s.t.
c
t
+
k
t
+1
≤
f
(
k
t
, n
t
)
k
0
given,
c
t
, k
t
+1
, n
t
, `
t
≥
0
`
t
+
n
t
= 1
Show that the set of feasible allocations is convex.
Solution:
1. Showing that the constraint set is convex. Deﬁne,
B
=
{{
c
t
, l
t
, k
t
+1
}
∞
t
=0
∈
`
∞
:
c
t
+
k
t
+1
≤
f
(
k
t
,
1

l
t
)
, c
t
≥
0 and
k
t
+1
≥
0
∀
t
}
Let
{
c
1
t
, l
1
t
, k
1
t
+1
} ∈ B
,
{
c
2
t
, l
2
t
, k
2
t
+1
} ∈ B
and
θ
∈
[0
,
1]
.
The following holds,
c
1
t
+
k
1
t
+1
=
f
(
k
1
t
,
1

l
1
t
)
(4)
c
2
t
+
k
2
t
+1
=
f
(
k
2
t
,
1

l
2
t
)
(5)
Then for 0
< θ <
1,
[
θc
1
t
+ (1

θ
)
c
2
t
] + [
θk
1
t
+1
+ (1

θ
)
k
2
t
+1
]
=
θf
(
k
1
t
,
1

l
1
t
) + (1

θ
)
f
(
k
2
t
,
1

l
2
t
)
(6)
Assuming f is concave as usual,
θf
(
k
1
t
,
1

l
1
t
) + (1

θ
)
f
(
k
2
t
,
1

l
2
t
)
≤
f
(
θk
1
t
+ (1

θ
)
k
2
t
, θ
(1

l
1
t
) + (1

θ
)(1

l
2
t
))
(7)
(6) and (7) imply,
[
θc
1
t
+ (1

θ
)
c
2
t
] + [
θk
1
t
+1
+ (1

θ
)
k
2
t
+1
]
≤
f
(
θk
1
t
+ (1

θ
)
k
2
t
, θ
(1

l
1
t
) + (1

θ
)(1

l
2
t
))
(8)
⇒ {
(1

θ
)
c
1
t
+
θc
2
t
,
(1

θ
)
l
1
t
+
θl
2
t
,
(1

θ
)
k
1
t
+1
+
θk
2
t
+1
} ∈ B
⇒ B
, the set of sequences
{
c
t
, l
t
, k
t
+1
}
∞
t
=0
that satisfy the constraints is
a convex subset of
R
∞
.
Problem 3
Consider the social planner’s problem (SPP)above with CobbDouglas
technology, partial depreciation and CRRA preferences. Derive the Euler equa
tion.
2