University of Warwick
EC9A2 Advanced Macroeconomic Analysis
Problem Set #3: AK models
Jorge F. Chavez∗
December 3, 2012
Problem 1
Consider a competitive economy, in which the level of technology, which is external to the firm,
is At = A(kt ) = ktβ where kt = Kt /Lt and β ∈ (0, 1]. All firms share the same technology. In
particular, production of firm j is:
α 1−α
Yj,t = At Kj,t
Lj,t
where α ∈ (0, 1), Kj,t and Lj,t are the quantities of capital and labor employed by firm j.
(a) Find the factor shares, show that kj,t = kt for all j, and that aggregate output is Yt =
∑
α 1−α
j Yj = A t Kt Lt
Solution. Firms solve:
{
}
α 1−α
max
At Kj,t
Lj,t − Rt Kj,t − wt Lj,t
2
(Kj,t ,Lj,t )∈R+
FONCs:
wrt Kj,t :
α−1 1−α
α−1
αAt Kj,t
Lj,t = αAt kj,t
= Rt
wrt Lj,t :
α −α
α
(1 − α) At Kj,t
Lj,t = (1 − α) At kj,t
= wt
Note that the fact that F is strictly concave and homothetic rules out corner solutions. In a competitive economy, firms re price takers. That is, all firms will face the same Rt and wt . Since firms
are not heterogeneous in any dimension, they all must choose the same kj,t = kt , ∀j.1
1
∗
e-mail:j.chavez-cotrado@warwick.ac.uk
If you would like to be more formal you can prove this result by contradiction. Suppose ∃i s.th. ki,t > kt .
α−1
Diminishing returns will imply that this firm will be willing to pay a lower return: αAt ki,t
< Rt = αAt ktα−1 .
Therefore this firm will be unable to rent the additional capital it requires being forced to reduce its demand
until it reaches kt .
1
EC9A2 (Fall 2012)
Problem Set # 3
Next, constant returns to scale in F imply that in equilibrium: F (Kt , Lt ) = F ′ (Kt )Kt + F ′ (Lt )Lt .
Therefore it is straightforward to check that:
Rt Kt
Yt
=
wt Kt
Yt
=
α−1
αAt kj,t
α L1−α
At Kj,t
j,t
Kt = α
α
(1 − α) At kj,t
α L1−α
At Kj,t
j,t
Lt = 1 − α
Finally, note that because all firms choose the same capital-labor ratio:
(i) the aggregate capital output ratio will also be kt :
∑
Kj,t
Kt
j Kj,t
kt =
= ∑
=
Lj,t
Lt
j Lj,t
(ii) they must be producing the same amount of output in per capital terms f (kt ) = At ktα .
Therefore:
Yt =
∑
Yj =
∑
j
(
Lj,t At ktα
=At
j
Kt
Lt
)α ∑
Lj,t =At Ktα L1−α
t
j
(b) Find the dynamical system kt+1 = ϕ(kt ), assuming that saving per worker is syt , s ∈ (0, 1).
Calculate limk→0 ϕ′ (kt ) and limk→∞ ϕ′ (kt ) for β + α < 1, β + α = 1 and β + α > 1.
Solution. Recall that At = ktβ . Then, the law of motion for kt+1 is:
(1 + n) kt+1 = syt + (1 − δ) kt
where yt =
At Ktα L1−α
t
Lt
kt+1 =
= ktβ ktα = ktα+β . Finally:
sktα+β + (1 − δ) kt
≡ φ (kt )
1+n
Next:
φ′ (kt ) =
s (α + β) ktα+β−1 + (1 − δ)
s (α + β) α+β−1 1 − δ
=
k
+
1+n
1+n t
1+n
(1)
When kt → 0:
lim φ′ (kt ) =
kt →0


+∞
[s (α + β) + (1 − δ)] / (1 + n)

(1 − δ) / (1 + n)
α+β <1
α+β =1
α+β >1
When kt → ∞:
lim φ′ (kt ) =
kt →+∞
Jorge F. Chávez


(1 − δ) / (1 + n)
[s (α + β) + (1 − δ)] / (1 + n)

+∞
α+β <1
α+β =1
α+β >1
2
EC9A2 (Fall 2012)
Problem Set # 3
(c) Show that for β = 1 − α the economy grows at a constant rate in the steady state2 if
s > δ + n. What is the steady state growth rate if s < δ + n?
Solution. When α + β = 1 and s > δ + n, the policy rule (1) becomes:
kt+1 =
(s + 1 − δ) kt
1+n
This can be rewritten as:
∆kt+1 =
sktα+β − (n + δ) kt
1+n
Recall that for a ∈ R small log(1 + a) ≈ a. This is equivalent to say that if b − 1 ∈ R is small enough,
then log(b) ≈ b − 1. Then taking logs to both sides of the above equality:
(
)
(
)
kt+1
(s + 1 − δ)
log kt+1 − log kt = log
= log
kt
1+n
Therefore:
⇒ γk =
(s + 1 − δ)
s − (n + δ)
−1=
1+n
1+n
Therefore, to have growth at a constant rate (which means that the path of {kt }∞
t=0 will be unbounded), we need the slope of φ(kt ) to be greater than 1 as is shown in figure 1b.
Finally, when s < δ + n the growth rate γk < 0 ⇔ φ′ (kt ) < 0 so that now the trivial steady state
k̄ = 0 sill become stable as shown in figure 1a.
(d) Find the economy’s steady state for β < 1 − α. Is it unique? Is it stable? How is it affected
by the saving rate?
Solution. When α + β < 1, the steady-state condition ∆kt+1 = 0 implies two constant steady-states:
)1/(1−α−β)
(
s
the trivial one k1 = 0 and a unique, non-trivial one k2 = n+δ
.
(i) For uniqueness it suffices to argue that:
φ′′ (kt ) =
s (α + β) (α + β − 1) α+β−2
kt
<0
1+n
which implies that φ(·) is strictly concave. This with the fact that φ(0) = 0 will imply that the
function φ(kt ) will cut from above the 45o line in exactly one point, which guarantees that the
non-trivial steady state is unique.
2
Define “steady-state” here.
Jorge F. Chávez
3
EC9A2 (Fall 2012)
Problem Set # 3
Figure 1: Long-run growth
(a) If
(s+1−δ)
1+n
<1
(b) If
45o
kt+1
(s+1−δ)
1+n
>1
45o
kt+1
ϕ2 (kt )
k2
ϕ1 (kt )
k1
k1
k2
k3
k2
k1
k0
k0
kt
k1
k2
kt
(ii) For stability, we need to check if |φ′ (k2 )| < 1:
φ′ (k2 ) =
=
=
=
=
s (α + β) α+β−1 1 − δ
+
k
1+n 2
1+n
[(
)1/(1−α−β) ]α+β−1
s (α + β)
s
1−δ
+
1+n
n+δ
1+n
s (α + β)
1+n
(
n+δ
s
)
+
1−δ
1+n
(α + β) n + (α + β) δ + 1 − δ
1+n
(1 + (α + β) n) − δ [1 − (α + β)]
1+n
< 1
where the last inequality follows from the fact that α+β < 1 which implies that δ [1 − (α + β)] >
0.
Finally, it is easy to show that
∂k2
∂s
>0
(e) Find the economy’s steady state for β > 1 − α. Is it unique? Is it stable? How is it affected
by the saving rate? How do you interpret this result?
Solution. The argument is analogous to the previous question. Now φ(·) is convex and |φ′ (kt )| > 1
which imply that the non-trivial steady state is also unique but unstable.
Jorge F. Chávez
4
EC9A2 (Fall 2012)
Problem Set # 3
Problem 2
Production is given by:
Yt = F (Kt , ht ) = BKtα h1−α
,
t
where ht+1 = h(et ) = ρet , is human capital and et is investment in human capital. The saving
rate is s ∈ (0, 1) (i.e., St = sYt ), investment is efficient, and physical capital fully depreciates at
the end of the period, (δ = 1):
(a) Find the dynamical system governing the evolution of output, Yt .
Solution. Before proceeding we need to realize two things. First, note that the stock of human
capital for t + 1 depends on the investment decision et which takes place at time t, and which is
going to be a function of sYt . More precisely the economy’s aggregate investment It = sYt must be
used to accumulate both physical capital and human capital. All this will mean that, unlike previous
models, now aggregate output Yt will be a state variable: Yt+1 will depend on Yt and thus we will
need to track the evolution of Yt+1 .
Second, the fact that investment is efficient means that the allocation decision for sYt must be done
in an efficient way (i.e. with some optimization criteria). In this case, the social planner (the one
that decides how much investment devote to accumulate human capital) will choose θt to maximize
α
1−α
Yt+1 = B(θt sYt ) (ρ (1 − θt ) sYt )
. That is:
{
}
α
1−α 1−α
max BsYt (θt ) (1 − θt )
ρ
0≤θt ≤1
FOCN for interior solution:3
α−1
α(θt )
1−α
(1 − θt )
α
−α
+ (1 − α) (θt ) (1 − θt )
(−1) = 0
Solving for θt we get:
1 − θt
1−α
=
θt
α
which implies that θt∗ = α, ∀t.
Finally, the law of motion for Yt+1 will be given by the value function Yt+1 (θt∗ ):
1−α
Yt+1 ≡ ξ (Yt ) = αa (1 − α)
ρ1−α sB Yt
|
{z
}
(2)
≡ψ
(b) What is the condition on the parameters that assures steady state growth?
Solution. From condition (2), taking logs and using the fact that for b − 1 ∈ R small enough log(b −
1) ≈ b is straightforward to show that we need ψ > 1
3
It is easy to show that corner solutions will be ruled out
Jorge F. Chávez
5
EC9A2 (Fall 2012)
Problem Set # 3
(c) Suppose that h(et ) = eβt , where β ∈ (0, 1). find the dynamical system governing the evolution
of output, Yt . Is there growth in the steady state?
Solution. Apply the same idea as in part (a). The social planner decides the share θt by solving:
{
}
α+β(1−α)
α
β(1−α)
max B(sYt )
(θt ) (1 − θt )
0≤θt ≤1
FONC for interior solution:
α−1
α(θt )
β(1−α)
(1 − θt )
[
]
−1
αθt−1 − β (1 − α) (1 − θt )
=0
Then:
β (1 − α)
α
=
θt
1 − θt
which implies:
θt∗ =
α
α + β (1 − α)
Finally, the law of motion of Yt+1 is:
(
Yt+1 = Bs
|
α+β(1−α)
α
α + β (1 − α)
)α (
{z
α
1−
α + β (1 − α)
≡λ
)β(1−α)
α+β(1−α)
Yt
}
Taking logarithms:
log Yt+1 = log λ + [α + β (1 − α)] log Yt
(3)
where α + β (1 − α) < 1. This implies that the growth rate of {Yt+1 }∞
t=0 will be decreasing overtime,
and in fact will tend to 0 in the long run. In other words, the economy will exhibit growth only
transitorily. To see this subtract (3) from its lagged version:
log Yt+1 − log Yt
γYt+1
Jorge F. Chávez
= [α + β (1 − α)] (log Yt − log Yt−1 )
= [α + β (1 − α)] γYt
6