Ps. 3
Exercise 1
a) a = aR + ω − c.
0
b)
State variable is a.
A recursive competitive equilibrium is a set of functions V (a), g (a), R such
that
1. V (a) solves the consumer's functional equation
n
0 o
V (a) =max u (c) + βV a
c
0
s.t.a = aR + ω − c
and a = g (a) is the associated policy function
2. Consistency is satised
0
g (0) = 0
c)
n
0 o
V (a) = max u (ct ) + βV a
ct
0
0
0
u (ct ) = βV a
0
0
0
V (a) = βV a R
Hence the resulting Euler equation
0
0
u (ct ) = βRu (ct+1 )
d)
The consistency condition together with the budget constraints implies ω = c
and by the Euler equation R = β1
Exercise 2
a)
The budget constraints are
I = c0 + qa
c1s = a + ω1s n1s
b)
maxm U = ln (c0 ) + β
c0 ,{ns }s=1
m
X
πs [ln (c1s ) + ln (1 − n1s )]
s=1
s.to c1s =
1
(I − c0 ) + ω1s n1s
q
substituting the budget constraint in the utility function we get
U = ln (c0 ) + β
m
X
s=1
πi ln
1
(I − c0 ) + ω1s n1s
q
1
+ ln (1 − n1s )
c)
with FOCs
m
1
βX
1
=
πs
c0
q i=s
c1s
ω1s
1
=
c1s
1 − n1s
b)
From the intratemporal condition
ω1s (1 − n1s ) = a + ω1s n1s
ω1s − a = 2ω1s n1s
ω1s − a
n1s =
2ω1s
therefore
c1s =
ω1s + a
2
and from the Euler equation
m
1
βX
2
=
πs
I − qa
q i=s
ω1s + a
5.
No, consumption will be dierent in each state of the world.
Exercise
3
a)
m
X
qs as + c0 = I
s=1
c1s = ns ωs + as
b)
at time t = 1, the agent chooses ns such that
maxm U = ln (c0 ) + β
c0 ,{ns }s=1
m
X
πs [ln (c1s ) + ln (1 − n1s )]
s=1
s.to
m
X
qs (c1s − ns ωs ) + c0 = I
s=1
c)
Using Lagrange we derive the FOCs:
1
=λ
c0
β
πs
= λqs
c1s
2
βπs
= qs ωs
1 − ns
Hence the Euler equation
β
πs
1
= qs
c1s
c0
Exercise 4
1. The resource constraint of each agent is a = aR + ω − c. The resource
constraint of the economy is
0
c1t + c2t = ω 1 + ω 2
2. Market clearing requires
a = a1t + a2t = 0
3. Since market clearing implies a1t = −a2t , we can use A1 as the aggregate
state variable (it is a state variable since the distribution of asset holding matters
for prices). Whenever marginal propensity to save out of wealth are not the same
across the individuals, dierent prices are required to make total savings be zero.
A recursive equilibium is a set of functions
quantities: g1 (a1 , A1 ), g2 (a2 , A1 ) , G (A1 )
Lifetime utility levels: V1 (a1 , A1 ) ,V2 (a2 , A1 )
Prices: q (A1 )such that
1. Vi (ai , A1 ) is the solution to consumer i's problem:
n
0 0 o
Vi (ai , A1 ) = max 0 ui ci + βi Vi ai , A1
ci ≥0,ai
0
s.to. ci + ai q (A1 ) = ai + ωi
0
A1 = G (A1 )
The solution to this functional equation delivers the policy function gi (ai , A1 )
2. Consistency
G (A1 ) = g1 (A1 , A1 )
−G (A1 ) = g2 (−A1 , A1 )
3