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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]
12 / 18
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), . . .
13 / 18
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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