Problem Set 5, Jeremy Lise - UCL

Problem Set 5, Jeremy Lise
November 21, 2007
Exercise 1
(a)
The Bellman Equation for this problem is
(b)
V (k t , h t ) =
max
{φ t,h t+1 ,k t+1 } {u(kα t (φ t h t ) 1−α − k t+1 ) + βV (k t+1 , h t+1 )
+λ[A(1 − φ t )h t − h t+1 ]}
The first order conditions for the control variables are
∂V (k t , h t )
∂k t+1
= 0 : c −γ
t = βV k (k t+1 , h t+1 )
∂V (k t , h t )
= 0 : λ = βV h (k t+1 , h t+1 )
∂h t+1
∂V (k t , h t )
= 0 : (1 − α) y t
c −γ
t = Ah t λ
∂φ t φ t
The Envelope Conditions for the state variables are
and
V k (k t , h t ) = α y t
c −γ
t
k t
V h (k t , h t ) = (1 − α) y t
c −γ
t + λA(1 − φ t )
h t
= (1 − α) y t
c −γ
t + (1 − α) y ( )
t 1 −
c −γ φt
t
h t h t φ t
1
= (1 − α) y t
c −γ
t
φ t h t

Use the Envelope Condition for k t in the first order condition for consumption
to get the standard consumption Euler Equation
( )
c −γ yt+1
t = αβ c −γ
t+1 (1)
k t+1
Combine the first order conditions for φ t and h t+1 and use the Envelope
Condition for h t to get the Euler Equation for human capital
y t+1
(1 − α) y t
c −γ
t = Ah t β(1 − α) c −γ
φ t φ t+1 h t+1
c −γ
t
= Aβ
(
yt+1
y t
t+1
) (
φt
φ t+1
) (
ht
h t+1
)
c −γ
t+1 (2)
(c) Along a balanced growth path (BGP) human capital h t grows at a
constant rate g h . From the production function for human capital we have
1 + g h = A(1 − φ t )
But as the left hand side of the equation is constant, so is the right hand
side. It follows that along a BGP, φ t = φ is constant.
(d) First consider equation (1)
( ) γ ct+1
= (1 + g c ) γ = αβ
c t
(
yt+1
k t+1
)
For g c to be constant along a BGP, the ratio y t+1 /k t+1 has to be constant.
But this implies g y = g k .
Now divide the resource constraint by k t
c t
k t
= y t
k t
− k t+1
k t
= y t
k t
− (1 + g k )
Along a BGP the right hand side has to be constant, so the ratio c t /k t has
to be constant, i.e. g c = g k .
Finally, divide the consumption goods production function by k t
( ) α ( ) 1−α ( ) 1−α
y t kt
= φ 1−α ht
= φ 1−α ht
k t k t k t k t
2

As g y = g k , it follows from this equation that g h = g k .
This establishes that g h = g k = g h = g.
In a last step we will solve for g. Take (2) along the BGP
( ) ( )
φ 1
(1 + g) γ = Aβ(1 + g)
= Aβ
φ 1 + g
It follows that
Exercise 3
g = (Aβ) 1/γ − 1
(a) As only households can hold bonds, in equilibrium aggregate bonds
need to be zero, i.e. households’ demand for bonds has to be equal to households’
supply of bonds. However given that all households are the same and
they all face the same aggregate shock, at given prices they all either want to
sell bonds or buy bonds. So in equilibrium (!) for the bond market to clear,
individual bond holdings have to be zero ( B t = 0 ) in ever period. But this
implies that the transversality condition for bonds holds trivially.
(a) Let’s solve the household problem first. The households will chose
current and future consumption and labor supply in order to maximize its
intertemporal utility function subject to its budget constraint, taking prices
as given.
The Bellman equation for the households problem is
[ c
1−θ
]
t
V (B t ) = max
C t,L t 1 − θ − φ L1+η t
1 + η + e−ρ V (W t L t + R t−1 B t−1 − C t )
The first order conditions are
and by the Envelope Condition
C −θ
t = e −ρ V ′ (B t )
φL η t = W t e −ρ V ′ (B t )
V ′ (B t−1 ) = R t−1 e −ρ V ′ (B t ) = R t−1 C −θ
t
3

where I have used the first order condition for consumption.
Combining these three equation we get a labor supply equation for the household
φL η t = W t Ct −θ
(3)
and a consumption Euler Equation
C −θ
t
Together with the household’s budget constraint
= e −ρ R t C −θ
t+1 (4)
W t L t + R t−1 B t−1 = C t + B t (5)
these two equations describe optimal household behavior at given prices.
Now let’s turn to the firm. The firm will maximize profits taking prices
as given. The firm’s profit function is
Π t = e zt N t − W t N t = (e zt − W t )N t
For a given W t the firm’s labor demand is as follows
N t = ∞ for W t < e zt
N t = [0, ∞) for W t = e zt
N t = 0 for W t > e zt
In equilibrium households and the firm will behave optimally given prices,
and all markets clear. We have three markets, the consumption goods market,
the bond market, and the labour market. By Walras’ Law we can ignore
one market clearing condition. Let’s take the consumption goods market.
As seen in (a), bond market clearing requires
Labor market clearing requires
B t = 0 (6)
N t = HL t (7)
But from the firm’s labor demand curve we can see that this immediately
implies
W t = e zt . (8)
Equations (3) to (8) characterize the competitive equilibrium of this economy.
4

(c) Now assume z t = 0 and derive the steady states for W, Y, C, N and R.
First, from (8) we have
W = 1
Using this and B t = 0 we get from (5) that
L = C
Now by (3)
φL η = L −θ ⇒ L =
( 1
φ) 1
η+θ
But this together with the labor market clearing condition (7) gives us steady
state output
( 1
1
η+θ
Y = H
φ)
Finally from the consumption Euler Equation (4) we have
R = e ρ
(d) Now let’s log-linearize equations (3) to (8).
Equation (3) is
φ(e lt L) η = e wt W (e ct C) −θ = φL η e ηlt = W C −θ e wt−θct
Using the steady state definition and taking logs on both sides we get
Equation (4):
l t = 1 η (w t − θc t ) (9)
(e ct C) −θ = e −ρ e rt R(e c t+1
C) −θ
Again using the steady state of R we get
c t+1 − c t = 1 θ (r t − ρ) (10)
Equation (5): Use the equilibrium condition B t = 0. Then
e wt W e lt L = e ct C
5

Equation (7):
Equation (8):
Finally output is
c t = w t + l t
e nt N = He lt L
n t = l t (11)
e wt W = e zt
w t = z t (12)
e y t Y = e zt Ne nt
y t = z t + n t (13)
(e)
Using the budget constraint and w t = z t we have
c t = z t + l t
Now plug this into the labor supply equation
ηl t = z t − θ(z t + l t )
It follows that labor is
l t =
( ) 1 − θ
z t (14)
θ + η
Now use the budget constraint again to get c t
( ) 1 + η
c t = z t (15)
θ + η
Output in this model has to be equal to consumption as there is no investment
( ) 1 + η
y t = z t (16)
θ + η
Finally the interest rate is
[ ] θ(1 + η)
r t = ρ + θ(c t+1 − c t ) = ρ +
(z t+1 − z t )
θ + η
6

where
[ ] θ(1 + η)
r t = ρ +
e t+1 −
θ + η
z t+1 = αz t + e t+1
1
[ θ(1 + η)(1 − α)
θ + η
]
z t (17)
To derive the variances, note that Var(z t ) = σ 2 /(1 − α 2 ). So
Var(w t ) =
σ2
1 − α 2
( ) 2 1 − θ σ 2
Var(l t ) =
θ + η 1 − α 2
( ) 2 1 + η σ 2
Var(c t ) = Var(y t ) =
θ + η 1 − α 2
{ [θ(1 ] 2 [ ] 2 ( ) } + η) θ(1 + η)(1 − α) 1
Var(r t ) =
+
σ 2
θ + η
θ + η 1 − α 2
(f) Real wage are procyclical in the model whereas they a acyclical in the
data.
Hours worked and productivity ( here z t ) are positively correlated in the
model, but are uncorrelated or slightly negatively correlated in the data.
Hours worked fluctuate less than output and consumption in the model (
−θ < η ), but are as volatile as output in the data, and more volatile than
consumption.
Output is as volatile as consumption in the model, but is more volatile than
consumption in the data.
1 I have changed notation here so as not to confound the persistence of the shock with
the time discount factor.
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