Appendix: Asset Pricing with Endogenously Uninsurable Tail
Risks
Hengjie Ai and Anmol Bhandari
January 30, 2017
1
Normalized Value functions
For a given stochastic process of {Λt }t let Vt (y, U ) be the value attained by sequential
problem (4) for some history of aggregate shocks g t and initial conditions y0 = y and
U0 = U .
Vt (y, U ) =
max
C,{U 0 (z 0 )}
(y − C) + Et Λt,t+1 Vt+1 y 0 (y, z 0 ), U 0 z 0
U = (1 − β) C
Vt+1
where the operator Mt
(U 0 )
1
1− ψ
+ βMt
1− 1 1−1 1
ψ
ψ
U 0
U 0 (z 0 ) ≥ U (z 0 )
y 0 (y, z 0 ), U 0 z 0 ≥ 0, f or all z 0 .
(22)
(23)
(24)
(25)
n
o 1
1−γ 1−γ
0
0
≡ Et U (z )
. Guess that Vt+1 (y 0 (y, z 0 ), U 0 (z 0 )) admits
a multiplicative decomposition
0
0
0
Vt+1 y (y, z ), U z
0
= vt+1
1
U (z 0 )
y 0 (y, z 0 )
y 0 (y, z 0 ).
0
0
Substituting in the right hand side of (22) and using y(y, z 0 ) = yeg +ε from equation (2)
(3) from the text, we can express the objective function as
U (z 0 )
C
g 0 +ε0
+ Et Λt,t+1 vt+1
e
y
1−
y
y 0 (y, z 0 )
Since the Mt (U 0 ) and hence the right hand side of (23) is a homogeneous of degree one in
0
C, U we can divide and multiply by y to obtain
"
0
1 # 11
1− 1
ψ
U
U g0 +ε0 1− ψ 1− ψ
C
e
= (1 − β)
+ βMt
y
y
y0
Next assuming U (z 0 ) = u(z 0 )y 0 (z 0 ) we can scale both the constraints (24) and (22) with
y 0 (z 0 ). Using definitions
u=
U
;
y
c=
C
,
y
the objective function in (22) and the constraints (23)-(25) can be expressed in normalized
consumption and normalized promised values and we obtain the recursion for vt ( Uy ) as in
problem (7) in text.
2
Proof of Proposition 1
Standard arguments from SLP imply that the value function v is continuous, strictly
decreasing, strictly concave and differentiable in the interior.
Applying the envelope
condition we obtain
∂
v ( u| φ, g) = −
∂u
1
(1 − β)
c( u|φ,g)
u
− 1 ,
(26)
ψ
and the first order condition with respect to continuation utility requires
0 0
−γ
1
β
γ− 1
c ( u| φ, g) ψ m ( u| φ, g) ψ eg +ε u0 u, g 0 , ε0
1−β
∂
≥ −Λ g 0 φ, g
v u0 u, g 0 , ε0 φ0 , g 0 ,
∂u
2
(27)
Monotonicity of the value function with respect to promised values means that the limited
commitment constraint () can be written as u0 (u, g 0 , ε0 ) ≤ ū (φ0 , g 0 ) for all g 0 , where ū (φ0 , g 0 )
is defined as v ( ū (φ0 , g 0 )| φ0 , g 0 ) = 0. Then above conditions together imply that for all
possible realizations of (g 0 , ε0 ),
"
0
eε c ( u0 ( u, g 0 , ε0 | φ, g)| φ0 , g 0 )
c ( u| φ, g)
#− 1 "
ψ
0
eε u0 ( u, g 0 , ε0 | φ, g)
m ( u| φ, g)
# 1 −γ
ψ
x (φ0 , g 0 )
≥
x (φ, g)
− 1 ψ
1
w (φ0 , g 0 ) ψ −γ
n (φ, g)
(28)
The complementary slackness condition implies that (27) has to hold with equality whenever
u0 (u, g 0 , ε0 ) < ū (φ0 , g 0 ) and u0 (u, g 0 , ε0 ) ≥ ū (φ0 , g 0 ).
Let E = {ε0 s.t the limited liability constraint equation () and () are slack}. For a given
m̃, using the first order condition in equation (28), for all ε0 ∈ E define an implicit function
ũ (u, ε0 , m̃, g 0 |, φ, g) such that z = ũ (u, ε0 , m̃|φ, g) solves
"
0
eε c (z|φ0 , g 0 )
c ( u| φ, g)
#− ψ1 "
0
eε z
m ( u| φ, g)
# ψ1 −γ
=
x (φ0 , g 0 )
x(φ, g)
− ψ1 w (φ0 , g 0 )
n (φ, g)
ψ1 −γ
,
(29)
where φ0 = Γ (g 0 |φ, g) with the property that
u0 (u, g 0 , ε0 |φ, g) = ũ u, ε0 , m(u|φ, g), g 0 |, φ, g .
Strict concavity of v(u|φ, g) with respect to u implies
(29) this yields
∂ ũ(u,ε0 ,m̃,g 0 |φ,g)
∂ε0
≤ 0 and
∂ ũ(u,ε0 ,m̃,g 0 |φ,g)
∂u
∂c(·|φ,g)
∂u
≥ 0. Together with equation
≥ 0. Since these properties hold for
an arbitrary m̃ we conclude that the policy rule for continuation values u0 (u, g 0 , ε0 |φ, g) is
decreasing in ε0 and increasing in initial promised values u.
Substituting the optimal policy for u0 , note that the left hand side of equation (29) is
monotonic and continuous in ε0 . Both c(u|φ, g) and u0 (u, g 0 , ε0 |φ, g) are bounded and hence
the left hand side approaches zero as ε0 → ∞ and approaches ∞ as ε → −∞. For a strictly
positive x(φ, g), by the Intermediate Value Theorem, there exists ε(u, g 0 |φ, g) such that
eε c (u(φ0 , g 0 )|φ0 , g 0 )
c ( u| φ, g)
− 1 ψ
eε u(φ0 , g 0 )
m ( u| φ, g)
1 −γ
ψ
=
3
x (φ0 , g 0 )
x(φ, g)
− 1 ψ
w (φ0 , g 0 )
n (φ, g)
1 −γ
ψ
,
and ε(u, g 0 |φ, g) such that
eε c (u(φ0 , g 0 )|φ0 , g 0 )
c ( u| φ, g)
− ψ1 eε u(φ0 , g 0 )
m ( u| φ, g)
ψ1 −γ
=
x (φ0 , g 0 )
x(φ, g)
− 1 ψ
w (φ0 , g 0 )
n (φ, g)
1 −γ
ψ
.
Monotonicity of u0 with respect to ε0 implies that the set E = [ε(u|g 0 |φ, g), ε(u|g 0 |φ, g)]
and for ε 6∈ E, the limited liability constraints () and () bind giving us u0 (u, g 0 , ε0 |φ, g) =
u(φ0 , g 0 ) for ε0 < ε(u|g 0 |φ, g) and u0 (u, g 0 , ε0 |φ, g) = u(φ0 , g 0 ) for ε0 < ε(u|g 0 |φ, g).
Lastly we turn to the monotonicity of the threshold ε(u, g 0 |φ, g) and ε(u, g 0 |φ, g) with
respect to u. The right hand side of equation (29) is independent of u but the equation
needs to be satisfied for all u. The definitions of ε(u|g 0 |φ, g) and ε(u|g 0 |φ, g) mean that
u0 (u, g 0 , ε(u|g 0 |φ, g)|φ, g) is constant with respect to u. Hence we have
∂u0 (u, ε(u|g 0 |φ, g), g 0 |φ, g) ∂u0 (u, ε(u|g 0 |φ, g), ε0 |φ, g) ∂ε(u|g 0 |φ, g)
+
=0
∂u
∂ε0
∂u
and
∂u0 (u, ε(u|g 0 |φ, g), g 0 |φ, g) ∂u0 (u, ε(u|g 0 |φ, g), ε0 |φ, g) ∂ε(u|g 0 |φ, g)
+
= 0.
∂u
∂ε0
∂u
Since u0 is decreasing with respect to ε0 and increasing with respect to u it follows that
∂ε(u|g 0 |φ,g)
∂u
3
≥ 0 and
∂ε(u|g 0 |φ,g)
∂u
≥0
Details of the negative exponential distribution
We assume that the density function of the idiosyncratic shock fgL (ε) takes the following
form:
f (ε) =
The integral of f (ε) is
ˆ
ε > εmax
0
λeλ(ε−ε
max )
ε ≤ εmax
y
f (ε) dε = eλ(y−ε
−∞
4
max )
.
In particular,
´∞
−∞ f
ˆ
(ε) dε = 1 is a proper density. Also,
y
eθε f (ε) dε =
−∞
In particular,
ˆ
λ −λεmax +(θ+λ)y
e
f or λ + θ > 0.
λ+θ
∞
eθε f (ε) dε =
−∞
λ θεmax
e
f or λ + θ > 0.
λ+θ
With θ = 1, the requirement E [eε ] = 1 requires that εmax = ln 1+λ
λ .
4
Proof for Proposition 3
We assume in Section 4 that the distribution f ( ε| gH ) is degenerate, and f ( ε| gL ) is the
negative exponential distribution with parameter λ:
f ( ε| gL ) =
0
ε > εM AX
λeλ(ε−εM AX ) ε ≤ εM AX
.
For later reference, we note that the moments of f ( ε| gL ) can be easily computed as:
ˆ
ε
eθt f ( t| gL ) dt =
−∞
λ −λεM AX +(θ+λ)ε
e
f or λ + θ > 0.
λ+θ
(30)
It is straightforward to use (30) to show that the assumption E [eε ] = 1 amounts to the
parameter restriction that εM AX = ln 1+λ
λ .
We first introduce some simplifying notations. Let ε=ε(u0 , gL ) be the cutoff level of ε
such that any lower levels of ε will lead to a binding limited commitment constraint. We
denote
uH
= u0 (u0 , gH ) ,
vH
= v u0 (u0 , gH ) , vL (ε) = v u0 (u0 , gL , ε) ;
= c u0 (u0 , gH ) , cL (ε) = c u0 (u0 , gL , ε) .
cH
uL (ε) = u0 (u0 , gH ) ;
Note that all above quantities are functions of u0 and we will use the notation ε(u0 ),
5
uH (u0 ), uL (u0 , ε), vH (u0 ), vL (u0 , ε), cH (u0 ), and cL (u0 , ε) to emphasize this dependence
whenever necessary.
Also, we use wH and wL to denote the capital owners’ utility
(normalized by aggregate output) at node H and L, respectively. In addition, let uFHB
and uFL B denote the utility-to-consumption ratio of an agent who consumes the aggregate
consumption in state gH and gL , respectively. That is, they are the normalized utility
associated with full risk sharing. The first best uFHB and uFL B are determined by
uFHB = egH uFHB
β
uFL B = egL uFL B
β
.
Also, we use uCD
to denote the normalized utility of an agent in an economy without risk
L
sharing (the Constantinides-Duffie economy). That is, it is utility-consumption ratio of an
agent who consumes αyt every period:
uCD
L
=
ˆ h
e{ε+gL } uCD
L
i1−γ
β
1−γ
.
f (ε|gL )dε
(31)
It is straightforward to show that as γ → 1 + λ, uCD
L → 0.
Optimal risk sharing We first prove the following lemma that uses the optimal risk
sharing condition (28) to express cH as a function of cL (ε), the consumption of the marginal
agent whose limited commitment constraint is just about to bind.
Lemma 1. (FOC for the marginal agent)
Given the consumption share of the capital owners, xH and xL , for all u0 , the normalized
consumption of the marginal worker with ε1 =ε(u0 ) must satisfy:
F B η
uL
cH (u0 )
xH
.
=
CD
(1+η)ε(u
)
0 c (u , ε (u ))
xL
e
ξuL
0
0
L
(32)
Proof. By Proposition ??, the optimal risk sharing condition (28) must hold with equality
for the marginal worker with the realization of ε at node L:
cH
eε cL (ε)
−1 uH
eε uL (ε)
1−γ
6
xH
=
xL
−1 wH
wL
1−γ
.
(33)
We can use the promise keeping constraint to represent continuation utility as functions of
consumption. For capital owners,
β
gH F B
= x1−β
H nH , where nH = (1 − α) e uH ,
wH
β
gL F B
wL = x1−β
L nL , where nL = (1 − α) e uL ,
(34)
where the computation of continuation utility nH and nL uses the fact that capital owners
are not exposed to idiosyncratic risks and that together they consume 1 − α fraction
of aggregate output. Because workers are not exposed to idiosyncratic risks at node H
and consume α fraction of aggregate output, their continuation utility at node H can be
computed similarly:
1−β β
uH = cH
mH ;
mH = αuFHB egH .
At node L, workers consume αyt for t = 2, 3, · · · and their normalized certainty equivalent
in period 2 is
ˆ
∞
mL (ε1 ) =
h
e
g 0 +ε0
αuCD
L
−∞
i1−γ
0
0
1
1−γ
f ε gL dε
= αξuCD egL ,
where we define ξ ∈ (0, 1) as:
ˆ
∞
ξ=
e
(1−γ)ε0
0
0
1
1−γ
f ε gL dε
.
−∞
Therefore,
uL (ε) = cL (ε)1−β mL (ε)β ;
gL
mL = αξuCD
L e ,
(35)
Now we use expressions in (34) and (35) to replace the continuation utilities in (33) and
simplify to get:
cH
[1+η]ε
e
cL (ε)
−Ω ξuCD
L
uFL B
−β(1−γ)
xH
=
xL
−Ω
,
where to simplify notation, we denote
Ω = 1 + (1 − β) (γ − 1) > 0, and η =
7
β (γ − 1)
.
Ω (γ)
(36)
We can therefore obtain (32) by raising both sides of equation (33) to their − Ω1 th power.
Market Clearing/Aggregation
Next, we exploit the market clearing condition to
express the total consumption of all workers as a function of cL (ε). Our main result is
summarized in the following lemma:
Lemma 2. (Market clearing)
Given the consumption share of the capital owners, xH and xL , for all u0 , the expected
consumption of a worker with promised utility u0 at node L is given by:
h
ε0
0
E e cL u0 , ε
i
=e
(1+η)ε(u0 )
cL (u0 , ε (u0 ))
λ −λεM AX +(λ−η)ε(u0 )
e
+ Φ (ε) ,
1+λ
(37)
where the function Φ (ε) is defined as:
Φ (ε) =
i
λ h −ηεM AX
e
− e−λεM AX +(λ−η)ε .
λ−η
(38)
Proof. The expected consumption of a worker with promised utility u0 at node L can be
computed as
ˆ
ε
ε0
ˆ
0
0
εM AX
e cL (ε) f ε gL dε +
−∞
0
eε cL ε0 f ε0 gL dε0 .
(39)
ε̄
The first order condition (28) implies that for all ε ≥ ε,
e−γε cL (ε)−1 uL (ε)1−γ = e−γε cL (ε)−1 uL (ε)1−γ .
(40)
gL β
uL (ε) = cL (ε)1−β αξuCD
.
L e
(41)
Using equation (35),
We can combine equations (40) and (41) to get:
e−γε cL (ε)−1+(1−γ)(1−β) = e−γε cL (ε)−1+(1−γ)(1−β) .
1
Raising both sides of the above equation to the − 1−1+(1−γ)(1−β)
th power and using the
8
definition of Ω and η, we have, for all ε ≥ε,
eε cL (ε) = e−ηε e(1+η)ε cL (ε) .
Now, we compute the integral in (39) as:
ˆ
ε
cL (ε)
0
eε f ε0 gL dε0 + e(1+η)ε cL (ε)
ˆ
−∞
=
εM AX
e−ηε f ( ε| gL ) dε
ε
i
λ −λεM AX +(1+λ)ε
λ h −ηεM AX
cL (ε̄) + e(1+η)ε cL (ε)
e
e
− e−λεM AX +(λ−η)ε ,
1+λ
λ−η
as needed.
General Equilibrium In general equilibrium, u0 = ū0 , and the market clearing
h 0
i
conditions imply that cH = 1 − xH and E eε cL (ε0 ) = 1 − xL . Using Lemma 2, we
can write
h
ε0
1 − xL = E e cL ε
0
i
(1+η)ε
=e
cL (ε)
λ −λεM AX +(λ−η)ε
e
+ Φ (ε)
1+λ
(42)
Using the above equation to replace e(1+η)ε cL (ε) and using 1 − xH to replace cH , we can
rewrite equation (32) as
xH 1 − xL
=
xL 1 − xH
F B η
uL
λ −λεM AX +(λ−η)ε
+ Φ (ε)
e
.
1+λ
ξuCD
L
(43)
Note that if γ = 1, then η = 0, and using the definition of Φ (ε) in (38),
λ −λεM AX +(λ−η)ε
1 λ(ε−εM AX )
e
+ Φ (ε) = 1 −
e
< 1.
1+λ
1+λ
It follows immediately that for γ = 1,
xH
xL
< 1 is counter-cyclical. As γ → 1 + λ, uCD
L → 0,
while all other terms on the right hand side of (43) remain bounded. We must have
By continuity, there exists γ̂ ∈ (1, 1 + λ) such that
9
xH
xL
xH
xL
→ ∞.
> 1 if and only γ > γ̂, as needed.
5
Proof of Proposition 4
Normalized value functions
First, note that starting from period two, all workers
consume a fraction α of firms’ output, and as a result, firms’ cash flow is (1 − α) yt for
t = 2, 3, · · · . In addition, capital owners consume 1 − α fraction of aggregate output.
Therefore, Given the assumption of unit IES for capital owners, the price-to-dividend ratio
of all firms’ cash flow is a constant,
1
1−β .
Therefore, at node H, the value of a firm (with promised utility u0 ) normalized by
output is
vH (u0 ) = 1 − cH (u0 ) + ΛH egH
1
(1 − α) ,
1−β
where we use ΛH for the Arrow-Debreu price of one unit of consumption goods in period
two measured in period-one numeraire. Because there is no uncertainty in period two, ΛH
can be computed from the consumption growth rate of capital owners:
ΛH = β
(1 − α) gH
e
xH
−1
,
where xH and 1 − α is the consumption share of capital owners at node H in period one
and period two, respectively. We have:
vH (u0 ) = 1 − cH (u0 ) +
β
xH .
1−β
(44)
Similarly,
vL (u0 , ε) = 1 − cL (u0 , ε) + ΛL egL
where ΛL = β
h
(1−αH )
xL gL
i−1
1
(1 − α) ,
1−β
. Therefore, similar to (44), we have
vL (u0 , ε) = 1 − cL (u0 , ε) +
β
xL ,
1−β
(45)
and
E [eε vL (u0 , ε)] = 1 − E [eε cL (u0 , ε)] +
10
β
xL .
1−β
(46)
At u0 = ū0 , using the market clearing condition, 1 − cH (ū0 ) = xH and 1 −
E [eε cL (ū0 , ε)] = xL , equations (44) and (46) imply
β
1
xH =
xH ,
1−β
1−β
β
1
E [eε vL (ū0 , ε)] = 1 − E [eε cL (ū0 , ε)] +
xL =
xL .
1−β
1−β
vH (ū0 ) = 1 − cH (ū0 ) +
Clearly,
vH (ū0 )
xH
>1
=
E [eε vL (ū0 , ε)]
xL
if and only if γ > γ̂.
Cross section of expected returns To characterize the dependence of
vH (u0 )
E[eε vL (u0 ,ε)] ,
note that in general,
cH (u0 ) =
xH
xL
ξuCD
L
uFL B
η
e(1+η)ε(u0 ) cL (u0 , ε (u0 ))
by Lemma 1 and
ε
E [e cL (u0 , ε)] = e
(1+η)ε(u0 )
cL (u0 , ε (u0 ))
λ −λεM AX +(λ−η)ε(u0 )
e
+ Φ (ε)
1+λ
by Lemma 2. Because at ε = ε (u0 ), the limited commitment constraint, vL (u0 , ε) = 0
β
β
xL by (45). To simplify notation, we denote θH = 1 + 1−β
xH
binds, cL (u0 , ε (u0 )) = 1 + 1−β
and θL = 1 +
β
1−β xL .
We then write
vH (u0 )
E[eε vL (u0 ,ε)]
as:
θH − φθL e(1+η)ε(u0 )
vH (u0 )
,
=
E [eε vL (u0 , ε)]
θL 1 − e(1+η)ε(u0 ) ω (ε (u0 ))
where we denote φ =
xH
xL
h
iη
ξuCD
L
θL ,
F
B
uL
and ω (ε) =
λ −λεM AX +(λ−η)ε(u0 )
+ Φ (ε)
1+λ e
to simplify
notation. By Proposition ??, ε (u0 ) is a strictly increasing function of u0 . Therefore, to
prove Proposition 4, it enough to show
∂ θH − φe(1+η)ε
> 0,
∂ε 1 − e(1+η)ε ω (ε)
11
which is given by the following lemma.
Lemma 3. There exists γ̃ ∈ (1, 1 + λ) such that γ > γ̃ implies that for all ε ∈ (−∞, εM AX ),
"
#
θH − φe(1+η)ε
∂
> 0.
∂ε 1 − e(1+η)ε ω (ε)
(47)
Proof. We can compute (47) as :
=
"
#
θH − φe(1+η)ε
∂
∂ε 1 − e(1+η)ε ω (ε)
−φe(1+η)ε (1 + η) 1 − e(1+η)ε ω (ε) + θH − Φe(1+η)ε e(1+η)ε [(1 + η) ω (ε) + ω 0 (ε)]
.
2
1 − e(1+η)ε ω (ε)
We focus on the numerator and simplify:
h
i h
i
−φe(1+η)ε (1 + η) 1 − e(1+η)ε ω (ε) + θH − Φe(1+η)ε e(1+η)ε (1 + η) ω (ε) + ω 0 (ε)
h
i
= θH (1 + η) ω (ε) + ω 0 (ε) − φ (1 + η) + e(1+η)ε ω 0 (ε)
It is therefore enough to show
h
i
θH (1 + η) ω (ε) + ω 0 (ε) − φ (1 + η) + e(1+η)ε ω 0 (ε) > 0
(48)
Using the expression of ω (ε), we can compute
i
λ h −ηεM AX
e
− e−λεM AX +(λ−η)ε
λ−η
i
λ −ηεM AX h
= (1 + η)
e
1 − e−(λ−η)εM AX +(λ−η)ε
λ−η
i
λ −ηεM AX 1 + λ h
= (1 + η)
e
1 − e−(λ−η)εM AX +(λ−η)ε
1+λ
λ−η
i
1+λ h
= (1 + η) e−(1+η)εM AX
1 − e−(λ−η)(εM AX −ε) > 0,
λ−η
(1 + η) ω (ε) + ω 0 (ε) = (1 + η)
where the last line uses the fact εM AX = ln 1+λ
λ . Also, the second term in equation (48)
12
can be written as
λ −λεM AX +(1+λ)ε
e
ω (ε) = (1 + η) 1 −
1+λ
h
i
= (1 + η) 1 − e−(1+λ)(εM AX −ε) .
(1+η)ε 0
(1 + η) + e
Therefore, to prove (48), it is enough to show that for all ε,
θH e−(1+η)εM AX
i
h
i
1+λ h
1 − e−(λ−η)(εM AX −ε) − φ 1 − e−(1+λ)(εM AX −ε) > 0.
λ−η
Because φ → 0 as γ → 1 + λ, we have θH e−(1+η)εM AX > φ for γ close enough to 1 + λ. We
complete the proof by making the following observation:
Define
i
1+λ h
1 − e−(λ−η)(εM AX −ε)
λ−η
g (ε) = 1 − e−(1+λ)(εM AX −ε) ,
f (ε) =
then f (ε) > g (ε) for all ε < εM AX .
To see this, note that f (εM AX ) = g (εM AX ) = 0. Aslo, f 0 (ε) < g 0 (ε) for all ε < εM AX ,
because
f 0 (ε) = − (1 + λ) e−(λ−η)(εM AX −ε)
g 0 (ε) = − (1 + λ) e−(1+λ)(εM AX −ε) .
6
Optimal contract with Gaussian shocks
In Section 5 we augmented the stochastic process for aggregate endowment with i.i.d
standard Gaussian shocks η. This extension introduces an addition complication in the
computation of equilibrium, because all equilibrium objects, in principle, could now depend
13
the an extra state variable η. However, we show below that the equilibrium prices and
the optimal contract satisfy a homogeneity property, and the presence of η shocks does not
increase the state space of the equilibrium value and policy functions. In particular,
Lemma 4. For all u, all (g 0 , ε0 ) and η10 6= η20 , u0 ( u, ε0 , g 0 , η10 | φ, g) = u0 ( u, ε0 , g 0 , η20 | φ, g).
Proof. The SDF when Yt follows (20) is
"
0
0
x (φ0 , g 0 ) eg +σ(g )η
Λ g , η φ, g = β
x (φ, g)
0
0
0
#− 1 "
ψ
0
0
w (φ0 , g 0 ) eg +σ(g )η
n (φ, g)
0
# 1 −γ
ψ
,
(49)
with w(·) and n(·) in equations (15) and (16) appropriately modified.
The optimality conditions (28) now are modified such that all possible realizations of
(ε0 , g 0 , η 0 ),
"
0
0
0
0
eg +σ(g )η +ε c ( u0 ( ε0 , g 0 , η 0 | φ, g)| φ0 , g 0 )
β
c ( u| φ, g)
#− 1 "
ψ
0
0
0
0
eg +σ(g )η +ε u0 ( ε0 , g 0 , η 0 | φ, g)
m ( u| φ, g)
# 1 −γ
ψ
≥ Λ g 0 , η 0 φ, g ,
(50)
and u0 ( ε0 , g 0 , η 0 | φ, g) < ū (φ0 , g 0 ) implies that ”=” must hold in (50), where m ( u| φ, g) is
the modified certainty equivalent defined in (9).
When u0 ( ε0 , g 0 , η 0 | φ, g) < ū (φ0 , g 0 ), (50) simplifies to
"
0
eε c ( u0 ( ε0 , g 0 , η 0 | φ, g)| φ0 , g 0 )
c ( u| φ, g)
#− 1 "
ψ
0
eε u0 ( ε0 , g 0 , η 0 | φ, g)
m ( u| φ, g)
# 1 −γ
ψ
x (φ0 , g 0 )
=
x (φ, g)
− 1 ψ
1
w (φ0 , g 0 ) ψ −γ
n (φ, g)
(51)
From (51), we can get u0 ( g 0 , η10 , ε0 | φ, g) = u0 ( g 0 , η20 , ε0 | φ, g) for any η10 and η20 since c is a
non-decreasing function in u0 . It means the first order condition does not depend on η 0
when u0 ( ε0 , g 0 , η 0 | φ, g) < ū (φ0 , g 0 ).
When u0 ( ε0 , g 0 , η 0 | φ, g) = ū (φ0 , g 0 ), we can get that u0 ( ε0 , g 0 , η 0 | φ, g) = ū (φ0 , g 0 ) for any
η 0 . Since ū (φ0 , g 0 ) does not depend on η 0 , the first order condition does not depend on η 0
when u0 ( ε0 , g 0 , η 0 | φ, g) = ū (φ0 , g 0 ). Since we characterize the normalized optimal contracting
problem by some equivalent first order conditions which do not depend on η and η 0 , our
14
assumption that z does not depend on η is correct. We can always find an equilibrium such
that η is not a state variable.
7
Computational Algorithm
Below we describe briefly our computation algorithm.
1. Start with an initial guess of the law of motion of x, Γx (g 0 |g, z). Given the law of
motion of x, compute the SDF Λ ( g 0 , η 0 | g, x) using equation (49).
2. With the SDF, we solve the following optimal contracting problem
ˆ
v (u|g, x) = max
c,{u0 (z 0 )}
0
(1 − c) + (1 − κ)
0
Λ g , η |g, x e
g 0 +σ(g 0 )η 0 +ε0
0
0
0
0
v u z |g , x Ω(dz |g)
h
1− 1 i 1−1 1
1− 1
ψ
ψ ,
s.t : u = (1 − β) c ψ + β Mg u0
1
ˆ h
1−γ
i1−γ
0
g 0 +σ(g 0 )η 0 +ε0 0
0
0
dΩ(z |g)
e
u z
Mg u = (1 − κ)
,
u0 z 0 ≥ u g 0 , f or all z 0 ,
v u0 z 0 g 0 , x0 ≥ 0, f or all z 0 .
This step yields i) the value function v ( u| g, x) and the cutoff value of ū (g, x)
that satisfies p ( ū (g, x)| g, x) = 0; and ii) the policy functions c ( u| g, x) and
u0 ( u, g 0 , ε0 | g, x), which specifies the law of motion of u0 .
3. Let φ (t) denote the summary measure at time t. In simulations, we approximate
the continuous distribution φ (t) by a finite-state distribution as follows. We choose
(t)
(t)
(t)
(t)
(t)
u1 , u2 , · · · , uN , where u1 = u (gt ) and uN = ū (gt , xt ). A density φ is characterized
+2
N +2
by a set of grid points {u∗ [n] (t)}N
n=1 and corresponding weights {φ [n] (t)}n=1 such
that
• u∗ [1] and u∗ [N + 1] are the boundaries where the limited commitment constraint
binds: u∗ [1] = u (gt ) and u∗ [N + 1] = ū (gt , xt ). u∗ [N + 2] = ᾱuF B (gt ) is the
restarting utility.
15
0
• {u∗ [j]}j=2,3,···N are the interior points: u∗ [j] ∈ (uj−1 , uj ), for j = 2, 3 · · · N, are
chosen appropriately to minimize the approximation error.
• φ [1] and φ [N + 1] are the income shares of agents with a binding limited
commitment constraint at u∗ [1] and u∗ [N + 1], respectively.
• {φ [j]}j=2,3,···N are the income shares of agents in the interior and φ [N + 2] is
the income share of agents who entry the economy.
Clearly, we should have
PN +2
j=1
φ [j] (t) = 1 for all t.
4. Starting with an initial distribution of u, denoted φ0 (u). For example, we can choose a
point mass at u0 (g) with uIM N (g) < u (g) < ū (g, x) for every x, g, solve the following
equation for x0 :
ˆ
φ0 (u) c ( u| g, x0 ) du = 1 − x0
Now
we
have
an
initial
+1
{φ [n] (0) , u∗ [n] (0)}N
n=1 ,
condition
(φ0 , x0 ),
and
φ0
is
represented
as
such that φ [m] (0) = 1, and u∗ [m] (0) = u0 (g0 ), where
m is defined by u0 (g) ∈ (um−1 , um ).
5. Having solved x0 , we use the law of motion of u0 ( u, g 0 , ε0 | g, x) to compute φ1 . Here we
+2
describe a general procedure to solve for {φ [n] (t + 1) ; u∗ [n] (t + 1) ; xt+1 }N
n=1 given
+2
{φ [n] (t) ; u∗ [n] (t) ; xt }N
n=1 . Note that the assumed law of motion gives a natural
candidate for xt+1 . We denote xt+1 = Γ ( xt | gt , gt+1 ).
(a) First, we approximate the distribution f ( ε| g) by a finite dimensional distribution
P
P
such that Jj fg [j] = 1 and Jj eεj fg [j] = 1, for g = gH , gL .
+2
(b) Given {φ [n] (t) , u∗ [n] (t)}N
n=1 for period t, conditioning on the realization of
aggregate state gt+1 , for each n = 1, 2, · · · , N + 2, we compute {φt+1 [n, j]}n,j .
The interpretation is that φt+1 [n, j] is the total measure of income share that
comes from agents with u∗ [n] (t) and with realization of εj , which is given by:
φt+1 [n, j] = (1 − κ)fgt+1 [j] φt [n] eεj ,
j = 1, 2 · · · , J.
Note that the continuation utility of these agents is u0 ( u∗ [n] (t) , gt+1 , εj | gt , xt ),
16
a fact that we will use below.
(c) We now compute {φt+1 [m]}m=1,2,···N +2 for the next period.
φt+1 [1] =
N
+2 X
J
X
φt+1 [n, j] I{u0 ( u∗ [n](t),gt+1 ,εj |gt ,xt )≤uM IN (gt+1 )} ,
n=1 j=1
φt+1 [2] =
N
+2 X
J
X
φt+1 [n, j] Inu0 ( u∗ [n](t),g
n=1 j=1
φt+1 [m] =
N
+2 X
J
X
φt+1 [n, j] Inu0 ( u∗ [n](t),g
n=1 j=1
φt+1 [N + 1] =
N
+2 X
J
X
t+1 ,εj |gt ,xt )∈
h
t+1 ,εj |gt ,xt )∈
(t+1)
o ,
(t+1) (t+1)
,u2
u1
(t+1)
um−1 ,um
o ,
m = 3, · · · N
φt+1 [n, j] I{u0 ( u∗ [n](t),gt+1 ,εj |gt ,xt )≥ū(gt+1 ,xt+1 )} ,
n=1 j=1
φt+1 [N + 2] = κ.
The interpretation is again that φ [1] and φ [N + 1] are the income shares of
agents with a binding limited commitment constraint at u∗ [1] and u∗ [N + 1],
respectively, and {φ [j]}j=2,3,···N are the income shares of agents in the interior.
φ [N + 2] is the income share of agents who entry the economy.
+2
(d) We need to update the vector normalized utilities {u∗ [n] (t + 1)}N
n=1 . Clearly,
we should have u∗ [1] (t + 1) = uM IN (gt+1 ), u∗ [N + 1] (t + 1) = ū (gt+1 , xt+1 )
and u∗ [N + 2] (t + 1) = ᾱuF B (gt+1 ) . For m = 2, we choose u∗ [2] (t + 1) ∈
(t+1) (t+1)
u1
, u2
such that the resource constraint holds exactly for u ∈
(t+1) (t+1)
u1
, u2
. That is, we pick u∗ [2] (t + 1) to be the solution (denoted u∗ ) to
N
+2 X
J
X
=
φt+1 [n, j] c u0 ( u∗ [n] (t) , gt+1 , εj | gt , xt ) gt+1 , xt+1 Inu0 ( u∗ [n](t),g
n=1 j=1
c( u∗ | gt+1 , xt+1 )φt+1 [2] .
t+1 ,εj |gt ,xt )∈
h
(t+1) (t+1)
For m = 3, · · · , N , we choose u∗ [m] (t + 1) ∈ um−1 , um
such that the
h
(t+1) (t+1)
resource constraint holds exactly for u ∈ um−1 , um
. That is, we pick
17
(t+1) (t+1
u1
,u2
u∗ [m] (t + 1) to be the solution (denoted u∗ ) to
N
+2 X
J
X
=
φt+1 [n, j] c u0 ( u∗ [n] (t) , gt+1 , εj | gt , xt ) gt+1 , xt+1 Inu0 ( u∗ [n](t),g
n=1 j=1
c( u∗ | gt+1 , xt+1 )φt+1 [m] .
t+1 ,εj |gt ,xt )∈
6. Up to now, we have described a procedure to simulate forward the economy. This
C ∞
allows us to compute the ”market clearing” xM
t+1 t=0 as follows:
C
xM
t+1 = 1 −
N
+2
X
c( u∗ [m] (t + 1)| gt+1 , xt+1 )φt+1 [m] .
(52)
m=1
Given the sequence of {gt }Tt=1 , we simulate the economy forward for T periods to
C T
obtain xM
. We divide the sample into four cases: gH → gH , gH → gL ,
t
t=0
gL → gH , gL → gL and use regression to update the law of motion of x. We go back
to step 1 to iterate.
Note the under the above procedure, given the sequence of {gt }Tt=1 , the sequence of
xt+1 that is used for computing decision rules is complete determined by (52). In the
simulation, we assume that xt+1 follows the perceived law of motion, based on which
agent make their decisions. We use the market clearing condition to update the actual
law of motion of x and iterate.
18
h
(t+1)
(t+1
um−1 ,um