University of Minnesota
Department of Economics
Econ 3102: Intermediate Macroeconomics
Handout 5
Just for completeness, this handout presents the deﬁnition of a competitive equilibrium (CE)
for a twoperiod economy, and characterizes its solution. To ease notation, I’ll use subindices
1 and 2 to identify the current and future periods, respectively.
1 Competitive equilibrium
Consider the following:
Deﬁnition 1.1.
A
competitive equilibrium
is a set of prices (
w
1
,w
2
,r
), currentperiod vari
ables (
C
1
,‘
1
,S,N
d
1
,I,T
1
) and futureperiod variables (
C
2
,‘
2
,N
d
2
,T
2
) such that given the
exogenous variables (
G
1
,G
2
,z
1
,z
2
,K
1
) we have:
1. Given (
w
1
,w
2
,r
), the representative consumer chooses (
C
1
,C
2
,‘
1
,‘
2
,S
) to solve
max
C
1
,C
2
,‘
1
,‘
2
,s
U
(
C
1
,‘
1
) +
U
(
C
2
,‘
2
)
subject to
C
1
+
S
=
w
1
(
h

‘
1
) +
π
1

T
1
C
2
=
w
2
(
h

‘
2
) +
π
2

T
2
+ (1 +
r
)
S
C
1
,C
2
≥
0
0
≤
‘
1
,‘
2
≤
h,
where
N
s
1
=
h

‘
1
,
N
s
2
=
h

‘
2
.
2. Given (
w
1
,w
2
,r
), the representative ﬁrm chooses (
N
d
1
,N
d
2
,I
) to solve
max
N
d
1
,N
d
2
,I
π
1
+
π
2
1 +
r
subject to
π
1
=
z
1
F
(
K
1
,N
d
1
)

w
1
N
d
1

I
π
2
=
z
2
F
(
K
2
,N
d
2
)

w
2
N
d
2
+ (1

d
)
K
2
K
2
= (1

d
)
K
1
+
I
N
d
1
,N
d
2
,I
≥
0
,
where
Y
1
=
z
1
F
(
K
1
,N
d
1
),
Y
2
=
z
2
F
(
K
2
,N
d
2
).
3. The government’s budget constraints are satisﬁed
G
1
=
T
1
+
B
G
2
+ (1 +
r
)
B
=
T
2
.
1