ISSN (Print) 0473-453X

Discussion Paper No. 1215

ISSN (Online) 2435-0982

GENERALIZED UZAWA GROWTH THEOREM

Gregory Casey
Ryo Horii

October 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

A Generalized Uzawa Growth Theorem∗
Gregory Casey† Ryo Horii‡
October 2023

Abstract
We prove a generalized, multi-factor version of the Uzawa steady-state
growth theorem. Balanced growth with capital-augmenting technical change
is possible when capital has a unitary elasticity of substitution with at least
one other factor of production. Thus, a neoclassical growth model with three ormore factors of production can be consistent with empirical evidence on boththe capital-labor elasticity of substitution and the declining price of investment relative to consumption. In a three-factor model calibrated to US data,
medium-run fluctuations in the investment price explain labor share movementsfrom 1960-2000, but not the subsequent fall in the labor share.

Keywords: Balanced Growth, Uzawa Steady-State Growth Theorem, Technological
Change, Land, Natural Resources
JEL Classification Codes E13, E22, O33, O41∗

An earlier version of this paper has been circulated under the title “A Multi-factor Uzawa Growth
Theorem and Endogenous Capital-Augmenting Technological Change.” Sections of the earlier paper
have been split into a separate paper entitled “Endogenous Capital-Augmenting Technical Change”.
We thank Daron Acemoglu, Been-Lon Chen, Robert Chirinko, Oded Galor, Andreas Irmen, Cecilia
Garcia-Peñalosa, Miguel Leon-Ledesma, Dean Scrimgeour, Esteban Rossi-Hansberg, Chris Tonetti,
Alain Venditti, Ping Wang, David Weil, and seminar participants at Aix-Marseille School of Economics, Brown University, Chulalongkorn University, Kobe University, Liberal Arts Macro Workshop, National Graduate Institute for Policy Studies, Shiga University, Society for Economic Dynamics, Osaka School of International Public Policy, Tongji University, for their helpful comments
and suggestions. All remaining errors are our own. This study was financially supported by the
JSPS Grant-in-Aid for Scientific Research (15H03329, 15H05729, 15H05728, 16K13353, 17K03788,
20H01477, 20H05631, 20H05633).
†
Department of Economics, Williams College, Schapiro Hall, 24 Hopkins Hall Dr., Williamstown,
MA, 01267. Email: gpc2@williams.edu.
‡
Correspondence: Institute of Social and Economic Research (ISER), Osaka University, 6-1 Mihogaoka, Ibaraki city, Osaka 567-0047, Japan. Email: horii@iser.osaka-u.ac.jp.

1

Introduction

The neoclassical growth model was developed to explain a set of stylized macroeconomic facts that can be classified under the umbrella of balanced growth (Solow, 1956,
1994). As conventionally understood, the Uzawa (1961) steady-state growth theorem
says that on the balanced growth path (BGP) of a neoclassical growth model, all technological change must be labor-augmenting, unless the aggregate production function
is Cobb-Douglas (Jones and Scrimgeour, 2008). This creates a significant problem
for the neoclassical growth model, because data from the United States strongly suggest that (i) there is capital-augmenting technical change on the BGP and (ii) the
aggregate production function is not Cobb-Douglas (see, e.g., Antras et al., 2004;
Grossman et al., 2017; Oberfield and Raval, 2021).
The standard neoclassical growth model assumes that there are only two factorsof production, labor and reproducible capital. In reality, there are many other factorsof production, including various types of land, energy, and other natural resources.
These factors do not fit well in the notion of capital in the neoclassical growth modelin that they cannot be readily accumulated (or reproduced) through savings. In thispaper, we derive conditions under which it is possible to make neoclassical models
consistent with the data by adding additional factors of production. We then show
how these results can be used to model the behavior of the economy along the BGP.
We start by proving a multi-factor version of the Uzawa (1961) steady-state growththeorem. When building macroeconomic models, researchers have incomplete knowledge of how the aggregate production function evolves over time due to technologicalchange. We show that, if an economy has a BGP, the Uzawa theorem provides
guidance on how to choose a simple representation of the ever-changing production
function. We call this the Uzawa Representation. The Uzawa Representation gives
the correct relationship between aggregate inputs and aggregate output on the BGP,
while capturing steady-state technological change through factor-augmenting terms
on inputs other than reproducible capital. The Uzawa Representation has the same
derivatives and elasticity of substitution (EoS) as the true production function. However, this representation implies that there is no capital-augmenting technical change,
which is at odds with the evidence.
To identify other possible representations that are a better match with data,1

we prove a generalized version of the multi-factor Uzawa theorem. The generalized
theorem demonstrates that there are a continuum of representations with capitalaugmenting technical change, as long as reproducible capital has a unitary EoS with atleast one other factor. From this broader class of Factor-Augmenting Representations,
it is possible to choose a representation that matches the empirically-observed speedof capital-augmenting technological progress. When we explicitly consider three ormore production factors, the factor-augmenting representations can be simultaneously
consistent with balanced growth, a non-unitary EoS between capital and labor, andcapital-augmenting technical change. We also provide conditions under which these
Factor-Augmenting Representations have the same derivatives and EoS as the true
production function. Therefore, even when the exact pattern of technological change
cannot be observed, factor-augmenting representations can be used as a production
function in economic analysis, as long as the economy follows a BGP in the long run.
We then discuss existing evidence on the elasticity of substitution between capitaland other factors of production. We highlight that long-run elasticities are relevant forunderstanding balanced growth. While inconclusive, existing evidence suggests thatenergy may have a unitary long-run EoS with capital (e.g., Koetse et al., 2008; Van derWerf, 2008). To the best of our knowledge, evidence on capital-land substitution islimited to sector-specific studies, where the EoS between land and structures is often
reported to be close to one (e.g., Epple et al., 2010; Ahlfeldt et al., 2015). We hopethat our theoretical results will spur further work estimating these elasticities, which
could make a significant contribution to the understanding of balanced growth.
We also demonstrate how to calibrate a three-factor production function to simultaneously match U.S. data on balanced growth, the negative trend in the relative
price of capital, and the non-unitary elasticity of substitution between capital andlabor. It is not possible to identify the true production function and the pattern oftechnological change just from those data. However, given that we know that theeconomy has a BGP, our propositions imply that the factor-augmenting representation constructed from the data matches the marginal properties of the true production
function near the BGP and can therefore be used in place of the production function
for a many of economic analyses.
As an example, we use our calibrated model to study how medium-run fluctuations2

in capital-augmenting technical change have influenced the labor share of income
over the last sixty years. Movements in capital-augmenting technical change are acommon explanation for labor share fluctuations (Grossman and Oberfield, 2022).
Our model provides a unique perspective on how the two are related in the vicinityof the BGP. The generalized Uzawa theorem implies that the labor share is constanton a BGP with a constant rate of capital-augmenting technological change, but thelabor share fluctuates when technology deviates from its BGP trend. This prediction
differs from those generated by two-factor production functions, which must imply
either that labor shares are always constant or that the labor share changes whenever
there is non-zero capital-augmenting technical change. We find that medium-run
deviations of capital-augmenting technology from the long-term trend explain labor
share movements from 1960-2000, but not the subsequent fall in the labor share.
Related Literature. This paper is related to a long literature on balanced growth
and the Uzawa steady-state growth theorem. Although the theorem is well known,
Uzawa (1961) does not provide a clear statement or proof of the theorem. A simple
and intuitive proof was proposed by Schlicht (2006) and updated by Jones and Scrimgeour (2008), Acemoglu (2008), Irmen (2016), and Grossman et al. (2017). With the
exception of Acemoglu (2008), the literature has been concerned only with whether
a particular production function can match the level of output on the BGP. We contribute to this literature in several ways. First, we extend the theorem to multiple
factors of production. Second, we prove a generalized version of the theorem thatprovides a set of representations from which an economist can choose the one thatmatches the data on capital-augmenting technological progress. Third, we deriveconditions under which representations have the same first-order derivatives and EoS
as the true production function.
As noted above, the existing literature has treated the Uzawa theorem as a restrictive condition. As a result, many studies have tried to explain why the economy
might endogenously conform to the two-factor version of the theorem. Acemoglu(2003) and Irmen and Tabaković (2017) provide models where capital-augmenting
technical change disappears in the long run, while Jones (2005) and Leon-Ledesma
and Satchi (2019) specify models that are Cobb-Douglas in the long-run. We build3

on these works by presenting a model that is consistent with data on both the existence of capital-augmenting technical change and the less-than-unitary long-run EoS
between capital and labor.
To the best of our knowledge, Grossman et al. (2017, 2021) provide the only otherattempt to square the Uzawa steady-state growth theorem with data on the EoS andthe capital-augmenting technical change. In their model, schooling is both laboraugmenting and capital-dis-augmenting. In this setting, they show that there is a
scope for additional capital-augmenting technological change. Our results indicatethat there is a wider scope for ways in which the neoclassical growth model can bemade to be consistent with the data. Indeed, their results can be understood as aparticular case of the two-factor Uzawa theorem (see subsection 5.2).
Our results also stress the importance of natural resources for understanding
macroeconomic outcomes. Historically, natural resources were only included in aggregate macroeconomic analyses when the research question under study was explicitly about those natural resources. For example, energy is generally only included
in growth models when studying the depletion of finite resources (e.g., Hotelling,
1931; Heal, 1976; Hassler et al., 2021) or climate change (e.g., Nordhaus and Boyer,
2003; Golosov et al., 2014). Our results suggest a much broader importance of nonreproducible factors. The neoclassical growth model was originally developed to explain the balanced growth facts (Solow, 1956, 1994). In order to explain a wider setof stylized facts, including the existence of capital-augmenting technological change,
the model must incorporate factors beyond reproducible capital and labor.
Roadmap. The remainder of the paper proceeds as follows. Section 2 discusses
the evidence motivating this study. Section 3 proves a multi-factor version of the
Uzawa steady-state growth theorem. In Section 4, we generalize the theorem, proving that neoclassical models can have a positive rate of capital-augmenting on the
BGP. Section 5 presents three applications of these results, focusing on simple casesand existing literature. Section 6 discusses existing evidence on the EoS between
capital and natural resources. Section 7 presents a simple calibration exercise andan application for labor share. Section 8 extends our results to investment-specifictechnological change. Section 9 concludes.

4

(a) Real Aggregate Variables per Capita(b) Relative Price of Investment

Figure 1: Balanced Growth with Capital-Augmenting Technical Change
Note: See Appendix Section D.1 for data sources.

2

Motivation and Stylized Facts

The neoclassical growth model, first developed by Solow (1956) and Swan (1956),
serves as a basis for much contemporary research in macroeconomics. Such models
are designed to explain a set of stylized facts, known as ‘balanced’ or ‘steady’ growth(Jones, 2016). The main stylized fact is that income per capita has grown at aconstant rate over long periods of time. Panel (a) in figure 1 presents U.S. datafrom 1960-2020, which clearly demonstrates this fact. In addition, it shows that real
investment, consumption and capital per capita have grown at roughly the same rate
as real income per capita over this period, capturing the notion of ‘balance’.1
The neoclassical growth model is founded on two building blocks. The first is theprocess of capital accumulation: capital is accumulated linearly from saved output.
This specification implies that capital ‘inherits’ the growth rate of output (Jones andScrimgeour, 2008). The second is a neoclassical aggregate production function thathas constant returns to scale (CRS) in all production factors. With a CRS production
function, balanced growth is achieved when all effective factors grow at the same rate
1

See Papell and Prodan (2014), Jones (2016), and others for longer time series and data fromother countries.

5FactorNatural Resources (incl. Land)
Land
Energy

Share
8%
5%
4%

SourceCaselli and Feyrer (2007)
Valentinyi and Herrendorf (2008)
Golosov et al. (2014)

Table 1: This table presents some estimates of U.S. factor shares for inputs other than reproducible
capital and labor. Definitions and methodologies vary. Natural Resources in Caselli and Feyrer(2007) encompasses land, energy, and other resources.
as output. Since capital grows at the same rate as output, this implies that there isno capital-augmenting technological progress. In the two-factor neoclassical growthmodel, the only exception is when the aggregate production function is Cobb-Douglas.
In this case, effective factors can grow at different speeds while keeping the growthrate of output constant. These results are widely known as the Uzawa theorem.
A long literature has estimated the elasticity between capital and labor in a twofactor production function and rejected the Cobb-Douglas specification. Most of
the papers in the literature argue that the elasticity is less than one (e.g., Antras
et al., 2004; Chirinko and Mallick, 2017; Oberfield and Raval, 2021). In addition,
there is evidence that capital-augmenting technical change has been occurring evenas the economy has exhibited signs of balanced growth (Grossman et al., 2017). As
shown in Appendix C.1, the speed of capital-augmenting technological change in the
neoclassical growth model can be measured by the rate of decline in the relative priceof investment, calculated as the ratio of the implicit price deflator for investment tothat for personal consumption expenditures in the NIPA statistics, with a one-period
lag. Panel (b) of Figure 1 shows that the price of investment goods, and equipment inparticular, has been falling relative to the price of consumption goods in the UnitedStates over at least the last half a century. A long literature demonstrates thatdeclining investment price (equivalently, capital-augmenting technological change)
explains a quantitatively significant portion of economic growth in the United States(e.g., Greenwood et al., 1997; Krusell et al., 2000).2
These findings create a puzzle. Given that the EoS between capital and labor is
not equal to one, the Uzawa theorem implies that any two-factor neoclassical growth2

See He et al. (2008) and Maliar and Maliar (2011) for discussions of the Uzawa steady-state
growth theorem in this context. Karabarbounis and Neiman (2014) show that declining investment
prices are a widespread phenomenon in cross-country data.

6

model that is consistent with balanced growth is necessarily at odds with evidence
on capital-augmenting technical change. Put differently, the standard neoclassicalgrowth model cannot explain the broader set of stylized growth facts that we observein the United States.
In this paper, we examine production functions with additional inputs, beyond
reproducible capital and labor. It is obvious that other factors – such as land, energy,
and other natural resources – exist in the production process. Table 1 collects someevidence on the importance of these factors in the United States. Broadly speaking,estimates suggest that non-reproducible factors other than labor account for about
8-9% of total factor payments. In standard two-factor neoclassical growth models,
those factors are often implicitly included in capital. However, this is not an adequate
treatment unless they can be linearly accumulated with saved output. This paper
shows that explicitly separating these factors from capital is key to solving the puzzle
raised by the Uzawa theorem.

3

A Multi-factor Uzawa Theorem

In this section, we prove that the steady-state growth theorem by Uzawa (1961)(hereafter, the Uzawa theorem) extends to multi-factor environments that explicitly
consider inputs beyond labor and reproducible capital. As shown by Solow (1956),
sustained economic growth requires the shape of the production function to changeover time, which economists usually call technological change. Given the existenceof a BGP, the Uzawa theorem provides a convenient representation of the evolutionof the production function. We stress the importance of making a clear distinction
between this representation and the true production function, for which we often have
limited information. In particular, we also prove a new set of propositions that clarify
the conditions under which the representation given by the Uzawa theorem matches
important properties of the true production function, implying that the representation
serves as a good approximation of the true production function in economic analysis.

7

3.1

Neoclassical Growth Model

The Uzawa theorem depends on two assumptions: (a) the economy is described by aneoclassical growth model, and (b) the economy has a balanced growth path (BGP).
We start with a description of a neoclassical growth model, which is defined broadly toincorporate a wide range of dynamic macroeconomic models. We consider a discretetime setting, where t = 0, 1, 2, . . ., but it is straightforward to consider the continuoustime equivalents of the results.
Definition 1. A multi-factor neoclassical growth model is an economic environment that satisfies:
1. Output, Yt , is produced from capital, Kt , and J ≥ 1 kinds of other inputs,{Xj,t }Jj=1 :3,4
Yt = F (Kt , X1,t , ..., XJ,t ; t).

(1)

In any t ≥ 0, it has constant returns to scale (CRS) in all inputs, Kt , X1,t , ..., XJ,t ,and each input has a positive and diminishing marginal product.
2. Capital, Kt , accumulates linearly with the saved output
Kt+1 = Yt − Ct − Rt + (1 − δ)Kt , K0 > 0,(2)where Ct > 0 is consumption, Rt ≥ 0 is expenditure other than capital investment or consumption, and δ ∈ [0, 1] is the depreciation rate. The term
Yt − Ct − Rt on the RHS represents the amount of saved output, or equivalently,
physical capital investment.
There are five points to note regarding Definition 1. First, production function
F (·; t) in (1) depends on t, capturing technological progress. Importantly, we place
no restrictions on how the shape of F (·; t) changes over time. As discussed below,
3

If we allow J = 0, the only constant-returns-to-scale production function is in the form of
Yt = AK,t Kt . Although it cannot satisfy decreasing marginal products of its input (Kt ), this AK
functional form is also subject to the Uzawa theorem, in the sense that AK,t must be constant on
the BGP (i.e., there may not be any technological change).
4
We follow Uzawa (1961) and Jones and Scrimgeour (2008) by including t as an argument in F .
Alternatively, We can write equation (1) as Yt = Ft (Kt , X1,t , ..., XJ,t ) to highlight that Ft changes
with t.

8

the Uzawa theorem provides insight into how to approximate the time dependence ofF (·; t) with standard factor-augmenting terms.
Second, if J equals 1 and X1,t is interpreted as labor, Lt , then equation (1) reduces
to a familiar two-factor neoclassical production function, Yt = F (Kt , Lt ; t). In addition, if we assume Lt grows exogenously, Definition 1 essentially coincides with thedefinition of a neoclassical growth model in Schlicht (2006) and Jones and Scrimgeour(2008), who provide a simple statement and proof of the two-factor Uzawa theorem.
Third, the only reason why capital, Kt , is distinguished from other production
factors X1,t , ..., XJ,t is that we explicitly specify its linear accumulation process (2).
From a theoretical viewpoint, Kt needs not to be limited to physical capital. Kt canbe any combination of factors that can be accumulated linearly with saved output.5
Fourth, the Uzawa theorem holds regardless of the evolution process for other
inputs. The Xj,t ’s can be either endogenous or exogenous. If some factors are endogenous, Rt term in (2) may include costs to enhance them. For example, the future
growth of labor may be dependent on child-raising costs, which could be included in
Rt .6 Technological change, represented by the last t term in the production function(1), can also be exogenous or endogenous. If R&D expenditures enhance technologies,
such expenditures would also be included in Rt in (2).
Fifth, equation (2) implies that period t final output and period t + 1 capital aremeasured in the same units. Since this is merely a choice of units, it does not limit theapplicability of our results. In models with investment-specific technological change(ISTC), one unit of output can be converted to increasingly many units of capital as
technology improves. Section 8 shows that all of our results can be translated to amodel with ISTC by a change of variables.

3.2

Balanced Growth Path

Now, we turn to the second requirement of the Uzawa theorem, the BGP.
Definition 2. A balanced growth path (BGP) in a multi-factor neoclassicalgrowth model is a path along which all quantities, {Yt , Kt , X1,t , ..., XJ,t , Ct , Rt }, grow
5

For example, in the pre-industrial Malthusian economy where population was proportional to
output (e.g., Galor, 2011; Li et al., 2016), labor could be included in Kt , not in Xj,t .
6
As highlighted in footnote 5, child-rearing costs could be part of investment in a Malthusian
economy.

9

at constant exponential rates for all t ≥ 0. On the BGP, we denote the growth factorof output by g ≡ Yt /Yt−1 , and the growth factors of any variable Zt ∈ { Kt , X1,t , ...,
XJ,t , Ct , Rt } by gZ ≡ Zt /Zt−1 . A non-degenerate balanced growth path is a BGP
with gK > 1 − δ.
From (2), condition gK > 1−δ means that physical capital investment Yt −Ct −Rt
is strictly positive along the BGP. The rest of the paper focuses on this non-trivial
case. We call it a non-degenerate BGP and simply mention it as a BGP when there
is no risk of confusion. Note that, while a BGP requires variables to grow at constantrates, it does not require them to grow at the same rate. Still, the following lemma
confirms that capital and consumption need to grow at the same speed as output tomaintain a BGP.
Lemma 1. On any non-degenerate BGP in a multi-factor neoclassical growth model,
the capital-output ratio Kt /Yt and the consumption-output ratio Ct /Yt are constantand strictly positive.
Proof. See Appendix A.2.
The proof utilizes the assumption of C0 > 0 from Definition 1. If R0 > 0, we cansimilarly show that Rt /Yt is constant.

3.3

Uzawa Representation and Its Properties

Having defined the neoclassical growth model and the BGP, we are ready to present
a multi-factor version of the Uzawa theorem.
Proposition 1. (A Multi-Factor Uzawa Theorem) Consider a non-degenerate
eX ,t ≡ (g/gX )t where
BGP in a multi-factor neoclassical growth model, and define Ajj

j = 1, ..., J. Then, on the BGP,eX1 ,t X1,t , ..., AeX ,t XJ,t ) holds for all t ≥ 0,
Yt = Fe(Kt , A
J(3)

where Fe(·) ≡ F (·; 0).
eX ,t ≡ (g/gX )t , the growth factor of AeX ,t Xj,t is g for
Proof. From the definition of Ajjj
all j. The growth factor of Kt is also g from Lemma 1. Therefore, all the arguments
10

in function Fe(·) are multiplied by g each period. This means that the RHS of (3)is multiplied by g each period since Fe(·) ≡ F (·; 0) has CRS. Note that in period 0,
equation (3) holds because it is identical with (1). Therefore, (3) holds for all t ≥ 0,
where both sides are multiplied by g in every period.
It is important to understand what the theorem does and does not imply. Recall
that the neoclassical production function F (·; t) in (1) is a time-varying functionthat potentially depends on t in complex ways. If the economy is on the BGP, theUzawa theorem says that there should be a simple representation of this dependenceof function F (·; t) on t, which holds at least along this particular BGP. We call thisrepresentation, which is given by (3), the Uzawa representation. It consists of a timeeX ,t terms. At t = 0, equation (3)invariant function Fe(·) and exponentially growing A
jcoincides with the true production function (1). The Uzawa representation illustrates
how the production function evolves from there as t changes.
However, caution is needed when interpreting Fe(·) as a production function beyond t = 0, because Proposition 1 only guarantees that the value of Fe(·) coincides
with that of the true production function F (·; t) exactly on a particular BGP. As isclear from the proof of the proposition, function Fe(·) contains no information about
what will happen when inputs deviate even slightly from the BGP. As a result, there
is no guarantee that the derivatives of function Fe(·), even on the BGP, are equal tothe derivatives of the production function F (·; t), apart from time t = 0. Without
further information, therefore, the Uzawa theorem has little use in economic analysis.
In the following two propositions, we extend the theorem by focusing on the conditions under which the Uzawa representation has the ‘correct’ marginal properties.
We start by looking at first-order derivatives.
Proposition 2. (Derivatives of the Uzawa representation) Let FZ (·; t) denotethe partial derivative of function F (·; t) with respect to its argument Z ∈ {Kt , X1,t , ..., XJ,t }.7
If the share of factor Z, i.e., sZ,t = FZ (·; t)Zt /Yt , is constant on a non-degenerate
BGP of a multi-factor neoclassical growth model, then the following holds on the BGP:∂ e
eX1 ,t X1,t , ..., AeX ,t XJ,t ) = FZ (Kt , X1,t , ..., XJ,t ; t) for all t ≥ 0.
F (Kt , AJ
∂Zt
7Appendix A.1 discusses the details regarding notation for derivatives.

11

(4)

Proof. See Appendix A.3.
If the factor shares are constant on the BGP, equation (4) says that Fe(·) has thesame derivatives as the true production function F (·; t) on the BGP. We can also
show that the elasticity of substitution (EoS) between capital and other production
factors in the Uzawa representation Fe(·) coincides with the EoS in the true production
function F (·; t) on the BGP, if the latter does not change over time.8 Let us first define
the EoS when there are more than two inputs.9
Definition 3. The Elasticity of Substitution between capital Kt and input Xj in
multi-factor neoclassical production function F (K, X1 , ..., XJ ; t) in (1) is defined byσKXj ,t = −d ln(Kt /Xj,t )d ln FK (Kt , X1,t , ..., XJ,t ; t)/FXj (Kt , X1,t , ..., XJ,t ; t),(5)

Yt ,X−j,t :const

where X−j,t ≡ {X1,t , ..., XJ,t }\Xj,t represents the inputs other than Kt and Xj,t .
Using this definition, we can show that the Uzawa representation has the correct
EoS if the true EoS is stationary on the BGP.
Proposition 3. (Elasticity of Substitution in the Uzawa Representation)
Let σeKXj ,t denote the EoS in the Uzawa representation, as in Definition 3. If the EoSof the true production function, σKXj ,t for some j ∈ {1, ..., J}, is constant over timeon the BGP, then σeKXj ,t = σKXj ,t holds for all t ≥ 0 on the BGP.
Proof. See Appendix A.4.

4

A Generalized Uzawa Growth Theorem

eX ,t as the factor Xj,t -augmenting technology term, Proposition 1 implies
By viewing A
jthat it is always possible to interpret the time variation of the true production function
8

To the best of our knowledge, Acemoglu (2008) is the only example of previous work considering
first-order properties implied by the Uzawa theorem. He looks at first-order conditions in the twofactor case, providing a special case of Proposition 2.
9
When there are more than two production factors, there are various ways to define the elasticity
of technical substitution. See Stern (2011) for a concise taxonomy. The elasticity in (5) is calculatedusing the inverse of the symmetric elasticity of complementarity (SEC), defined in Stern (2010),
which has a desirable property of symmetry between the two variables.

12F (·; t) on the BGP in terms of exponential augmentation of production factors. In
addition, Propositions 2 and 3 show that the Uzawa representation (3) is ‘correct’ interms of its marginal properties. It is tempting to conclude that there should be notechnological change that enhances the productivity of capital on the BGP, because
eK,t term in (3). This reasoning is insufficient because Proposition 1 does
there is no A
not establish uniqueness. As a result, it does not rule out the existence of better
representations of the true production function.
In this section, we prove a further generalized version of the Uzawa theorem thatallows for representations with capital-augmenting technical change. We explore thepossibility that the true production function has more than one factor-augmenting
representation and identify a condition under which there will be a representation thatmatches the data on the capital-augmenting technological change shown in Section
2. To satisfy this condition while remaining consistent with empirical evidence, it isessential to include factors of production beyond labor and reproducible capital in
the aggregate production function.

4.1

Factor-Augmenting Representation and Factor Substitution

We start by defining a factor-augmenting representation.
Definition 4. A Factor-Augmenting Representation of the true production function (1) is a combination of a time-invariant constant-returns-to-scale function F (·)
and the growth factors of factor-augmenting technologies γK > 0 and γXj > 0,j ∈ {1, ..., J}, such that the paths of output and inputs on a BGP satisfy
Yt = F (AK,t Kt , AX1 ,t X1,t , ..., AXJ ,t XJ,t ) holds for all t ≥ 0,

(6)

where AK,t = (γK )t and AXj ,t = (γXj )t .
Our objective is to find a factor-augmenting representation F (·) that matches
a wider set of properties of the BGP, including the evidence of capital-augmenting
technological change. By comparing (3) with (6), it is clear that the Uzawa representation is a special case of a factor-augmenting representation. As we explain below,13(3) assumes that all effective factors grow at the same rate of g, while (6) permits
different growth rates among different effective factors. In other words, the Uzawa
representation hypothesizes that there is no factor substitution taking place when theeconomy grows along the BGP. The homothetic expansion of every effective input isthe simplest interpretation of a steadily growing economy, but it does not necessarily
constitute the best description of reality.
To see this, suppose that every effective input, including effective capital, grows at
the same speed as the output. Recall that physical capital is already growing at thesame speed as output on the BGP (Lemma 1). Then, there is no room for additional
capital-augmenting technological progress to further augment its effectiveness. As
discussed in Section 2, however, there is clear evidence that the productivity of capital,
measured in terms of output as in our model, has steadily been increasing on the BGP.
Thus, the interpretation of the BGP as being a homothetic expansion of every input
is at odds with a well-established stylized fact.
Motivated by this contradiction, we now consider a broader range of possibilities
in which effective inputs grow at different constant rates. To have balanced growthwith non-homothetic expansion of production factors, it is necessary to further restrict the possible functional forms of the factor-augmenting representation. Before
moving to formal propositions, we provide a heuristic discussion that highlights the
key intuition. Suppose that the true production function can be represented in thefactor-augmenting way (6) on the BGP with the correct derivatives and EoS. Then,the growth rate of output can approximately be written as follows:10
g ≡ Yt+1 /Yt ≈ sk,t γK gK +JXsXj ,t γXj gXj ,

(7)

j=1

where sk,t ≡ FK (Kt , X1,t , ..., XJ,t ; t)Kt /Yt is the share of capital at time t and similarly
for sXj ,t .
Equation (7) says that the growth rate of the output is the weighted average of thegrowth rates of different effective factors, where the weights are factor shares. When
10

To obtain this decomposition, we performed a Taylor-expansion on the RHS of (6) for t + 1
with respect to every effective factor around the period t values. Then, we divided the result by the
RHS of (6) for t. The Taylor expansion is exact when the variables in t and t + 1 are sufficiently
close, or equivalently, in continuous time.

14the effective factors grow at different speeds, γK gK and the γXj gXj ’s are different.
Specifically, let us assume that effective capital grows faster than output due to Kaugmenting technological change (γK gK > g). Then there must be at least one
effective factor that is growing slower than output. Let us say that this factor is X1(i.e.,γX1 gX1 < g) and that all the other effective factors are growing at the same rate
as output. Then, dividing the factor augmenting representation (6) by Yt gives
1=FAK,t Kt AX1 ,t X1,t,
, constants .
Yt
Yt

(8)

In this form, it is evident the growing effective capital-output ratio AK,t Kt /Ytpermits the production of unit output with the shrinking effective X1 -output ratio
AX1 ,t X1,t /Yt . In other words, factor substitution is occurring.
Now, let us check if this ongoing factor substitution is consistent with the definitionof the BGP. On the BGP, output grows at a constant rate, g, which means that the
RHS of (7) must also be constant. Given γK gK > γX1 gX1 , the RHS of (7) only remains
constant when the factor shares, sk,t and sX1 ,t , do not change over time. This happens
if and only if the EoS of F (·) between K and X1 , defined in Definition 3, is equal toone. To summarize, for K-augmenting technological change to happen on the BGP
in a factor-augmenting representation, the functional form of F (·) needs to have aunitary EoS between capital and some other factor.11 In this case, it is possible tohave balanced growth even when effective capital grows faster than output.
Once we obtain a factor-augmenting representation, we hope to use it as an approximation of the true production function. In particular, as in Proposition 3, therepresentation is especially useful if the EoS of F (·) matches that of the true production function. This is only possible when the true production function F (·; t)
has a unitary EoS between capital and some other factor, because we already knowthat F (·) must have a unitary EoS. As discussed in section 2, there is a great dealof evidence suggesting that the EoS between capital and labor is different than one.
However, our definition of the neoclassical growth model allows for any number of
11

In the Uzawa representation, γK gK = γXj gXj holds for all j. Because the production function
PJ
is assumed to have CRS (which guarantees sK,t + j=1 sXj ,g = 1), the RHS is always constant.
Therefore, we can use F (·; 0) as the Uzawa representation without checking its EoS properties (seeProposition 1) at the cost that it cannot accommodate the possibility of K-augmenting technologicalchange.

15

inputs. Once we consider the realistic case with more than two factors of production,
it becomes more likely that at least one input has a unitary EoS with capital.

4.2

A Steady-State Growth Theorem with K-augmenting Technical Change

Here, we formally construct a function that can be used as a basis for a factoraugmenting representation. Consider factors of production other than capital, {X1,t , ..., XJ,t },and suppose that some of them are substitutable with capital, Kt , with unitary elasticity in the period 0 production function, F (·; 0). Without loss of generality, wereorder these factors so that the first j ∗ ∈ {1, ..., J} of them can be substituted withcapital with the unitary EoS.
If capital is substitutable with j ∗ other factors with a unitary elasticity, we
can interpret them as if they are combined together in the Cobb-Douglas fashion
to form an intermediate input. The intermediate input, which we call the capital composite, will then be one argument in the final production function. Using
the share of factors in period 0, sK,0 ≡ FK (K0 , X1,0 , ..., XJ,0 ; 0)Kt /Yt and sXj ,0 ≡FK (K0 , X1,0 , ..., XJ,0 ; 0)Xj,t /Yt , we define period-0 relative shares within the capitalcomposite:∗

α = sK,0 /(sK,0 +jX∗

sXj ,0 ),ξj = sXj ,0 /(sK,0 +

j=1

j
X

sXj ,0 ).

(9)

j=1

Using these relative shares, we can represent the production function in a nested
form:12

 Y∗

j
ξ
j
α
F (k, x1 , ..., xJ ) ≡ Fb k
xj , xj ∗ +1 , ..., xJ .
j=1

(10)

Q∗ ξ
The first argument of the Fb(·), m = k α jj=1 xjj , represents the capital composite,which combines capital and the other j ∗ factors that have a unitary EoS with capital.
Capital composite m is an argument in the outside function Fb(·), along with other12

In this section, we use lowercase letters k, x1 , ..., xJ to denote variables, while uppercase letters
Kt , X1,t , ..., XJ,t are the BGP values, unless otherwise noted. Also, with a slight abuse of notation,
here we define function F (·) by (10), while F (·) was previously used in a factor-augmenting representation. This abuse will be resolved in Proposition 4, which shows F (·) in (10) actually constitutes
a factor-augmenting representation.

16

factors xj ∗ +1 , ..., xJ . The shape of the outside function Fb(·) is defined using theperiod-0 production function F (·; 0):13
Fb(m, xj ∗ +1 , ..., xJ ) ≡ F

Y ∗
j
j=1

ξ
Xj,0j

−1/α

!
m1/α , X1,0 , ..., Xj ∗ ,0 , xj ∗ +1 , ..., xJ ; 0 .
(11)

The first argument of Fb(·), m, collects the j ∗ relevant inputs and combines them withcapital in the first argument. As a result, function Fb(·) has j ∗ fewer arguments thanF (·; 0). Note that the RHS of (11) includes the BGP values Xj,0 , J = 1, . . . , j ∗ , which
are treated as constants. Changes in the xj,0 terms only matter through m.
As the following lemma shows, the nested representation, F (·) with Fb(·), approximates the true production function around the BGP in period 0.
Lemma 2. (Nested representation of the production function at t = 0)
a. F (K0 , X1,0 , ..., XJ,0 ) = F (K0 , X1,0 , ..., XJ,0 ; 0).
b. For any Z ∈ {K, X1 , ..., XJ }, F Z (K0 , X1,0 , ..., XJ,0 ) = FZ (K0 , X1,0 , ..., XJ,0 ; 0).
c. For any j = 1, ..., j ∗ , σ KXj ,0 = σKXj ,0 , where σ KXj ,0 is the EoS of function
F (k, x1 , ..., xJ ) between k and xj , evaluated at the period-0 BGP.
d. Functions Fb(m, xj ∗ +1 , ..., xJ ) and F (k, x1 , ..., xJ ) have constant returns to scale.
Proof. See Appendix A.5.
Properties a, b, and c respectively confirm that the nested representation F (·)
matches the period-0 true production function, F (·; 0), in terms of the level of inputs
and output, the first derivatives for any input, and the EoS between K and any other
input Xj , when the function is evaluated around the period-0 BGP.14 Property dconfirms the CRS property.
Thanks to the CRS property, the nested representation can be used not only
for period 0, but also for representing how the production function evolves fromthere along the BGP. The following proposition establishes that, with the nested
F (·; 0) needs to satisfy σKXj ,0 = 1 for j = 1, ..., j ∗ . Other than that, the following analysis only
requires that the local properties of F (·; 0) are known to researchers. See Remarks 1-3 for related
discussions in the context of the multi-factor Uzawa theorem.
14
Namely, when {k, x1 , ..., xJ } are at the period-0 BGP values {K0 , X1,0 , ..., XJ,0 }.
13

17

representation F (·), there are multiple ways to represent the technological change ina factor-augmenting fashion.
Proposition 4. (A Generalized Uzawa Growth Theorem) Suppose that σKXj ,0 =1 for j = 1, ..., j ∗ . On a non-degenerate BGP, let γK > 0 and γXj > 0, j ∈ {1, ..., j ∗ },be any combination that satisfies the technology condition(γK g)αYj ∗j=1(γXj gXj )ξj = g.

(12)

For j ∈ {j ∗ + 1, ..., J}, let γXj = g/gXj . With γK and each γXj , define AK,t = (γK )tand AXj ,t = (γXj )t . Also, define function F (·) by (9) and (10). Then, on the BGP,
Yt = F (AK,t Kt , AX1 ,t Xj,t , ..., AXJ ,t XJ,t ) for all t ≥ 0.

(13)

Proof. See Appendix A.6.
Note that (13) constitutes a factor augmenting representation, as defined by Definition 4.15 Thus, Proposition 4 characterizes the set of factor-augmenting representations of the true production function along the BGP. When there is no factor thatis substitutable with capital with a unitary elasticity at time 0 (i.e., j ∗ = 0), then
Proposition 4 becomes identical to Proposition 1.16 However, given that there aremany factors of production in reality, it seems plausible that at least one of them issubstitutable with capital with a unitary elasticity (j ∗ ≥ 1). In this case, there areseveral aspects of the proposition that warrant further discussion.
First, unlike Proposition 1, the generalized theorem implies that there is a continuum of representations. Factor-augmenting terms, γK and γXj for j = 1, ..., j ∗ , canbe any combination that satisfy condition (12). This enables applied researchers topick the representation that is most consistent with data on technical change. The
Uzawa representation is a special case of the factor-augmenting representation with
γK = 0.
15

Recall that function F (·) has CRS from Lemma 2.
Pj ∗If j ∗ = 0, condition α + j=1 ξj = 1 in Lemma 2 implies α = 1. Then, condition (12) reduces
to γK = 1, which means AK,t = 1 for all t. Then, (13) becomes identical to (3).
1618

Second, condition (12) implies that the amount of effective capital composite,
Mt = (AK,t Kt )α

Yj ∗
j=1

(AXj ,t Xj,t )ξj ,must grow at the same speed of output, g. This result can be expressed in a log-linear
form:
α log γK +

Xj ∗
j=1

ξj log γXj = (1 − α) log g −

Xj ∗
j=1

ξj log gXj .

(14)

When the growth rates of the factor-augmenting technologies are exogenous, thislog-linear condition may seem restrictive. In a model where the direction of technical change is endogenous, however, this condition can be endogenously satisfied as
the economy converges to the BGP. If the speed of capital-augmenting technologicalchange is slower than required in (14), then the effective capital composite gradually
becomes scarcer relative to the other effective factors. The scarcity will induce firms
to do more R&D to enhance the capital-augmenting technology rather than enhancing
other factors. As a result, condition (14) will necessarily be satisfied in the long run
as long as the economy converges to a BGP. We build such a model in a companion
paper (Casey and Horii, 2023). The results suggest that the log-linear condition does
not impose any extra restrictions.
Third, similar to the original Uzawa theorem (Proposition 1), equation (13) is
not a functional relationship. It only states that the level of inputs and outputs inthis representation match those of the true production function on the BGP. The
following propositions establish that, under conditions similar to Propositions 2 and3, the factor-augmenting representation (13) gives the correct first derivatives andthe correct EoS between capital and other factors around the BGP.
Proposition 5. (Derivatives of the Factor-Augmenting Representation) Suppose that σKXj ,0 = 1 for j = 1, ..., j ∗ . If the share of factor Zt ∈ {Kt , X1,t , ..., XJ,t },i.e., sZ,t = FZ (·; t)Zt /Yt , is constant on a non-degenerate BGP of a multi-factor
neoclassical growth model, the following holds on the BGP:∂F (AK,t Kt , AX1 ,t X1,t , ..., AXJ ,t XJ,t ) = FZ (Kt , X1,t , ..., XJ,t ; t) for all t ≥ 0. (15)∂Zt
Proof. See Appendix A.7.

19

Proposition 6. (The EoS of the Factor-Augmenting Representation) Suppose that σKXj ,0 = 1 for j = 1, ..., j ∗ and let σ KXj ,t denote the EoS in the factoraugmenting representation
F (AK,t Kt , AX1 ,t X1,t , ..., AXJ ,t XJ,t ). If the EoS of the true production function, σKXj ,tfor some j ∈ {1, ..., J}, is constant over time on the BGP, then σ KXj ,t = σKXj ,t holds
for all t ≥ 0 on the BGP.
Proof. See Appendix A.8.

4.3

Practical Use and Comparison to Uzawa Theorem

Propositions 4-6 demonstrate that the factor-augmenting representation captures key
elements of the true production function and is potentially useful for economic analysis. When developing a dynamic macroeconomic model, researchers need to take astand on how to represent technical change. In other words, they need to decide how
the shape of the production function will evolve over time. This is a challenging taskthat can influence the results, especially in a quantitative setting.
Given the requirement that the model should have a BGP, the propositions provide guidance in choosing a suitable representation of the evolution of the production
function. As shown in equation (1), the definition of the neoclassical growth modelallows the aggregate production function to evolve in any way. The factor-augmenting
representation captures this evolution only through factor-augmenting terms. Proposition 4 demonstrates that the representation matches the level of all the key variableson the BGP, recreating an important set of stylized facts. Proposition 5 implies thatthe representation has the correct derivatives, and therefore factor shares, as long asfactor shares are constant on the BGP. Relatedly, Proposition 6 says that the Uzawa
representation has the correct EoS between capital and other variables, as long asthat elasticity is constant. Thus, the Uzawa representation can be useful as a local
approximation of the true function around the BGP.
These properties echo that of the Uzawa representation, as previously shown in
Propositions 1-3. However, the original Uzawa theorem explains the balanced growthby homothetic expansion of every effective production factor. In other words, theUzawa representation hypothesizes that no factor substitution is taking place along
the BGP. As a result, it requires the productivity of capital to stay constant. Our
20generalized theorem in Proposition 4 clarifies that the Uzawa theorem is only a single
possibility out of a continuum of possible factor-augmenting representations, as longas the production function allows factor substitution on the BGP (i.e., at least one
factor of production has a unitary EoS with capital).17 Every candidate representation
can explain the observed quantities on the BGP, but they differ in the rates of factoraugmenting technological progress among different production factors. Given that
other properties are the same, it is useful to choose the candidate representation thatmatches the rate of capital-augmenting technological progress observed in data.
In Propositions 4-6, we construct the factor-augmenting representation based on
the period-0 production function. Period 0 can be chosen freely by the researcher aslong as the economy is on the BGP in that period. Still, it may suggest that thecomplete shape of the production function at some point in time must be known.
However, this assumption is made for the sake of clarity and is not necessary. The
following remarks state that we can apply the propositions even when only the local
properties of the period-0 production function are known.18
Remark 1. The proof of Proposition 4 holds as long as Fb(·) in (10) is any CRS
functionthat matches the level ofinputs and output in period 0: i.e.,

Q
∗
ξ
α
Fb K0 j X j , Xj ∗ +1,0 , ..., XJ,0 = Y0 .
j=1

j,0

Remark 2. The proof of Proposition
as long as Fb(·) matches
 the first deriva 5 holdQ∗ξtive of F (·; 0) in period 0, i.e., ∂ Fb K α j X j , Xj ∗ +1,0 , ..., XJ,0 = FZ (K0 , X1,0 , ..., XJ,0 ; 0)
∂Z0

0

j=1

j,0

for Z = Kt , X1 , ..., XJ , in addition to the condition in Remark 1.
Remark 3. The proof of Proposition 6 holds as long as the EoS of Fb (·) between
capital and all other inputs evaluated in period 0 match that of F (·; 0), in addition to
the conditions in Remark 1.
Thus, when building models for macroeconomic research, economists can pick a
CRS production function (e.g., a CES production function) and calibrate its parameters to match the level, derivatives (or factor shares), and the EoS from the data at aparticular point in time, which can be regarded as time-0 BGP values. Then, the representation will continue to match these moments on the BGP, as long as the factor
17

The Uzawa representation, where γK = 1 and γXj = g/gXj for all j, satisfies condition (14).
Therefore, it is a special case of the factor-augmenting representation.
18
Similar remarks apply to Propositions 1-3.

21

shares and the EoS are stationary. In addition, according to the data on technological change, the economist can calibrate the speed of factor-augmenting technologicalchange, γK and γXj ’s, using constraint (12) or, equivalently, its log-linear version (14).

5

Three Examples

In this section, we explain the use and implications of the generalized Uzawa theorem in three concrete settings. We choose examples to explore the simplest way
to make neoclassical models consistent with aggregate data on the relative price ofcapital and the EoS between capital and labor. In subsection 5.1, we explain why
a standard neoclassical economy with only two factors cannot accomplish this goal.
Then, subsection 5.2 discusses the approach taken by Grossman et al. (2017) as a
special case of the 2-factor neoclassical environment. Finally, subsection 5.3 showsthat the conflict between data and neoclassical models can be resolved when including factors of production beyond labor and reproducible capital. These examples will
also illustrate how the theorem can be applied to other settings.

5.1

Standard 2-Factor Neoclassical Growth Model

Suppose that the true production function uses only two kinds of inputs, capital, Kt ,and labor, Lt , i.e., Yt = F (Kt , Lt ; t). The production function F (·; t) depends ontime due to the technological change. Then, Proposition 1 says that, on any BGP
with positive investment, technological change can always be represented as Yt =Fe(Kt , AL,t Lt ). If these two factors are substitutable with a unitary elasticity (σKL =1), Proposition 4 shows there are other possible factor-augmenting representations of
the same BGP:19
Yt = A(AK,t Kt )α (AL,t Lt )1−α , where A > 0 is a constant,(16)

eL,t Lt )1−α as a special case. Given
which includes an Uzawa representation Yt = AKtα (Athe growth factors of output and labor on the BGP, condition (12) implies that any
When there are two factors (J = 1) and they are substitutable with a unitary elasticity (j ∗ = 1),equation (13) in Proposition 4 implies that Yt = Fb (AK,t Kt )α (AL,t Lt )1−α . Because function Fb(·)
has CRS and has only one argument, we can write Fb(x) = Ax for some A > 0, which gives (16).
19

22

combination of γK = AK,t+1 /AK,t and γL = AL,t+1 /AL,t is consistent with the BGPα,as long as they satisfy γK(γL gL )1−α = g 1−α . By rewriting (16) as Yt = At Ktα L1−αt
1−αwhere total factor productivity (TFP), At , is given by At ≡ AAαK,t AL,t
, it is clear thatvarious combinations of capital- and labor-augmenting technological changes give thesame rate of growth for TFP and, therefore, output.
This result confirms the widely understood version of the Uzawa theorem: ona BGP, all technological progress must be labor-augmenting, unless the production
function is Cobb-Douglas. As we have seen in Section 2, this theoretical result is incontradiction with two stylized facts: (i) the productivity of capital has been steadily
increasing, and (ii) the EoS between capital and labor is less than one, ruling out the
Cobb-Douglas production function. No standard two-factor production function canreconcile these two stylized facts on a BGP.

5.2

Inclusion of Schooling in a Two-Factor model

Grossman et al. (2017) propose a possible solution to this contradiction by including
schooling, st ≥ 0, in a standard two-factor production function. Their result canbe understood intuitively in terms of our analytical framework. While they start
their analysis from a factor-augmenting representation, it is worthwhile to consider
an underlying time-varying true production function in the form of (1):20
Yt = F (Kt , Lt ; t) = F s (D(st )a Kt , D(st )−b Lt ; t),(17)

where a > 0, b > 0, D(·) ∈ [0, 1], and D0 (·) < 0. With D(st ) terms, the RHS of(17) specifies the production function beyond the general form F (Kt , Lt ; t). When st
increases, the multiplier D(st )a on Kt shrinks, raising the marginal product of capital.
The opposite holds for labor. In this way, Grossman et al. (2017) specify a certain
type of complementarity between schooling and capital.
Note that st is not a production factor in the neoclassical sense, because theproduction function has CRS only in capital and labor. Still, as the D(st ) term
20

They considered not only factor-augmenting technological progress, but also investment-specifictechnological change. Definition 1 can include both cases as we will show in Section 8. In a subsequent study (Grossman et al., 2021), they used human capital instead of schooling as an argumentin the production function otherwise similar to (17). Our interpretation also applies to this paper.

23

changes over time, it affects the amount of output produced from given quantities ofKt and Lt . This is a particular form of technological change, and we can consider
D(st ) as being included in the t term of F (Kt , Lt ; t), as in the middle part of (17).
Therefore, it falls within the definition of a two-factor neoclassical growth model (i.e.,
Definition 1 with J = 1).
From Proposition 1, this production function has an Uzawa representation Yt =eL,t Lt ) with A
eL = (g/gL )t on a BGP, where both effective factors Kt andFe(Kt , AeL,t Lt grow at the same speed as output. The production function in GrossmanA
et al. (2017) can be interpreted in the following way, keeping the multiplier D(st )term in the expression:
eL,t Lt ) = Fe(AK,t D(st )a Kt , AL,t D(st )−b Lt ).
Yt = Fe(Kt , A(18)

Comparing the arguments in the RHS to those in the middle, we immediately obtain
eL,t D(st )b on the BGP. Because the multiplier D(st )a
AK,t = D(st )−a and AL,t = A
shrinks as st increases, the capital-augmenting technology AK,t must grow so as toexactly offset the shrinking D(st )a term. Conversely, the labor-augmenting term AL,teL,t , because the multipliershould grow slower than that in the Uzawa representation A
D(st )−b is also augmenting labor.21 In this sense, there is no overall growth in capital
productivity in the Grossman et al. (2017) formulation.
Within the limits of the two-factor Uzawa theorem, Grossman et al. (2017) propose
a new interpretation of the production function, which provides the first possible
solution to the contradiction raised by the Uzawa theorem. In their formulation, it isimportant that schooling enters the production function precisely in the form of (18),where the same function D(st ) appears both before capital and labor, with powers of
opposite signs. In addition, the functional form of D(st ) and the dynamic path st in
equilibrium must be specified such that D(st ) shrinks exponentially over time.
Future empirical work could inform the understanding of long-run economic growth
21
From these observations, the main result of (Grossman et al., 2017, proposition 2) can easily be−aobtained as follows. Taking the growth factor of the both sides of AK = D(st )−a gives γK = gD.
−1/a
From this, we obtain a discrete-time equivalent of their Proposition 2(ii): gD = γK . Notethat Grossman et al. (2017) assumed Lt = D(st )Nt , which means gL = gD gN . Because effective−b
1−blabor AL,t D(st )−b Lt in (18) must grow at the same rate as output, g = γL gD
gL = γL gD
gN =(b−1)/aγL γKgN , which is a discrete time equivalent of their proposition 2(i).

24by testing whether the formulation (17) is consistent with data. In this paper, we
propose a wider class of functions that are consistent with balanced growth. The next
subsection discusses a particularly simple example.

5.3

A Simple Three-Factor Model with Natural Resources

As shown in Section 2, a significant portion of GDP is paid to production factors that
do not fit well in the notion of Kt or Lt . Thus, it is natural to consider production
functions with more than two factors. Adding these additional factors makes it possible to reconcile neoclassical models with the data. While labor cannot be substituted
by capital with unitary elasticity (σKL 6= 1), Proposition 4 only requires that there isa single production factor satisfies this requirement. In this case, there exist factoraugmenting representations of the production function that have capital-augmenting
technological change (γK > 1).
Let us consider the simplest extension of the standard neoclassical production
function,t
Yt = Ft (Kt , Lt , Xt ; t), where Xt = X0 gX
for all t, X0 > 0, gX > 0.

(19)

Here, we have a third production factor Xt , which is either growing (gX > 1), shrinking (gX ∈ (0, 1)), or constant (gX = 1). One example of such a factor is land. In thatcase, gX represents the growth factor of the available land space. If the total area ofavailable land is constant, gX would be one on the BGP. Another example is fossil
fuels. Any kind of natural resource, or a collection of natural resources (includingland), is a candidate for Xt .
Among many candidates for the third production factor, we focus on those thathave a unitary EoS with capital: σKX = 1. Then, Proposition 4 implies that, along a
non-degenerate BGP, technological change can be represented in a factor-augmenting
fashion:
Yt = Fb

AK,t Kt

αAX,t Xt

1−α, AL,t Lt , α ∈ (0, 1),(20)

αwhere the growth factor of technology variables must satisfy γL = g/gL and γK(γX gX )1−α =25

g 1−α . As in (14), the latter condition can be written in a log-linear form:
log γK =1−α(log γL + log gL − log γX − log gX ) .
α(21)Thus, there must be a positive capital-augmenting technological change on a BGP(γK > 1), as long as the economy is growing faster than the effective input of the third
factor (g = γL gL > γX gX ). We will calibrate this simple three-factor model to U.S.
data in Section 7. Before doing so, we discuss empirical evidence on the elasticity of
substitution between capital and the third factor Xt .

6

The EoS: Linking Theory and Empirics

In this section, we discuss existing empirical evidence on the elasticity of substitution
between capital and other factors, and the link between these estimates and economic
theory. We start by highlighting the difference between short- and long-run elasticity
and then discuss existing empirical work on capital-labor, capital-energy, and capitalland substitution.

6.1

Short- and Long-run EoS: Theory

When the relative factor prices change, firms try to adjust relative inputs to the
production function, substituting one factor with another. The EoS measures theextent of this activity. The ability of firms to substitute between production factors
depends on the time horizon. In the short run, the ability of factor substitution istypically limited, because firms face various constraints that are alleviated over time.
Put differently, there is a difference between the short-run EoS, which is affected bythese constraints, and the long-run EoS, which is not.
Leon-Ledesma and Satchi (2019) show that it is the long-run EoS that is relevant
for the Uzawa theorem, because the theorem is concerned only with outcomes along
the balanced growth path. They consider an aggregate production function in theform of Yt = F short (AK,t Kt , AL,t Lt ; θ), where technology parameter θ > 0 is taken as
given in the short run. In this formulation, the EoS of function F short (·; θ), given thevalue of θ, provides the short-term (or constrained) EoS between Kt and Lt . In the26

long run, firms will choose θ optimally. Therefore, the long-run production function is
given as the solution to F long (AK,t Kt , AL,t Lt ) ≡ maxθ F short (AK,t Kt , AL,t Lt , θ). They
have theoretically shown that the EoS of F long (·) is higher than that of F short (·; θ),consistent with the finding in the empirical literature. They also show that, to obtain
the BGP with capital-augmenting technological change, the long-term EoS between
capital and labor needs to be one, but there is no restriction on the short-run EoS.22
In other words, it is the long-run EoS that determines compliance with the two-factor
Uzawa growth theorem.
In our theory, we consider the neoclassical production function (1) without shortterm restrictions, such as θ above. Therefore, F (Kt , X1,t , ..., XJ,t ; t) in (1) and its
factor-augmenting representation Yt = F (AK,t Kt , AX1 ,t Xj,t , ..., AXJ ,t XJ,t ) in (13) areunconstrained, long-run production functions. As with most neoclassical growth models, we implicitly assume that production function F includes all substitution possibilities, given the currently available set of technologies in the economy.23 Differently
from Leon-Ledesma and Satchi (2019), we focus on the case where the long-term (orunconstrained) EoS between Kt and Lt is less than one, which fits better the findingsin the empirical literature. To do so, we look for some other factor Xt such that EoS
between Kt and Xt in this long-term aggregate production function is one.

6.2

Long-run EoS: Empirical evidence

The distinction between the short-run and long-run EoS is important for interpreting
empirical estimates. Analyses with time-series and panel data looking at the immediate reaction of factor use to relative price changes will estimate the short-run (constrained) elasticity, which may be very different than the long-run (unconstrained)
elasticity. To estimate long-run elasticities, it is often necessary to use cross-sectional
22

To be consistent with the BGP, they have shown that production function Yt = F short (AK,t Kt ,AL,t Lt ; θ) needs to take the form of Fe(At θα−1 Kt , At θα Lt ), where At is the general level of technology. Since Fe(·) is CRS, we can show that Yt = At Ktα L1−α
maxθ>0 Fe((θLt /Kt )α−1 , (θLt /Kt )α ).
t
The maximized value in the latter expression, a∗ = maxθ>0 Fe((θLt /Kt )α−1 , (θLt /Kt )α ) becomes
constant because the maximand depends only on the technology-adjusted factor intensity θLt /Kt ,which can be set to any positive value adjusting θ. Therefore, the long-term (or unconstrained)
production function is Cobb-Douglas: Yt = a∗ At Ktα L1−α.
t
23
If Θt is the available set of technology in time t and F short (Kt , X1,t , ..., XJ,t ; t, θ)
is the production function given technology θ, then F (Kt , X1,t , ..., XJ,t ; t) = maxθ∈Θt
F short (Kt , X1,t , ..., XJ,t ; t, θ).

27

data or adopt other approaches that separate short- and long-run variation. In thiscase, it is possible to observe the degree of factor substitution after short-run constraints have dissipated. In the following sections, we briefly review the existing evidence on substitution between capital and labor (Section 6.2.1), capital and energy(Section 6.2.2), capital and land (Section 6.2.3).
6.2.1

Capital and LaborIn Section 2 and other places, we mentioned that existing evidence suggests that
capital and labor have an EoS that is different than one. This is true for the long-run
EoS. Chirinko (2008) provides a helpful summary of the literature. He stresses thedistinction between the short- and long-run EoS and the fact that empirical evidence
suggests that the long-run EoS is less than one. This result is confirmed by more
recent work (e.g., Chirinko and Mallick, 2017; Oberfield and Raval, 2021).24
6.2.2

Capital and Energy

There is a long literature on the economic implications of changes in energy prices. A
key finding in this literature is that the long-run EoS between energy and non-energy
inputs is significantly higher than the short-run EoS, which is near zero (e.g., Berndt
and Wood, 1975; Griffin and Gregory, 1976; Pindyck and Rotemberg, 1983; Atkeson
and Kehoe, 1999). Indeed, the existing evidence is consistent with a long-run EoS
between capital and energy of one.
Koetse et al. (2008) perform a meta-study of existing literature on substitution
between capital and energy. They find country-level, cross-sectional elasticities thatare very close to one in North America. The estimates for Europe are around 0.8,
but do not reject unitary elasticities. This is the most direct test of the long-run
elasticity, as highlighted in the previous section.
Van der Werf (2008) estimates nested CES production functions. When he adopts
a nesting structure consistent with the existence of a capital composite, he finds elasticities very close to one, although alternate nesting structures yield higher goodness-offit measures. This estimation is done in panel data and does not distinguish between
24

Karabarbounis and Neiman (2014) is an interesting exception. Their estimate of the long-run
elasticity is above one.

28

short- and long-run elasticities.
Hassler et al. (2021) study a structural model of fossil fuel energy use and directed
technical change. Similarly to Leon-Ledesma and Satchi (2019), they stress the rolethat endogenous technology plays in creating a difference between short- and long-run
elasticities and derive similar conditions under which the long-run elasticity between
energy and other factors is equal to one. In their estimated model, they assume anesting structure that is different from that implied by the generalized Uzawa theorem
and stress the lack of precision in their long-run estimates. Still, their estimatessuggest only a modest change in the energy expenditure share following a significant
change in trend energy prices, implying that the implied elasticity is not far from one.
Our reading of the empirical evidence is as follows. Energy use (or perhaps somesubset of energy) is a reasonable candidate for the third factor X that has a unitary
EoS with capital. However, more research is needed, both to estimate the relevant
elasticity and to determine the appropriate nesting structure. We hope that ourtheoretical results will spur new empirical work, which could significantly improve
our understanding of balanced growth.
6.2.3

Capital and Land

Land is an intuitive candidate for the third factor, because it has a relatively large
factor share. Unfortunately, the existing literature on capital-land substitution ismore limited than the evidence regarding energy. In the urban economics literature,
it is common to estimate the EoS between land and other inputs from cross-sectional
data on the production of housing services, and the estimated elasticities are often
close to one (e.g., Epple et al., 2010; Ahlfeldt and McMillen, 2014; Ahlfeldt et al.,2015). While this result is suggestive, the EoS estimated in the urban literature does
not necessarily coincide with the macro EoS, because the urban literature only examines the elasticity in housing, while the macro elasticity is affected by the elasticity
in all sectors and the reallocation between sectors induced by changes in relative
factor prices (Oberfield and Raval, 2021). Markandya and Pedroso-Galinato (2007)
use aggregate data to examine many different nesting structures that include land.
In several specifications, they find an EoS between capital and land that is close to29

one.25 Our interpretation of the evidence on the capital-land EoS is similar to ourinterpretation for the capital-energy EoS. It seems plausible that the capital-land EoS
is close to one, but further research is needed. We hope that our theoretical results
will spur this empirical work.

7

Quantitative Exercise

7.1

Calibration

In this section, we show how the generalized Uzawa theorem can be used to calibrate
the three-factor neoclassical growth model from Section 5.3, assuming perfect competition. We will obtain a factor-augmenting representation that is simultaneously
consistent with the data on balanced growth, capital-augmenting technical change,
and the elasticity of substitution between labor and capital. Our calibration uses
annual U.S. data from 1960-2020. Appendix D.2 contains details on data sources.
Table 2 lists the calibration targets and the parameters that are taken directly fromthe data.
This calibration exercise assumes that the third factor X is a Cobb-Douglas combination of land and fossil fuel energy, which we refer to as ‘natural resources.’ Wealso assume that the EoS between K and X is one. As highlighted in Definition 1, thecommon structure of the neoclassical growth model only specifies the accumulation
process for capital (2). The determinants of the supply of other factors, Lt and Xt ,depend on the application. We calibrate the model without relying on specifications
for the supply of other factors, demonstrating that our approach could be used in awide range of settings.26
The generalized Uzawa theorem (Proposition 4) implies that, along a non-degenerate
BGP, technological change can always be represented as in equation (20). The functional form of Fb(·) in this equation is given by (11), which depends on the shape of25

Some of their estimates for the capital-energy EoS are close to one, while others are lower.
For example, energy could be available in finite supply and costlessly extracted from the environment as in Hotelling (1931) and Hassler et al. (2021), or it could be extracted from the environment
using the final good as in Casey (forthcoming). In the latter case, extraction costs would be acomponent of Rt in the Definition 1. It is common to model land as an endowment that is renewed
every period, but it would also be reasonable to take into account the fact that some expenditure isnecessary to keep land in a usable state.
26

30

Table 2: Calibration Targets and Parameters Taken from Data
TargetsL
sX
K0 /Y0σKL
gL − 1
gfossil − 1
γL − 1γK − 1
Y0 , L0 , X0

Value
Description
Source62.5%
Labor share
NIPA
9%Natural resource share
NIPA
2.9
Capital-output ratio
NIPA
0.6
EoS between L and K
Oberfield and Raval (2021)
1.02%
Population growth
NIPA
0.85%
Fossil fuel energy growth
EIA
1.93% Labor-augmenting tech. change
NIPA
0.80% Capital-augmenting tech. change
NIPA
1
Initial variables
Normalization

the true aggregate production function on the BGP at some point in time, F (·; 0).
Even when we do not know the exact shape of F (·; 0), Remarks 1-3 indicate that we
can use a CES production function instead of F (·; 0), as long as it matches the levels,
first derivatives (or shares), and the EoS. Therefore, in this exercise, we consider a
CES function for Fb(·, ·) and calibrate its parameters to match the U.S. long-run data.
Equation (20) now can be written as

n
o −1
 −1
−1
Yt = ηKX (AK,t Kt )α (AX,t Xt )1−α  + ηL (AL,t Lt ) 
,

(22)

t
t
AK,t = γK, AX,t = γX
, AL,t = γLt ,where α ∈ (0, 1) is the share of physical capital in the capital composite,  > 0 is theelasticity of substitution between labor and the capital composite, and ηKX and ηLare distribution parameters. Here, we assume  6= 1, since  = 1 would imply that theentire production function is Cobb-Douglas, which is inconsistent with the observed
long-term elasticity of substitution between capital and labor.
Our data indicate that sL = 62.5% was the average labor share of income overthis period. For the share of natural resources, sX , we use the results from Table 1,
which reports that the shares of land and energy are 4% and 5%, respectively. Thus,sX = 4% + 5% = 9%. This pins down the share of capital, sK = 1 − sL − sX = 28.5%,
and the relative weights inside the capital composite, α = sK /(sK + sX ) = 0.76.
Lemma 1 implies the BGP growth rate of capital, gK , always coincides with thegrowth factor of output, g. The definition of balanced growth (Definition 2) implies31

that the growth factors of the other inputs, gL and gX , as well as the growth factors
technologies, γL , γK , and γX , are constant. In this environment, the capital composite,(AK,t Kt )α (AX,t Xt )1−α , grows at a factor of (γK g)α (γX gX )1−α , and effective labor,
AL,t Lt , grows at factor of γL gL . The generalized Uzawa theorem (Proposition 4 withj ∗ = 1) requires that these two growth factors coincide with g on the BGP:
g = (γK g)α (γX gX )1−α = γL gL ,(23)which is equivalent to equation (21) in Section 5.3. From the U.S. data, we takethe annual growth rate of real GDP per capita to be (γL − 1) = 1.93% and the
population growth rate to be (gL − 1) = 1.02%. This gives a GDP growth rateof (g − 1) = γL gL − 1 = 2.95%. Average growth rates are measured by fitting an
exponential trend through the data using ordinary least squares.
The growth rate of natural resources, gX , is a geometric mean of the growth factorsof land and fossil energy, where the weights are the expenditure shares. We assume
that land is constant and take the growth rate of fossil energy to be (gfossil −1) = 0.85%
per year, based on data from the Energy Information Administration (2023). Thus,
54(gX − 1) = 1 9 (gfossil ) 9 − 1 = 0.38%.
With this information, we can use equation (23) to pick the growth rates of capitaland natural resource-augmenting technological change, {γK , γX }. It is customary tochoose γK = 1 (constant AK,t ) — because this is the only choice consistent with balanced growth in two-factor neoclassical growth models without Cobb-Douglas production (see Section 5.1) — and to exclude the third variable X. In our setting, we
can choose γK to be consistent with data. In particular, the growth rate of capitalaugmenting technical change can be chosen to match the fall in the relative price ofcapital, yielding (γK − 1) = 0.80%. Then, from the technology condition (23), we
obtain (γX − 1) = 0.04%.27
Next, we show how to calibrate , assuming that a researcher has a target value
27
As explained in Appendix C, the fall in the relative price of investment can be used to measure
the growth rate of capital-augmenting technological change, because the neoclassical growth model(Definition 1) is built on the assumption that final goods in period t can be converted to Kt+1
linearly. This property does not hold for other production factors, implying that we can not use
price data to calibrate γX directly. For example, a standard Hotelling (1931) model would imply
that changes in the price of energy reflect changes in scarcity rents, rather than energy-augmenting
technical change. Therefore, we use condition (23) to calibrate γX .

32

for the elasticity of substitution between capital and labor. In Appendix B.1, we
calculate the elasticity of substitution between K and L in function (22) according
to Definition 3, which yields
σKL =sK /sL + 1≡ Σ(, α, sK /sL ).
(sK /sL + α)−1 + (1 − α)(24)Equation (24) shows that σKL = Σ(, α, sK /sL ) is an increasing function , withΣ(0, α, sK /sL ) = 0, Σ(1, α, sK /sL ) = 1, and lim→∞ Σ(, α, sK /sL ) = (sK /sL +1)/(1−α). This means the researcher can pick  > 0 to match any target value of σKL between
0 and (sK /sL + 1)/(1 − α). Specifically,=sK /sL + α−1(sK /sL + 1)σKL− (1 − α)if 0 ≤ σKL <sK /sL + 1
.
1−α

(25)

We use σKL = 0.6 (see, e.g., Chirinko, 2008; Oberfield and Raval, 2021), which gives28
 = 0.56.
Lastly, we discuss how to calibrate distribution parameters. We consider someperiod t = 0 in which the economy is on a BGP. Appendix B.2 shows that the factor
shares and the distribution parameters on the BGP are related by
sK
ηKX =
α



K0α X01−α
Y0

 1−

,


ηL = sL

L0
Y0

 1−

.

(26)

The values of ηKX and ηL in the above equations reflect the choice of units. To
set units for labor and natural resources, we normalize L0 = X0 = 1 without lossof generality. Since the process of capital accumulation is given by (2), capital andoutput must be measured in the same units.29 In the data, the capital-to-output ratio
is K0 /Y0 = 2.9. So, we normalize Y0 = 1 and K0 = 2.9. This gives ηKX = 0.71 andηL = 0.63.
28

While we think that this is a reasonable calibration of , it should be noted that most estimatesof σKL assume a two-factor production function. Our results suggest that there is considerable value
in re-estimating substitution elasticities in multi-factor settings that include natural resources.
29
Strictly speaking, the amount of capital in a given period, Kt , should be measured in the unitof the previous period’s output, Yt−1 . However, there is no simple way to do this. The standard
convention in the calibration of the neoclassical growth model is to measure simply the nominal
value of the capital stock and output in the same year to attain the capital-output ratio. We follow
this convention. This compromise only affects the CES distribution parameters, ηKX and ηL .

33

Table 3: Calibrated Parameters
Parameter Value
Description

0.56
EoS b/w capital composite and L
α0.76 Capital share in capital composite
ηL
0.63
CES distribution parameter
ηKX
0.71
CES distribution parametergX − 1
0.38%
Natural resource (X) growthγX − 1
0.04%
X-augmenting tech. changeThis completes the simple calibration exercise. The calibrated parameter values
are shown in Table 3. We explained how a researcher could obtain a representationof the evolution of production function in the form of (22), where all the parameters, {α, , ηKX , ηL , γK , γX , γL } are specified. This representation can be used as an
approximation of the true production function around the BGP. Unlike standard twofactor production functions, it simultaneously matches the declining price of capitalon the BGP (γK > 1) and the non-unitary elasticity of substitution between laborand capital (σKL 6= 1).

7.2

Explaining movements in the labor share

Here, we present a simple application of the calibrated model from the previous
section. We use it to study how fluctuations in capital-augmenting technical change(as reflected in the relative price of investment) have influenced the labor share overthe last sixty years. Our model provides a new perspective on this question. In astandard CES production function, the labor share is constant only when AK,t isconstant, and a long-term increase in AK,t would imply that the labor share willcontinue to rise (if σKL < 1) or fall (if σKL > 1). In a standard Cobb-Douglas
production function, the labor share is always constant. In our model, by contrast,
the labor share is constant when AK,t grows along the BGP at a rate determinedby the log-linear technology condition (21). Any deviation of capital-augmenting
technology from its BGP path will generate fluctuations in the labor share.
In the following, let AK,t represent the BGP value of capital-augmenting technicalchange at time t, and aK,t denote its actual value, whether or not it is on the BGP.
To focus on the impacts of a deviation of aK,t from AK,t , we assume that all other34

technologies and production factors follow their respective balanced growth paths. In
Appendix B.3, we show that the labor share in period t implied by our calibrated
model is
sL,t =!−1α(−1)/
1
aK,t
−1
+1
,
sL
AK,t

(27)

where sL (without a timescript) is the labor share on the BGP.
It is straightforward to take this expression to the data. In our calibration,
α = sK /(1 − sL ) and  in (25) depend only on factors shares and the estimate ofσKL . Therefore, the calculation of (27) does not require much information other thanthe deviation of capital-augmenting technology from its BGP path, aK,t /AK,t . The
relative price of investment can be used to measure this deviation. In Appendix C.1,
we derive
aK,t
=
AK,t



rpit−1
RP It−1

−1
,

(28)

where rpit−1 is the relative price of investment and consumption in the NIPA statistics
at time t − 1, and RP It−1 is the BGP value. The lag between the measurement of
capital-augmenting technical change and the relative price of investment reflects thefact that the efficiency and quantity of period t capital depends on investment inperiod t − 1. These results imply that the period length matters for our results. We
calculate results using period lengths of one year and five years.30
The results are shown in Figure 2. Panel (a) shows the relative price of investment
from the NIPA data and its exponential trend estimated via ordinary least squares,
which we take to be the BGP level. From (28), comparing the data and trend yields
a measure of how much capital-augmenting technical change has deviated from its
BGP. As noted in the previous section, the relative price of investment fell at an
average rate of 0.80% over this period. The actual data lie below the trend prior to1975 and spike well above the trend in the early 1980s. Toward the end of the period,
the data lie close to the long-run trend.
Panel (b) shows the movements in the labor share in the BEA data and those30
For the calculation with five-year periods, we first compute the deviation of the actual relative
price of investment from its BGP value for each year. Then, we take averages over five-year periods,
starting with 1960-1964. For each of these five-year periods, we compute (28) using data from theprevious period. Finally, we plug back into (27). The markers in the figure show the first year ofeach five-year period.

35

(a) Investment Price(b) Labor Share

Figure 2: The Relative Price of Investment and the Labor Share.
Note: See Appendix D.3 for data sources.

implied by the model. When the relative price of investment is above its BGP level,
the ratio of the effective capital composite to the effective labor is below its BGPlevel. Note that  < 1 means that the labor share declines when labor becomesabundant relative to the effective capital composite. Therefore, an upward deviationof the relative price of investment from its BGP level implies a lower labor share in
our model. The opposite occurs when the relative price of investment is below its
BGP level. Thus, consistent with the data, the model correctly predicts that thelabor share was above its long-run level early in the 1960s and 1970s and fell below
its long-run level in the 1980s. Towards the end of the period, however, the data andmodel diverge. There was a fall in the labor share starting in the early 2000s eventhough the relative price of investment has been near its long-run trend.
Our results imply that changes in the relative price of investment can explain
swings in the labor share prior to 2000, but not the precipitous drop in this century.
These findings suggest that other potential explanations for the changing labor share,like market power or trade, are more likely explanations (Grossman and Oberfield,2022).

36

8

Investment-Specific Technological Change

Throughout the paper, we have considered standard neoclassical models where one
unit of final output in period t can be transformed into one unit of capital at time t+1,
as specified by (2). The following lemma shows that our theory can accommodate
models where qt > 0 units capital can be produced from a unit of final output,
where qt is investment-specific technology (IST). The growth of qt over time is called
investment-specific technical change (ISTC).
Lemma 3. (Investment-Specific Technological Change) Consider an economic
environment where output Yt and capital KtIST are determined by
Yt = F IST (KtIST , X1,t , ..., XJ,t ; t),
IST
Kt+1= (Yt − Ct − Rt )qt + (1 − δ IST )KtIST ,

(29)
(30)

where F IST (·) has CRS and positive and diminishing returns to all inputs. If thegrowth factor of IST, gq = qt /qt−1 , is constant,31 this environment can be transformed
into (1) and (2) in Definition 1 through a change of variables of Kt ≡ KtIST /qt−1 andδ = (δ IST + gq − 1)/gq , as well as a redefinition of the production function,
F (Kt , X1,t , ..., XJ,t ; t) = F IST (qt−1 Kt , X1,t , ..., XJ,t ; t).

(31)

Proof. See Appendix A.9.
Note that the linearity of the accumulation function equation is still preservedin (30), which is a key assumption of the neoclassical growth model in Definition
1. In Lemma 3, the change of variables effectively normalizes the unit of capital sothat capital Kt ≡ KtIST /qt−1 is always measured in terms of the previous period’s
final good, as in (2). The depreciation rate after this normalization, δ, should behigher than δ IST , because positive investment-specific technological change decreases
the value of older capital.
All the results in this paper can be applied to models with ISTC by re-interpretingthe variables and the production function using Lemma 3. When the transformed
31

We assume that gq is constant for simplicity. This condition is not necessary if we extend
Definition 1 to allow the depreciation rate to change over time. As long as we focus on the BGP,
the results will not be affected.

37

production function (31) is used in Definition 1, it already includes a type of capitalaugmenting technological change, because qt−1 , which multiplies Kt on the RHS,
grows over time. This component of capital-augmenting technological change is often
called embodied technological change. Even when the model has ISTC, the aggregate production function (29) may exhibit further capital-augmenting technologicalchange that is not included in the growth of qt . Such technological change is called
disembodied technological change. When we convert the variables and the production
function using Lemma 3, the sum of embodied and disembodied technological change
will show up as overall capital-augmenting technological change in production function (1). Therefore, the original two-factor Uzawa theorem in the presence of ISTC
says that the sum of these technological changes must be zero on the BGP, unless the
production function is Cobb-Douglas.
We also note how the sum of these technological changes can be obtained from
data. Appendix C.2 shows that the relative price of investment in the NIPA statistics
measures the sum of the rates of embodied and disembodied technological changes
when capital accumulation is given by (30).32 Given that the ratio is falling, theoriginal two-factor Uzawa theorem requires any model with positive ISTC to adopt
a Cobb-Douglas aggregate production function (e.g., Greenwood et al., 1997, 2000).
This paper presents a way to relax this limitation.

9

Conclusion

The relative price of investment has been falling in the U.S. for long periods of time,
indicating the existence of capital-augmenting technological change on the balanced
growth path. Due to the Uzawa steady-state theorem, however, this fact could not
be incorporated into macroeconomic models that use empirically relevant values forthe elasticity of substitution between capital and labor. This paper presents a generalized Uzawa theorem, which demonstrates how this limitation can be overcome bytaking the realistic step of adding additional inputs, such as land and energy, into the32

Since the NIPA data only gives γK · gq , we need another source of information either on theembodied technological change, gq , or disembodied technological change, γK , to calibrate the modelwith ISTC. Accordingly, we would have obtained a different value for γK in the calibration of
Subsection 7.1 if we used accumulation equation (30), rather than (2). This difference demonstrates
that the result of the calibration depends on the specification of the accumulation process.

38

production function.
On a BGP, all effective factors of production grow at the same rate as output,
unless there is a unitary elasticity between two or more factors. As noted by Jonesand Scrimgeour (2008), the linear accumulation process implies that “capital inherits
the trend in output.” On a BGP where all effective factors grow proportionally to
output, there should be no capital-augmenting technological change, because capitalis already growing at the same rate as output. The generalized Uzawa growth theorem considers a hybrid of capital and other non-accumulable inputs, which we callthe capital composite. There is a unit EoS between the components of the capitalcomposite. If the capital composite would grow slower than output in the absenceof technological change, capital composite-augmenting technical change is necessary
for a BGP. Thus, capital-augmenting technical change, which is one component ofcomposite-augmenting technical change, is compatible with balanced growth and a
non-unitary EoS between capital and labor.
What if there is no Xj with σKXj = 1?
For a production factor to be a part of the capital composite, the long-run EoS
between capital and this factor must be unity, since otherwise factor shares and thegrowth rate cannot be kept constant simultaneously. We discussed energy and landas candidates in Section 6.2, and there are a wide range of other factors used in
production. However, it is still an open question whether there is a production factor
that has unit-elastic substitution with capital. If future research determines there isno such factor, the generalized Uzawa Growth theorem would then imply that the
speed of capital-augmenting technological change must be exactly zero on any BGP
and the puzzle would persist.
In this case, it would be necessary to question the assumptions of the theorem.
A remarkable property of the generalized Uzawa theorem is that it depends on very
few assumptions: (i) the economy can be expressed by a neoclassical growth model,and (ii) there is a BGP. The NIPA data strongly suggests the existence of the BGP.
Therefore, we can narrow down the concern to the assumptions in the neoclassicalgrowth model, as given in Definition 1. The definition consists of two parts, theaggregate production function (1) and the capital accumulation equation (2). In the39

Uzawa theorem, the latter is only utilized in Lemma 1, which shows that K/Y must
be constant in the BGP. The result of this lemma is clearly visible in the NIPA
data depicted in Figure 1(a). Therefore, this accumulation equation seems to do no
additional harm.
The remaining suspect is the aggregate production function: Yt = F (Kt , X1,t , ...,
XJ,t ; t). It assumes that there is a mapping from aggregate inputs to the aggregate output. This is not a weak assumption. While the NIPA statistics show that theprice of investment is declining relative to consumption, it does not mean that various
kinds of equipment are becoming cheaper proportionally. If newer capital goods havemore margins for cost reductions and quality improvements, their quality-adjusted
prices will fall faster. Moreover, new kinds of capital goods are continually introduced through R&D, whereas old goods disappear from the market. This cycle also
contributes to the fall in the quality-adjusted price of investment in the NIPA statistics. However, in the framework of the neoclassical growth model, we need to map thestatistics to the model assuming that all kinds of capital goods can be aggregated into
one variable. The same can be said for the left-hand side of the production function,i.e., aggregated output Yt . When the composition of inputs and outputs evolves overtime, it is not obvious whether we can define a functional relation over the aggregate
variables. If there is no Xj with σKXj = 1, it might suggest that aggregation createdthe problem. Exploring disaggregated models seems important in this respect.33

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44Online Appendix“A Generalized Uzawa Growth Theorem ”
by Gregory Casey and Ryo Horii
October 2023

A

Proofs of Propositions and Lemmas

A.1

Notation for derivatives

Unless mentioned otherwise, FK (·; t) denotes the partial derivative of function F (·; t)with respect to its first argument, whereas FXj (·; t) denotes the partial derivative of
F (·; t) with respect to its (1 + j)th argument. The same applies to other functions,such as Fe(·).
∂
Following the convention in economics, ∂Kand ∂X∂j,t represent the partial derivat
tives with respect to variables Kt and Xj,t , respectively. For example, if Fe(·) is the

production function, ∂X∂j,t Fe(·) gives the marginal product of factor Xj,t .
Note that these two definitions are different when the argument of function is nota single variable. For example, using the chain rule, we have
∂ e
eX1 ,t X1,t , ..., AeX ,t XJ,t ). (A.1)
eX1 ,t X1,t , ..., AeX ,t XJ,t ) = A
eX ,t FeX (Kt , A
F (Kt , A
j
j
J
J
∂Xj,t

A.2

Proof of Lemma 1

t+1
Using the notation in Definition 2, equation (2) can be written as K0 gK= Y0 g t −t
C0 gCt − R0 gRt + (1 − δ)K0 gK. Dividing all terms by g t and rearranging them gives

Y0 = C0 (gC /g)t + R0 (gR /g)t + K0 (gK + δ − 1)(gK /g)t .

(A.2)

Because all three terms on the right hand side (RHS) of (A.2) are non-negative
exponential functions of t, every one of them needs to be constant for the sum of
all the terms to become constant (Y0 ). For the first term C0 (gC /g)t to be constant,
gC = g must hold since C0 > 0 from Definition 1. This means Ct /Yt = C0 /Y0 > 0.
For the third term (gK + δ − 1)(gK /g)t to be constant, gK = g must hold since K0 > 0
A-1

and gK > 1 − δ. This implies Kt /Yt = K0 /Y0 > 0. If R0 > 0, gR = R must hold sinceotherwise the second term cannot be constant.

A.3

Proof of Proposition 2

Because the production function in period 0 is F (·; 0) ≡ Fe(·), we can write the shareof factor Z in period 0 asZ0
sZ,0 = FeZ (K0 , X1,0 , ..., XJ,0 ) ,
Y0

(A.3)

where FeZ (·) represents the derivative of function Fe(·) with respect to its argument(see Appendix Section A.1). Note that, since function Fe(·) has constant returns toscale, its partial derivative function FeZ (·) must be homogeneous of degree 0 (SeeTheorem M.B.1 in Mas-Colell et al., 1995). Therefore, the value of FeZ (·) will beunchanged when all of its arguments are multiplied by the same factor g t = Yt /Y0 =eX ,t Xj,t /Xj,0 . (Here we used gK = g from Lemma 1.) Applying this for
Kt /K0 = Aj(A.3) gives
eX1 ,t X1,t , ..., AeX ,t XJ,t )sZ,0 = FeZ (Kt , A
J

Z0
.
Y0

In addition, because the effective amount of production factors and the output grow
eZ,t Zt /Yt holds on the BGP. (In the case of Zt = Kt , weat the same speed, Z0 /Y0 = A
eK,t ≡ 1.) Therefore,
define A
eX1 ,t X1,t , ..., AeX ,t XJ,t )AeZ,tsZ,0 = FeZ (Kt , AJ

eX1 ,t X1,t , ..., AeX ,t XJ,t ) Zt
Zt
∂ Fe(Kt , A
J
=
,
Yt
∂Zt
Yt
(A.4)

where the validity of the second equality is guaranteed by the chain rule.34 Recall
that we assumed that the share is constant over time, which means
sZ,0 = sZ,t = FZ (Kt , X1,t , ..., XJ,t ; t)
By comparing (A.4) and (A.5), we obtain (4).
34

See Appendix A.1.

A-2

Zt
.
Yt

(A.5)

A.4

Proof of Proposition 3

As in Definition 3, the EoS between Kt and Xj , j ∈ {1, ..., J}, in the Uzawa RepreeX1 ,t X1,t , ..., AeX ,t XJ,t ) is defined as
sentation Fe(Kt , AJ

d ln(Kt /Xj,t )σeKXj ,t = −d lneX ,t X1,t ,...,AeX ,t XJ,t )FeK (Kt ,A
1
J
e
e
e
eX ,t XJ,t )AX ,t FX (Kt ,AX ,t X1,t ,...,A
j

1

j

J

.



(A.6)

Yt ,X−j,t :const

We used (A.1) for calculating the marginal product of Xj in the denominator. Note
that, in addition to output Yt and other production factors X−j,t , we keep technologies
eX1 ,t , ..., AeX ,t fixed when calculating the EoS.
A
J

In this proof, we evaluate the value of (A.6) on the BGP. This means Yt and
X−j,t are their BGP values, but we still need to consider (infinitesimally) small perturbations of Kt and Xj,t from these BGP values. To make this distinction, let
Yt , Kt , X1,t , ..., XJ,t denote the specific BGP values, and k and xj the variables to be
perturbed. Then, (A.6) can be written as35

d ln(k/xj )σeKXj ,t = −d lneX ,t X1,t ,...,AeX ,t xj ,...,AeX ,t XJ,t )FeK (k,A1
jJ
eX ,t FeX (k,AeX ,t X1,t ,...,AeX ,t xj ,...,AeX ,t XJ,t )
A
j

j

1

j

J

.


eX ,t X1,t ,...,AeX ,t xj ,...,AeX ,t XJ,t )=Yt
Fe(k,A1
j
J
k=Kt ,xj =Xj,t(A.7)
eX1 ,t X1,t , ..., AeX ,t xj , ..., AeX ,t XJ,t ) = Yt says that k and xj need to
Condition Fe(k, AjJmove to ensure that this equality is satisfied. The other conditions k = Kt , xj = Xj,t
say that, after the differentiation is complete, the EoS is evaluated at the BGP values.
eX ,t xj . Then, kNow, consider a change of variables: k 0 = g −t k and x0j = g −t Ajt 0eX ,t )x0 . Specifically, k/xj in thein (A.7) is replaced by k = g k and xj is by (g /A
j
j
0
0
e
numerator becomes AX ,t k /xj . In the denominator,tjeX1 ,t X1,t , ..., AeX ,t xj , ..., AeX ,t XJ,t ) = FeK (g t k 0 , g t X1,0 , ..., g t x0j , ..., g t XJ,0 )FeK (k, A
jJ= FeK (k 0 , X1,0 , ..., x0j , ..., XJ,0 ),35

We omit condition “X−j,t : const” because Xj,t ’s BGP values, not variables.

A-3

eX ,t ≡ g t Xj,0 /Xj,t and the homogeneity of degree 0
where we used the definition of Aj36
eproperty of the FK (·) function. Similarly,
eX1 ,t X1,t , ..., AeX ,t xj , ..., AeX ,t XJ,t ) = FeX (k 0 , X1,0 , ..., x0 , ..., XJ,0 ).
FeXj (k, AjjjJeX1 ,t X1,t , ..., AeX ,t xj , ..., AeX ,t XJ,t ) =Note that, using the CRS property of Fe(·), condition Fe(k, AjJ
Yt can be simplified as
Fe(g t k 0 , g t X1,0 , ..., g t x0j , ..., g t XJ,0 ) = g t Fe(k 0 , X1,0 , ..., x0j , ..., XJ,0 ) = Yt .
Since Yt = g t Y0 , the condition reduces to Fe(k 0 , X1,0 , ..., x0j , ..., XJ,0 ) = Y0 . The pointof evaluation, k = Kt , becomes g t k 0 = Kt , or k 0 = g −t Kt = K0 . Similarly, xj = Xj,teX ,t Xj,t = Xj,0 . Therefore, (A.7) can be expressed in terms of k 0
becomes x0j = g −t Aj

and x0j as follows:eX ,t k 0 /x0j )d ln(AjσeKXj ,t = −d ln

FeK (k0 ,X1,0 ,...,x0j ,...,XJ,0 )e
AX ,t FeX (k0 ,X1,0 ,...,x0 ,...,XJ,0 )
j

(A.8)

Fe(k0 ,X1,0 ,...,x0j ,...,XJ,0 )=Y0
k0 =K0 ,x0j =Xj,0

j

j

.



eX ,t fixed when calculating the EoS. We canRecall that we keep technology AjeeX ,t k 0 /x0j ) = d(ln(k 0 /x0j ) + ln AeX ,t ) =eliminate AX ,t from the numerator from d ln(A
j

j

j

0

d ln(k /x0j ).

eX ,t in the denominator can also be eliminated (orIn the same way, Ajreplaced by AXj ,0 ≡ 1). Finally, using Fe(·) ≡ F (·; 0), (A.8) can be written asσ
eKXj ,t = −d ln

d ln(k 0 /x0 ) F (k0 ,X ,...,xj0 ,...,XJ,0 ;0)j
FXj (k0 ,X1,0 ,...,x0j ,...,XJ,0 ;0)
K

1,0

.



(A.9)

F (k0 ,X1,0 ,...,x0j ,...,XJ,0 ;0)=Y0
k0 =K0 ,x0j =Xj,0Then, comparing with Definition 3, it turns out that the RHS of (A.9) exactly
matches the definition of σKXj ,0 , evaluated at the period-0 BGP. Since it is assumed
that σKXj ,t does not change over time, we have σ
eKXj ,t = σKXj ,0 = σKXj ,t .
36

For the homogeneity of degree 0 property, see the proof of Proposition 2 in appendix A.3.

A-4

A.5

Proof of Lemma 2

Proof of part a
By substituting K0 , X1,0 , ..., XJ,0 into (10) and then using (11),


Yj ∗
ξ
j
α
F (K0 , X1,0 , ..., XJ,0 ) = Fb K0
Xj,0 , Xj ∗ +1,0 , ..., XJ,0
j=1

=F

Y ∗
j
j=1

!
−1/α 
1/α
∗
Y
j
ξ
ξ
Xj,0j
K0α
Xj,0j
, X1,0 , ..., XJ,0 ; 0
j=1= F (K0 , X1,0 , ..., XJ,0 ; 0) .
Proof of part b
Q∗ξ
Let M0 = K0α jj=1 Xj,0j denote the amount of capital composite m in period 0, andFbM (·) denote the derivative of function Fb(·) with respect to its first argument. By
differentiating both sides of (11) by m with the chain rule and substituting the period0 BGP values for k, xj ∗ +1 , ..., xJ ,FbM (M0 , Xj ∗ +1,0 , ..., XJ,0 ) = FK (K0 , X1,0 , ..., XJ,0 ; 0)
= FK (K0 , X1,0 , ..., XJ,0 ; 0)

Y ∗
j
j=1

ξ
Xj,0j

−1/α

1 (1−α)/α
M
α 0

K0
,
αM0
(A.10)

where the last equality follows from the definition of M0 = K0α

Qj ∗

ξj
j=1 Xj,0 .

Now,consider the case of Z = K. By differentiating both sides of (10) by k with the chain
rule and substituting the period-0 BGP values,
F K (K0 , X1,0 , ..., XJ,0 ) = FbM (M0 , Xj ∗ +1,0 , ..., XJ,0 ) αM0 /K0= FK (K0 , X1,0 , ..., XJ,0 ; 0),A-5where the last equality is from (A.10). Similarly, for the case of Z = Xj , wherej ∈ {1, ..., j ∗ },F Xj (K0 , X1,0 , ..., XJ,0 ) = FbM (M0 , Xj ∗ +1,0 , ..., XJ,0 ) ξj M0 /Xj,0ξj K0= FK (K0 , X1,0 , ..., XJ,0 ; 0).
α Xj,0(A.11)

Note that, from the definitions of α and ξj in (9), ξj /α = sXj ,0 /sK,0 . Therefore,(A.11) becomesFK (K0 , X1,0 , ..., XJ,0 ; 0)FXj (K0 , X1,0 , ..., XJ,0 ; 0)Xj,0 K0= FXj (K0 , X1,0 , ..., XJ,0 ; 0).
FK (K0 , X1,0 , ..., XJ,0 ; 0)K0 Xj,0Finally, consider the case of Z = Xj , where j ∈ {j ∗ + 1, ..., J}. Similarly to theproof of part a, we can confirm that F (K0 , X1,0 , ..., Xj ∗ ,0 , xj ∗ +1 , ...xJ ) = F (K0 , X1,0 , ..., Xj ∗ ,0 , xj ∗ +1 , ..., xJ
for any xj ∗ +1 , ..., xJ . This means that they are identical functions of xj ∗ +1 , ..., xJ ,and have the same derivatives with respect to these variables. Therefore, for j ∈{j ∗ + 1, ..., J}, we have F Xj (K0 , X1,0 , ..., XJ,0 ) = FXj (K0 , X1,0 , ..., XJ,0 ; 0).
Proof of part c
The EoS for function F (·) between capital and factor j, evaluated at the period-0
BGP, is defined asd ln(k/xj )σ KXj ,0 = −d ln

F K (k,X1,0 ,...,xj ,...,XJ,0 )F Xj (k,X1,0 ,...,xj ,...,XJ,0 ),

(A.12)F (k,X1,0 ,...,xj ,...,XJ,0 )=Y0
k=K0 ,xj =Xj,0

where k and xj are variables to be perturbed and Y0 , K0 , X1,0 , ..., XJ,0 are the period-0
BGP values.
Let us first examine σ KXj ,0 for the case of j ∈ {1, ..., j ∗ }. In this case, factors
Xj ∗ +1,0 , ..., XJ,0 are fixed at the BGP values. Using (10), function F (k, X1,0 , ..., xj , ..., XJ,0 )can be written as Fb(m, Xj ∗ +1,0 , ..., XJ,0 ), where m is the amount of capital composite,ξ 0
ξ Q
defined as m = k α xjj j 0 ∈{1,...,j ∗ }\j Xj 0j,0 . Using the chain rule, its derivative with

A-6

respect to k becomes∂ b
F (m, Xj ∗ +1,0 , ..., XJ,0 )∂k∂m= FbM (m, Xj ∗ +1,0 , ..., XJ,0 )∂km= FbM (m, Xj ∗ +1,0 , ..., XJ,0 )α .
k

F K (k, X1,0 , ..., xj , ..., XJ,0 ) =Similarly, FXj (k, X1,0 , ..., xj , ..., XJ,0 ) = FbM (m, Xj ∗ +1,0 , ..., XJ,0 )ξj xmj . Substituting these
into (A.12) gives

σ KXj ,0 = −d ln(k/xj ).
xd ln ξαj kj F (k,X1,0 ,...,xj ,...,XJ,0 )=Y0

(A.13)

k=K0 ,xj =Xj,0Since α and ξj are constant parameters, the denominator can be simplified as d ln ((α/ξj )(xj /k)) =d (ln(α/ξj ) + ln(xj /k)) = d ln(xj /k). Using this, (A.13) gives σ KXj ,0 = 1. Recall that
σKXj ,0 = 1 because j ∈ {1, ..., j ∗ }. Therefore, σ KXj ,0 = σKXj ,0 holds.
Next, we examine σ KXj ,0 for the case of j ∈ {j ∗ + 1, ..., J}. In this case, equations(10) and (11) imply
F (k, X1,0 , ..., xj , ..., XJ,0 ) = F (k, X1,0 , ..., xj , ..., XJ,0 ; 0),for any k > 0 and xj > 0. Therefore, the EoS of function F (k, X1,0 , ..., xj , ..., XJ,0 )between k and xj is identical with that of function F (k, X1,0 , ..., xj , ..., XJ,0 ; 0). Thismeans σ KXj ,0 = σKXj ,0 .

A-7

Proof of part d
Let us first consider the CRS property of function Fb(m, xj ∗ +1 , ..., xJ ). We multiply
every argument by an arbitrary factor of λ > 0. From (11),Fb(λm, λxj ∗ +1 , ..., λxJ )
!
Y ∗
−1/α
j
ξ
= F λ1/α
Xj,0jm1/α , X1,0 , ..., Xj ∗ ,0 , λxj ∗ +1 , ..., λxJ ; 0(A.14)

j=1= λF(1−α)/α

λ

Y ∗
j

j=1

ξ
Xj,0j

−1/α

!

m1/α ,

Xj ∗ ,0X1,0, ...,, xj ∗ +1 , ..., xJ ; 0 ,λλwhere the last equality comes from the CRS property of the period-0 true production
function F (·; 0). (All the arguments are divided by λ.) Our objective it to show thatthe last line of (A.14) coincides with λFb(m, xj ∗ +1 , ..., xJ ). Using (11), this desiredcondition can be written as
F

Y ∗
j

=F

j=1

ξ

−1/α

Xj,0j

λ(1−α)/α!
m1/α , X1,0 , ..., Xj ∗ ,0 , xj ∗ +1 , ..., xJ ; 0

Y ∗
j

j=1

ξ

Xj,0j

−1/α

!
∗
X
X
1,0
j ,0
m1/α ,, ...,, xj ∗ +1 , ..., xJ ; 0 .
λλ(A.15)In the following, we establish this equality by focusing on the isoquants of function
F (·; 0).
Recall that we defined j ∗ such that the period-0 true production function F (k, x1 , x2 , . . . , xJ ; 0)satisfies σKXj = 1 for j = 1, ..., j ∗ . For concreteness, let us focus on capital k andx1 . From Definition 3, σKX1 = 1 means that equation d ln(FK /FX1 )/d ln(k/x1 ) = −1
holds when the output and other inputs are kept constant.37 In other words, thisdifferential equation is satisfied on the isoquant curve in the k-x1 space. Integrating
equation d ln(FK /FX )/d ln(k/x1 ) = −1 gives ln(FK /FX ) = − ln(k/x1 ) + ξe1 , where11ξe1 is a constant of integration. Taking the exponential of the both sides gives
FK /FX1 = (exp ξe1 )(x1 /k).
37

(A.16)To minimize notation we omit the arguments of the functions FK (k, x1 , ..., xJ ; 0) andFX1 (k, x1 , ..., xJ ; 0).

A-8

From the definition of the isoquant curve, the amount of output must be constant:dY = FK dk + FX1 dx1 = 0. Rearranging and using (A.16), we have the slope of theisoquant curve as dx1 /dk = −FK /FX = −(exp ξe1 )(x1 /k). Integrating this differential
1

equation by separation of variables gives ln k = −(1/ exp ξe1 ) ln x1 + ye1 , where ye1 isanother constant of integration.38 By taking the exponential,−1/ exp ξe1

k = (exp ye1 )x1.

(A.17)

Equation (A.17) defines an isoquant curve with two parameters, ye1 and ξe1 . The
value ξe1 can be pinned down by the factor share. Using (A.16), the relative sharebetween k and x1 is written as kFK /x1 FX1 = exp ξe1 . The result does not dependon k or x1 , which means that the relative share is constant on the isoquant curve.
Also, notice that the value of ξe1 must be the same across all isoquant curves, sinceotherwise they intersect with each other, which is impossible by the definition of theisoquant curve. From (9), we know that the relative share in period 0 is α/ξ1 . Using
these, the isoquant curve (A.17) can be written as−ξ /αk = (exp ye1 )x1 1 .

(A.18)The remaining parameter ye1 specifies the location of the isoquant curve. Now,
−1/α
Q ∗
ξj
j
X
m1/α andconsider a particular isoquant curve that goes through k =
j,0
j=1
−1/α
Q ∗
ξj
ξ1 /α
j
m1/α X1,0. From (A.18), we canX
x1 = X1,0 , which means exp ye1 =
j,0
j=1
−1/α
Q ∗
ξj
j
0
ξ1 /α
m1/α
confirm that this isoquant curve also goes through k = λ
j=1 Xj,038This integration can be done by separation of variables. Rearranging the equation dx1 /dk =−(exp ξe1 )(x1 /k), we have (1/k)dk = −(1/ exp ξe1 )(1/x1 )dx1 . Integrating both sides of this equaRRR
tion
separately gives (1/k)dk = −(1/ exp ξe1 ) (1/x1 )dx1 . Since (1/k)dk = ln k + constant andR(1/x1 )dx = ln x1 + constant, we obtain the result in the text.

A-9and x01 = X1,0 /λ.39 Since the output is the same on an isoquant curve, we have
F

Y ∗
j
j=1

=F

λξ1 /α

ξ

−1/α

Xj,0j
Y ∗
j

!
m1/α , X1,0 , ..., Xj ∗ ,0 , xj ∗ +1 , ..., xJ ; 0
ξ

j=1

−1/α

Xj,0j

!
X
1,0
, ..., Xj ∗ ,0 , xj ∗ +1 , ..., xJ ; 0 .
m1/α ,
λBy repeating this operation for j = 2, ..., j ∗ and using(A.19)Pj ∗j=1 ξj = 1 − α from (9), we

obtain (A.15). This establishes the CRS property of function F (m, xj ∗ +1 , ..., xJ ).
Next, we prove the CRS property of function F (k, x1 , ..., xJ ). From (10),


α

Yj ∗

ξj



F (λk, λx1 , ..., λxJ ) = Fb (λk)(λxj ) , λxj ∗ +1,0 , ..., λxJ,0j=1
Yj ∗ ξ
jα= Fb λk
xj , λxj ∗ +1,0 , ..., λxJ,0
j=1

 Y∗
j
ξj
α
b
= λF kxj , xj ∗ +1,0 , ..., xJ,0j=1

= λF (k, x1 , ..., xJ ).
The second equality utilizes α +Pj∗

j=1 ξj = 1 from (9), whereas the third equality isfrom the CRS property of function Fb(·).

A.6

Proof of Proposition 4

Using (10), the RHS of equation (13) can be written as


Yj ∗
α
ξj
b
F (AK,t Kt )(AXj ,t Xj,t ) , AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t .
j=1(A.20)

The first argument of function Fb(·) represent the effective amount of capital composite
on the BGP. It is multiplied by g each period from condition (12). Also, all theother arguments of Fb(·) are multiplied by g each period because it is assumed thatγXj = g/gXj for j ∈ {j ∗ + 1, ..., J}. Since Fb(·) has CRS from property d of Lemma 2,(A.20) is multiplied by g each period.
This can be confirmed by substituting k 0 and x01 into (A.18). It yields the same exp ye1 as in
the previous sentence.
39

A-10

Also, the LHS of (13), Yt , is multiplied by g every period by the definition of the
BGP. In period 0, (13) holds from property a of Lemma 2. Therefore, (13) holds for
all t ≥ 0.

A.7

Proof of Proposition 5

The proof relies on Lemma 2, but otherwise it proceeds similarly to the proof for
Proposition 2. Let us first consider the case of Zt = Kt . Using property b of Lemma
2 and (10), the share of factor K in period 0 can be written as (10),K0
Y0K0= F K (K0 , X1,0 , ..., XJ,0 )
Y0


∗
Y
j
∂ b
K0
ξj
α
F K0
.
=
Xj,0 , Xj ∗ +1,0 , ..., XJ,0
j=1
∂K0
Y0

sK,0 = FK (K0 , X1,0 , ..., XJ,0 ; 0)

(A.21)Let FbM (·) be the derivative of function Fb(·) with respect to its first argument.
Note that, in (A.21), the first argument is the capital composite in period 0, M0 =Q∗ξ
K0α jj=1 Xj,0j . Using the chain rule, (A.21) becomes
dM0 K0αM0sK,0 = FbM (M0 , Xj ∗ +1,0 , ..., XJ,0 )= FbM (M0 , Xj ∗ +1,0 , ..., XJ,0 ), (A.22)
dK0 Y0Y0where the second equality follows from dM0 /dK0 = αM0 /K0 .
Recall that Fb(·) has CRS from Lemma 2, and therefore its derivative FbM (·) is aQ∗homogeneous function of degree 0. Let Mt = (AK,t Kt )α jj=1 (AXj ,t Xj,t ) denote theeffective amount of capital composite in period t. From condition (12), Mt grows by afactor of g every period. The same applies to the effective amounts of factors not in thecapital composite: AXJ ,t Xj,t for j = j ∗ +1, ..., J. Therefore, when we consider function
FbM Mt , AX ∗ ,t Xj ∗ +1,t , ..., AX ,t XJ,t , every argument is multiplied by g every period,
j +1

Jwhich does not change the value of FM (·) over time due to homogeneity of degree 0.
Therefore, (A.22) can be written as αM0
sK,0 = FbM Mt , AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t
.
Y0

A-11

(A.23)

Note that, because Mt and Yt grow at the same speed, the last term can be transformed as αM0 /Y0 = αMt /Yt = (αMt /Kt )(Kt /Yt ). In addition, αMt /Kt in thelatter expression represents dMt /dKt , which can be confirmed by differentiatingQ∗
Mt = (AK,t Kt )α jj=1 (AXj ,t Xj,t ) by Kt . Therefore, (A.23) becomes dMt Kt
sK,0 = FbM Mt , AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,tdKt Yt


Yj ∗
∂ b
Kt
α
=
F (AK,t Kt )(AXj ,t Xj,t ), AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t
j=1
∂Kt
Yt
Kt
∂
F K (AK,t Kt , AX1 ,t X1,t , ..., AXJ ,t XJ,t ) ,=∂Kt
Yt(A.24)
where the second equality uses the chain rule, and the third is from the definition offunction F (·) in (10). Note that the share of capital is the same in period t and 0,
which implies
sK,0 = sK,t = FK (Kt , X1,t , ..., XJ,t ; t)

Kt
.
Yt

(A.25)

By comparing (A.24) with (A.25), we obtain (15) for the case of Zt = Kt . The proofof the proposition for the case of Zt = Xj,t , j ∈ {1, ..., j ∗ } proceeds exactly the same
way as above, with only the modification that Kt is replaced by Xj,t and α by ξj .
Finally, the case of Zt = Xj,t , j ∈ {j ∗ + 1, ..., J}, can be confirmed in a simieX ,t in thelar way as in Proposition 2, because the value of AX ,t is the same as Ajj

Uzawa theorem. In particular, we use Fb(Mt , AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t ) instead ofeX1 ,t X1,t , ..., AeX ,t XJ,t ), and define FbX (·), j ∈ {j ∗ , ..., J}, as the derivative ofFe(Kt , AjJ∗bfunction F (·) with respect to its (j − j + 1)th argument.40 Except for these slight
modifications, the proof proceeds exactly as in Appendix A.3.
40 b
FXj (·) needs to be defined this way because j ∗ arguments are eliminated from function Fb(·) in

definition (11). Also, note that similarly to function Fe(·), function Fb(·) has a CRS property from
Lemma 2.

A-12

A.8

Proof of Proposition 6

Similarly to Definition 3, the EoS σ KXj ,t on the BGP is defined asd ln(k/xj )σ KXj ,t = −d lnAK,t F K (AK,t k,AX1 ,t X1,t ,...,AXj ,t xj ,...,AXJ ,t XJ,t )AXj ,t F Xj (AK,t k,AX1 ,t X1,t ,...,AXj ,t xj ,...,AXJ ,t XJ,t ).


F (AK,t k,AX1 ,t X1,t ,...,AXj ,t xj ,...,AXJ ,t XJ,t )=Yt
k=Kt ,xj =Xj,t(A.26)
where Yt , Kt , X1,t , ..., XJ,t indicate the BGP values, and k and xj are the variables tobe perturbed.41
Let us first consider the case of j ∈ {1, ..., j ∗ }. In this case, factors Xj ∗ +1,t , ..., XJ,t
are fixed at the BGP values. Now, we simplify the denominator of (A.26), particularly
focusing on the fraction inside ln(·). Using (10), function F (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )can be written as Fb(m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t ), where m is the effective amountQof capital composite, m = (AK,t k)α (AXj ,t xj )ξj j 0 ∈{1,...,j ∗ }\j (AXj0 ,t Xj 0 ,0 )ξj0 . Note thatdm/dk = αm/k. Using these properties and the chain rule, we have
AK,t F K (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )∂F (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )
=
∂k
∂ b
=
F (m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t )∂km= FbM (m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t )α .
k

(A.27)

AXj ,t F Xj (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )m= FbM (m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t )ξj .
xj(A.28)

Similarly,

41

In definition (A.26), condition F (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t ) = Yt means thatk and xj are perturbed so that output Yt is unchanged from the BGP value. Condition k =Kt , xj = Xj,t says that the EoS is evaluated at the BGP values. It is also important to keep inmind the notation for derivatives: AK,t F K (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t ) is the partial
derivative of F (·) with respect to k, which corresponds to FK (·; t) in Definition 3.

A-13

Substituting (A.27) and (A.28) into (A.26) gives

σ KXj ,t = −d ln(k/xj ).
xd ln ξαj kj F (AK,t k,AX ,t X1,t ,...,AX ,t xj ,...,AX ,t XJ,t )=Yt
1j(A.29)

J

k=Kt ,xj =Xj,tSince α and ξj are constant parameters, the denominator can be simplified as d ln ((α/ξj )(xj /k)) =d (ln(α/ξj ) + ln(xj /k)) = d ln(xj /k). Using this, (A.29) gives σ KXj ,t = 1. Recall that
σKXj ,0 = 1 because j ∈ {1, ..., j ∗ }, and that σKXj ,t does not change over time on the
BGP. Therefore, σKXj ,t = σKXj ,0 = 1 = σ KXj ,t holds.
Next, we examine σ KXj ,t for the case of j ∈ {j ∗ + 1, ..., J}. Similarly to the proofof Proposition 3, consider a change of variables: k 0 = g −t k and x0j = g −t AXj ,t xj .
Then, k in (A.26) is replaced by k = g t k 0 and xj is replaced by (g t /AXj ,t )x0j . In the
numerator, k/xj becomes AXj ,t k 0 /x0j . In the denominator, by the same operations asin (A.27), AK,t F K (·) can be written asm
FbM (m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )α ,kwhere m = (AK,t k)α

Qj ∗

ξj 0
j 0 =1 (AXj 0 ,t Xj 0 ,0 ) .

(A.30)The definition of m does not include xj∗

because j ∈ {j + 1, ..., J} means that xj is not a part of capital composite. Instead,
AXj ,t xj appears in (A.30) as the (j − j ∗ + 1)th argument of the Fb(·) function. UsingQ∗
xj = (g t /AXj ,t )x0j , AXj ,t xj can be written as g t x0j . Since Mt = (AK,t Kt )α jj=1 (AXj ,t Xj,t )grows by a factor of g every period, the capital composite m can also be written as
m = g t k 0 /Kt
where m0 = (k 0 )α

Qj ∗

α

j 0 =1 (AXj ,0 Xj 0 ,0 ).
0αMt = (k 0 /K0 ) g t M0 = g t m0 ,Other effective factors also grow by a factor of

A-14

g: AXj0 ,t Xj 0 ,t = g t Xj 0 ,0 for j 0 ∈ {j ∗ + 1, ..., J}\j. Using these, (A.30) becomest0gm
FbM (g t m0 , g t Xj ∗ +1,0 , ..., g t x0j , ..., g t XJ,0 )α t 0
gkm0= FbM (m0 , Xj ∗ +1,0 , ..., x0j , ..., XJ,0 )α 0k
∂ b 0
= 0 F (m , Xj ∗ +1,0 , ..., x0j , ..., XJ,0 )∂k= F K (k 0 , X1,0 , ..., x0j , ..., XJ,0 ),where the first equality is from the homogeneity of degree 0 property of the FbM (·)
function, the second equality is from the chain rule and dm0 /dk 0 = αm0 /k 0 , and thelast equality is from the definition of F (·) in (10).
Likewise, AXj ,t F Xj (·) in the denominator of (A.26) can be expressed in terms of0

k and x0j as∂
F (AK,t k, AX1,t ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )∂xj∂ b
F (m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )=∂xj
dAXj ,t xj= FbXj (m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )dxj= FbX (g t m0 , g t Xj ∗ +1,0 , ..., g t x0 , ..., g t XJ,0 )AX ,t
j

j

j

0

= FbXj (m , Xj ∗ +1,0 , ..., x0j , ..., XJ,0 )AXj ,t=∂ b 0
F (m , Xj ∗ +1,0 , ..., x0j , ..., XJ,0 )AXj ,t∂x0j= AXj ,t F Xj (k 0 , X1,0 , ..., x0j , ..., XJ,0 ).
The definition in (A.26) evaluates F̄ (·) and its arguments at their BGP values.42
42

I.e., the conditions that are written to the right of “|”.

A-15

We also need to re-write these conditions in terms of their period-0 values. Note that
F (AK,t k,AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )= Fb(m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t )= Fb(g t m0 , g t Xj ∗ +1,0 , ..., g t x0j , ..., g t XJ,0 )AXj ,t= g t Fb(m0 , Xj ∗ +1,0 , ..., x0j , ..., XJ,0 )AXj ,t= g t F (k 0 , X1,0 , ..., x0j , ..., XJ,0 ).
Therefore, condition F (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t ) = Yt can be substituted by
F (k 0 , X1,0 , ..., x0j , ..., XJ,0 ) = Y0 .
The point of evaluation, k = Kt , becomes g t k 0 = Kt , or k 0 = g −t Kt = K0 . Similarly,
xj = Xj,t becomes x0j = g −t AXj ,t Xj,t = Xj,0 . Using all these results, (A.26) can be
expressed in terms of k 0 and x0j as follows:d ln(AXj ,t k 0 /x0j )σ KXj ,t = −d ln

F K (k0 ,X1,0 ,...,x0j ,...,XJ,0 )AXj ,t F Xj (k0 ,X1,0 ,...,x0j ,...,XJ,0 ).



(A.31)

F (k0 ,X1,0 ,...,x0j ,...,XJ,0 )=Y0
k0 =K0 ,x0j =Xj,0

We can eliminate constant AXj ,t from the numerator because d ln(AXj ,t k 0 /x0j ) =d(ln(k 0 /x0j ) + ln AXj ,t ) = d ln(k 0 /x0j ). In the same way, AXj ,t in the denominator canalso be eliminated. Also, recall that AK,0 = AXj ,0 = 1. Then, comparing (A.31) with
(A.26), it turns out that the RHS of (A.31) coincides with σ KXj ,0 . From Lemma 2,
σ KXj ,0 = σKXj ,0 holds in period 0. In addition, it is assumed that σKXj ,0 does notchange over time. Therefore, σ KXj ,t = σ KXj ,0 = σKXj ,0 = σKXj ,t .

A.9

Proof of Lemma 3

Note that δ = (δ IST + gq − 1)/gq means 1 − δ IST = (1 − δ)gq = (1 − δ)qt /qt−1 .
ISTDividing equation (30) by qt and using the above result, we have Kt+1 = Kt+1
/qt =(Yt − Ct − Rt ) + (1 − δ IST )KtIST /qt = (Yt − Ct − Rt ) + (1 − δ)(qt /qt−1 )KtIST /qt =(Yt − Ct − Rt ) + (1 − δ)Kt , which coincides with (2).
A-16

Using KtIST = qt−1 Kt , production function (29) can be written as Yt =F IST (qt−1 Kt , X1,t , ..., XJ,t ; t).

It is a CRS function of Kt , X1,t , ..., XJ,t and de-pends on time both through the shape of F IST (·; t) and through the growth of qt .
Therefore, we can define a new production function (31), F (Kt , X1,t , ..., XJ,t ; t) ≡F IST (qt−1 Kt , X1,t , ..., XJ,t ; t), where dependence of F (·; t) on t includes the effect from
qt−1 . From the assumptions on F IST (·), function F (·; t) obviously satisfies the required
marginal product properties in (1).

B

Calibration Details

B.1

Deriving the Elasticity of Substitution in a Three-Factor
CES function (24)In this proof, we write (22) as Yt = Ft (Kt , Lt , Xt ) and drop time subscripts forconvenience.43 Also, in Appendix B, we use FK , FL and FX to denote the derivativeof F (K, L, X) with respect to K, L and X, respectively. From Definition 3, theelasticity of substitution (EoS) between capital and labor for function F (K, L, X) is
defined as
σKL = −d ln(K/L).
d ln(FK /FL ) Y,X: const(B.1)

Total differentiation of Y = F (K, L, X) gives
FK dK + FX dX + FL dL = dY.
Noting that σKL is defined with the constraint that X and Y are kept constant(dX = dY = 0), the latter can be re-written as
dL = −

FK
dK.
FL

43

(B.2)

Function (22) depends also on technologies AK,t , AL,t , and AX,t . This dependence is included
in Ft , which changes with time. However, because we are calculating the elasticity of substitutionof production function at a given t, technologies are fixed in this calculation. Therefore, we omit t
from Ft for simplicity.

A-17

Therefore, the numerator of (B.1) can be written as

d ln

K
L


= d (ln K − ln L)
1
1 FK
+
dK
K L FL
KFK dK
=
1+
.
LFL
K
=

Since we assume perfect competition, KFK /LFL is the ratio of capital share sK tolabor share sL . Therefore,

d ln

K
L




=

sK
+1
sL



dK
.
K

(B.3)

Next, we calculate the denominator. By taking the partial derivatives in (22), theratio of marginal products turns out to be
−1

1

−1

FK
ηKX α (AL L)  (AK K)α  (AX X)(1−α) =.
FLηL AL K

(B.4)

Again noting that the EoS is defined for dX = dY = 0, the total differentiation of
the log of (B.4) becomes

d ln

FK
FL


=
=
=
=
=

n
o
1
α −1
−1


d ln L + ln K
1 dL
 − 1 dK
− 1−α
 L

K


1 FK
 − 1 dK
dK − 1 − α
−
L FL

K


1 KFK
 − 1 dK
−
+1−α
 LFL

K



sK
1
dK
−
+α
+ (1 − α).
sLKSubstituting (B.3) and (B.5) into (B.1) gives (24) in the main text.

A-18

(B.5)

B.2

Factor shares and distribution parameters (26)The partial derivatives of (22) with respect to Lt and Kt are:
−1

1


FL,t = ηL (AL,t Lt )  L−1
t Yt

(B.6)

1

FK,t = ηKX α (AK,t Kt )α (AX,t X)1−α Kt−1 Yt  .

(B.7)

With perfect competition, factors shares in period t are given by sL,t = Lt FL,t /Yt andsK,t = Kt FK,t /Yt . First order conditions (B.6) and (B.7) implies that their values are −1
AL,t Lt 
sL,t = ηL
Yt
"
# −1

(AK,t Kt )α (AX,t X)1−αsK,t = αηKX
.
Yt


(B.8)
(B.9)

Note that the above result holds even when the economy is not on the BGP (sL,t canbe different from its BGP value sL ).
We calibrate the model to a starting period t = 0 when the economy is on a BGP.
From (22), AK,0 = AX,0 = AL,0 = 1. Also, sL,0 = sL and sK,0 = sK . Then, evaluating(B.8) and (B.9) at t = 0 gives equation (26) in the main text.

B.3

Deviation of the labor share from the BGP value

In this section, we derive (27), the labor share when the capital-augmenting technology aK,t deviates from its BGP path AK,t . As explained in Subsection 7.2, we assume
that other technologies and inputs (AX,t , AL,t , Kt , Xt and Lt ) follow the BGP path.
Then, from (22), the output deviates from the BGP value Yt and now becomes

n
o −1
 −1
−1
α
1−α 

yt = ηKX (aK,t Kt ) (AX,t Xt )+ ηL (AL,t Lt ).

A-19

(B.10)

Substituting (B.10) into (B.8) gives the labor share when aK,t deviates from AK,t .
−1ηL (AL,t Lt ) sL,t =

−1

−1

ηKX ((aK,t Kt )α (AX,t Xt )1−α )  + ηL (AL,t Lt ) !−1 −1 α −1ηKX (AK,t Kt )α (AX,t Xt )1−α 
aK,t
=
+1
.
ηL
AL,t Lt
AK,t(B.11)

In contrast, if all variables, including AK,t , follow the BGP, dividing (B.9) by (B.8)
gives
ηKX
ηL



(AK,t Kt )α (AX,t Xt )1−αAL,t Lt

 −1

=

sK /α
,
sL

(B.12)

where sK and sL are the shares on the BGP. Note that the LHS of (B.12) is the sameas the first term in the last line of (B.11). Note also that sK /α = 1 − sL since α
is defined as sK /(sK + sX ) = sK /(1 − sL ). Hence, the RHS of (B.12) is (1/sL ) − 1.
Therefore, by substituting (B.12) into (B.11), we obtain (27) in Section 7.2.

C

Mapping data on investment prices to Kaugmenting technical change in the model

In this appendix, we show how changes in the relative price of investment in the
NIPA statistics can be used to measure capital-augmenting technical change in the
neoclassical growth model. In Section C.1, we explain the mapping between the
NIPA data and the standard neoclassical growth model, as defined by Definition 1.
In Section C.2, we explain how the result changes when the data is mapped to aneoclassical growth model with investment-specific technical change, as discussed in
Section 8. Both results follow from the property that capital can be accumulated
linearly from the previous period’s output, specified either by (2) or (30).

C.1

Relative Price of Investment in the NIPA Statistics Interpreted in the Standard Neoclassical Growth ModelIn this section, we prove that the speed of capital-augmenting technological change in
the neoclassical growth model can be measured by the rate of decline in the relative
A-20

investment price in the NIPA statistics with a one-period lag. This result will be given
in equation (C.4), which is a basis for the discussion in Section 2. We then use thisresult to show how to measure the deviation of the capital-augmenting technologicalchange from its BGP path, which leads to equation (28) in Section 7.2.
We start by discussing the relative prices of investment and consumption in themodel. The unit of the consumption good is defined so that the period utility is atime-invariant function of Ct /Lt . This implies that one unit of consumption in themodel always increases the argument of the utility function by one. The process ofcapital accumulation (2) implies that one unit of the final good can be used as either
one unit of the consumption good or one unit of the investment good, implying thatthe price of the investment good is also equal to one. In addition, one unit of theinvestment good in period t becomes one unit of capital in period t + 1. Therefore,
Kt+1 is measured in the same units as Ct . One unit of investment in period t will
increase effective capital in t + 1 by AK,t+1 efficiency units.
Now, we explain the mapping from relative prices in the NIPA statistics to variables in the model. First, we consider the mapping for the consumption good. In the
NIPA statistics, the prices and quantities are adjusted for quality changes. Therefore, any improvement in the quality of consumption goods shows up as an increasein quantity. When the statistics are mapped to the model, this means that one unitof consumption in the NIPA statistics always increases the argument of the utility
function by the same amount. Let this amount be χC . Since one unit of consumption
in the model increases the argument of the same utility function by one, one unit ofconsumption in the NIPA statistics should equal χC units of consumption in the neoNIPA
classical growth model. Next, let PC,tbe the dollar price of one unit of consumption

in the NIPA statistics, and Pt$ be the dollar price of one unit of consumption in the
neoclassical growth model. Then, since one unit of consumption in the NIPA is χCtimes that in the model, we have
NIPA
PC,t
= Pt$ χC .

(C.1)

Similarly, we can derive the mapping for the investment good. The NIPA statistics are quality-adjusted also for investment goods. So, as before, any quality improvements will appear as quantity increases. As a result, one unit of investment in
A-21

the statistics should increase the next period’s effective capital by a time-invariant
amount, which we denote with χI . Recall that one unit of the investment good in period t in the model increases the next period’s effective capital by AK,t+1 . Therefore,
one unit of investment in NIPA should be the same as χI /AK,t+1 units of investmentin the model. Since the price of investment is the same as the price of consumption
in the model, the dollar price of investment in the model is given by Pt$ . Then, since
one unit of investment in NIPA is χI /AK,t+1 times that in the model, the dollar priceof one unit of investment in the NIPA statistics is
NIPA
=
PI,t

Pt$ χI.
AK,t+1(C.2)

Combining (C.1) and (C.2), the relative price of investment from the NIPA data,
denoted by RP It , can be mapped to the model by
NIPA
PC,t
χC
RP It = NIPA =.
χI AK,t+1
PI,t

(C.3)

Because χC and χI do not depend on time,
AK,t+1
=
AK,t



RP It
RP It−1

−1
.

(C.4)

The above equation means that the speed of capital-augmenting technological change
in the neoclassical growth model is given by the speed of the fall in the relative investment price in the NIPA statistic with a one-period lag. This property is mentioned
in Section 2. If the economy is on a BGP, equation (C.4) implies that RP It changes
at a constant rate gRP I = 1/γK . This is the BGP path for RP It .
Now, let us consider the situation in which capital-augmenting technical change
deviates from its BGP value. Below, we use AK,t to represent the BGP value, andaK,t to denote the value of capital-augmenting technical change at time t, whether
or not it is on the BGP. Our analysis that leads to (C.3) does not assume that theeconomy is on the BGP. Therefore, the relative price of investment in this case canalso be given byrpit = (χC /χI )A-22

1
aK,t+1,(C.5)

where rpit represents the relative price of investment that would be observed even
if the level of capital augmenting technical change, aK,t , deviates from its balanced
growth level, AK,t . Dividing (C.5) by (C.3) and substituting t by t − 1 gives (28).

C.2

Relative Price of Investment Interpreted in the Modelwith Investment-Specific Technological Change

In the neoclassical growth model with IST, as shown in (30), one unit of consumption
IST
in period t + 1.
good Ct in period t can be converted to qt units of capital Kt+1

(Similarly to Lemma 3, we put superscript IST on capital to show that it is now
measured in different units than in the standard neoclassical growth model). So, qt
represents investment-specific technology in period t, which may improve over time.
IST
at period t is 1/qt . Suppose that, in
Then, the price of investing in one unit of Kt+1

addition the IST, there is also capital-augmenting technology AIST
K,t+1 in period t + 1
production. Then, one unit of investment in period t will increase effective capital int + 1 by AIST
K,t+1 efficiency units.
Now, we explain how the relative price of investment in the NIPA statistics canbe represented by variables in the model with IST. Since there is no change in thedefinition of consumption goods, the dollar price of consumption goods in the NIPA
can still be represented by (C.1). As noted in the previous subsection, the NIPA
statistics are also quality-adjusted for investment goods, and one unit of investmentin the statistics should increase the next period’s effective capital by a time-invariant
amount. Let us denote this by χIST
I . As shown above, one unit of the investment
good in period t in the model increases the next period’s effective capital by AIST
K,t+1 .
IST
Therefore, one unit of investment in NIPA should be the same as χISTI /AK,t+1 units

of investment in the model. Since the price of investment is 1/qt times that of theconsumption in the model, the dollar price of investment in the model is given by
IST
Pt$ /qt . Then, since one unit of investment in the NIPA is χISTI /AK,t+1 times that in

the model, the dollar price of one unit of investment in the NIPA statistics can be
represented as
NIPA
PI,t
=

$
χIST
Pt$
χIST
I
I Pt
·
=
.
qt
qt AK,t+1
AIST
K,t+1

(C.6)

Combining (C.1) and (C.6), the relative price of investment in the NIPA now
A-23becomes
RP It =

NIPA
PC,t
χC
= IST
.
NIPA
PI,t
χI qt AIST
K,t+1

(C.7)

Because χC and χIST
do not depend on time,
I
AIST
K,t+1
· IST =
qt−1 AK,t
qt



RP It
RP It−1

−1
.

(C.8)

When the model includes IST, the rate of decline in the relative price of investmentin the NIPA statistics in period t is the sum of the rate of investment-specific technological change in period t and the rate of capital-augmenting technological change inperiod t + 1. In the literature on IST, qt /qt−1 is called the rate of embodied technoIST
logical change, while AIST
K,t+1 /AK,t is that of disembodied technological change. Bothtechnologies improve the process in which saved output in period t contributes to the
production t + 1. The sum of both improvements shows up as the fall in the relative
price of investment in the NIPA statistic. In effect, the model with IST divides thisprocess into two parts. When we define the combination of the two technology as
AK,t+1 = qt AIST
K,t+1 in (C.7) and (C.8), then we are back to the standard neoclassicalgrowth model and obtain (C.3) and (C.4).

D

Data Sources and Definitions

Most data are originally from the Bureau of Economic Analysis (BEA) and retrieved
from FRED, Federal Reserve Bank of St. Louis, https://fred.stlouisfed.org.
We reference these series by their codes in FRED. Energy use data are from theEnergy Information Administration (2023).

D.1

Figure 1

Panel (a) show several real aggregate variables per capita for the United States.
• Real GDP (GDPCA), real investment (GPDICA), and real personal consumption expenditures (PCECCA) are originally from the National Income and Product Accounts (NIPA) collected by Bureau of Economic Analysis(BEA).
A-24

• The nominal capital stock is the current cost net stock of fixed assets(K1TTOTL1ES000). It is taken from the fixed asset tables collected by the
BEA.
• The GDP deflator (A191RD3A086NBEA) is set to 100 in 2012. It is originallyfrom NIPA.
• The real capital stock is the ratio of the nominal capital stock and the GDP
deflator.
• Population (B230RC0A052NBEA) includes resident population and armed
force overseas. It is originally from NIPA.
• Real variables per capita are calculated by dividing the relevant aggregate
quantity by population.
Panel (b) shows the relative price of investment and consumption. All data are
originally from NIPA and indexed to be equal to 100 in 2012.
• We use three different price deflators for gross private investment: all(A006RD3A086NBEA), non-residential (A008RD3A086NBEA), and equipment(Y033RD3A086NBEA).
• The price deflator for consumption (DPCERD3A086NBEA) covers all personal consumption expenditures.
• To measure the relative price of investment, we divide the price deflator forinvestment by the price deflator for consumption.

D.2

Calibration in Section 7• The data for real GDP, population and the relative price of investment
are the same as shown in Section D.1.
• Nominal GDP (GDPA) is from NIPA.
• The relative price of investment comes from Figure 1 and cover all investment.
A-25

• Labor compensation compensation of employees (A033RC1A027NBEA) plus
proprietors’ income with inventory valuation and capital consumption adjustments (A041RC1A027NBEA). Both series are originally from NIPA.
• The labor share of income is the ratio of labor compensation and nominal
GDP.
• The capital-output ratio is the ratio of nominal capital and nominal GDP.
• Total fossil fuel energy use is the total consumption of fossil fuels measuredin BTUs. It is taken from the Energy Information Administration (EIA)’s
Annual Review. See ‘Total Fossil Fuels Consumption’ in Section 1.1 of https://www.eia.gov/totalenergy/data/annual/.

D.3

Figure 2• Panel (a) uses data on the relative price of investment from Section D.2.
• Panel (b) uses the labor share of income measure from Section D.2.

A-26