Risk Sharing and Financial Amplification

Luigi Bocola Guido LorenzoniStanford, Minneapolis Fed, Northwestern and NBER

and NBER

SED Mexico City

June 2018

Question

• Large literature on financial frictions in macro

• Logic of financial amplification in these models builds on two blocks

1 Investment capacity of “specialists” depend on dynamics of net worth

2 Net worth sensitive to shocks affecting valuation of their assets

net worth = assets - liabilities

If value of assets drops 10% and leverage is 2, net worth drops 20%

• Second block relies on liabilities being rigid: non-state-contingent debt

• Why are specialists taking these risks?

1 / 18

Why risk exposure?

• Not many explanations. Many macro models just assume it

• Problematic for two reasons

1 No amplification once we allow for state contingent claims (Krishnamurthy,2003; Di Tella, 2017; Cao, 2017; Calstrom et al., 2016)

2 Challenging to build models of macroprudential regulation of financial risks

• Our idea: insurance is costly in general equilibrium

• When balance sheet of specialists compromised→ incomes go down foreveryone

• This makes it costly to insure these shocks ex-ante

2 / 18

This paper

Start from simplest model with two agents: entrepreneurs and consumers

• Neoclassical structure with limited commitment for entrepreneurs

• Agents can trade full set of state contingent claims

• Macro spillover: when kt goes down, wages of consumers decline

Main results

• Calibrated model can feature amplification comparable to standardincomplete market economy

• Key for results: consumers sufficiently risk averse and their incomesdecline in a crisis

• Competitive equilibrium constrained inefficient: insuring bad shocks“too costly” for entrepreneurs

3 / 18

Overview of the talk

1 The model

2 Non-amplification results in the literature

• Risk neutral consumers

• No macroeconomic spillovers

3 Numerical simulations

• Amplification with risk averse consumers and macroeconomic spillovers

4 Welfare analysis

Environment

• Time is discrete t = 0, 1, . . .

• An history is st = (s0, s1, . . . , st), where st is a Markov process withtransition probability π(st+1|st)

• Two agents: consumers and entrepreneurs

• Consumers: work for final good firms

• Entrepreneurs: accumulate capital, rent it to final good firms

• Technology to produce final goods

Y(st) =(u(st)K

(st−1))α L (st)

1−α

u(st) is a shock to the “quality” of capital

• All markets competitive

4 / 18

Financial Markets

• Full set of one period ahead contingent claims at prices q(st+1|st)

• A(st, st+1) are claims of consumers toward state (st, st+1)

• B(st, st+1) are promised payments of entrepreneurs toward state (st, st+1)

• Market clearingA(st, st+1) = B(st, st+1)

• Limited enforcement of debt contracts for entrepreneurs

• After renting capital, entrepreneurs can default on the payments b(st)

• If they default, they loose a fraction θ of capital

• Default entails no exclusion from financial markets

5 / 18

Consumers

• Consumers have Epstein-Zin preferences,

V(st) = max

{(1− β)c(st)1−ρ + β

[E(V(st, st+1)

1−σ|st)] 1−ρ1−σ

} 11−ρ

• Budget constraint

c(st) +∑st+1

q(st+1|st)a(st, st+1) = w(st) + a(st)

• Optimality conditions for contingent claims

q(st+1|st) = βπ(st+1|st)

(c(st, st+1)

c(st)

)−ρ[E (V(st, st+1)

1−σ|st)] 1

1−σ

V(st, st+1)

σ−ρ

6 / 18

Entrepreneurs

• Entrepreneurs have CRRA preferences

E0

[∑st

βtece(st)1−γ − 1

1− γ

]

and discount factor βe ≤ β• The beginning of period net worth for the entrepreneur is

n(st) = [r(st) + (1− δ)]u(st)k(st−1)− b(st)

• The budget constraint is

ce(st) + k(st) = n(st) +∑st+1

q(st+1|st)b(st, st+1) (λ(st))

• Entrepreneur has no incentives to default in state (st, st+1) if

b(st, st+1) ≤ θ(1− δ)u(st+1)k(st) (µ(st, st+1))

7 / 18

Entrepreneurs’ optimality

• Optimality for state contingent bonds

q(st+1|st)− βeπ(st+1|st)

(ce(st)

ce(st, st+1)

)γ= βeµ(st, st+1) (ce(st))

γ

• Optimality for capital

Est

[βe

(ce(st)

ce(st,st+1)

)γ [r(st+1) + (1− δ)

]u(st+1)

]1− (1− δ)θ

∑st+1

µ(st, st+1)u(st+1)= 1

• Consumption policy is linear in net worth

ce(st) = [1− κe(st)]n(st)

with κe(st) = βe if γ = 1

8 / 18

Overview of the talk

1 The model

2 Non-amplification results in the literature

• Risk neutral consumers

• No macroeconomic spillovers

3 Numerical simulations

• Amplification with risk averse consumers and macroeconomic spillovers

4 Welfare analysis

Non-amplification I: Linear utility for consumers

Suppose σ = ρ = 0, and γ = 1. Then, the change in net worth between anytwo states st and (st, st+1) is bounded below by βe/β

• Why? From risk sharing condition

q(st+1|st) ≥ βeπ(st+1|st)

(Ce(st)

Ce(st, st+1)

)γ⇒ β ≥ βe

N(st)

N(st, st+1)

• When consumers have linear utility, entrepreneurs use contingentmarkets to hedge risk. Fall in net worth after bad shocks bounded

• Result in the spirit of Krishnamurthy (2003)

9 / 18

Non-amplification II: No macroeconomic spillovers

Suppose σ = ρ = γ, and assume that the technology to produce final goodsis Y(st) = Au(st)K(st−1) with u(st) iid. Then, the dynamic response oflog(Kt+1) to a ut shock is equivalent to the one in the first best

• Why? In a stationary equilibrium, C(st) = κA(st) and Ce(st) = κeN(st)

β

[B(st)

B(st, st+1)

]γ≥ βe

[N(st)

N(st, st+1)

]γwhich implies that B(st, st+1) increases with u(st+1)

• Important assumption: no wages for consumers

• Result in the spirit of Di Tella (2017), but in discrete time

10 / 18

Overview of the talk

1 The model

2 Non-amplification results in the literature

• Risk neutral consumers

• No macroeconomic spillovers

3 Numerical simulations

• Amplification with risk averse consumers and macroeconomic spillovers

4 Welfare analysis

Numerical illustration

• We deviate from non-amplification results in two dimensions

• Allow households to be more risk averse than entrepreneurs

• Allow for macroeconomic spillovers (through wages)

• Compare competitive equilibrium of complete market economy toeconomy with non-state-contingent bonds

• b(st, st+1) = b(st) ∀ st+1

• b(st) ≤ θ(1− δ)uk(st)

• Main results

• The two models can feature comparable degrees of financial amplification

• Both ingredients necessary

11 / 18

Calibration

ValueCapital income share α = 0.330Capital depreciation δ = 0.025Discount factor, consumers β = 0.990Inverse IES, consumers ρ = 1.000Discount factor, entr. βe = 0.988Inverse IES, entrepreneurs γ = 1.000Capital quality in low state uL = 0.850Probability of bad shock πL = 0.025Fraction of assets θ = 0.674

• {πL, uL} chosen to obtain “rare” and “large” shocks

• {βe, θ} to obtain a leverage of 3 and “spreads” of 25bp in deterministicsteady state for complete markets model

• Sensitivity on consumers’ risk aversion, σ ∈ [0, 50]

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Entrepreneurs’ balance sheet and amplification

Risk neutral consumers

First Best Incomplete markets Complete marketsEntrepreneurs’ balance sheet

Mean(Bt,H) 9.40 15.11Mean(Bt,L) 9.40 12.31Mean(Kt/Nt) 1.75 2.76Stdev(Nt) 26.09% 5.88%

Financial amplificationStdev(Yt) 2.84% 3.82% 2.89%Acorr(Yt) 0.96 0.98 0.96

• Payments from entrepreneurs to households higher after good shocks→Net worth more stable

• Virtually no financial amplification

13 / 18

Entrepreneurs’ balance sheet and amplification

CRRA consumers (log utility)

First Best Incomplete markets Complete marketsEntrepreneurs’ balance sheet

Mean(Bt,H) 9.42 15.20Mean(Bt,L) 9.42 12.51Mean(Kt/Nt) 1.75 2.78Stdev(Nt) 26.07% 6.81%

Financial amplificationStdev(Yt) 2.81% 3.83% 2.91%Acorr(Yt) 0.96 0.98 0.96

• No financial amplification with complete markets (Di Tella (2017) resultholds approximately)

• Confirms quantitative findings by Cao (2017), Calstrom et al. (2016), . . .

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Entrepreneurs’ balance sheet and amplification

As σ increases, complete markets model approaches incomplete market

0 20 40 608

10

12

14

16

0 20 40 601.5

2

2.5

3

0 20 40 600

0.1

0.2

0.3

0.4

0 20 40 602.5

3

3.5

4

4.5

0 20 40 600.95

0.96

0.97

0.98

0.99

Bad state

Incomplete markets

Good state

First best

Incomplete markets

Complete markets

IRFs

13 / 18

Why entrepreneurs choose riskier balance sheet?

βπs

(Ct

Ct+1,s

)[Et(V1−σ

t+1,s

)] 11−σ

Vt+1,s

σ−1

= qt,s ≥ βeπsNt

Nt+1,s

0 10 20 30 40 500.8

1

1.2

1.4

1.6

0 10 20 30 40 501.009

1.0095

1.01

1.0105

0 10 20 30 40 500.6

0.7

0.8

0.9

0 10 20 30 40 500.028

0.03

0.032

0.034

0.036

Low u state

High u state

• As σ increases, consumers discount more heavily low u states

• Low u states associated to more persistent declines in wages (GE effect)14 / 18

The role of wages

• Two departures from non-amplification result

1 Consumers more risk averse than entrepreneurs

2 Consumers’ wages decline after bad shocks

• Ingredient 1 necessary (essentially no amplification with σ = 1)

• To isolate ingredient 2, consider version of the model with fixed wages

• Consumers earn fixed income W from abroad

• Entrepreneurs pay wages to hand-to-mouth agents

• Compare benchmark CM market model with one with fixed wages

15 / 18

The role of wagesConsumers’ risk aversion set to σ = 30

0 50 100 150-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

BenchmarkFixed wage

0 50 100 150-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

First best

• Essentially no amplification with fixed wages

• Both ingredients necessary for results

15 / 18

Overview of the talk

1 The model

2 Non-amplification results in the literature

• Risk neutral consumers

• No macroeconomic spillovers

3 Numerical simulations

• Amplification with risk averse consumers and macroeconomic spillovers

4 Welfare analysis

Welfare

• A planner has full set of taxes, but faces same collateral constraints

• At t = 1 temporary shock u1 ∈ {uL, uH}. From t = 2 on, ut = uH

• Assume that constraint binds in L but not in H in competitive equilibrium

• Study one shot policy intervention at date 0

• Reduce BL1 , increase BH

1 , keep constant C0,Ce,0,K1

• No additional resources for entrepreneurs,∑

s={L,H} qs1Bs

1 ≤ 0

• Can the planner obtain a Pareto improvement? Yes

• Entrepreneur can gain because marginal utility higher when constraint binds

• Households can gain because of higher future wages in L

16 / 18

Pareto improvement

• Consider a special case• ρ = 0 → No movement in interest rate from t = 2 on

• β = βe → Simplifies some expressions

• Effect on consumers’ welfare at date 0 proportional to∑s={L,H}

πs (Vs1)−σ

(1−

∞∑t=2

βt ∂Wst

∂Ns1

)dBs

1

• Effect on entrepreneurs’ welfare at date 0∑s={L,H}

πs

(−u′(Cs

e,1) +

∞∑t=2

βtu′(Cse,t)∂Ws

t

∂Ns1

)dBs

1

• Pareto improvement possible because∞∑

t=2

βt

(1−

u′(CLe,t)

u′(Cse,1)

)∂WL

t

∂NL1> 0

17 / 18

Conclusions

• Common view: need incomplete markets to generate quantitativelymeaningful financial amplification

• Revisited common view

• Macro spillovers and consumers’ risk aversion make hedging “costly"

• Complete market economy can feature comparable degrees of financialamplification than economy with state uncontingent debt

• Hedging bad states is “too costly” in competitive equilibrium

• In progress: quantification of mechanism and optimal policy in a modelwith purely financial shocks and more realistic propagation

18 / 18

Financial amplification: risk neutral consumers

Impulse response functions after a negative shock

• Ergodic mean and high debt (75th percentile)

0 50 100 150-0.6

-0.4

-0.2

0

ErgodicHigh debt

0 50 100 150-0.06

-0.04

-0.02

0

0 50 100 150-0.6

-0.4

-0.2

0

0 50 100 150-0.06

-0.04

-0.02

0First best

Financial amplification: risk averse consumers (σ = 30)Impulse response functions after a negative shock

• Two initial conditions: Ergodic mean and high debt (75th percentile)

0 50 100 150-0.6

-0.4

-0.2

0

ErgodicHigh debt

0 50 100 150-0.06

-0.04

-0.02

0

0 50 100 150-0.6

-0.4

-0.2

0

0 50 100 150-0.06

-0.04

-0.02

0First best

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