This brief note revisits the proof of the Steady-State Growth Theorem, first provided by Uzawa (1961). We provide a clear statement of the theorem and a new version of Uzawa's proof that makes the intuition underlying the result more apparent.

Charles I. Jones

Dean Scrimgeour

English

Research Papers in Economics

NBER WORKING PAPER SERIES
THE STEADY-STATE GROWTH THEOREM:
A COMMENT ON UZAWA (1961)
Charles I. Jones
Dean Scrimgeour
Working Paper 10921
http://www.nber.org/papers/w10921
NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
November 2004
The  views expressed herein are those of the author(s) and do not necessarily reflect the views of the National
Bureau of Economic Research. 
 
© 2004 by Charles I. Jones and Dean Scrimgeour.  All rights reserved. Short sections of text, not to exceed
two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is
given to the source.  The Steady-State Growth Theorem: A Comment on Uzawa (1961)
Charles I. Jones and Dean Scrimgeour
NBER Working Paper No. 10921
November 2004
JEL No. E1, O4
ABSTRACT
This brief note revisits the proof of the Steady-State Growth Theorem, first provided by Uzawa
(1961). We provide a clear statement of the theorem and a new version of Uzawa's proof that makes
the intuition underlying the result more apparent.
Charles I. Jones
University of California, Berkeley
Department of Economics
549 Evans Hall, #3880
Berkeley, CA 94720-3880
and NBER
chad@econ.berkeley.edu
Dean Scrimgeour
University of California, Berkeley
Department of Economics
549 Evans Hall, #3880
Berkeley, CA 94720-3880
dscrim@econ.berkeley.edu2 CHARLES I. JONES AND DEAN SCRIMGEOUR
1. INTRODUCTION
The Steady-State Growth Theorem says that if a neoclassical growth
model exhibits steady-state growth, then technical change must be la-
bor augmenting, at least in steady state.1 It did not escape the attention
of economists, either in the 1960s or more recently, that this is a very
restrictive theorem. We often want our models to exhibit steady-state
growth, but why should technical change be purely labor-augmenting?
The induced-innvoation literature associated with Fellner (1961), Kennedy
(1964),Samuelson(1965),andDrandakisandPhelps(1966)explicitlypon-
dered this question without achieving a clear answer. Recently, Acemoglu
(2003) and Jones (2004) have returned to this puzzle.
Perhaps surprisingly, then, given its importance in the growth literature,
we have been unable to nd a clear statement and proof of the theorem.
Uzawa (1961) is typically credited with the proof of the result,2 and there
is no doubt that he proved the theorem. However, Uzawa is primarily con-
cerned with showing the equivalence of Harrod-neutral technical change
(i.e. technical change that leaves the capital share unchanged if the inter-
est rate is constant) and labor-augmenting technical change, formalizing
the graphical analysis of Robinson (1938). It is of course, only a small and
well-knownsteptoshowthatsteady-stategrowthrequirestechnicalchange
to be Harrod neutral. But again, the modern reader of Uzawa will be struck
by the absence of a statement and direct proof of the steady-state growth
theorem.
Barro and Sala-i-Martin (1995, Chapter 2) come closest to providing a
clear statement and proof of the theorem. However, their statement of the
1It is sometimes added that an alternative is for the production function to be Cobb-
Douglas, at least in steady state. But this is really subsumed in the original version of
the theorem since technical change can always be written in the labor-augmenting form in
steady state if the production function is Cobb-Douglas.
2For example, see Barro and Sala-i-Martin (1995) and Solow (1999).THE STEADY-STATE GROWTH THEOREM 3
theorem is more restrictive: if technical change is factor augmenting at a
constant exponential rate, then steady-state growth requires it to be labor-
augmenting. This leaves open the door, however slightly, to the possibility
that there might be some perverse non-factor augmenting twist of technical
change that could be consistent with steady-state growth.
This comment lls the gap in the literature. We provide a clear statement
and proof of the steady-state growth theorem. The inspiration for the proof
is Uzawa (1961), but we present the crucial steps in a slightly different way
that allows the economic intuition for the proof to come through.
2. STATING THE THEOREM
The steady-state growth theorem applies to the one-sector neoclassical
growth model. We begin by dening the model precisely and then dening
a balanced growth path.
Definition 2.1. A neoclassical growth model is given by the follow-
ing economic environment:
Yt = F(Kt;Lt;t); (1)
_ Kt = Yt   Ct   Kt; K0 > 0;   0; (2)
and
Lt = L0ent; L0 > 0; n  0: (3)
The production function F satises the standard neoclassical properties:
constant returns to scale in K and L, positive and diminishing marginal
products of K and L, and the Inada conditions that the marginal product of
a factor input goes to zero as that input goes to innity and goes to innity
as the input goes to zero.
A balanced growth path in the neoclassical growth model is dened as a
situationinwhichallquantitiesgrowatconstantexponentialrates(possibly4 CHARLES I. JONES AND DEAN SCRIMGEOUR
zero) forever. We follow the usual convention of also refering to this as a
steady state.
Finally, we will dene FKK=Y to be the capital share and FLL=Y to
be the labor share. As usual, the two factor shares sum to a value of unity,
by Euler's theorem. We follow standard notation in denoting y  Y=L
and k  K=L, and we will use an asterisk superscript to denote a variable
along the steady-state path.
Withthesedenitions,wecannowpresenttheSteady-StateGrowthThe-
orem:
Theorem 2.1 (The Steady-State Growth Theorem, Uzawa 1961). If
aneoclassicalgrowthmodelpossessesasteadystatewithconstant,nonzero
factorsharesand _ y
t=y
t = g > 0,thenitmustbepossiblealongthesteady-
state path to write the production function as Y 
t = G(K
t ;AtLt), where
_ At=At = g and where G is a neoclassical production function.
Before presenting the theorem, we pause to make several remarks. First,
the restriction to the case of positive factor shares is primarily intended
to rule out AK style models. Second, as is well-known, in the case of
Cobb-Douglasproduction, capital-andlabor-augmentingtechnicalchange
are equivalent. One sometimes sees the theorem interpreted as saying
that technical change must be labor-augmenting or the production function
must be Cobb-Douglas. This is equivalent to the statement of the theorem
as given.THE STEADY-STATE GROWTH THEOREM 5
3. PROVING THE THEOREM
This proof largely follows Uzawa (1961) in spirit. It differs in that we
provide more economic intuition, highlight the key steps of the proof more
clearly, and ll in some details.3
The capital-output ratio is a key variable throughout the proof, so we de-
ne x  K=Y . We also make the standard denition f(k;t)  F(k;1;t).
The proof now follows.
1. Therststepoftheproofistorewritetheproductionfunctioninterms
of the capital-output ratio: yt = (xt;t). Intuitively, this step is readily
understood by drawing the production function in (k;y) space: for each
ray through the origin  that is for each capital-output ratio  there is a
unique level of output per worker on that ray.4
2. Next, we note that the elasticity of yt with respect to xt satises a
familiar property:
@ logyt
@ logxt
=
(xt;t)
1   (xt;t)
(4)
3The only substantive innovation in the proof is in writing the key differential equation
in (4) below in terms of the elasticity of output with respect to the capital-output ratio. This
produces a familiar equation in a way that Uzawa's consideration of the marginal product
of capital does not.
4Formally, we can use the inverse function theorem to justify this step. The capital-
output ratio depends only on kt and t, since yt is a function of kt and t: xt = kt=yt =
k=f(kt;t)  h(kt;t). Wecanapplytheinversefunctiontheoremtoshowthatthisfunction
can be inverted:
@h(kt;t)
@kt
=
1
f(k;t)
 
ktfk(kt;t)
f(kt;t)2
=
1
f(kt;t)

1  
fk(kt;t)kt
f(kt;t)

6= 0 8kt;
where the last step follows from the fact that the labor share is strictly between zero
and one. Therefore, by the inverse function theorem, h
 1( ;t) exists, and we can write
kt = h
 1(xt;t). Finally, we can substitute this result into the production function to get
yt = f(kt;t) = f(h
 1(xt;t);t)  (xt;t).6 CHARLES I. JONES AND DEAN SCRIMGEOUR
where (xt;t)  fkkt=yt is the capital share. This equation says that the
elasticity of output with respect to the capital-output ratio is equal to the
ratio of the capital and labor shares. Such an equation is well-known in the
case of Cobb-Douglas production, where it has been exploited by Mankiw,
RomerandWeil(1992),KlenowandRodriguez-Clare(1997),andHalland
Jones (1999), among others. Equation (4) shows that this property holds
more generally.5
3. Now comes the key step of the proof. From this point on, we assume
the economy is on a balanced growth path. Because the capital share is
constant in steady state, the right side of equation (4) is invariant over time.
Then, since yt = (xt;t), we can write this equation as
@ log(x;t)
@ logx =
(x)
1   (x)
; (5)
where we use an asterisk to indicate a quantity along a balanced growth
path.
Because the right-hand side of this equation does not depend on time,
this partial differential equation can be solved to yield6
log(x;t) = a(t) +
Z (x)
1   (x)
dx
x (6)
for some function a(t). And therefore
y
t = (x;t) = A(t) (x); (7)
where A(t)  exp(a(t)) > 0 and  (x)  exp
R (x)
1 (x)
dx
x

.
Thisisthecrucialresult. We'veshownthattheeffectsoftandx canbe
separated. This implies, for example, the familiar result that y
t=At = x is
5To derive this equation, begin with kt = ytxt. This implies that dkt=kt = dyt=yt +
dxt=xt. Multiply through by y=dy to get
dktyt
dytkt = 1 +
dxtyt
dytxt, which can be rearranged to
yield the desired result.
6This can be readily veried by differentiating the solution.THE STEADY-STATE GROWTH THEOREM 7
constant along a balanced growth path, where At  A(t). Since y
t grows
at rate g by assumption, it must therefore be the case that _ At=At = g as
well.
4. To conclude the proof, note that k = xy, so that
k
t
At
=
y
t
At
  1

y
t
At

 ~ G 1

y
t
At

: (8)
Inverting7, we have
y
t
At
= ~ G

k
t
At

(9)
and therefore
Y 
t = AtLt ~ G

K
t
AtLt

 G(K
t ;AtLt): (10)
And this proves the key result: technical change is labor-augmenting along
thebalancedgrowthpath. Finally,sinceY 
t = F(K
t ;Lt;t) = G(K
t ;AtLt),
it must be the case that G satises the standard neoclassical properties as
well.
4. DISCUSSION
Here's the one paragraph version of the proof. The crux of the proof is
step 3 above. To begin, we notice that the familiar Cobb-Douglas property
also holds more generally: the elasticity of output per worker with respect
7To show invertibility, differentiate:
~ G
 1(z) = z 
 1(z)
)
d ~ G
 1
dz
=  
 1(z) + z
d 
 1(z)
dz
=  
 1 +
z  
d 
dx

> 0 8z > 0
as   = y=A is always positive and d =dx is also always positive. So ~ G() exists.8 CHARLES I. JONES AND DEAN SCRIMGEOUR
to the capital-output ratio is (x;t)=(1   (x;t)). Then, the fact that the
capital share must be constant in steady state means that the production
function must be factorable. That is, it must be possible, at least in steady
state, to write the production function as yt = A(t) (x). But this means
that y=A and k=A must be constant as well, and one can really just look
at the production function y=A = F(k=A;1=A;t) to see that this requires
technical change to be labor augmenting. The intuition that is closest to the
spirit of this proof, then, is that technical change must be labor-augmenting
in order to keep the capital share constant.
A related intuition can be obtained by looking at the marginal product of
capital. Because both the capital-output ratio and the capital share must be
constant in steady state, we know the marginal product of capital must be
constant as well. This marginal product is F1(K;L;t) = F1(k=y;1=y;t)
since the marginal product is homogeneous of degree zero in the factor
inputs. Since k=y is constant in steady state and y grows at a constant
exponentialrate,technicalchangemustexactlyoffsetthefactthateffective
labor is falling at the rate of growth of y. That is, technical change must
be labor-augmenting. If this were not the case, then the marginal product
of capital would trend over time.
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Labor- and Capital-Augmenting Technical Change,

Neutral Inventions and the Stability of Growth Equilibrium, Review of Economic Studies,

The Classi of Inventions, Review of Economic Studies,

Two Propositions in the Theory of

Why Do Some Countries Produce So Much More Output per Worker than Others?,

The Steady-State Growth Theorem: A Comment on Uzawa (1961)

The Steady-State Growth Theorem: A Comment on Uzawa (1961)

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