Douglas Hibbs
Macro Theory 2004 04 07
Neoclassical Growth Theory with
Exogenous Saving
(Solow-Swan)
1
Stylized Facts
(Q=N ) > 0; steady secular growth
(K=N ) > 0; physical capital to labor ratio grows
@Q
seems to fall with development, that is, the marginal product
r = F 0K
@K
of capital is lower in mature industrial/post-industrial economies than in less
developed ones that are growing (an important quali…cation)
K
;
Q
capital to output ratio is relatively constant in mature economies
wN rK
; Q ;factor
Q
2
i (Q=N );
shares are relatively constant
2
i
(Q=N ) ; (where i indexes national economies) are too "big"
to be consistent with the standard neoclassical model
Politics, policies and institutions are increasingly recognized as the key to
explaining the broad international and intertemporal patterns in growth and
development
1
2
The Basic Neoclassical Model
(1)
Q = F (A; K; N )
where A is the exogenously given, commonly available state of technology, K is
proprietary (rivalrous) capital input, and N is proprietary labor input. K and N
are homogenous in productive quality.
Properties:
F ( ) is twice di¤erentiable and homogeneous of degree 1 (h = 1) in N and
K inputs:
K F 0 (K) + N F 0 N = h F (A; K; N ) ;
F (A;
K;
N) =
F (A; K; N ) ;
h=1
all > 0
For K > 0; N > 0;the function F ( ) has positive and diminishing marginal
products:
F 0 (K) > 0; F 0 (N ) > 0;
F 00 (K) < 0; F 00 (N ) < 0
Inada conditions on input extremities:
lim
lim
K ! 0F 0 (K) = 1;
N ! 0F 0 (N ) = 1
lim
lim
K ! 1F 0 (K) = 0;
2.1
(N ) ! 1F 0 (N ) = 0
The Intensive Form
To …nd steady-states with growing technology, we must assume A is embodied in
N (or is a disembodied productivity ’shifter’, as in the Cobb-Douglas function1 .
1
See Barro and Sala-i-Martin (2004), appendix 1.5.3 for a proof. Note that the famous CobbDouglas production function
Q = AK N 1
satis…es the labor-augmenting technological progress condition, as it could just as well be written,
1
e = A1
with A
Q = K (AN )
e N1
= AK
, which just re-scales A:
2
So I express the model as
(2)
Q = F (K; A N )
The degree 1 homogeneity of F ( ) allows the model to be written:
K
;1
N
K
= N F
;1
N
= N f (k)
(3)
Q = N F
Hence the model in "intensive" (per "e¤ective" unit of labor input) form is:
(4)
q = f (k)
Q
K
where q AN
; k AN
;and f F (k; 1) :
Note that the intensive form representation has the same properties as the
model in basic form, concavity and Inada conditions at the extremities:
f 0 (k) > 0;
f 0 (0) = 1;
f 00 (k) < 0
f 0 (1) = 0
For example, f 0 (k) = F 0 (K) :
F 0 (K)
K
@ AN F AN
;1
@F (K; AN )
=
@K
@K
@ [AN f (k)] @k
@ [AN f (k)] @ K (AN )
=
=
@k
@K
@k
@K
1
0
0
= AN f (k) (AN ) = f (k) :
[Graph q = f (k) ]
2.2
Input Laws of Motion
2.3
Labor
N
=n
N
N (t) = N (0) ent ;
3
N (0) = 1:
(5)
1
2.4
Technology/Knowledge/Productivity
A_
=g
A
A (t) = A (0) egt ;
2.5
A(0) = 1:
The Fundamental Dynamic Equation (capital)
The economy is closed, so:
(6)
Q=C +I
S = I = (Q
(7)
C) :
The equation of motion for the aggregate stock of physical capital is:
K_ = I
K
K_ = sQ
K
S
Q
where s
Now …nd k_
I
Q
and 0 <
d[K (AN )
dt
1
]
< 1:
:
1
dK @ K (AN )
_k = @ K (AN )
+
@K
dt
@AN
= (AN ) 1 K_ K (AN ) 2 (AN )
= (AN )
= sq
= sq
1
(8)
(9)
(sQ
K)
k
1
d (AN )
dt
(10)
(AN )
AN
k
k (g + n)
( + g + n) k
k_ = sf (k)
( + g + n) k:
(11)
The curvature of production, along with the Inada conditions, insure that there
is a unique stead-state level of capital intensity per worker, k ; at which k_ = 0
satisfying sq = ( + g + n) k:
[Graphs: sf (k); ( + g + n)k; k; k ; phase diagram k_ wrt k:]
4
2.6
Output Growth Per E¤ective Worker
Given q = f (k); the growth rate of output per e¤ective worker is
1 dq
1 @f (k) dk
=
q dt
q @k
dt
1 0 _
=
f (k)k
q
q_
q
(12)
which upon multiplication of the right-side by kk yields
q_
f 0 (k) k k_
=
q
q
k
(13)
=
k
k_
k
where k denotes capital’s share of output,
product at every instant.2
At k = k and, hence, at
is therefore zero:
k_
k
f 0 (k) k
;
q
if capital commands its marginal
= 0; the growth rate of output per e¤ective worker
q_
(k = k ) = 0:
q
(14)
We calibrate "prosperity", however, in terms of output per worker. De…ne the
per worker quantities:
Q
K
q~
;
k~
:
N
N
By de…nition
!
d ln K (AN ) 1
k_
1 dk
d ln k
K_
A_
N_
(15)
k
k dt
dt
dt
K A N
=
k~
k~
(16)
g
2
Note that since we know that f 0 (k) = F 0 (K), the capital share is the same in the generic
form of the model as in the intensive form:
0
0
(K=N )
K
f 0 (k) k
= F (K)
= F (K)
= K:
k =
q
(Q=N )
Q
5
and
1 dq
q dt
q_
q
=
q~
q~
d ln q
dt
d ln Q (AN )
dt
1
Q_
Q
A_
A
N_
N
!
(17)
(18)
g
where recall g is the growth rate of A(t):
Given
q_
=
q
we know
q~
q~
g =
k
q~
=
q~
k
k_
k
k
0
1
~
@k
k~
k~
+ (1
k~
gA
(19)
k )g:
And since consumption per worker is
c~ = (1
s) q~
(20)
it grows at the same rate as output for any exogenously give savings rate:
c~
1
=
(1
c~
c~
=
s) q~
q~
q~
(21)
At steady-state k = k ; kk = qq = 0; the implication is that output and capital
and consumption per unit labor input grow at the rate of exogenous technological
progress, g, and are therefore independent of parameters of production and rates of
saving, depreciation and labor force growth (and anything else, including politics,
policies and institutional arrangements):
q~
k~
c~
j(k=k ) = j(k=k ) = j(k=k ) = g:
q~
c~
k~
6
(22)
3
The Level of Output Per Worker at SteadyState
At steady-state, all we can say about the (technology-conditional) convergent level
of output per worker is
q~
= A (t) q
= A (t) f (k )
where k will be some function of the given parameters ; ; g and n: To say more
we need to specify F ( ) –the functional form of production.
In the Cobb-Douglas case, the convergent level of output per worker would be
q~ jA(t) = A(t)k ;
2 (0; 1) :
(23)
k it is readily derived from the …rst-order di¤erential equation for capital, evaluated at k_ = 0. With Cobb-Douglas production we have
k = sk
(24)
( + g + n) k
which at k = 0 implies
sk
(25)
= ( + g + n) k :
Solving for k gives steady-state capital per e¤ective worker as
s
k =
( + g + n)
1
(1
)
:
(26)
q~ jA(t) = A (t) q = A (t) k
(27)
Convergent output worker is then
s
( + g + n)
= A(t)
(1
)
We shall revisit this result when considering the relative incomes of economies
(the wealth and poverty of nations).
7
4
E¤ects of Changes in the Rate of Saving
We aim to learn how a change in the exogenous rate of saving a¤ects the steadystate values of capital, output and consumption per worker.
Steady-State Capital per E¤ective Worker (k ):
We already know that when k = k ; k = 0: So
(28)
sf (k ) = ( + g + n) k :
Therefore we can …nd
dk
ds
by implicit di¤erentiation:
d [sf (k )]
d [( + g + n) k ]
=
ds
ds
f (k ) + s f 0 (k )
dk
dk
= ( + g + n)
ds
ds
f (k ) = [( + g + n)
Solving for
dk
ds
(29)
s f 0 (k )]
(30)
dk
:
ds
(31)
gives
dk
f (k )
=
> 0:
ds
[( + g + n) s f 0 (k )]
(32)
The response of the steady-state capital stock is unambiguously positive because s f 0 ( k ) is always less than ( + g + n), though it need not be less than f 0 (
k ); given the possibility of ’under saving’in the Solow model. [See Graph on the
"golden rule" introduced ahead in the lecture]
Steady-State Output per E¤ective Worker (q ):
dq
ds
df (k )
ds
dq
dk
= f 0 (k )
> 0:
(33)
ds
ds
The response of steady-state output is unambiguously positive since both f 0 (k )
and dk
are positive.
ds
Steady-State Consumption per E¤ective Worker (c ):
Recall C = (Q
I); and s
I
Q
=
S
:
Q
Let c
8
C
AN
Q
= ( AN
I
):
AN
Hence,
c = (q sq)
c = (1 s) q
c = (1 s) f (k)
As noted repeatedly, at steady-state sf (k ) = ( + g + n) k ; which implies
that at steady-state any given saving rate may be expressed:
s(k ) =
( + g + n) k
:
f (k )
(34)
Therefore steady-state consumption per e¤ective worker is
c = (1
h
c = (1 (
c = [f (k )
s) f ( k )
i
+g+n)k
)
f( k )
f (k )
( + g + n) k ] :
The response of c to a change in s is then
dc
ds
= f 0 (k )
= [f 0 (k )
dk
ds
( + g + n)
( + g + n)]
dk
ds
(35)
dk
? 0:
ds
dk
ds
is unambiguously positive, so the sign of the response of c to a change in
s depends on whether the marginal product of capital at the new steady-state
capital stock that is generated by a change in s is greater or less than the slope
of the break-even investment schedule, ( + g + n). The sign of dc
is therefore
ds
uncertain.
We can say, however, that c is maximized at a saving rate generating a
steady-state capital stock, k ; such that f 0 (k ) = ( + g + n) ; which is known
as the "golden rule" (of capital accumulation) saving rate.3 Note that in the
Solow-Swan neoclassical model saving behavior is given exogenously, so there is no
endogenous/model-driven reason why s would happen take a value that maximizes
c :[ Graphs to illustrate]
3
This result appears to have been discovered independently and more or less simultaneoulsy
(as often happens in science) by Phelps, AER (1961), Swan in Berrill ed. Economic Development (1964), Desrousseaux, Annales des Mines (1961), Joan Robinson, RES (1962) and von
Weizs•
acker, Wachstum, Zins und Optimale Investitionsquote (1962), and probably others.
9
4.1
The Elasticity of q with Respect to the Saving Rate
Some insight is gained by expressing the response of q to s as an elasticity:
s
:
q
dq
ds
We already know that
dq
ds
= f 0 (k )
dk
ds
f (k )
> 0:
[( + g + n) s f 0 (k )]
= f 0 (k )
Hence,
s
q
dq
s
=
f 0 (k )
ds
f (k )
which after substitution for s(k ) =
s
q
dq
( + g + n) k
=
ds
f (k )
f (k )
(( + g + n) s f 0 (k ))
( +g+n)k
f (k )
s
q
dq
ds
yields
2
1
f (k )
f 0 (k ) 4
f (k )
( + g + n) ( +g+n)k
f (k )
Recall that the capital share of income is
elasticity of saving may be expressed
k
=
f 0 (k) k 4
.
f (k)
( + g + n) k
[( + g + n) ( + g + n)
=
k
=
(1
k
)
(36)
f 0 (k )
3
5:
(37)
Therefore the output
k
]
(38)
:
The proportional response of output per e¤ective worker to a proportional
change in the saving rate is then
dq
=
q
(1
k
k
)
ds
s
:
(39)
It follows, for example, that if the physical capital share of income takes a
"traditional" value of around 1=3; the output elasticity is around 1=2: Doubling
4
As before, note
k
f 0 (k) k
q
=
~
k
~ k
~
f 0 (k)
q~
=
10
K
f 0 (K) K
:
Q
the savings rate for physical capital accumulation will yield a permanent increase
in output per e¤ective worker by half as much.
The concept of "capital," and the e¤ect of capital’s income share on the response of output to saving rates, are important to evaluation of the neoclassical
model’s consistency with patterns in relative incomes observed through time and
space. I shall revisit this point in subsequent lectures.
Finally, since q~ = A(t) q ; the elasticity of output per worker to a change
in the saving rate is of course identical to elasticity of output per e¤ective worker
derived above, because technological progress is exogenous and does not respond
to investment, i.e. dA(t)
= 0:
ds
s
q~
5
d~
q
ds
s
d [A(t) q ]
A(t) q
ds
s
dq
=
A(t)
+q
A(t) q
ds
s
dq
A(t)
+0
=
A(t) q
ds
s dq
:
=
q
ds
(40)
dA(t)
ds
Transitional Dynamics
We already know that on the balanced growth path (the long-run, steadystate growth path) capital, output, and consumption growth rates per worker in
the Solow-Swan model are determined by the exogenous growth rate of technical
progress (growth of ’knowledge’), g; and therefore are independent of the saving
rate (and anything else). This result is both important and disappointing, in that
we have a theory of growth that yields the conclusion that economic growth is
driven by exogenous growth of technology, the sources of which are unexplained.
Along with alleged inconsistencies of the Solow-Swan model with important stylized facts, this de…ciency is a principal motivation of ’new growth theory’, which
aims to make A(t) endogenous.
Yet the Solow-Swan model does have interesting implications about the con_
vergence of economies to steady-states. In this section we investigate how gk kk ,
q_
c
gq
and gc
respond to movements in in the capital stock, k; along its
q
c
transition to unique steady to steady-state equilibrium, k (s) :
11
5.1
Capital Transition to Steady-State
From the fundamental dynamic equation for capital accumulation we know
gk =
sf (k)
k
(41)
( + g + n) :
Therefore
@gk
sf 0 (k)
=
@k
k
=
s
sf (k)
k2
(f (k)
(42)
f 0 (k) k)
< 0:
k2
The derivative is negative because f 0 (k) k (income going to capital) is less than
f (k) (total income). It follows that k converges to k at a rate that declines with
with k. [Show graph]
_
Since qq_ = k kk ; an implication is that "capital poor" economies should grow
faster than "capital rich" economies when savings rates (and other given parameters of the model) are common.
Although the Solow-Swan model is lodged in a closed economy environment,
it would seem to follow also that investment incentives should favor capital poor
economies and, consequently, that capital (human and physical) should ‡ow from
rich to poor economies. Alas, by and large it does not when ’economies’are taken
to be nations. In fact investment ‡ows (particularly in the form of ‡ows of high
skill workers) in many cases go in the opposite direction – from economies with
low capital endowments to economies with high endowments. I return to these
issues later in the lectures on convergence and growth and development empirics.
For now let us examine the transitional dynamics of output analytically.
5.2
Output Transition to Steady-State
We know already that gq =
gq =
k
gk and that gk =
k
sf (k)
k
12
sf (k)
k
( + g + n)
( + g + n) : Hence
(43)
which after substitution for
gq =
k
gives
f 0 (k) k
f (k)
= sf 0 (k)
sf (k)
k
( + g + n)
( + g + n)
f 0 (k) k
:
f (k)
(44)
Now …nd how the output per e¤ective worker growth rate, gq ; responds to
movements in k :
8
9
f 00 (k) k
f 0 (k)
00
sf
(k)
(
+
g
+
n)
(
+
g
+
n)
>
<
=
f (k)
f (k) >
@gq
=
(45)
>
>
@k
[f 0 (k)]2 k
:
;
+ ( + g + n) [f (k)]2
=
=
8
>
>
<
>
>
:
f 00 (k)
h
( + g + n)
( + g + n)
h
8
00
>
< f (k) s
>
:
s
f 0 (k)
f (k)
h
1
( + g + n)
( + g + n)
f 0 (k)
f (k)
[1
k
f (k
i
f 0 (k) k
f (k)
k
f (k
9
>
>
=
i >
>
;
i 9
>
=
k]
>
;
f (k)
k
Multiplying the …rst term on the right-side by f (k)
(= 1) we obtain
k
h
i 9
8
f (k)
k
00
>
( + g + n) >
f (k) f (k) s k
=
@gq <
(46)
=
>
>
@k
;
:
f 0 (k)
( + g + n) f (k) [1
k]
Since gk = s f (k)
k
( + g + n) ;the equation above can be written
8
9
k
00
>
>
f
(k)
g
k
=
f (k)
@gq <
=
:
>
>
@k
: ( + g + n) f 0 (k) [1
;
k]
f (k)
(47)
What can we now say about the response of gq to movements in k; which is a
principal object of the exercise? The second right-side term in the equation above
13
q
for @g
is always < 0 because f 0 (k) > 0 and 0 <
@k
determined by the …rst right-side term.
k
< 1: So the sign of
@gq
is
@k
In the …rst right-side term f 00 (k) < 0; by the assumption of diminishing returns
(concavity) of the neoclassical production function. So the sign of the …rst term
is determined by the sign of gk : If capital is below the steady-state value implied
by the prevailing savings rate, that is, if sfk(k) > ( + g + n) and therefore k < k ;
then gk > 0: In this case the growth rate of output, gq ; unambiguously slows as
the capital stock, k; rises and, therefore, gq slows also as q rises.
q
If gk is negative because k > k ; then the sign of @g
is indeterminate. If the
@k
economy is in the vicinity of k = k ; however, we know gk will be small which
q
implies that the second right-side term in the @g
function dominates, and so gq
@k
falls as k and q rise.
5.3
Consumption Transition to Steady-State
Recall c = (1
s)q: Hence
gc
c
(1
=
c
(1
= gq :
s)q_
s)q
(48)
gc and gq therefore have identical transitional dynamics.
_
Finally remember that x
e = A(t) x; x ) fk; q; cg ; A
= g, and therefore
A
gxe = gx + g: Because technology is exogenous in the neoclassical model, the growth
of A(t) is not a¤ected by capital formation (or anything else in the model environment). Hence the transitional dynamics of per worker growth rates gxe; with respect
to k are identical to the transitional dynamics of per e¤ective worker growth rates
gx with respect to k:
6
6.1
The Speed of Convergence when k is near k
Capital
When k is in the vicinity of k we can obtain a useful approximation of the
speed of convergence by taking a …rst-order Taylor expansion of k_ = sf (k)
14
( + g + n) k at k ' k :5 The …rst-order approximation is
k_ ' sf (k )
which since [sf (k )
( + g + n) k + [sf 0 (k )
k_
k
k )
(49)
( + g + n) k ] = 0; simpli…es to
k_ ' [sf 0 (k )
Hence for gk
( + g + n)] (k
( + g + n)] (k
k ):
(k
k )
(50)
we obtain
gk ' [sf 0 (k )
( + g + n)]
Since the given rate of saving is s(k ) =
for gk is
( +g+n)k
f (k )
( + g + n) k 0
f (k )
f (k )
gk '
' [( + g + n) (
(k
' [( + g + n) (1
k
(51)
:
; the …rst-order approximation
( + g + n)
1)]
k
k
(k
k )
k
(52)
k )
k
(k
)]
k)
k
Because (k k k) is very nearly equal to (ln k
ln k) when k is not too far from
k; the approximation to the growth rate of capital may be written:
gk ' [( + g + n) (1
5
k
)] (ln k
ln k) ;
(53)
Recall Taylor’s nth-order polynomial expansion formula for approximating a function, here
the fundamental dynamic equation for capital, k, with …nite derivatives of all orders when evaluated at some value, here k = k :
k_ =
8
>
>
>
>
>
<
>
>
>
>
>
:
sf (k )
2
_
( + g + n) k +
1 @ k
+ 2!
(k
(@k )2
2
k ) + ::: +
1 @ k_
1! @k
1 @ n k_
n! (@k )n
(k
(k
+Rn+1
The remainder (due to Lagrange) is Rn+1 (x) =
number between k and k :
15
k )
n
k )
9
>
>
>
>
>
=
>
>
>
>
>
;
1
@nk
(n+1)! (@x)n
(k
n+1
k )
; where x is some
The growth rate of capital per e¤ective worker, gk
equation in ln k(t) which has solution
d ln k
;
dt
ln k(t) = ln k + e t [ln k (0) ln k ]
= e t ln k (0) + 1 e t ln k
where k(0) > 0 is an initial condition and
6
( + g + n) (1
k ).
is a di¤erential
(54)
is a composite parameter that denotes
Express the solution as [ln k(t) ln k ] = e t [ln k (0) ln k ] ; and let gapk (t)
denote the left-side, that is, the deviation of log capital per e¤ective worker, ln k(t);
from its its steady-state, ln k . Note that
d [gapk (t)]
=
dt
(55)
gapk (t)
1
d [gapk (t)]
=
gapk (t)
dt
(56)
Hence ln k(t) converges to its steady-state value ln k at a rate per unit time
equal to (now commonly know as "beta convergence"). The convergence rate,
; depends importantly on the magnitude of of k ; the larger k the slower the
implied rate of convergence to steady-state. This will become important later
when we consider the consistency of the neoclassical model with data.
Looking at the solution in the form of the second line of (54),
ln k (t ) = e
t
ln k (0) + 1
e
t
ln k ;
note that the ’half life’(the time date t at which ln k(t) is halfway from the initial
condition ln k(0) to its steady-state destination value ln k ) satis…es the relations
e
t
= 1
e
t = ln
6
t
1
= ;
2
1
2
I work with log capital (and below with log output) because log quantities relate more
naturally to neoclassical growth theory empircs, which we take up later. The results all pass
through, however, to k and q: For example, the solution to the di¤erential equation for dk
dt ;
k_ ' [( + g + n) (1
k );
k )] (k
is
k(t) = k + e t [k(0) k ] :
16
t = ln 2 =) t = ln 2= = 0:69=
Therefore, if we had a reliable estimate of we could calculate in the same
fashion the time it would take the capital stock to converge to any fraction of its
steady-state value, ln k ; from some starting date value, ln k(0):
6.2
Output
The …rst-order approximation for the speed of output convergence shows that q
approaches q at that same rate as capital approaches its steady-state value. This
can be established using the same basic approach used above to obtain the convergence rate of capital. We know that q = f (k) ; and that k_ = sq ( + g + n) k,
and so
q_ = f 0 (k) k_
(57)
= f 0 (k) [sq
( + g + n) k] :
Taking a …rst-order approximation around q ' q we obtain
8
9
0
( + g + n) k ] =
< f (k ) [sq
q_ '
: @ff 0 (k ) [(sq ( +g+n)k )]g
;
+
(q q )
@q
"
q_ ' 0 + s f 0 (k )
'
h
s f 0 (k )
' [s f 0 (k )
( + g + n) f 0 (k )
( + g + n) f 0 (k ) (f 0 (k ))
( + g + n)] (q
Substituting the given rate of saving s(k
yields
q_ '
( + g + n) k
q
' [( + g + n)
' [( + g + n) (
;q )
=
( +g+n)k
q
1)] (q
17
i
(q
(q
q )
(59)
q )
into the above equation
( + g + n)
( + g + n)] (q
k
1
#
q ):
f 0 (k )
k
1
@q
@k
(58)
q )
q )
(q
q )
(60)
q_ '
where, as before,
(q
= [( + g + n) (1
k
(61)
q)
)].
q_
q
The growth rate of output per e¤ective worker, gq
given by
(q
gq '
gq '
q)
(q
q
is therefore
(62)
q
(ln q
where I make use of the fact that
from ln q .
q)
d ln q
;
dt
(63)
ln q)
' (ln q
ln q) when ln q is not too far
ln q
gq d dt
is a linear di¤erential equation for ln q which for given initial condition
ln q(0) has a solution in exactly the same form as the corresponding solution for
ln k :
ln q(t) = ln q + e
= e
t
t
[ln q (0)
ln q (0) + 1
ln q ]
e
t
(64)
ln q :
When q is not too far from q ; output therefore exhibits the same convergence
properties as capital, no matter what the functional form of the underlying neoclassical production function and, consequently, earlier remarks about capital’s
rate of convergence apply here as well. Also remember that consumption obeys
the same dynamics as output, and so its (identical) convergence properties can be
deduced from:
ln c(t) = ln c + e t [ln c (0) ln c ] :
(65)
c Douglas A. Hibbs, Jr.
18