using elasticities to derive optimal bankruptcy exemptions · using elasticities to derive optimal...

Using Elasticities to DeriveOptimal Bankruptcy Exemptions

Eduardo Davila

NYU Stern

Structural reforms in the wake of recovery: Where do we stand?Bank of SpainJune 18 2015

Eduardo Davila (NYU Stern) Optimal Bankruptcy Exemptions 1 / 16

Motivation

Question

How large should bankruptcy exemptions be?

• Exemption: dollar amount borrower gets to keep if he does not repay

• Substantial variation on exemptions across regions/time

• This paper1. Characterizes the optimal bankruptcy exemption and dW

dm

• Very generally

2. As a function of a few measurable sufficient statistics• Calibrates optimal exemption


Main argument

Risk AverseBorrowers

Risk NeutralLenders

Borrow from

First Best: Full Insurance(((((((((((((hhhhhhhhhhhhhFirst Best: Full InsuranceKey Friction: Debt contract (baseline model)

Incomplete markets (extension)

Second Best: Bankruptcy exemption provides partial insurance

Exemption ↑

Marginal Benefit:More consumption

when bankrupt

Marginal Cost:Higher interest rates

Effects on Borrowing,Default Decision,

Labor Supply,Moral Hazard, etc

Optimality

Do not matterdirectly

Only throughsufficient statistics

OptimalExemption

m∗

Sufficient statistic logic(CAPM analogy)


Main argument


Risk NeutralLenders

Borrow from

First Best: Full Insurance

(((((((((((((hhhhhhhhhhhhhFirst Best: Full InsuranceKey Friction: Debt contract (baseline model)



Exemption ↑


when bankrupt




Optimality



OptimalExemption

m∗



Main argument


Risk NeutralLenders

Borrow from

First Best: Full Insurance

(((((((((((((hhhhhhhhhhhhhFirst Best: Full Insurance

Key Friction: Debt contract (baseline model)Incomplete markets (extension)


Exemption ↑


when bankrupt




Optimality



OptimalExemption

m∗



Main argument


Risk NeutralLenders

Borrow from

First Best: Full Insurance(((((((((((((hhhhhhhhhhhhhFirst Best: Full Insurance

Key Friction: Debt contract (baseline model)Incomplete markets (extension)


Exemption ↑


when bankrupt




Optimality



OptimalExemption

m∗



Main argument


Risk NeutralLenders

Borrow from

First Best: Full Insurance(((((((((((((hhhhhhhhhhhhhFirst Best: Full InsuranceKey Friction: Debt contract (baseline model)



Exemption ↑


when bankrupt




Optimality



OptimalExemption

m∗



Outline of the talk

1. Baseline model• Positive analysis• Welfare analysis ⇒ dW

dm and m∗ (main results)

2. Extensions

3. Calibration

4. Conclusion


Environment (baseline model)• Two dates t = 0, 1

• Risk averse borrowers

maxB0

U (C0) + βE[max

U(CD1), U(CN1)]

,

C0 = y0 + q0 (B0,m)B0 CN1 = y1 −B0

CD1 = min y1,m

1. Stochastic endowment y1 (assets), cdf F (·), support on[y1, y1

]

2. Debt contract (key friction)

3. Constant bankruptcy exemption: m dollars4. Regularity conditions on F (·) and preferences5. Equilibrium: borrowers internalize q0 (B0,m)

• Risk neutral lenders

• Required return 1 + r∗, fraction δ deadweight loss in bankruptcy• Zero profit




maxB0

U (C0) + βE[max

U(CD1), U(CN1)]

,

C0 = y0 + q0 (B0,m)B0 CN1 = y1 −B0

CD1 = min y1,m


]

2. Debt contract (key friction)3. Constant bankruptcy exemption: m dollars4. Regularity conditions on F (·) and preferences5. Equilibrium: borrowers internalize q0 (B0,m)

• Risk neutral lenders

• Required return 1 + r∗, fraction δ deadweight loss in bankruptcy• Zero profit




maxB0

U (C0) + βE[max

U(CD1), U(CN1)]

,

C0 = y0 + q0 (B0,m)B0 CN1 = y1 −B0

CD1 = min y1,m


]


• Risk neutral lenders• Required return 1 + r∗, fraction δ deadweight loss in bankruptcy• Zero profit




maxB0

U (C0) + βE[max

U(CD1), U(CN1)]

,

C0 = y0 + q0 (B0,m)B0 CN1 = y1 −B0

CD1 = min y1,m


]

2. Debt contract (key friction)

3. Constant bankruptcy exemption: m dollars4. Regularity conditions on F (·) and preferences5. Equilibrium: borrowers internalize q0 (B0,m)





maxB0

U (C0) + βE[max

U(CD1), U(CN1)]

,

C0 = y0 + q0 (B0,m)B0 CN1 = y1 −B0

CD1 = min y1,m


]

2. Debt contract (key friction)3. Constant bankruptcy exemption: m dollars

4. Regularity conditions on F (·) and preferences5. Equilibrium: borrowers internalize q0 (B0,m)





maxB0

U (C0) + βE[max

U(CD1), U(CN1)]

,

C0 = y0 + q0 (B0,m)B0 CN1 = y1 −B0

CD1 = min y1,m


]

2. Debt contract (key friction)3. Constant bankruptcy exemption: m dollars4. Regularity conditions on F (·) and preferences

5. Equilibrium: borrowers internalize q0 (B0,m)





maxB0

U (C0) + βE[max

U(CD1), U(CN1)]

,

C0 = y0 + q0 (B0,m)B0 CN1 = y1 −B0

CD1 = min y1,m


]




Borrowers’ problem

• Two economic decisions

• Period t = 1: Default (given B0)• Period t = 0: Borrowing B0


• Two economic decisions• Period t = 1: Default (given B0)

• Period t = 0: Borrowing B0


• Two economic decisions• Period t = 1: Default (given B0)• Period t = 0: Borrowing B0




C1

y1

45

y1 y1




C1

y1

45

y1 y1B0

CN1 = y1 −B0




C1

y1

45

y1 y1B0

CN1 = y1 −B0

mCD1 = miny1,m




C1

y1

45

y1 y1B0

CN1 = y1 −B0

mCD1 = miny1,m

m m+B0

Default Repayment




C1

y1

45

y1 y1B0

CN1 = y1 −B0

mCD1 = miny1,m

m m+B0

ForcedDefault

StrategicDefault



U ′ (C0)

[q0 (B0,m) +

∂q0 (B0,m)

∂B0B0

]= β

∫ y1

m+B0

U ′ (y1 −B0) dF (y1)



U ′ (C0)

[q0 (B0,m) +

∂q0 (B0,m)

∂B0B0

]= β

∫ y1

m+B0

U ′ (y1 −B0) dF (y1)

• Marginal Benefit: funds raised at t = 0 (accounting for price impact)• Marginal Cost: repayment at t = 1 only if no default



U ′ (C0)

[q0 (B0,m) +

∂q0 (B0,m)

∂B0B0

]= β

∫ y1

m+B0

U ′ (y1 −B0) dF (y1)

• Marginal Benefit: funds raised at t = 0 (accounting for price impact)• Marginal Cost: repayment at t = 1 only if no default• Characterizes equilibrium borrowing (as a function of m)

B0(m)



U ′ (C0)

[q0 (B0,m) +

∂q0 (B0,m)

∂B0B0

]= β

∫ y1

m+B0

U ′ (y1 −B0) dF (y1)

• Marginal Benefit: funds raised at t = 0 (accounting for price impact)• Marginal Cost: repayment at t = 1 only if no default• Characterizes equilibrium borrowing (as a function of m)

B0(m)

dB0

dmR 0 (income, substitution and direct effects)

Lenders’ interest rate schedule

• Risk neutral pricing

q0 (B0,m) =δ∫m+B0

my1−mB0

dF (y1) +∫ y1m+B0

dF (y1)

1 + r∗

• Properties

∂q0 (B0,m)

∂B0< 0 and

∂q0 (B0,m)

∂m< 0

• More borrowing ⇒ Higher interest rates

• Higher exemptions ⇒ Higher interest rates


Main Result 1: Marginal Change (directional test)

• Social welfare W (m) is given by borrowers utility




Marginal change in exemption

dW

dm= U ′ (C0)

∂q0

∂mB0

︸︷︷︸Marginal Cost: moreexpensive borrowing

+

∫ m+B0

mβU ′

(CD1)dF (y1)

︸︷︷︸Marginal Benefit: more

consumption when bankrupt

• Intuition• dB0

dm and changes in default decision do not appear• Borrowing and default are done optimally




Marginal change in exemptiondWdm

U ′ (C0)C0= −Λεr,m︸︷︷︸

Marginal Cost

+1

m

Πm

CD1

C0︸︷︷︸Marginal Benefit





U ′ (C0)C0= −Λεr,m︸︷︷︸

Marginal Cost

+1

m

Πm

CD1


Λ ≡ q0B0

y0 + q0B0Leverage

εr,m ≡∂ log (1 + r)

∂mInterest rate sensitivity





U ′ (C0)C0= −Λεr,m︸︷︷︸

Marginal Cost

+1

m

Πm

CD1


Πm

CD1

C0≡∫ m+B0

m

CD1C0

βU ′(CD1)

U ′ (C0)dF (y1) Price-Consumption ratio





U ′ (C0)C0= −Λεr,m︸︷︷︸

Marginal Cost

+1

m

Πm

CD1


Πm

CD1

C0≡∫ m+B0

m

CD1C0

βU ′(CD1)


• Cash Flow and Discount Rate effects





U ′ (C0)C0= −Λεr,m︸︷︷︸

Marginal Cost

+1

m

Πm

CD1


Πm

CD1

C0≡∫ m+B0

m

CD1C0

βU ′(CD1)


• Same formula for P/D ratios as in Consumption-Based AP





U ′ (C0)C0= −Λεr,m︸︷︷︸

Marginal Cost

+1

m

Πm

CD1


Πm

CD1

C0≡∫ m+B0

m

CD1C0

βU ′(CD1)


• CRRA Utility:

Πm

CD1

C0= β

∫ m+B0

m

(CD1C0

)1−γ

dF (y1)


Main Result 2: Optimal Exemption

• dWdm = 0 (under regularity conditions) ⇒ m∗

• Remarks

1. High Λεr,m and LowΠmCD

1 C0

⇒ Low m (and viceversa)2. Variables are endogenous and observable

• Sufficient statistic logic (CAPM analogy)• Similar to optimal taxation problems

3. Exact expression (no approximations)




Optimal exemption

m∗ =

ΠmCD1 C0

Λεr,m

• Remarks


1 C0







Optimal exemption

m∗ =

ΠmCD1 C0

Λεr,m

• Remarks


1 C0

⇒ Low m (and viceversa)

2. Variables are endogenous and observable






Optimal exemption

m∗ =

ΠmCD1 C0

Λεr,m

• Remarks


1 C0





Figure: Social welfare

65 70 75 80 85 9088.92

88.94

88.96

88.98

89

89.02

89.04

89.06

89.08

89.1

89.12

W (m)

Exemption m

Welfare


Extensions

1. Endogenous labor income and effort choice (moral hazard) M. Hazard

2. Non-pecuniary losses Utility loss

3. Epstein-Zin utility Epstein-Zin

4. Multiple contracts Multiple

5. Heterogeneous borrowers Heterogeneous


Extensions






• Social welfare function W =∫λ (i)W (i) dG (i)

m∗ =

EG,A[

Πm,iCD1i

C0i

]

EG,A [Λiεri,m]

• EG,A[·] are cross sectional averages (for active borrowers)• Observed heterogeneity: exclusion• Unobserved heterogeneity: pooling and exclusion


Extensions






6. Price taking borrowers Price taking

7. Bankruptcy exemptions contingent on aggregate risk Aggregate Risk

8. Endogenous income: labor wedges and aggregate demand Agg. demand

9. Dynamics Dynamics


Calibrationm∗ =

βπm

(CD

1C0

)1−γ

Λεr,m

• Baseline model with CRRA

• Further assumptions• Fully secured collateralized credit ⇒ εrcollat,m = 0• Borrowers own a house (to use variations in homestead exemptions)

• Validity of calibration• m∗ assumes that right hand side variable are constant ⇒

approximation error• dW

dm more accurate and useful for policymaker (simple test)


Calibrationm∗ =

βπm

(CD

1C0

)1−γ

Λεr,m

Parameter/Variable Value SourceCD1C0

Consumption change 0.9 Filer, Filer 05 PSID

Λ Leverage 0.0775 Livshits, MacGee, Tertilt AER07

πD Default probability (ch.7) 0.008 ”

πm|D No-Asset bankruptcy 0.1 Lupita ABILawReview12

εr,m Credit spread sensitivity 2.5 · 10−7 Gropp, Scholz, White QJE97

β Discount factor 0.96γ Risk aversion 1, 5, 10, 20, 50


Calibrationm∗ =

βπm

(CD

1C0

)1−γ

Λεr,m








• Debt to personal income q0B0

y0= 8.4%⇒ Λ


Calibrationm∗ =

βπm

(CD

1C0

)1−γ

Λεr,m








• πm = πD︸︷︷︸0.008

×πm|D︸︷︷︸0.1

(zero recovery by unsecured creditors ≈ 90%)


Calibrationm∗ =

βπm

(CD

1C0

)1−γ

Λεr,m








• Increasing m by $100,000, increases credit spread by 250 basis points


Calibrationm∗ =

βπm

(CD

1C0

)1−γ

Λεr,m









Calibration

m∗ =

βπm

(CD

1C0

)1−γ

Λεr,m

• Optimal exemption m∗ (in dollars):

γ = 1 (log) γ = 5 γ = 10 γ = 20 γ = 50

m∗ 39,638 60,416 102,314 293,435 6,922,079

• References• MA,NV homestead exemption: $500, 000; NJ none• Average exemption US: $70, 000

• Size of welfare gains at $70, 000 using dWdm formula ($10, 000 change)

γ = 1 (log) γ = 5 γ = 10 γ = 20 γ = 50dWdm

U ′(C0)C0-0.0084% -0.0027% 0.0089% 0.062% 1.897%


Calibration dWdm

U ′ (C0)C0= −Λεr,m +

1

mβπm

(CD1C0

)1−γ

• Optimal exemption m∗ (in dollars):

γ = 1 (log) γ = 5 γ = 10 γ = 20 γ = 50

m∗ 39,638 60,416 102,314 293,435 6,922,079

• References• MA,NV homestead exemption: $500, 000; NJ none• Average exemption US: $70, 000

• Size of welfare gains at $70, 000 using dWdm formula ($10, 000 change)

γ = 1 (log) γ = 5 γ = 10 γ = 20 γ = 50dWdm

U ′(C0)C0-0.0084% -0.0027% 0.0089% 0.062% 1.897%


Literature

1. Sufficient Statistics/Public Economics: Diamond 98, Saez 01,Shimer-Werning 07, Chetty 09, Arkolakis et al 12

2. General Equilibrium with Incomplete Markets: Zame 93,Dubey-Geanakoplos-Shubik 05

3. Quantitative Literature:• Structural: Chatterjee-Corbae-Nakajima-Rios-Rull 07,

Livshits-MacGee-Tertilt 07,• Microeconometric: Gross-Souleles 02, Fay-Hurst-White 02, Fan-

White 03, Severino-Brown-Coates 14

4. Security Design: Ross 76, Allen-Gale 94, Duffie-Rahi 95

5. Optimal Contracting: many papers


Conclusion

1. This paper has characterized optimal bankruptcy exemptions• As a function of a few sufficient statistics• For a wide range of environments• Sensible calibration (measurement is challenging)

2. New paper on mortgage design (with John Campbell)• Optimal recourse• ARM vs FRM


Figures: Interest rate schedule

25 30 35 40 450.72

0.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

q0 (B0,m)

Loan Size B0

Price

65 70 75 80 85 900.882

0.884

0.886

0.888

0.89

0.892

0.894

0.896

0.898

0.9

0.902

q0 (B0,m)

Exemption m

Price


Figures: Borrowers’ choices

25 30 35 40 4587

87.5

88

88.5

89

89.5

J (B0,m)

Loan Size B0

Welfare

65 70 75 80 85 9027

28

29

30

31

32

33

34

35

B0 (m)

Exemption m

LoanSize


Extensions



Extensions


• Same expression for m∗ if disutility of labor is separable (frictionlesslabor markets)

• Different characterization of default region: does not matter for m∗

(insight applies more generally)• Intuition: optimality


Extensions




Extensions



• Same expression for m∗

• Effects work through spreads in εr,m and default regions• Implicit assumption: no renegotiation• Easy to add externalities/internalities


Extensions





Extensions





Πm

CD1

C0≡(Q

C0

)γ− 1ψ

β

∫ m+B0

m

(CD1C0

)1−γ

dF (y1)

• Where Q is the certainty equivalent of t = 1 consumption


Extensions





Πm

CD1

C0≡(Q

C0

)γ− 1ψ

β

∫ m+B0

m

(CD1C0

)1−γ

dF (y1)

• Where Q is the certainty equivalent of t = 1 consumption• Corrected Stochastic Discount Factor


Extensions






Extensions





• Optimal exemption

m∗ =

ΠmCD1

C0∑Jj=1 Λjεrj ,m

Λj ≡q0jB0j

y0 +∑Jj=1 q0jB0j


Extensions






m∗ =

ΠmCD1

C0∑Jj=1 Λjεrj ,m

Λj ≡q0jB0j

y0 +∑Jj=1 q0jB0j

• Complete markets as special case ⇒ m∗ = 0


Extensions







Extensions







m∗ =

EG,A[

Πm,iCD1i

C0i

]

EG,A [Λiεri,m]

• EG,A[·] are cross sectional averages (for active borrowers)• Observed heterogeneity: exclusion• Unobserved heterogeneity: pooling and exclusion


Extensions







• Euler equation

U ′ (C0) q0 = β

∫ y1

φm+B0

U ′(CN1

)dF (y1)


Extensions







• Euler equation

U ′ (C0) q0 = β

∫ y1

φm+B0

U ′(CN1

)dF (y1)


m∗ =

ΠmCD1

C0

Λεr,mwhere εr,m =

d log(1 + r)

dm

• Full GE effect


Extensions








• Aggregate shocks ω ∈ Ω, we can condition exemptions on those

m∗ (ω) =

Πm(ω)CD1 C0

Λεr,m(ω), ∀ω


Extensions









m∗ =

ΠmCD1 C0

+ΠN

τ(ω)

dY (ω)dmm

C0

Λεr,m

• Where 1 + τ(ω) = w1(ω)A is the labor wedge


Extensions









9. Dynamics Dynamics

m∗ =

∑Tt=1

Πm,tCDt C0∑T−1

t=0 ΠN ,t gtΛtεrt,m• Simple formula interpreted as steady state• Income process subsumed in sufficient statistics (permanent vs.

transitory shocks, health shocks, family shocks, etc...)


Sign of dB0

dm

sign

(U ′′ (C0)

∂q0

∂mB0

[q0 +

∂q0

∂B0B0

]+ U ′ (C0)

[∂q0

∂m+

∂2q0

∂B0∂mB0

]

+ βU ′ (m) f (m+B0))

Back to text


m∗ in (almost) closed form

m∗ =

(β (1 + r∗)

Υ + δ (1−Υ)

) 1γ

C0

Υ =f (m∗ +B0)B0

F (m∗ +B0)− F (m∗)

• Υ = 1 for uniform (measure of curvature)

Back to text


Moral Hazard/Elastic Labor Supply

• Borrowers’ problem

maxC0,C1y1 ,B0,ξy1 ,N0,N1y1 ,a

U (C0, N0; a) + βEa [U (C1, N1)]

s.t. C0 = y0 + w0N0 + q0B0

CN1 = y1 + w1NN1 −B0; CD1 = min y1,m

• Two new optimality conditions

∂U

∂a(C0, N0; a) + β

∫V (C1, N1) fa (y1; a) dy1 = 0

w0∂U

∂C(C0, N0; a) = − ∂U

∂N(C0, N0; a)

• Same m∗

Back to text


Moral Hazard/Elastic Labor Supply: DefaultDecision

U (·)

Income

U (m)

m y1y1 y1

U (y1; 0) V(y1;B

b0, w1

)

U (m)


Non-Pecuniary Losses

• Assumes that renegotiation is not possible

• Non-pecuniary loss: U (φC), where φ ∈ [0, 1]

• Same m∗

• Only default region changes

if y1 < φm+B0 Default

Back to text


Borrowers Internalize Price Response

• Borrowers optimality condition

U ′ (C0)

[q0 +

∂ql0∂B0

Bb0

]= β

∫ y1

φm+Bb0

U ′(CN1)dF (y1)


m∗ =

ΠmCD1 C0

Λ(εr,m + εB0,mεql0,B0

)

where εB0,m ≡dB0B0dm , εql0,B0

≡∂ql0q0

∂Bb0B0

Back to text


Epstein-Zin

• Only difference: stochastic discount factor

• m∗ doesn’t change but

Πm

CD1

C0≡(Q

C0

)γ− 1ψ

β

∫ m+B0

m

(CD1C0

)1−γ

dF (y1)

• Q is the certainty equivalent of consumption

Q ≡(∫ m

y1

(y1)1−γ dF (y1) +

∫ m+Bb0

m(m)1−γ dF (y1) +

∫ y1

m+B0

(y1 −B0)1−γ dF (y1)

) 11−γ

,

Back to text


Multiple Arbitrary Contracts

• Budget constraints

C0 = y0 +

J∑

j=1

q0j (B01,...B0J ,m)B0j

CN1 = y1 +

J∑

j=1

max −zj (y1)B0j , 0 −J∑

j=1

max zj (y1)B0j , 0

CD1 = min

y1 +

J∑

j=1

max −zj (y1)B0j , 0 ,m

• Weighted sum of elasticities

m∗ =

ΠmCD1 C0∑J

j=1 Λjεrj ,m

Back to text


Multiple Arbitrary Contracts

C1

y1

m

Dyy1 y1Dm

CN1 = y1 +∑J

j=1 max −zj (y1)B0j , 0 −∑J

j=1 max zj (y1)B0j , 0

CD1 = miny1 +

∑Jj=1 max −zj (y1)B0j , 0 ,m

y1 +∑J

j=1 max −zj (y1)B0j , 0

maxCD1 , CN1

45

NEduardo Davila (NYU Stern) Optimal Bankruptcy Exemptions 28 / 16

Heterogeneous borrowers (observed heterogeneity)

q0i (B0i,m) =

11+r∗ , B0i ≤ 0δ∫m+B0im

y1i−mB0i

dFi(y1i)+∫ y1im+B0i

dFi(y1i)

1+r∗ , B0i > 0

IA (m) = i|B0i (m) > 0 , Active borrowers

IN (m) = i|B0i (m) = 0 , Inactive borrowers

Back to text



0

q0 (B0,m)

B0

11+r∗

∫ y1m

dF (y1)

1+r∗

0

J (B0,m)

B0

i ∈ IA(m)

i ∈ IN (m)

Back to text



q0i (B0,m) =

11+r∗ , B0i ≤ 0∫IA(m)

q0i(B0,m)∫IA(m) dG(i)

dG (i) , B0i > 0,

where

q0i (B0,m) =δ∫m+B0

my1i−mB0

dFi (y1i) +∫ y1im+B0

dFi (y1i)

1 + r∗

Back to text


Aggregate Risk

• Now we have aggregate shocks ω ∈ Ω, we can condition theexemption level on those


m∗ (ω) =

Πm(ω)CD1 C0

Λεr,m(ω), ∀ω

Back to text


Dynamics

• Borrowers maximize maxE[∑T

t=0 βtU (Ct)

]

• Recursively VND,0 (B−1, y0;m) =maxB0 U (C0) + βE [max VND,1 (B0, y1;m) , VD,1 (y1;m)]


m∗ =

∑Tt=1

ΠmCDt C0∑T−1

t=0 ΠND gtΛtεrt,m

• Easy to allow for default before bankruptcy, wage garnishments,exclusion period

Back to text


Evolution filings US

Back to text

Parameters


Expression for welfare

W (m) =

U (y0 + q0 (B0 (m) ,m)B0 (m)) +

+β[∫my1U (y1) dF (y1) +

∫m+B0(m)m U (m) dF (y1) +

∫ y1m+B0(m) U (y1 −B0 (m)) dF (y1)

],


using elasticities to derive optimal bankruptcy exemptions · using elasticities to derive optimal...

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