A Macroeconomic Model with Financial Panics · Model Overview I Simple New Keynesian model with...
Transcript of A Macroeconomic Model with Financial Panics · Model Overview I Simple New Keynesian model with...
A Macroeconomic Model with Financial Panics
Mark Gertler, Nobuhiro Kiyotaki, Andrea Prestipino
NYU, Princeton, Federal Reserve Board1
March 2018
1The views expressed in this paper are those of the authors and do notnecessarily reflect the views of the Federal Reserve Board or the FederalReserve System
What we do
I Incorporate banks and banking panics in simple macro model
I Broad goal:
I Develop framework to understand dynamics of recent financialcrisis
I Specific goal:
I Characterize sudden/discrete nature of financial collapse in fall2008
I No observable large exogenous shockI Gorton (2010), Bernanke (2010): Bank runs at heart of
collapse
I Explore qualitatively and quantitatively:
I Spillover of crisis to real activityI Role of monetary policy and macro-prudential policy
Motivation
GDP Growth, Credit Spreads, and Broker Liabilities during the Financial Crisis
-3
-2
-1
0
1
2
3
4
5
6
2004 2005 2006 2007 2008 2009 2010-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1. GDP Growth and Credit Spreads
Nominal GDP GrowthBAA-10 Year Treasury Spread
2.5
3.0
3.5
4.0
4.5
5.0
2004 2005 2006 2007 2008 2009 2010
2. Broker Liabilities
Lehman failure
Lehman failure
Model Overview
I Simple New Keynesian model with investment
I Banks intermediate funds between households and productivecapital
I Hold imperfectly liquid long term assets and issue short termdebt →
I Vulnerable to panic failure of depositors to roll over short termdebt
I Based on GK (2015) and GKP (2016)I In turn based on Cole/Kehoe(2001) self-fulfilling sovereign
debt
I Households may directly finance capital, but less efficient atmargin than banks
Evolution and Financing of Capital
I End of period capital St vs. beginning Kt
St = Γ( ItKt
)Kt + (1− δ)Kt
Γ′ > 0, Γ′′ < 0
I St → Kt+1:
Kt+1 = ξt+1St
ξt+1 ≡ ”capital quality” shock
I Sbt intermediated by banks; Sh
t directly held by households
St = Sbt + Sh
t
Household and Bank intermediation
I If Sht /St > γ, (utility) cost to household of direct finance
ς(Sht ,St) = χ
2 (Sht
St− γ)2St
I Marginal rate of return on intermediated capital
Rbt+1 = ξt+1
Zt+1+(1−δ)Qt+1
Qt
I Marginal rate of return on directly held capital
Rht+1 = 1
1+ ∂ζ(·)∂Sht
1Qtλt
Rbt+1
with∂ς(·)∂Sh
t= max
{χ(S
ht
St− γ), 0
}For Sh
t /St > γ, increasing marginal cost of direct finance
Household and Bank Intermediation
���������
�� ���� ��� ��� ����� � � ��
Qt St
Nt
Dt
HOUSEHOLDSCAPITAL
����������� �������������! �"����#�$��%
NO BANK RUN EQUILIBRIUM
Qt St
St
BANK RUN EQUILIBRIUM
HOUSEHOLDStQ* St
CAPITAL
St
h
b
BankersI Bankers exit with exogenous probability 1− σI Objective
Vt = EtΛt,t+1[(1− σ)nt+1 + σVt+1]
I Net worth nt accumulated via retained earnings - no newequity issues
nt+1 = Rbt+1Qts
bt − Rt+1dt if no run
= 0 if run
I Balance sheetQts
bt = dt + nt
Deposit Contract
Rt+1 ≡ deposit rate; Rt+1 ≡ return on depositspt ≡ run probability; xt+1 < 1 ≡ recovery rate
I Deposit contract: (One period)
Rt+1 =
{Rt+1 with prob. 1− ptxt+1Rt+1 with prob. pt
Limits to Bank Arbitrage
I Moral Hazard Problem:
I After banker borrows funds at t, it may divert fraction θ ofassets for personal use.
I If bank diverts, creditors can
I recover the residual funds andI shut the bank down.
I ⇒ Incentive constraint (IC)
θQtsbt ≤ Vt
Solution
I Endogenous leverage constraint:
Qtsbt ≤ φtnt
φt depends on aggregate state only
I Note: nt ≤ 0⇒ bank cannot operate (key for run equilbria)
Bank Runs
I Self-fulfilling ”bank run” equilibrium (i.e. rollover crisis)possible iff:
I A depositor believes that if other households do not roll overtheir deposits, the depositor will lose money by rolling over.
I Condition met iff banks’ net worth nt goes to zero during a run
I nt = 0 → bank would divert any new deposit
Existence of Bank Run Equilibrium
I Forced liquidation → Q∗t < Qt
Q∗t = Et{(Λt,t+1ξt+1(Zt+1 + (1− δ)Qt+1)} − χ(Sht
St− γ)
1
λt
evaluated at Sht
St= 1.
I Run equilibrium exists if
xt =ξt(Zt + (1− δ)Q∗t )Sb
t−1
RtDt−1< 1
or equivalently if ξt < ξRt
xt(ξRt
)=ξRt (Zt + (1− δ)Q∗t )Sb
t−1
RtDt−1= 1
Run Equilibrium
I Run at t + 1 if : (i) A run equilibrium exists (ii) A sunspotoccurs
I Assume sunspot occurs with probability κ.
I → The time t probability of a run at t + 1 is
pt = Pr t{ξt+1 < ξRt+1} · κ
Production, Pricing and Monetary Policy (Standard)
I Production, resource constraint and Q relation for investment
Yt = AKαt L
1−αt
Yt = Ct + It + GQt = Φ( It
Kt)
I Monopolistically comp. producers with quadratic costs ofnominal price adjustment (Rotemberg)
I Monetary policy: simple Taylor rule
Rnt =
1
β(Pt
Pt−1)κπ(Θt)
κy
Response to a Capital Quality Shock: No Run Case
0 20 40 60-2
-1.5
-1
-0.5
0
% "
from
SS
Capital Quality
Capital QualityRun Threshold
0 20 40 600
0.005
0.01
Leve
l
Run Probability
0 20 40 60-20
-10
0
10
% "
from
SS
Bank Net Worth
0 20 40 600
5
10
15
% "
from
SS
Leverage Multiple: ?
0 20 40 60-6
-4
-2
0
2
% "
from
SS
Investment
0 20 40 60-1.5
-1
-0.5
0
0.5
% "
from
SS
Output
0 20 40 60
Quarters
0
100
200
300
Leve
l Ann
ual B
asis
Poi
nts Excess Return: ER b-Rfree
0 20 40 60
Quarters
300
350
400
450
Leve
l Ann
ual B
asis
Poi
nts Policy Rate
0 20 40 60
Quarters
-10
0
10
20
Leve
l Ann
ual B
asis
Poi
nts Inflation
Baseline No Financial Fricitons
Response to a Capital Quality Shock (1 std): No Run Case
Response to a Sequence of Shocks: Run VS No Run
2 20 40 60-2
-1
0
1
% "
from
SS
Capital Quality
Capital QualityRun ThresholdInitial Threshold
2 20 40 600
0.05
0.1
0.15
Leve
l
Run Probability
2 20 40 60-100
-50
0
% "
from
SS
Bank Net Worth
2 20 40 60-500
0
500
1000
1500
% "
from
SS
Leverage Multiple: ?
2 20 40 60-40
-20
0
20
% "
from
SS
Investment
2 20 40 60-8
-6
-4
-2
0
% "
from
SS
Output
2 20 40 60
Quarters
0
500
1000
1500
2000
Leve
l Ann
ual B
asis
Poi
nts Excess Return: ER b-Rfree
2 20 40 60
Quarters
-200
0
200
400
600
Leve
l Ann
ual B
asis
Poi
nts Policy Rate
2 20 40 60
Quarters
-100
-50
0
50
Leve
l Ann
ual B
asis
Poi
nts Inflation
RUN (Run Threshold Shock and Sunspot) NO RUN (Run Threshold Shock and No Sunspot)
Response to a Sequence of Shocks: Run VS No Run
Financial Crisis: Model vs Data
5
0
2510
Data
Model
Model No Run
Lehman Brothers
2004 2007q3 2008q4 2016
5
0
1
2
3
4
2004 2007q3 2008q4 2016 2004 2007q3 2008q4 2016
2004 2007q3 2008q4 2016 2004 2007q3 2008q4 2016 2004 2007q3 2008q4 2016
-25
-50
-75
-100
0
-10
-20
-30
-40
-50
Bear Sterns
0
-5
-10
5
0
-5
-10
0
-5
SHOCKS: -.3% -.6% -.5% -.8% -.7%2007Q4 2008Q1 2008Q2 2008Q3 2008Q4
1. Investment 2. XLF index and Net Worth 3. Spreads (AAA-Risk Free)
4. GDP 5. Labor (Hours) 6. Consumption
Conclusion
I Incorporated banking sector within conventional macro model
I Banks occasionally exposed to self-fulfilling rollover crises
I Crises lead to significant contractions in real economic activity
I Model captures qualitatively and quantitatively
I Nonlinear dimension of financial crises
I The broad features of the recent recent collapse
I Next steps:
I Macroprudential policy (Run Externality)
I Lender-of-last resort policies
Conditions for Bank Run Equilibrium
I We can simplify existence condition for BRE:
xt = Rb∗t
Rt· φt−1
φt−1−1 < 1
with
Rb∗t = ξt [Zt+(1−δ)Q∗
t ]Qt−1
; φt−1 =Qt−1Sb
t−1
Nt−1
I Likelihood BRE exists decreasing in Q∗(·) and increasing inφt−1
I φt−1 countercyclical → likelihood BRE exists iscountercyclical.
Run Equilibrium Threshold
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
¡1
¤
B
Negative Capital Quality shock
ANo Run-EquilibriumPossible
Run-EquilibriumPossible
RUN THRESHOLD
Non-Linearities (or Lack Thereof) due to OccasionallyBinding Constraints
- 1% 0 + 1%
Capital Quality Shock
-40
-20
0
20
40
%"
Bank Net Worth
- 1% 0 + 1%
Capital Quality Shock
6
7
8
9
10
Leve
l
Leverage multiple: ?
- 1% 0 + 1%
Capital Quality Shock
0
0.5
1
1.5
2
2.5
Leve
l (A
nnua
l %)
Excess Returns: ERb-R
- 1% 0 + 1%
Capital Quality Shock
-15
-10
-5
0
5
10
%"
Investment
- 1% 0 + 1%
Capital Quality Shock
-3
-2
-1
0
1
2
%"
Price of Capital
- 1% 0 + 1%
Capital Quality Shock
2
3
4
5
6
7
Net
Lev
el (
Ann
ual %
)
Real Interest Rate: Rfree
Constraint Binds Constraint Slack
Fig. 5. Non-Linearities due to Occasionally Binding Constraints
Non-Linearities From Runs
0r 0 1%
Capital Quality Shock
-100
-50
0
50
%"
Net Worth
0r 0 1%
Capital Quality Shock
0
5
10
15
20
25
Leve
lLeverage
0r 0 1%
Capital Quality Shock
0
5
10
15
20
25
Leve
l (A
nnua
l %)
Excess Returns: ERb-R
0r 0 1%
Capital Quality Shock
-10
-5
0
5
%"
Price of Capital
0r 0 1%
Capital Quality Shock
-10
-5
0
5
Net
Lev
el (
Ann
ual %
)
Real Interest Rate: Rfree
0r 0 1%
Capital Quality Shock
-30
-20
-10
0
10
%"
Investment
No Sunspot Sunspot Run Threshold: 0r= - 0.9%
Fig. 5. Non-linearities from Runs
Calibration
Parameter Description Value Target
Standard Parametersβ Impatience .99 Risk Free Rateγh Risk Aversion 2 Literatureϕ Frish Elasticity 2 Literatureε Elasticity of subst across varieties 11 Markup 10%α Capital Share .33 Capital Shareδ Depreciation .025 I
K = .025η Elasticity of q to i .25 Literaturea Investment Technology Parameter .53 Q = 1b Investment Technology Parameter -.83% I
K = .025G Government Expenditure .45 G
Y = .2ρjr Price adj costs 1000 Slope of Phillips curve .01κπ Policy Response to Inflation 1.5 Literatureκy Policy Response to Output .5 Literature
Financial Intermediation Parameters
σ Banker Survival rate .93 Leverage QSb
N = 10
ζNew Bankers Endowments
as a share of Capital.1% % ∆ I in crisis ≈ 35%
θ Share of assets divertible .23 Spread Increase in Crisis = 1.5%
γThreshold for
HH Intermediation Costs.432 Sb
S = .5
χ HH Intermediation Costs .065 ERb −R = 2% Annualκ Sunspot Probability .15 Run Probability 4% Annual
σ(εξ) std of innovation to capital quality .75% std Outputρξ serial correlation of capital quality .7 std Investment
1
Table 1
Households
I Within each household, 1− f ”workers” and f ”bankers”
I Workers earn wages
I Bankers manage financial intermediaries and pay dividends
I Perfect consumption insurance within the family
I Bankers have finite expected horizons
I With i.i.d. prob. 1− σ, a banker exits next period.
I ⇒ expected horizon = 11−σ (Run leads to earlier exit)
I Replaced by new bankers who receive start-up transfer fromthe family
Household Optimization
Choose {Cht , L
ht ,Dt ,S
ht } to maximize
Ut = Et
∞∑i=0
βi[
lnCht+i − 1
1+ϕ(Lht+i )1+ϕ − χ
2 (Sht+i
St+i− γ)2St+i
]
s.t.
C ht + Dt + QtS
ht = wtL
ht + RtDt−1 + ξt [Zt + (1− δ)Qt ]S
ht−1 + Πt − Tt
Optimal Household Asset Demands
Λt,t+1 ≡ βiCht /C
ht+1; ∗ ≡ conditional on run; − ≡ conditional on
no run
I Deposits:
{(1− pt)E−t (Λt,t+1) + ptE
∗t (Λt,t+1xt+1)} · Rt+1 = 1
I Capital:
Et{Λt,t+11
1+ ∂ζ(·)∂Sht
1Qtλt
Rbt+1} = 1
Run Probability pt
I Run at t + 1 if : (i) A run equilibrium exists (ii) A sunspotoccurs
I Condition (i) satisfied if
xt+1 =ξt+1(Zt+1 + (1− δ)Q∗t+1)Sb
t
Rt+1Dt
< 1
I Assume sunspot occurs with probability κ.
I → The time t probability of a run at t + 1 is
pt = Pr t{xt+1 < 1} · κ
I Pr t{xt+1 < 1} countercyclical → pt countercyclical
Response to a Sequence of Shocks in Flex Price Economy:Run VS No Run
2 20 40 60-6
-4
-2
0
2
% "
from
SS
Capital Quality
Capital QualityRun ThresholdInitial Threshold
2 20 40 600
0.05
0.1
Leve
l
Run Probability
2 20 40 60-100
-50
0
% "
from
SS
Bank Net Worth
2 20 40 60-500
0
500
1000
1500
% "
from
SS
Leverage: ?
2 20 40 60-30
-20
-10
0
10
% "
from
SS
Investment
2 20 40 60-4
-3
-2
-1
0
% "
from
SS
Output
2 20 40 60
Quarters
0
500
1000
1500
2000
Leve
l Ann
ual B
asis
Poi
nts Excess Return: ER B-Rfree
2 20 40 60
Quarters
-1000
-500
0
500
Leve
l Ann
ual B
asis
Poi
nts Natural Rate
2 20 40 60
Quarters
-3
-2
-1
0
1
% "
from
SS
Consumption
RUN (Off-Equilibrium) NO RUN
Response to the Same Sequence of Shocks in Flex Price Economy: Run VS No Run