+1 )S b t R t+1 D t < 1 Assume sunspot occurs with probability κ. The time t probability of a run at t + 1 is p t = Pr t {x t+1 < 1} κ Pr t {x t+1 < 1} countercyclical p t countercyclical

29 Level Annual Basis Points Level Annual Basis Points Level Response to a Sequence of Shocks in Flex Price Economy: Run VS No Run Response to the Same Sequence of Shocks in Flex Price Economy: Run VS No Run RUN (Off-Equilibrium) NO RUN 2-2 Capital Quality Capital Quality Run Threshold Initial Threshold.1.5 Run Probability -5 Bank Net Worth Leverage:? 1 Investment Output Excess Return: ER B -R free 5 Natural Rate 1 Consumption Quarters Quarters Quarters

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