1. No category

Macroeconomic Theory, Fall 2014
Instructor: Dr. Yulei Luo
SEF, HKU
October 2014
Suggested Solution to Problem Set 2
1. [10 points] Consider the following Ramsey-Cass-Koopmans model with …scal
policy. First, we assume that the private sector (households-…rms) is modeled
as a representative agent who maximizes its lifetime utility and has access to
a technology to produce consumption goods using capital and labor as inputs.
The single good in the economy can either be consumed or saved in the form
of physical capital. In addition, the government in the economy collects taxes
on output and consumption for …nancing its spending. The government does
not print any money or does not issue bond. Therefore, the single-period
government budget constraint is
gt =
y
t yt
+
c
t ct ; t
0;
(1)
where gt is government spending, yt is the output tax rate, ct is the consumption tax rate, yt is output, and ct is consumption. For simplicity, we assume
that yt and ct are constant over time, i.e., yt = y and ct = c . With population growth equal to n, the resource constraint for the representative agent
is
y
(1 + n) kt+1 = (1
) kt + (1
(1 + c ) ct ;
(2)
t ) yt
where yt = f (kt ) = kt ,
is to
2 (0; 1). The objective of the representative agent
max
fct ;kt+1 g
where
1
X
t=0
1
t ct
1
1
;
(3)
> 0 and subject to (2) and given k0 .
(a) Derive the Euler equation for this RCK model with …scal policy.
Solution: Given the balanced government budget, the resource constraint
facing the representative agent can be written as:
(1 + n) kt+1 = (1
y
t ) yt
The Lagrangian for this problem is
(
1
X
c1t
1
y
t
+ t [(1
L=
t ) yt
1
t=0
1
c
(1 +
(1 +
c
) ct + (1
) ct + (1
) kt :
) kt
(4)
)
(1 + n) kt+1 ] :
The FOCs w.r.t. ct and kt+1 yields:
ct
(1 + n)
c
= (1 +
t
=
)
t;
(5)
y
t)
(1
kt+11 + 1
t+1
and the TVC is
t
lim
t!1
t kt+1
= 0:
(6)
Combining the two FOCs yields the following Euler equation:
(1 + n) ct
=
y
t)
(1
kt+11 + 1
ct+1 :
(7)
(b) Find the steady state values of k and c.
Solution: In the steady state, we have kt+1 = kt = k and ct+1 = ct = c.
Using (7), we can pin down:
k=
1=(1
y
t)
(1
(1 + n) =
(1
)
:
)
Using the resource constraint, we can pin down:
i
1 h
y
c=
k
(n
+
)
k
:
(1
)
t
1+ c
(8)
(c) After linearizing the two-di¤erence equation system around the steady
state, show that the model economy is saddle-point stable in the neighborhood
of the steady state.
Solution: We can now linearize the two di¤erence equation system around
the steady state:
#"
#
"
#"
# "
1
1+ c
e
e
k
1 0
kt+1
t
1+n
=
;
(9)
e
ct
Q 1
e
ct+1
0
1
| {z }| {z } |
{z
}| {z }
J
u
v
M
where x
et+1 = xt+1 x (x = c; k) and Q = 1 1+n (1
0. The above system can be rewritten as
"
# "
#"
1
1+ c
e
kt+1
1+n
=
c
1
e
ct+1
Q 1 + Q 1+
1+n
| {z } |
{z
}|
u
jbI
Kj = 0 =)
K=J
1M
1+ c
1+n
1
b
1
2
Q
b
1
y
t)
(
#
e
kt
:
e
ct
{z }
v
c
Q 1+
1+n
=0
1) k
2
c>
(10)
b2
1+Q
1+ c
1
1
b + = 0 =)
+
1+n
trace (K) = b1 + b2 = 1 + Q
det (K) = b1 b2 =
1
1
1+ c
+ > 2;
1+n
>1
Hence, the discriminant should be positive because
= trace (K)2
4 det (K) =
1+Q
1+ c
1
+
1+n
2
4
1
>0
which means that both roots are real. Also, because det = 1 > 1 and
trace > 2; the two roots must individually be positive. We can also judge
the magnitudes of the two roots as follows:
jbI
Kj = 0 () p (b) = (b
b1 ) (b
p (1) = (1
=
b1 ) (1 b2 ) = 1
1+ c
< 0:
Q
1+n
b2 ) = 0 =)
trace (K) + det (K)
This can only be true if one root (say b1 ) is less than 1 and the other root is
greater than 1: We can then conclude (and con…rm the predictions of the PD)
that the equilibrium is saddle-point.
2. [6 points] (a) Find the unique invariant unconditional probability distribution
for the Markov chain with transition probabilities described by
2
3
0:7 0:2 0:1
6
7
P = 4 0:3 0:4 0:3 5 :
(11)
0:1 0:1 0:8
Solution: (a) The unique invariant unconditional probability distribution is
= [0:4000 0:2857 0:3143] :
(b) Given the following AR(1) process:
yt+1 = yt + "t+1 + c;
where
(12)
2 (0; 1), and
Et ["t+1 ] = 0; Et "2t+1 =
3
2
; Et ["t+k "t+k+1 ] = 0:
(13)
Compute (i) the unconditional mean and variance of y: E [yt ] and var [yt ],
and (ii) the conditional expectation and variance of yt+k based on the time-t
information set: Et [yt+k ] and vart [yt+k ], for k 1.
Solution: (i)
E [yt ] =
2
c
; var [yt ] =
1
2
1
:
(ii)
Et [yt+k ] =
k
yt +
0
k 1
X
vart [yt+k ] = @
j=0
k 1
X
j
c;
j=0
1
2j A
2
; for k
1:
3. [5 points] Consider the following AK model with distortionary taxes:
max
fct ;kt+1 g
where
1
X
1
t ct
1
t=0
1
;
(14)
ct + Tt ;
(15)
> 0 and the resource constraint is
kt+1 = (1
) Akt
where is the tax rate on capital income levied by the government and Tt is
a lump-sum transfer from the government. Assume that the government runs
balanced budget:
Tt = Akt :
(16)
Derive the growth rate of consumption in terms of model parameters when the
economy is on the balanced growth path and show that this growth rate is a
decreasing function of the tax rate, .
Solution: Note that the agent takes the transfer from the government (Tt ) as
given. The Euler equation is
ct
=
(1
) Act+1 ;
(17)
where (1
) A is the after-tax return return on capital. The growth rate of
consumption can thus be written as
ct+1
= [ (1
) A]1= :
ct
It clearly shows that
@
ct+1
ct
@
4
< 0:
(18)
4. [9 points] In the optimal growth models discussed in class, we assume that
the agent does not value leisure and labor supply is inelastic and …xed. Here
we relax this assumption by allowing the agent to decide how to split his or
her time between enjoying leisure and working. Speci…cally, we assume that
the agent need to solve the following optimization problem:
max
fct ;lt ;kt+1 g
1
X
t
u (ct ; ht ) ;
t=0
subject to
kt+1 = (1
) kt + f (kt ; ht )
ct ;
given k0 , where kt is capital, ct is consumption, ht measures the units of labor
used to produce goods, 1 ht measures the units of leisure, and the production
function and the utility function are de…ned as
f (kt ; ht ) = kt ht1
and u (ct ; ht ) = ln (ct ) + B ln (1
ht ) ;
(19)
respectively. Here B is some positive constant.
(a) Find the …rst order conditions for ct ; lt ; and kt+1 .
Solution:Set up the Lagrangian:
"1
X
t
L=E
ln (ct ) + B ln (1 ht ) +
t
(1
) kt +
kt ht1
ct
t=0
kt+1
#
(20)
where t 0 denote the multiplier on the resource constraint, (??), at time t.
The FOCs w.r.t. ct , lt , and kt+1 are
1
ct
B
1
ht
t
=
t;
=
t (1
=
Et
(21)
"
)
1
kt
ht
+
;
(22)
kt+1
ht+1
1
!
t+1
#
:
(23)
(b) Using all the …rst-order conditions together with the resource constraint
to derive the steady state values of ct ; lt ; and kt+1 .
Solution: Combining two intra-temporal …rst-order conditions, (21) and (22),
gives
B
1
kt
= (1
)
:
(24)
1 ht
ct
ht
5
;
By the de…nition of steady sate, the steady state values of ct ; ht ; and kt+1
must satisfy
B
1
1
(1
c
=
h
1 =
k
h
)
1
;
k
h
+
1
0 =
(25)
!
1
k+k h
;
(26)
c:
(27)
From (26), we can determine
1
k
1
=
h
1=(
+
1)
1
:
(28)
Substituting it into (27), we have
c
1
=
k
1
k
h
and
Combining the expressions for
1
c
1
=
h
+
1
h
=
1
c
k
gives
1
1
Substituting the expressions for
c
+
k
h
1
and
c
k
1
1
B
:
1
1=(
+
1)
1
(29)
into (25) gives
=(
+
1)
1
:
(30)
Using (29) and (30), we can determine h:
1
h
h
1
=
=
h =
"
1
+
1
1
(
"
1+ 1
1
1
1
+
1
1
#
1
1
+
1
B
1
B
#
=)
B
1
)
1
;
which means that
@h
< 0:
@B
Using (31), (28), and (29), we can easily determine k and c.
6
1
1
1
1
+
1
(31)
(c) What is the e¤ect of an increase in B on the steady state value of capital?
Explain brie‡y about the economic intuition behind your result about the
e¤ect of B on the steady state capital stock.
Solution: Since
k=
and
@h
@B
1
1
1=(
+
1
1)
h;
< 0, an increase in B will reduce the steady state level of capital stock:
@k
< 0:
@B
The intuition behind this result is that an increase in B means that the agent
values leisure more and supplies less labor to the labor market; consequently
the economy will produce less goods and services and then lead to less capital
stock and consumption in the steady state.
7