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Macroeconomics I, UPF
Professor Antonio Ciccone
SOLUTIONS PROBLEM SET 5
1
The Solow AK model with transitional dynamics
Consider the following Solow economy: production is determined by
Y = F (K; L) = AK + K L1
Population grows at rate n, capital depreciates at rate :Consumers save a fraction s of their income. Moreover, sA > + n:
(a)
The intertemporal resource constraint is given by K_ = sF (K; L)
_
_
K L
d K
(L) = K
K, or, using k_ = dt
L
L L ; expressed per capita terms:
k_
and hence
= sf (k)
= (sA
k_
= sA
k
( + n)k
n)k + sk
( + n) + sk
1
(1)
_
Clearly, as t ! 1 (i:e: k ! 1) : kk = sA ( + n) > 0, by assumption. This
Solow economy features a constant positive growth rate in the very long run.
Because of this perpetual growth dynamics no balanced growth path exists. The
result is due to the fact that the returns to capital are decreasing but are never
lower than A; even in the very long run. In consequence, net savings are always
positive as capital accumulation is always advantageous.
(b)
Because of decreasing marginal product of capital the growth rate
of capital must decrease as the economy accumulates more capital (see (1)).
Output per worker, YL = Ak + k , will therefore also approach a constant
growth rate that is lower than the initial growth rate. For two economies that
di¤er only in their initial levels of capital, the poorer country will enjoy higher
initial growth rates of both capital and output per worker than the rich country.
Hence, there is convergence in growth rates in the very long-run but not in levels
since because of the perpetual growth dynamics the initially richer country will
have a higher capital stock per worker - and thus higher output per worker - ad
in…nitum.
1
2
Ramsey-Cass Koopmans and the AK model
Consider an economy where the production possibilities are described by Y = F (K ) = AK ,
where A is a positive constant. Consumption is maximized intertemporally by
in…nitely-lived consumers, which leads to the well-known Euler equation:
c(t)
_
= [r
c(t)
]
(K)
(a)
The capital market equilibrium implies M P K := dFdK
= A =
r+ , r = A
. The real interest rate is a constant in this model because the marginal product of capital is constant. Thus consumption growth in
equilibrium is
c(t)
_
= [r
] = [A
] =: c :
(2)
c(t)
(b)
A constant real interest rate implies a constant consumption - capital
ratio and hence a constant consumption - output ratio. The policy function for
consumption has the following form: C(t) = cY (t); where c is some positive
constant. Hence, in a balanced growth path the growth rate of consumption
will be equal to the growth rates of k and y; that is, c = k = y :
(c)
By (2) an unexpected fall in leaves r = A
unchanged and
induces an increase in the growth rates of consumption, capital and output in
the new balanced growth path. From the capital accumulation equation we see
_
K(t)
C(t)
that K(t)
=A
due to a fall in must lead to
K(t) = . An increase in
C(t)
a fall in the consumption - capital ratio, K(t)
, which is to say that the savings
rate has to increase. To see this e¤ect more directly, notice that we can rewrite
the capital accumulation equation as
_
AK(t) C(t)
K(t)
=
AK(t)
AK(t)
|
{z
}
_
K(t)
K(t)
,
=:
AK(t)
K(t)
= sA
,s=
+
;
A
s
where s is the savings rate. Thus, an increase in the growth rate, , increases the
savings rate, as well. The same qualitative results materialise if the production
function is subject to decreasing returns to capital. However, in this RamseyCass-Koopmans AK model there are no transitional dynamics. The savings rate
immediately jumps to its new permanent level as in the Solow model.
2
3
Growth through knowledge externalities in a
Ramsey-Cass-Koopmans model
Consider the following production function:
Y = K (AL)1
E¢ ciency A is determined by the size of the aggregate capital stock, K. Population growth is zero. Consumption is determined by the dynamic maximization,
leading to the usual Euler equation:
c(t)
_
= [r
c(t)
]:
(a)
In the decentralized equilibrium …rms maximize their pro…ts subject
~ = rt + ; i.e. the externality is not taken into account
to costs which yields f 0 (k)
by the marginal …rm. The capital accumulation equation for capital per e¤ective
labour is unchanged
:
k~t
= f (k~t ) ct
k~t
= k~t
ct
k~t :
The amount of capital on the balanced growth path is determined by the
c_ = 0 locus:
~
f 0 (k)
=
+
, k~
1
= +
(3)
1
1
) k~BGP =
:
+
The social planner, on the contrary, takes into account the interaction between the size of the capital stock K and A. That is, the planner chooses a
capital market equilibrium that re‡ects the total marginal product of capital:
K
1
~ 1 :
Y = F (K; AL) = KL1
since A = K and therefore y~
AL = L = kL
Thus,
@ y~
= L1 :
@ k~
which implies f 0 (k) = L1 = + : Hence,
1
k~SP = =
L
1
+
1
1
:
(4)
Comparing the capital stock in the decentralized market equilibrium (3) and
the social planner allocation (4) we observe that since 2 (0; 1) ; k~SP > k~BGP :
Because …rms do not take into account the link between technology and the
3
capital stock in the decentralized equilibrium there is less capital than in the
social planner optimal allocation.
(b)
Yes, the capital stock in (a) depends on the size of the labour force.
The marginal product of capital depends on the size of the labour force and
so does the capital stock. If the externality is determined by capital intensity,
i.e. A = k, instead, the production function facing the social planner reduces to
1
= K: Hence, y~ = k~ and the marginal product
Y = K (AL)1 = K ( K
L L)
0 ~
of capital is f (k) = 1: It does not depend on labour, therefore the capital stock
is independent of the size of the labour force.
4
Lucas (1988) model of human capital
Consider the following economy: individuals spend a fraction (1-u) of their lifetimes in accumulating human capital and the remainder (u) in production. The
workforce input in production can hence be described as uLh, where L represents
total labour force, and h is the level of human capital per capita:
1
Y = K [uLh]
:
(Physical) capital depreciates with rate : Human capital accumulation is proportional to the amount of time spent in education:
h_
= (1
h
u):
Consumption is determined by dynamic maximization of in…nitely-lived households, leading to the usual Euler equation:
c(t)
_
= [r
c(t)
(a)
]:
Output per capita:
Y
=: f (k) =
L
K
L
1
1
(uh)
= k (uh)
The intertemporal resource constraint is, K_ = F (K; L)
capita terms:
k_ = f (k) ( + n) k c
:
K
C; or, in per
where c := C=L:
K
: Then, as in the standard Ramsey-Cass-Koopmans
(b)
De…ne k~ := uhL
~
model, there exists a balanced growth path for k~ where dk=dt
=0: Implying that
on a balanced growth path
d ~
d
(k) =
dt
dt
K
uhL
=
4
d
dt
k
uh
= 0:
(5)
_
Since h=h
= (1
h: Thus,
u); in order for (5) to be ful…lled, k must be proportional to
k_
h_
= = (1 u):
k
h
Moreover, since the capital market is in equilibrium capital earns its marginal
1
kt
= rt + : On a balanced growth path k
product: M P K = f 0 (k) =
uht
and h grow at the same (constant) rate therefore the physical-to-human capital
ratio, k=uh; is constant along a balanced growth path. Hence, the real interest
rate is constant, implying that consumption, physical and human capital all
_
_
grow at the same constant growth rate in equilibrium: c=c
_ = k=k
= h=h
=
(1 u): The only way policymakers can in‡uence long-run growth is to increase
the accumulation of human capital.
5
Learning-by-doing
Consider a standard Solow model without depreciation and without population
growth ( = n = 0): The aggregate production function is Y = K (AL)1 ;
2 (0; 1):
Substitute
1
A_
Y
K (AL)
=
=
= k~ L = a
A
A
A
into
:
k~
= sk~ 1 a
k~
to get
:
k~
= k~
k~
s
k~
L :
Both s and L are (positive) constants. Therefore, the long-run growth rate of
:
~
capital per e¤ective worker must be zero, i.e. kk~ = 0: Hence, the long-run level
of k~ is:
s
k~BGP = :
L
The balanced growth path growth rate of income per capita is:
y_
y
= Lk~ = s L1
:
BGP
Finally, using y = A~
y , the long-run level of income per capita is given by:
yBGP = A
5
s
L
:
6
Learning-by-doing again
Now assume an economy as described in problem 5.5, but A_ = Y
< 1:
where 0 <
(a)
K (AL)1
A_
=
A
A
=A
1
L k~
=a
where, as usual, k~ = K=AL: The two-dimensional system governing the dynamics of capital per e¤ective worker and technology is given by:
:
a_
a
:
k~
=
= sk~
k
In (k; a) space the
a_
a
= 0 and
a_
a
k~
+(
k
k_
k
1
1)a
a:
= 0 isoclines yield the following two curves
=
0)a=
=
0 ) a = sk~
1
(1
)
sk~
1
:
k~
k
1
Graphically, the system can be represented as in …gure 1. The balanced growth
path is globally stable as can be seen from the two red lines depicting possible
paths of the system.
k - isocline
a - isocline
6
7
Endogenous growth, ideas and capital
Consider the following endogenous growth model: both capital and labour can be
used either in production of goods or in research and development. Fractions a K
and a L of capital respectively labour are used in R&D. The production function
hence is:
1
Y = [(1 aK ) K] [(A(1 aL ) L]
while new ideas are generated according to the following R&D process:
A_ = B [aK K] [aL L] A
where B, ; , are positive constants. Consumers save a fraction s of their
income; depreciation is equal to zero, population grows at rate n.
(a)
A_
= B [aK K] [aL L] A 1 :
gA :=
A
Hence, taking logs and derivatives
g_ A
= gK + (
gA
1)gA + n:
(6)
K
For gg_ K
observe that since = 0 the capital accumulation equation simpli…es to
_
K
_
K = sY: Thus, gK = K
= sY
K ; where Y is given above and
g_ K
=(
gK
(b)
1)gK + (1
) gA + (1
)n:
(7)
Isoclines [( + ) < 1] :
g_ A
gA
=
n
0 ) gK =
+
(1
)
gA
| {z }
>1
g_ K
gK
=
0 ) gK = n + gA
In this case we obtain convergence since the slope of the gA isocline is smaller
than the slope of the gK isocline (by assumption).
(c)
See …gure 2 for a graphical illustration.
K
which yields
On the balanced growth path, gg_ Aa = gg_ K
gA =
+
1
n:
The equation for gA depends only on the structural parameters of the R&D
sector.
7
gK
gA isocline
gK isocline
gA
(d)
Assumptions:
+
g_ A
gA
1 and n = 0: The isoclines are now:
=
0 ) gK =
(1
)
gA
| {z }
1
g_ K
gK
=
0 ) gK = gA
Two possibilities may be distinguished:
8
1. if
(1
)
= 1 ) the isoclines are identical.
2. if
(1
)
6= 1 ) only point of intersection is the origin.
Population growth and technological change
(Kremer 1993)
In a now famous QJE paper, Michael Kremer set out to explain why between
1 Million B.C. and 1990, a larger world population was going together with a
faster rate of world population growth. His idea was that this could be explained
by more people generating more ideas. To show this, he postulated that the
aggregate production function during this period was
Yt = At Lt T 1
where T indicates (a …xed amount of ) land, A total factor productivity, L world
population and Y world output.
8
(a)
(Y =L)t = y = At Lt 1 T 1 : Population adjusts such that income
per person is equal to y all the time. Hence,
:
y
=a+(
y
1)n = 0 ) n =
a
1
:
Population growth is constant, as well. The level of population at any point in
time is Lt = L0 exp(nt) where n = n(a) is given above.
_
A
(b)
If A
= L; the rate of population growth is no longer constant and
the relationship with total world population at any point in time is:
:
A_ t
y
=
+(
y
At
1)nt = 0 ) nt =
1
Lt :
The total population at any time t can be found by solving the above di¤erential
equation which yields
L0 (1
)
Lt =
:
(1
) L0 t
_
A
(c)
If A
= LA'
population is:
1
; ' < 1; the relationship of population growth and
:
A_ t
y
=
+(
y
At
1)nt = 0 ) nt =
1
Lt A'
t
1
:
Since ' < 1; population growth is inversely correlated with technology. The
total population is
L0 (1
)
Lt =
:
Rt
(1
) L0 0 A( )d
9
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