Risk and Risk-Sharing in Two-Country...
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Transcript of Risk and Risk-Sharing in Two-Country...
Risk and Risk-Sharing in Two-Country Models
David Backus (NYU), Chase Coleman (NYU),
Axelle Ferriere (EUI), & Spencer Lyon (NYU)
Gerzensee | October 23, 2015
October 26, 2015
Overview
Two paths to variable Pareto weights
I Capital market frictions
I Recursive preferences
Plan of attack
I Two-country model + recursive preferences
I Home bias in consumption, stochastic volatility
Intellectual debts
I Colacito and Croce, Kollmann, Tretvoll
I Anderson; Collin-Dufresne, Johannes, and Lochstoer
1 / 1
Recursive preferences
Time aggregator
Ujt = V [cjt , µt(Ujt+1)] = [(1− β)cρjt + βµt(Ujt+1)ρ]1/ρ
Certainty equivalent function
µt(Ujt+1) =[Et(U
αjt+1)
]1/α
Features
I If cjt = c is constant ⇒ Ujt = c
I V , µ both homogeneous of degree one (hd1)
I Intertemporal substitution: IES = 1/(1− ρ) > 0
I Risk aversion: RA = 1− α > 0
I Traditional additive preferences if α = ρ
3 / 1
Recursive preferences (continued)
Intertemporal marginal rate of substitution
mjt+1 = β
(cjt+1
cjt
)ρ−1 ( Ujt+1
µt(Ujt+1)
)α−ρ
Epstein-Zin term is white noise plus risk adjustment
logUt+1 = Et(logUt+1) +[
logUt+1 − Et(logUt+1)]
logµt(Ut+1) = α−1 log Et(eα log Ut+1)
= Et(logUt+1)
+ α−1[
log Et(eα log Ut+1)− Et(α logUt+1)
]
4 / 1
Two-country model: technology
Production of intermediate goods
yjt = f (kjt , zjt) =[(1− η)kνjt + ηzνjt
]1/ν
y1t = a1t + a2t
y2t = b1t + b2t
Armington aggregator for final goods
c1t + i1t = h(a1t , b1t) =[(1− ω)aσ1t + ωbσ1t
]1/σ
c2t + i2t = h(b2t , a2t)
Capital stocks
kjt+1 = (1− δ)kjt + ijt
6 / 1
Armington aggregator: final goods frontier
0.0 0.2 0.4 0.6 0.8 1.0
Final goods in country 2
0.0
0.2
0.4
0.6
0.8
1.0
Final goods
in c
ountr
y 1 share = 1/2
elasticity = 1/2
elasticity = 2
elasticity = 5
7 / 1
Two-country model: shocks
Productivities[log z1t+1
log z2t+1
]=
[1− γ γγ 1− γ
] [log z1t
log z2t
]+
[vt
1/2w1t+1
v1/2w2t+1
]
Conditional variance (“volatility”)
vt+1 = (1− ϕv )v + ϕvt + τw3t+1
Innovations {w1t ,w2t ,w3t} independent standard normal
8 / 1
Pareto problem
Bellman equation [st = (kjt , zjt , vt)]
J(Ut , st) = max{c1t ,Ut+1}
V{c1t , µt [J(Ut+1, st+1)]
}s.t. V
{c2t , µt(Ut+1)
}≥ Ut (λt)
plus resource constraints and shocks
Notation
I J is agent 1’s utility, U is agent 2’s utility (“promised utility”)
I λt is (relative) Pareto weight
Fundamental tradeoff
I Give you more today (c2t)
I Give you more in the future (µt(Ut+1))
10 / 1
Pareto problem: Pareto weight
First-order conditions
cρ−11t /p1t = λ∗t c
ρ−12t /p2t
β [Jt+1/µt(Jt+1)]α−ρλ∗t+1 = λ∗t β [Ut+1/µt(Ut+1)]α−ρ
Additive case (α = ρ)
λ∗t+1 = λ∗t
Otherwise
log λ∗t+1 − log λ∗t = (α− ρ) [white noise + risk adjustment]
11 / 1
Pareto problem: consumption
First-order conditions (repeated)
cρ−11t /p1t = λ∗t c
ρ−12t /p2t
β [Jt+1/µt(Jt+1)]α−ρλ∗t+1 = λ∗t β [Ut+1/µt(Ut+1)]α−ρ
Consumption and real exchange rate
et = p2t/p1t = λ∗t (c2t/c1t)ρ−1
12 / 1
Computation
Method adapted from Collin-Dufresne, Johannes, and Lochstoer
I Global projection method
I Implemented in Julia for speed
I State changed from Ut to sat = a1t/y1t or λ∗t
P2C2E
14 / 1
Dynamics of the Pareto weight
0 50 100 150 200 250
time (t)
0.010
0.005
0.000
0.005
0.010
∆lo
gλ∗
AdditiveRecursive
15 / 1
Consumption and real exchange rate
0.04 0.03 0.02 0.01 0.00 0.01 0.02 0.03
log(c2/c1 )
0.08
0.06
0.04
0.02
0.00
0.02
0.04
0.06
0.08
0.10
Log
Exc
hang
e R
ate
(log
(p2/p
1))
AdditiveRecursive
16 / 1
Responses to impulse in productivity in country 2 (blue first)
10 20 30 40−1
0
1volatility
10 20 30 400
0.01
0.02xi
10 20 30 40−0.01
−0.005
0lambda
%
10 20 30 40−2
0
2c1, c2
%
10 20 30 40−0.1
0
0.1a, b
%
time10 20 30 40
−1
0
1J, U
%
time
Student Version of MATLAB
17 / 1
Responses to impulse in volatility (blue first, red additive)
10 20 30 400
0.5
1x 10
−5 volatility
10 20 30 40−1
0
1xi
%
10 20 30 40−4
−2
0x 10
−4 lambda
%
10 20 30 40−5
0
5x 10
−5 c1, c2
%
10 20 30 40−5
0
5x 10
−5 a, b
%
time10 20 30 40
−0.02
−0.01
0J, U
%
time
Student Version of MATLAB
18 / 1
Stability of the Pareto weight
What we know
I Colacito and Croce: If ρ = σ = 0, stable by theorem
I Colacito and Croce, Tretvoll: With some other parametervalues, solutions seem stable
Open question
I What configurations of parameter values generate stability?
I Hint at problem: log Pareto weight close to martingale
I Hint at solution: shape of Pareto frontier (J vs U) reflectsfinal goods frontier
22 / 1