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