ISSN (Print) 0473-453X

Discussion Paper No. 1157

ISSN (Online) 2435-0982

A Generalized Uzawa Growth Theorem
and Capital-Augmenting
Technological Change
Gregory Casey
Ryo Horii

January 2022

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

A Generalized Uzawa Growth Theorem andCapital-Augmenting Technological Change∗
Gregory Casey† Ryo Horii‡
January 2022

Abstract
We prove a generalized, multi-factor version of the Uzawa steady-state growth theorem. The theorem implies that neoclassical growth models need at least three factorsof production to be consistent with empirical evidence on both the capital-labor elasticity of substitution and the existence of capital-augmenting technical change. We alsobuild and calibrate a three-factor endogenous growth model with directed technicalchange and show that it converges to a balanced growth path that is consistent withthe empirical evidence. Our results indicate that natural resources including land anddirected technical change play a central role in explaining balanced growth.

Keywords: Balanced Growth, Uzawa Steady-State Growth Theorem, Directed Technical
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.” The authors are grateful to Been-Lon
Chen, Oded Galor, Andreas Irmen, Cecilia Garcia-Peñalosa, Dean Scrimgeour, 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(GRIPS), Shiga University, Society for Economic Dynamics, Osaka School of International Public Policy(OSIPP), 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1Introduction

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 changemust be labor-augmenting, unless the aggregate production function is Cobb-Douglas (Jones,
2005; Jones and Scrimgeour, 2008). This creates a significant problem for the neoclassicalgrowth model, because data from the United States strongly suggest that (i) there is capitalaugmenting technical change on the BGP and (ii) the aggregate production function is not
Cobb-Douglas (see, e.g., Antras et al., 2004; Oberfield and Raval, 2014; Grossman et al.,2017).
The standard neoclassical growth model assumes that there are only two factors of production, labor and reproducible capital. In reality, there are many other factors of 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 model in that they cannot bereadily accumulated (or reproduced) through savings. In this paper, we examine whether
incorporating more factors of production makes it possible for neoclassical models to beconsistent with the empirical regularities mentioned above.
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 technological change. 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, suggesting that it can be useful for some economic analyses. But, thisparticular representation cannot match evidence on capital-augmenting technical change,
which is observed in the data. If we interpret this representation as the true production
function, the puzzle arises as discussed above. By distinguishing the true production function
and the Uzawa representation, however, our approach does not rule out the existence of otherrepresentations that might have more accurate descriptions of technological change.
To identify other possible representations that are a better match with data, we prove2

a further generalized version of the Uzawa theorem. The generalized theorem demonstrates
that there are a continuum of representations with capital-augmenting technical change, aslong as reproducible capital has a unitary EoS with at least one other factor. From thisbroader class of Factor-Augmenting Representations, it is possible to choose a representationthat matches the empirically observed speed of capital-augmenting technological progress.
In other words, the factor-augmenting representations can be simultaneously consistent withbalanced growth, a non-unitary EoS between capital and labor, and capital-augmenting
technical change. We also provide conditions under which these Factor-Augmenting Representations have the same derivatives and EoS as the true production function. Thus, these
representations are more suitable for economic analyses than standard specifications thatare based on the neoclassical growth model.
To demonstrate the practical importance of theorems, we formulate and analyze a model
economy with three factors of production and endogenous directed technical change. We
provide a micro foundation for the model such that the resultant aggregate production function has the form of a Factor-Augmenting Representation that fits the Generalized Uzawa
theorem. The model exhibits a non-unitary EoS between capital and labor, but nonethelesshas a BGP with a positive rate of capital-augmenting technical change, a result that is notpossible with existing models.1 We also calibrate the model to U.S. data and numericallyshow that the economy converges to the BGP from a wide range of initial conditions. The
global stability result has a particular theoretical importance, because it demonstrates thatthe direction of technological change endogenously conforms to a required condition embedded in the generalized Uzawa theorem. If technological change were exogenous, a knife-edge
condition would be required. In this sense, the model suggests that endogenous directed
technical change is essential in explaining balanced growth in neoclassical models.
Existing work on the Uzawa theorem tends to emphasize its restrictive nature, interpreting the theorem as giving the only possible form of technological progress in the neoclassicalgrowth model. In contrast, our results should be useful in applied macroeconomic research.
The Generalized Uzawa theorem provides guidance on how to choose functional forms for theaggregate production function and technical change that together are a good representationof the true production function along the BGP. In particular, we provide representations
that, along the BGP, have the correct speeds of technological progress, levels of inputs andoutput, first order derivatives and elasticities of substitution between factors of production.
Our results are relevant for a wide range of studies. Growth-, development-, and business1

Grossman et al. (2017) is the sole exception, which we discuss in more detail below.

3cycle accounting analyses all highlight the need to distinguish between different types of technology when describing macroeconomic outcomes (e.g., Greenwood et al., 1997; Krusell et al.,
2000; Fisher, 2006; Caselli and Feyrer, 2007; Hsieh and Klenow, 2010). Our results allow
such analyses to be conducted with production functions that are simultaneously consistent
with balanced growth facts, capital-augmenting technical change, and the EoS between capital and labor. Previous approaches can only be consistent with two out of these three. Thisis especially important for understanding the relationship between technology and inequality, because labor- and capital-augmenting technical change have different impacts on factor
shares when the EoS is different than one. For example, Karabarbounis and Neiman (2014)
suggest that capital augmenting technical change is responsible for the decline in the labor
share in many countries over the last several decades. On a separate note, we find that thethree factor model with endogenous and directed technical change converges slowly to thesteady state, when compared to standard neoclassical growth models. This is because of theslow dynamics of technology and the existence of fixed factors of production. The transition
dynamics in the standard neoclassical model are too fast to explain patterns observed indata (King and Rebelo, 1993; Banerjee and Moll, 2010). These slow dynamics may also haveimportant implications for transition dynamics following exogenous shocks (Leon-Ledesma
and Satchi, 2019).
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 matchthe 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 versionof theorem that stresses the difference between the true production function and representations of the production function. Third, we derive conditions 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 theorem. These studies frequently use models
with directed technical change. In particular, Acemoglu (2003) and Irmen and Tabaković

4

(2017) provide models where capital-augmenting technical change disappears in the longrun, while Jones (2005) and Leon-Ledesma and Satchi (2019) specify models that are CobbDouglas in the long-run. We build on these works by presenting an endogenous growthmodel that converges to a BGP that is consistent with data on both the existence of capitalaugmenting technical change and the less-than-unitary EoS between capital and labor. Ourmodel specification builds on the work of Irmen (2017) and Irmen and Tabaković (2017).
To the best of our knowledge, Grossman et al. (2017) provide the only other attempt
to square the Uzawa steady state growth theorem with data on the EoS and the capitalaugmenting technical change.

In their model, schooling is both labor-augmenting andcapital-dis-augmenting. In this setting, they show that there is a scope for additional capitalaugmenting technological change. Our results indicate that there is a wider scope for ways inwhich the neoclassical growth model can be made to be consistent with the data on EoS andcapital-augmenting technical change. Indeed, their results can be understood as a particularcase 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) or climate change (e.g.,
Golosov et al., 2014). Our results suggest a much broader importance of non-accumulable
factors: they must be incorporated into models of economic growth in order to recreate thebalanced growth facts that originally motivated aggregate growth modeling (Solow, 1956,
1994). As noted above, our results also indicate that directed technical change, as in Acemoglu (2002), is essential to explaining balanced growth. There is a growing literature thatcombines directed technical change and natural resources to ask resource-related questions(e.g., Acemoglu et al., 2012; André and Smulders, 2014; Hassler et al., forthcoming).2 Our
results suggests that these models could serve as the basis for a much wider set of analyses.
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 theorem, proving that neoclassical models canhave a positive rate of capital-augmenting on the BGP. Section 5 presents three applicationsof these results, focusing on simple cases and existing literature. Section 6 introduces a2

See also, Smulders and De Nooij (2003), Di Maria and Valente (2008), and Lemoine (2020).

5

full economic model with endogenous direction of technological change. Section 7 calibrates
the model to the U.S. economy and demonstrates that the model economy endogenously
converges to a BGP with positive capital-augmenting technical change. Section 8 discusses
the broader implications of our findings.

2

Empirical Motivation

The neoclassical growth model, first developed by Solow (1956) and Swan (1956), is thecentral building block of much contemporary research in economic growth. Such models aredesigned 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 a constant rate overlong periods of time. Panel (a) in figure 1 presents U.S. data from 1950-2012, which clearly
demonstrate this pattern.3 It also demonstrates that other macroeconomics aggregates havegrown at similar rates to GDP, capturing the notion of balance.
To explain these facts, the neoclassical growth model focuses on an aggregate production
function that has constant returns to scale (CRS) in two factors, reproducible capital andlabor.4 The ability of the neoclassical growth model to provide a simple explanation for thesefacts has led to its widespread adoption (Jones and Romer, 2010). The model, however, relies
on some strong assumptions, including those described by the Uzawa (1961) steady-state
growth theorem. As conventionally stated, the Uzawa theorem requires that on a BGP, alltechnological progress must be labor-augmenting, unless the aggregate production function
is Cobb-Douglas.
Given the restrictive nature of these conditions for balanced growth, it is natural to
ask whether they are consistent with data. A long literature has estimated the elasticity
between capital and labor in a two-factor production function and rejected the Cobb-Douglas
specification. Most of papers in the literature argue that the elasticity is less than one. For
example, Oberfield and Raval (2014) estimate the macro elasticity of around 0.7 using firmlevel micro data, and Antras et al. (2004) estimates an elasticity of 0.6 directly from macro
time-series data.
Panel (b) of Figure 1 demonstrates that the relative price of investment goods has been
3

See Papell and Prodan (2014), Jones (2016), and others for longer time series and data from othercountries.
4
Intuitively, the key to explaining balanced growth is that capital is reproducible (i.e., it is accumulated
from saved output). Thus, capital ‘inherits’ the constant growth rate of output, implying that the capitaloutput ratio will be constant in the long-run (Jones and Scrimgeour, 2008). Their joint growth rate is thendetermined by population growth and technological progress.

6

120

Price Index (1960=100)

Billions of $2012 (log scale)

100,000
10,000
1,000
100
10
C
1

1960

1970

1980

I

K
1990

80
60
40
20

Y
2000

100

All
2010

0
1960

2020

(a) BGP1970

Non-Residential
1980

1990

Equipment
2000

2010

2020

(b) K-Augmenting Technology

Figure 1: Balanced Growth with capital-augmenting technical change. This figure presents some of themain features of balanced growth in the United States. Panel (a) demonstrates that real output, investment,
consumption and the capital stock have grown at roughly constant rates over long periods of time. Theseempirical patterns summarize the notion of balanced growth. Panel (b) demonstrates that the price ofinvestment goods, and equipment in particular, have been falling relative to the price of consumption goodsin the United States. This indicates capital-augmenting technical change has been occurring along the BGP.
See appendix B for details on data sources.

falling in the United States. This is a type of capital-augmenting technical change. Intuitively, in a setting with perfect competition, decreases in the relative price of capital goods
reflect improvement in the efficiency of the investment goods sector. As a result, one unit of
the final (consumption) good can be transformed into increasingly more units of effective capital goods. In standard neoclassical growth models, however, we typically choose to measure
the quantity of capital in the same units as output. The declining relative price of capital
goods, therefore, should imply that the efficiency of a given unit of capital has increased.5 A
long literature demonstrates that declining investment prices are a quantitatively importantsource of growth in the United States (e.g., Greenwood et al., 1997; Krusell et al., 2000).6
As a result, there is broad consensus that capital-augmenting technical change has been
pervasive in the United States over at least the last half a century, even as the economy
5

Depending on how we measure the amount of capital, the declining relative price of capital goods canalso be interpreted as investment-specific technological change (IST). Lemma 1 in Section 3 formalizes thenotion that these two types of technological change are equivalent to each other. This result is also discussed
by Grossman et al. (2017), who demonstrate this equivalence in a proof of the Uzawa steady-state growththeorem.
6
See He et al. (2008) and Maliar and Maliar (2011) for discussions of the Uzawa steady state growththeorem in this context. Karabarbounis and Neiman (2014) show that declining investment prices are awidespread phenomenon in cross-country data. They use this fact to estimate the EoS between capital andlabor and find a value that is greater than one.

7

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 capitaland labor. Definitions and methodologies vary. Natural Resources in Caselli and Feyrer (2007) encompasses
land, energy, and other materials.

exhibited signs of balanced growth.
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 growth model that is
consistent with balanced growth is necessarily at odds with evidence on capital-augmenting
technical change. Put differently, the standard neoclassical growth model cannot explain the
broader set of stylized growth facts that we observe in the United States.
In this paper, we examine production functions with additional factors of production,
beyond reproducible capital and labor. It obvious that other factors – such as land, energy,
and other natural resources – exist in the production process. Table 1 collects some evidence
on the importance of these factors in the United States.7 Broadly speaking, estimates suggest that non-reproducible factors other than labor account for about 8-9% of total factor
payments. In particular, Caselli and Feyrer (2007) estimate the factor share of all natural
resources and find 8%. Meanwhile, work in climate economics investigates the factor shareof energy only, disregarding land. Golosov et al. (2014) summarize this evidence as pointing
to a 4% share for energy. Finally, Valentinyi and Herrendorf (2008) focus only on land andfind a share of 5%.

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 growthrequires the shape of the production function to change over time, which economists usuallycall technological change. Given the existence of a BGP, the Uzawa theorem provides aconvenient representation of the evolution of the production function. We stress the im7

Estimating factor shares for inputs other than labor is notoriously difficult and often requires structural
assumptions. Our intention is not to endorse any particular estimate. Instead, we simply note that there is
ample evidence that factors other than labor and reproducible capital play a non-negligible role in production.

8

portance 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 setof 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.
Neoclassical Growth Model
The Uzawa theorem depends on two assumptions: (a) the economy is described by a neoclassical growth model, and (b) the model has a balanced growth path (BGP). We start witha description of a neoclassical growth model, which is defined broadly to incorporate a wide
range of dynamic macroeconomic models. For readability and consistency with the following
sections, we consider a discrete time setting, where t = 0, 1, 2, . . ., but it is straightforward
to consider the continuous-time equivalents of the results.
Definition 1. A multi-factor neoclassical growth model is an economic environmentthat satisfies:
1. Output, Yt , is produced from capital, Kt , and J ≥ 1 kinds of other inputs, {Xj,t }Jj=1 :8,9
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 , andeach input has a positive and diminishing marginal product.
2. The amount of capital, Kt , evolves according toKt+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 (e.g., R&D inputs), and δ ∈ [0, 1] is the depreciation rate. The term
Yt − Ct − Rt on the RHS represents physical capital investment.
8

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).
9
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.

9

There are a number of points to note regarding Definition 1. First, production function
F (·; t) in (1) depends on t, capturing technological progress. Importantly, we place norestrictions on how the shape of F (·; t) changes over time. As discussed below, the Uzawa
theorem provides insight about how to approximate the time dependence of F (·; t) withstandard 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 the definition of aneoclassical growth model in Schlicht (2006) and Jones and Scrimgeour (2008), who provide
a simple statement and proof of the two-factor Uzawa theorem.
Third, we allow for the Rt term in (2). If we set Rt = 0, equation (2) is in line withthe previous definitions of the Uzawa theorem. This generalization is not essential for the
proof of the Uzawa Theorem, but it accommodates the possibility of endogenous growth,which we examine in later sections. In production function (1), any technological change iscaptured by the last t term. If we think technology can be affected by the R&D expenditure,then such expenditure would be included in Rt in (2). Similarly, the evolution of factors
Xj,t , including population Lt , can be either exogenous or dependent on particular types of
expenditure, such as child-raising costs. Such costs are also included in Rt .
Fourth, the only reason why capital, Kt , is distinguished from other production factors
X1,t , ..., XJ,t is that we explicitly specify its accumulation process as in (2), which guarantees
that Kt can be accumulated linearly with the output. From theoretical viewpoint, Kt needs
not to be limited to physical capital.10 We will show that the Uzawa theorem holds regardlessof the evolution process for other inputs. The Xj,t ’s can be either endogenous or exogenous.
Lastly, we measure the amount of capital by its value in terms of final output. Specifically,equation (2) implicitly normalizes the unit of period t + 1 capital so that period t final
output can always be converted to the same units of period t + 1 capital. Note that this ismerely a choice of units, and therefore should not limit the applicability of our results. For
example, Greenwood et al. (1997) and Grossman et al. (2017) consider investment-specifictechnological change, which enables more capital to be produced from a unit of final output.
The following Lemma shows that, by change of variables, Definition 1 can accommodate
such a case.
10

It can be any combination of factors that can be accumulated linearly with the output. 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 .

10

Lemma 1. Consider an economic environment where physical capital K̆t accumulates according toK̆t+1 = (Yt − Ct − Rt )qt+1 + (1 − δ̆)K̆t ,(3)

where qt > 0 is the investment-specific technology. The production function is given by
Yt = F̆ (K̆t , X1,t , ..., XJ,t ; t), which has CRS and positive and diminishing returns to all inputs.
If the growth factor of investment-specific technological change gq = qt+1 /qt is constant,11
this environment fits Definition 1 through a change of variables of Kt ≡ K̆t /qt and δ =(δ̆ + gq − 1)/gq .
Proof. See Appendix A.2.
The change of variables effectively normalizes the unit of capital so that capital Kt ≡K̆t /qt is always measured in terms of the previous period’s final goods. The depreciation
rate, δ, after this normalization should be higher than δ̆, because positive investment specifictechnological change decreases the value of older capital. Thus, the new value of δ must accommodate changes in both the numerator (physical depreciation) and denominator (fallinginvestment price) of Kt ≡ K̆t /qt . In the rest of the paper, we describe the economy in termsof Definition 1, where any investment-specific technological change is included in the changeof the shape of the production function F (·; t).
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 neoclassical growthmodel is a path along which all quantities, {Yt , Kt , X1,t , ..., XJ,t , Ct , Rt }, grow at constant
exponential rates for all t ≥ 0. On the BGP, we denote the growth factor of output byg ≡ 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 ofconfusion. Note that, while a BGP requires variables to grow at constant rates, it does not
11

We assume gq to be 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 beaffected.

11require them to grow at the same rate. Still, the following lemma confirms that capital andconsumption need to grow at the same speed as output to maintain a BGP.
Lemma 2. On any non-degenerate BGP in a multi-factor neoclassical growth model, thecapital-output ratio Kt /Yt and the consumption-output ratio Ct /Yt are constant and strictly
positive.
Proof. See Appendix A.3.
The proof utilizes the assumption of C0 > 0 from Definition 1. If R0 > 0, we can similarly
show that Rt /Yt is constant.
Uzawa Representation and Its Properties
Having defined the neoclassical growth model and the BGP, we are ready to present a multifactor version of the Uzawa Theorem.
Proposition 1. (A Multi-Factor Uzawa Theorem) Consider a non-degenerate BGP
eX ,t ≡ (g/gX )t where j = 1, ..., J.
in a multi-factor neoclassical growth model, and define Ajj

Then, on the BGP,eX1 ,t X1,t , ..., AeX ,t XJ,t ) holds for all t ≥ 0,
Yt = Fe(Kt , A
J(4)

where Fe(·) ≡ F (·; 0).
eX ,t ≡ (g/gX )t , the growth factor of AeX ,t Xj,t is g for all j.
Proof. From the definition of Ajjj
The growth factor of Kt is also g from Lemma 2. Therefore, all the arguments in function
Fe(·) are multiplied by g each period. This means that the RHS of (4) is multiplied by g eachperiod since Fe(·) ≡ F (·; 0) has CRS. Note that in period 0, equation (4) holds true because
it is identical with (1). Therefore, (4) holds for all t ≥ 0, where both sides are multiplied byg in every period.
It is important to understand what the theorem does and does not imply. Recall thatthe neoclassical production function F (·; t) in (1) is a time-varying function that potentiallydepends on t in complex ways. If the economy is on the BGP, the Uzawa theorem says that
there should be a simple representation of this dependence of function F (·; t) on t, whichholds at least along this particular BGP. We call this representation, which is given by (4),the Uzawa representation. It consists of a time-invariant function Fe(·) and exponentiallyeX ,t terms. At t = 0, equation (4) coincides with the true production function (1).
growing A
j
12

The Uzawa representation illustrates how the production function evolves from there as tchanges.
However, caution is needed when interpreting Fe(·) as a production function for beyondt = 0, because Proposition 1 only guarantees that the value of Fe(·) coincides with that of thetrue production function F (·; t) exactly on a particular BGP. As is clear from the proof ofthe 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 derivativesof function Fe(·), even on the BGP, are equal to the 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 theorem by focusing on the conditions underwhich 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) denote thepartial derivative of function F (·; t) with respect to its argument Z ∈ {Kt , X1,t , ..., XJ,t }.12 If
the share of factor Z, i.e., sZ,t = FZ (·; t)Zt /Yt , is constant on a non-degenerate BGP of amulti-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 , A
J
∂Zt

(5)

Proof. See Appendix A.4.
If the factor shares are constant on the BGP, equation (5) says that Fe(·) has the same
derivatives as the true production function F (·; t) on the BGP. We can also show that theelasticity 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.13 Let us first define the EoS when there are
more than two inputs.14
12

Appendix A.1 discusses the details regarding notation for derivatives.
To the best of our knowledge, Acemoglu (2008) is the only example of previous work considering firstorder properties implied by the Uzawa Theorem. He looks at first-order conditions in the two-factor case,providing a special case of Proposition 2.
14
When there are more than two production factors, there are various ways to define the elasticity oftechnical substitution. See Stern (2011) for a concise taxonomy. The elasticity in (6) is calculated using theinverse of the symmetric elasticity of complementarity (SEC), defined in Stern (2010), which has a desirable
property of symmetry between the two variables.
13

13

Definition 3. The Elasticity of Substitution between capital Kt and input Xj in multifactor 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),

(6)

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 have the following result.
Proposition 3. (Elasticity of Substitution in the Uzawa Representation) Let σeKXj ,tdenote the EoS in the Uzawa representation, as in Definition 3. If the EoS of the true
production function, σKXj ,t for some j ∈ {1, ..., J}, is constant over time on the BGP, thenσeKXj ,t = σKXj ,t holds for all t ≥ 0 on the BGP.
Proof. See Appendix A.5.
Benefits and Limitations of the Uzawa Theorem
eX1 ,t X1,t , ..., AeX ,t XJ,t )Propositions 1-3 demonstrate that the Uzawa representation Fe(Kt , AJ
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 task that can
influence the results, especially in a quantitative setting.
Given the requirement that the model should have a BGP, the multi-factor Uzawa theorem provides guidance in choosing an effective 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 Uzawa representation
captures this evolution only through factor-augmenting terms. Proposition 1 demonstrates
that the Uzawa representation matches the level of all the key variables on the BGP, recreating an important set of stylized facts. Proposition 2 implies that the Uzawa representation
has the correct derivatives, and therefore factor shares, as long as factor shares are constanton the BGP. Relatedly, Proposition 3 says that the Uzawa representation has the correct
EoS between capital and other variables, as long as that elasticity is constant. Thus, theUzawa representation can be useful as a local approximation of the true function around the
BGP.

14

The Uzawa representation is based on function Fe(·). In Propositions 1-3, we set Fe(·) =F (·; 0), which may suggest that the period-0 production function must be known to theresearcher. However, this assumption is made for the sake of clarity and is not necessary. The
following remarks state that we can apply propositions even when only the local propertiesof the period-0 production function are known.
Remark 1. The proof of Proposition 1 holds as long as Fe(·) is any CRS function thatmatches the level of inputs and output in period 0 of the BGP; i.e., F̃ (K0 , X1,0 , ..., XJ,0 ) = Y0 .
Remark 2. The proof of Proposition 2 only requires Fe(·) to match the first derivative of
F (·; 0) in period 0 of the BGP, i.e., F̃Z (K0 , X1,0 , ..., XJ,0 ) = FZ (K0 , X1,0 , ..., XJ,0 ; 0), in addition to the condition in Remark 1.
Remark 3. The proof of Proposition 3 only requires the EoS of Fe(·) evaluated in the period
0 of the BGP to match that of F (·; 0), in addition to the conditions in Remark 1.
Thus, when building models for macroeconomic research, economists can construct the
Uzawa representation that is consistent with BGP data, even if they do not know the wholeshape of period-0 true production function, F (.; 0). They can simply pick any CRS production function (e.g., a CES production function) and calibrate it to match the level, derivatives(or factor shares), and the EoS from the data at a particular point in time, which can be
regarded as time-0 BGP values. Then, Uzawa representation will continue to match these
moments on the BGP, as long as the factor shares and the EoS are stationary.
Correctly modeling the level of output, factor prices, factor shares, and EoS between
inputs is likely to be sufficient to answer many research questions in macroeconomics. In
this sense, the Uzawa theorem is a positive result, highlighting the usefulness — rather thanthe limitations — of neoclassical growth models. Still, this particular representation may
miss other important properties of the true production function. One of such properties isthe direction of the technological change. The next section further generalizes the Uzawa
growth theorem to overcome this limitation.

4

A Generalized Uzawa Growth Theorem

eX ,t as the factor Xj,t -augmenting technology terms, Proposition 1 implies
By viewing the Ajthat it is always possible to interpret the time variation of the true production function F (·; t)on the BGP in terms of exponential augmentation of production factors. It is tempting toconclude that there should be no technological change that enhances the productivity of15

eK,t term in (4). This reasoning is insufficient,capital on the BGP, because there is no A
because Proposition 1 does not establish uniqueness. As a result, it does not rule out theexistence of better representations of the true production function.
In this section, we prove a further generalized version of the Uzawa theorem that allows
for representations with capital-augmenting technical change. We explore the possibility
that the true production function has more than one factor-augmenting representation andidentify a condition under which there will be a representation that matches the data on thecapital-augmenting technological change shown in Section 2. To satisfy this condition whileremaining consistent with empirical evidence, it is essential to include factors of production
beyond labor and reproducible capital in the aggregate production function.
Factor-Augmenting Technological Change and Factor Substitution
We start by defining a factor-augmenting representation. Note that we still do not know thefunctional form of F AUG (·) below. Rather, the following definition sets out the goal of thissection.
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 AUG (·) and thegrowth factors of factor-augmenting technologies γK > 0 and γXj > 0, j ∈ {1, ..., J}, suchthat the paths of output and inputs on a BGP satisfy
Yt = F AUG (AK,t Kt , AX1 ,t X1,t , ..., AXJ ,t XJ,t ) holds for all t ≥ 0,

(7)

where AK,t = (γK )t and AXj ,t = (γXj )t .
By comparing (4) with (7), it is clear that the Uzawa representation is a special caseof factor-augmenting representation. As we explain below, (4) assumes that all effective
factors grow at the same rate of g, while (7) permits different growth rates among different
effective factors. In other words, the Uzawa representation hypothesizes that there is no
factor substitution taking place when the economy grows along the BGP. The homothetic
expansion of every effective input is the 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 thesame speed as the output. Recall that physical capital is already growing at the same
speed as output on the BGP (Lemma 2). Then, there is no room for additional capitalaugmenting technological progress to further augment its effectiveness. As discussed in
16Section 2, however, there is clear evidence that the productivity of capital, measured in termsof output as in our model, has steadily been increasing on the BGP. Thus, the interpretationof the BGP as being a homothetic expansion of every input is at odds with a well-establishedstylized fact.
Motivated by this contradiction, we now consider a broader range of possibilities inwhich effective inputs grow at different constant rates. To have balanced growth with nonhomothetic expansion of production factors, it is necessary to further restrict the possiblefunctional forms of the factor-augmenting representation. Before moving to formal propositions, we provide a heuristic discussion that highlights the key intuition. Suppose that thetrue production function can be represented in the factor-augmenting way (7) on the BGP
with the correct derivatives and EoS. Then, the growth rate of output can approximately be
written as follows:15
g ≡ Yt+1 /Yt ≈ sk,t γK gK +JXsXj ,t γXj gXj ,

(8)

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 (8) says that the growth rate of the output is the weighted-average of the growth
rates of different effective factors, where the weights are factor shares. When the effective
factors grow at different speeds, γK gK and γXj gXj ’s are different. Specifically, let us assume
that effective capital grows faster than output due to K-augmenting 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 (7) by Yt gives
1=F

AU G




AK,t Kt AX1 ,t X1,t,
, constants .
Yt
Yt

(9)

In this form, it is evident the growing effective capital-output ratio AK,t Kt /Yt permits
production of unit output with the shrinking effective X1 -output ratio AX1 ,t X1,t /Yt . In otherwords, factor substitution is occurring.
Now, let us check if this on-going factor substitution is consistent with the definition of
15

This decomposition is obtained by Taylor-expanding the RHS of (7) for t + 1 with respect to every
effective factor, around the period t values for the variables, and dividing the result by the RHS of (7) fort. The Taylor expansion is exact when the variables in t and t + 1 are sufficiently close, or equivalently, in
continuous time.

17the BGP. On the BGP, output grows at a constant rate, g, which means that the RHS of (8)must also be constant. Given γK gK > γX1 gX1 , the RHS of (8) only remains constant when
the factor shares sk,t and sX1 ,t do not change over time. This happens if and only if the EoSof F AU G (·) between K and X1 , defined in Definition 3, is equal to one. To summarize, for
K-augmenting technological change to happen on the BGP in a factor-augmenting representation, the functional form of F AU G (·) needs to have a unitary EoS between capital andsome other factor.16 In this case, it is possible to have balanced growth even when effective
capital grows faster than output.
Once we obtain a factor-augmenting representation, we hope to use it as an approximationof the true production function. In particular, as in Proposition 3, the representation isespecially useful if the EoS of F AU G (·) matches that of the true production function. Thisis only possible when the true production function F (·; t) has a unitary EoS between capitaland some other factor, because we already know that F AU G (·) must have a unitary EoS.
As discussed in section 2, there is a great deal of evidence suggesting that the EoS between
capital and labor is different than one. However, our definition of neoclassical growth modelallows for any number of 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 withcapital.
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 factor-augmenting
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, we reorder these factors sothat the first j ∗ ∈ {1, ..., J} of them can be substituted with capital with the unitary EoS.
If capital is substitutable with j ∗ other factors with unit elasticity, we can interpretthem 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
16

In the Uzawa representation, γK gK = γXj gXj holds for all j. Because the production function is
PJ
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 (see Proposition 1) at thecost that it cannot accommodate the possibility of K-augmenting technological change.

18

period-0 relative shares within the capital composite:∗∗

α = sK,0 /(sK,0 +jX

sXj ,0 ),ξj = sXj ,0 /(sK,0 +jX

sXj ,0 ).

(10)

j=1

j=1

Using these relative shares, we can represent the production function in a nested form:17
 Y∗

j
ξ
j
α
xj , xj ∗ +1 , ..., xJ .
F (k, x1 , ..., xJ ) ≡ Fb k

(11)

j=1

The first argument of the Fb(·), m = k α

Qj ∗

ξj
j=1 xj , 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 other factors xj ∗ +1 , ..., xJ .
The shape of the outside function Fb(·) is defined using the period-0 production function
F (·; 0):18
Fb(m, xj ∗ +1 , ..., xJ ) ≡ F

Y ∗
j

j=1

ξ

−1/α

Xj,0j

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

(12)

The first argument of Fb(·), m, collects the j ∗ relevant inputs and combines them with capitalin the first argument. As a result, function Fb(·) has j ∗ fewer arguments than F (·; 0). Notethat the RHS of (12) 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.
17
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.
18
F (·; 0) needs to satisfy σKXj ,0 = 1 for j = 1, ..., j ∗ . Other than that, the following analysis only requires
the local properties of F (·; 0) are known to researchers. See Remarks 1-3 for a related discussions in thecontext of the multi-factor Uzawa theorem.

19Lemma 3. (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.6.
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, thefirst derivatives for any input, and the EoS between K and any other input Xj , when thefunction is evaluated around the period-0 BGP.19 Property d confirms 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 from there along the BGP.
The following proposition establishes that, with the nested representation F (·), there are
multiple ways to represent the technological change in 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.

(13)

For j ∈ {j ∗ + 1, ..., J}, let γXj = g/gXj . With γK and each γXj , define AK,t = (γK )t and
AXj ,t = (γXj )t . Also, define function F (·) by (10) and (11). Then, on the BGP,
Yt = F (AK,t Kt , AX1 ,t Xj,t , ..., AXJ ,t XJ,t ) for all t ≥ 0.

(14)

Proof. See Appendix A.7.
Note that (14) constitutes a factor augmenting representation, as defined by Definition
4.20 Thus, Proposition 4 characterizes the set of factor-augmenting representations of the
19
20

Namely, when {k, x1 , ..., xJ } are at the period-0 BGP values {K0 , X1,0 , ..., XJ,0 }.
Recall that function F (·) has CRS from Lemma 3.

20

true production function along the BGP. When there is no factor that is substitutable withcapital with unit elasticity at time 0 (i.e., j ∗ = 0), then Proposition 4 becomes identical to
Proposition 1.21 However, given that there are many factors of production in reality, it seems
plausible that at least one of them is substitutable with capital with unit elasticity (j ∗ ≥ 1).
In this case, there are several aspects of the proposition that warrant further discussion.
First, unlike Proposition 1, the generalized theorem implies that there is a continuum ofrepresentations. Namely, factor-augmenting terms, γK and γXj for j = 1, ..., j ∗ , can be any
combination that satisfy condition (13). This enables applied researchers to pick 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.
Second, condition (13) 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. By taking logs, it 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 .

(15)

When the growth rates of the factor-augmenting technologies are exogenous, this log-linear
condition is restrictive. In a model where the direction of technical change is endogenous,
however, this condition can be endogenously satisfied once the economy converges to the
BGP. We will examine this possibility in sections 6 and 7.
Third, similar to the original Uzawa Theorem (Proposition 1), equation (14) is not afunctional relationship. It only states that the level of inputs and outputs in this representation match those of the true production function on the BGP. The following propositions
establish that, under conditions similar to Propositions 2 and 3, the factor-augmenting representation (14) gives the correct first derivatives and the correct EoS between capital andother factors around the BGP.
Proposition 5. (Derivatives of the Factor-Augmenting Representation) Supposethat 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
Pj ∗If j ∗ = 0, condition α + j=1 ξj = 1 in Lemma 3 implies α = 1. Then, condition (13) reduces toγK = 1, which means AK,t = 1 for all t. Then, (14) becomes identical to (4).
21

21

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.
∂Zt

(16)

Proof. See Appendix A.8.
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 factor-augmenting representation
F (AK,t Kt , AX1 ,t X1,t , ..., AXJ ,t XJ,t ). If the EoS of the true production function, σKXj ,t forsome j ∈ {1, ..., J}, is constant over time on the BGP, then σ KXj ,t = σKXj ,t holds for allt ≥ 0 on the BGP.
Proof. See Appendix A.9.
Summary and Comparison to Uzawa Theorem
It is informative to contrast the results in this section with those in Section 3. The Uzawa
theorem (Proposition 1) shows that if the economy exhibits balanced growth, as observed inmany countries, there always exists a representation of the evolution of the production funceX1 ,t X1,t , ..., AeX ,t XJ,t ). This simple representation, called the Uzawa Represention, Fe(Kt , AJ

tation, explains the balanced growth by homothetic expansion of every effective production
factor. In other words, the Uzawa representation hypothesizes that no factor substitution istaking place along the BGP.
While the Uzawa representation matches the behavior of the true, often unknown, production function around the BGP (Propositions 2 and 3), it fails to explain one criticalaspect of growth. In the Uzawa representation, the productivity of capital does not improve,eK term. Our generalized theorem in Proposition 4 clarifies that the
because there is no A
Uzawa theorem is only a single possibility out of a continuum of possible factor-augmenting
representations, as long as the production function allows factor substitution on the BGP(which requires that at least one factor of production has a unitary EoS with capital).22
Every candidate representation can explain the observed quantities on the BGP, but theydiffer in the rates of factor-augmenting technological progress among different production
factors. So, it is possible to choose a candidate representations that matches the rate ofcapital-augmenting technological progress observed in data. Given the evidence of positive
22

The Uzawa representation, where γK = 1 and γXj = g/gXj for all j, satisfies condition (15).

22capital-augmenting technological change, the Uzawa representation will be ruled out as an
appropriate representation of technological change.
Propositions 5 and 6 guarantee that, if factor shares and the EoS are stationary, thechosen factor-augmenting representation will have correct derivatives and EoS. Thus, therepresentation constitutes a local approximation of the actual production function along the
BGP, similar to the Uzawa representation. Therefore, it should be at least as useful as the
Uzawa representation in any economic analysis. Indeed, the factor-augmenting representation will be more useful in many applications, especially when the questions at hand require
understanding the evolution of productivity. In the reminder of the paper, we explain theuse of new propositions in several concrete settings.

5

Three Simple Examples

So far, we have presented our results in as general a setting as possible. To incorporate theseresults into neoclassical models suitable for economic analysis, it is necessary to specify the
production factors included in the production function. This section presents three examples
that explore the simplest way to make neoclassical models consistent with aggregate dataon the relative price of capital and the EoS between capital and labor. In subsection 5.1,
we explain why a standard neoclassical economy only with two factors cannot accomplish
this goal. Then, subsection 5.2 discusses the approach taken by Grossman et al. (2017) asa special case of the 2-factor neoclassical environment. Finally, subsection 5.3 shows thatthe conflict between data and neoclassical models can be resolved when including factors of
production beyond labor and reproducible capital. Throughout this section, we describe only
the production side of the economy and do not specify the source of technological change.
We develop a full macroeconomic model with endogenous technological change in Section 6.

5.1

Standard 2-Factor Neoclassical Growth Model

Suppose that the true production function uses only two kinds of inputs, capital, Kt , andlabor, Lt , i.e., Yt = F (Kt , Lt ; t). The production function F (·; t) depends on time due to thetechnological 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 ). However, if these two
factors are substitutable with unit elasticity (σKL = 1), Proposition 4 shows there are other23

possible factor-augmenting representations of the same BGP:23
Yt = A(AK,t Kt )α (AL,t Lt )1−α , where A > 0 is a constant,(17)

eL,t Lt )1−α as a special case. Given thewhich includes an Uzawa representation Yt = AKtα (Agrowth factors of output and labor on the BGP, condition (13) implies that any combinationof γK = AK,t+1 /AK,t and γL = AL,t+1 /AL,t is consistent with the BGP as long as they satisfy
α, where the TFP At is givenγK(γL gL )1−α = g 1−α . By rewriting (17) as Yt = At Ktα L1−αtby At ≡ AAαK,t A1−α
L,t , it is clear that various combinations of capital- and labor-augmenting
technological changes give the same rate of growth for TFP and, therefore, output.
This result confirms the widely understood version of the Uzawa theorem: on a BGP, alltechnological progress must be labor-augmenting, unless the production function is CobbDouglas. As we have seen in Section 2, this theoretical result is in contradiction 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 can reconcile 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 can be understood
intuitively in terms of our analytical framework. While they start their analysis from afactor-augmenting representation, it is worthwhile to consider an underlying time-varying
true production function in the form of (1):24
Yt = F (Kt , Lt ; t) = F s (D(st )a Kt , D(st )−b Lt ; t),(18)

where a > 0, b > 0, D(·) ∈ [0, 1], and D0 (·) < 0. With D(st ) terms, the RHS of (18)specifies the production function beyond the general form F (Kt , Lt ; t). When schooling 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) specified a certain type ofcomplementarity between schooling and capital.
When there are two factors (J = 1) and they are substitutable with unit elasticity (j ∗ = 1), equation(14) 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 (17).
24
They considered not only factor-augmenting technological progress, but also investment-specific technological change. Definition 1 can include both cases as we have shown in Lemma 1.
23

24

Note that st is not a production factor in the neoclassical sense, because the production function has CRS only in capital and labor. Still, as the D(st ) term changes overtime, it affects the amount of output produced from given Kt and Lt . This is a particularform of technological change, and we can consider D(st ) as being included in the t termof F (Kt , Lt ; t), as in the middle part of (18). Therefore, it falls within the definition of a
two-factor neoclassical growth model (i.e., Definition 1 with J = 1).
eL,t Lt )From Proposition 1, this production function has an Uzawa representation Yt = Fe(Kt , A
eL = (g/gL )t on a BGP, where both effective factors Kt and A
eL,t Lt grow at the samewith A
speed as output. The production function in Grossman 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(19)Comparing the arguments in the RHS to those in the middle, we immediately obtain AK,t =eL,t D(st )b on the BGP. Because the multiplier D(st )a shrinks as st
D(st )−a and AL,t = A
increases, the capital-augmenting technology AK,t must grow so as to exactly offset theshrinking D(st )a term. Conversely, the labor-augmenting term AL,t should grow slower thaneL,t because the multiplier D(st )−b is also augmentingthat in the Uzawa representation A
labor.25 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 is importantthat schooling enters the production function precisely in the form of (19), where the samefunction D(st ) appears both before capital and labor, with the 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 our understanding of long-run economic growth by
testing whether the formulation (18) is consistent with data. In this paper, we propose
a wider class of functions that are consistent with balanced growth. The next subsection
25

From these observations, the main result of (Grossman et al., 2017, proposition 2) can easily be obtained−aas follows. Taking the growth factor of the both sides of AK = D(st )−a gives γK = gD
. From this we−1/aobtain a discrete time equivalent of their Proposition 2(ii): gD = γK . Note that Grossman et al. (2017)
assumed Lt = D(st )Nt , which means gL = gD gN . Because effective labor AL,t D(st )−b Lt in (19) must grow(b−1)/a−b
1−b
at the same rate as output, g = γL gD
gL = γL gD
gN = γL γK
gN , which is a discrete time equivalentof their proposition 2(i).

25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 unitaryelasticity (σKL 6= 1), Proposition 4 only requires that there is a single production factor
satisfies this requirement. In this case, there exist factor-augmenting 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.

(20)

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 that case, gX
represents the growth factor of the available land space. If the total area of available land
asymptotes to an upper bound in the long run, then gX would be one on the BGP. Any kindof natural resources, or a collection of natural resources (including land), can be a candidate
for Xt . If Xt is non-renewable, gX ∈ (0, 1) will likely hold, while a renewable energy source(e.g., sunlight) could have gX = 1.
Among many candidates for the third production factor, we focus on those that havea unitary EoS with capital: σKX = 1. For concreteness, we refer to this factor natural
resources. Then, Proposition 4 implies that, along a non-degenerate BGP, the technologicalchange can be represented in a factor-augmenting fashion:Yt = Fb

AK,t Kt

αAX,t Xt

1−α, AL,t Lt , α ∈ (0, 1),(21)

αwhere the growth factor of technology variables must satisfy γL = g/gL and γK(γX gX )1−α =g 1−α . As in (15), the latter condition can be written in a log-linear form:
log γK =1−α(log γL + log gL − log γX − log gX ) .
α(22)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 =26

γL gL > γX gX ).
This finding raises an important question: even if there is a factor with σXj K = 1, will
the rates of technological change γK , γX , and γL be determined so as to satisfy the log-linear
condition (22)? If their values are exogenously given, then this is a knife-edge case. If growth
rates are endogenous, however, this need not pose any additional restrictions on the model.
In Section 6, we develop a growth model with endogenous and directed technical change,
where γK , γX , and γL are endogenously chosen. We will confirm that, on the BGP, condition(22) is satisfied. In Section 7, we calibrate a version of the model to moments from the longterm U.S. data and show that the BGP with positive capital-augmenting technical change
is both locally and globally stable. These two sections jointly demonstrate that regardlessof the initial state of technologies, condition (22) is always satisfied in the long run.

6

A Full Model with Directed Technological Change

So far, we have discussed the implications of the generalized Uzawa theorem focusing on the
production sector. In this section, we develop a complete endogenous growth model where thedirection of technological progress is determined by profit-maximizing firms. We show thatthe log-linear technology condition (22) is endogenously satisfied on the BGP. We base thissection on a streamlined version of the model of tasks developed by Irmen (2017) and Irmen
and Tabaković (2017) and expand it to incorporate three production factors. There are two
benefits from our specification. First, we can analyze intentional R&D within a perfectlycompetitive economy, which fills a gap between the standard neoclassical growth model(perfectly competitive) and standard endogenous growth theory (imperfect competition).
Second, our model of tasks will be scale independent, which implies that the model has a
BGP even when the amount of labor is changing.

6.1

The Model

There are non-overlapping generations of representative firms, each of which exists for only
one period. A representative firm performs two types of tasks, M-tasks and N-tasks. The
number of M-tasks, as well as that of N-tasks, determines the amount of final output. The
M-tasks require effective capital AK,t Kt and effective natural resource AX,t Xt as inputs,
where AK,t and AX,t are the representative firm’s capital-augmenting and labor-augmenting

27

technologies (explained in detail below). The number of M-tasks it can complete is given by
Mt = (AK,t Kt

αAX,t Xt

1−α, α ∈ (0, 1).

(23)

We refer to the RHS as the amount of the capital composite, which combines effective capitaland effective natural resources (including land) with unit elasticity. An N-task uses only
effective labor, AL,t Lt , where AL,t is the labor-augmenting technology of the representative
firm. The number of N-tasks is simply
Nt = AL,t Lt .

(24)

By performing Mt and Nt tasks, the representative firm produces
Yt = Fb(Mt , Nt ) = Fb

AK,t Kt

αAX,t Xt

1−α, AL,t Lt(25)

units of output, where Fb(·) is a standard neoclassical production function that has CRS andsatisfies the Inada conditions.26
Now, we explain how the factor-augmenting technologies {AK,t , AX,t , AL,t } are determined. Technical knowledge can be kept within the firm for only one period, after which it
becomes public. Thus, the representative firm at time t can freely use the technology of theperiod t − 1 firm, {AK,t−1 , AX,t−1 , AL,t−1 }. In addition, the period t firm can improve eachof factor-augmenting technologies through R&D. We assume that tasks are differentiatedand require separate R&D investments.27 To enhance the capital-augmenting technology for
an M-task by a factor of γK,t ≡ AK,t /AK,t−1 ≥ 1, the firm need to invest iK (γK,t ) units offinal goods. The firm faces similar choices when improving AX,t for each M-task, and also
when enhancing AL,t for each N-task. The R&D cost functions for natural resources andlabor-augmenting technologies are defined accordingly as iX (·) and iL (·).
It is reasonable to think that the marginal cost of improving the technology is small when
the size of innovation is small, but it becomes increasingly expensive when aiming for bigger
innovations.28 To capture this, we assume R&D cost functions iK (·), iX (·) and iL (·), have
26

There are two ways to represent the production function in intensive forms: f (M/N ) = Fb(M/N, 1) andh(N/M ) = Fb(1, N/M ). We assume that both f (·) and h(·) satisfy the Inada conditions.
27
From the symmetry of tasks within each group (M or N) and from the convexity of the R&D costfunctions as assumed in (26), it is always optimal to choose the same levels of AK,t , AX,t , and AL,t acrossindividual tasks. Therefore, we omit subscripts for technologies for individual tasks.
28This can be explained by congestion in R&D activities. When many researchers are devoted to improvements in the same task at the same time, some of them will end up inventing the same innovation. The28

the following properties:iZ (γZ ) > 0, i0Z (γZ ) > 0, i00Z (γZ ) > 0 for all γZ ≥ 1,
iZ (1) = 0, i0Z (1) = 0, iZ (∞) = ∞ for Z = K, X, L.

(26)

Note that the R&D costs must be incurred for each of the Mt and Nt tasks. This means
the total R&D cost for the representative firm is the sum of
RK,t = Mt · iK (AK,t /AK,t−1 ),RX,t = Mt · iX (AX,t /AX,t−1 ),(27)

RL,t = Nt · iL (AL,t /AL,t−1 ).
The objective of the representative firm is to maximize the single period profit net of
R&D costs, because it lives only for one period and its knowledge will become public next
period. By taking the output in each period as numéraire, the period profit is given by
πt = Fb(Mt , Nt ) − RK,t − RX,t − RL,t − rt Kt − τt Xt − wt Lt ,(28)

where rt , τt , and wt are interest rate, payment for a unit of natural resources (e.g., landrent), and wage rate, respectively.
We keep the demand side of the economy as standard as possible. There is a representative
household. The size of the representative household (i.e., population) evolves according to29
Lt = L0 gLt , L0 > 0, gL > 1 − δ : given.

(29)

As in the Ramsey-Cass-Koopman model, the period utility of the household is given by the
product of the number of household members and the per capita period felicity function:
ut = Lt u(Ct /Lt ),(30)

where Ct /Lt > 0 is per capita consumption. We assume the felicity function u(·) takes therisk of duplication become more prominent as R&D inputs increase, which makes the R&D cost functioniK (·) convex. See Horii and Iwaisako (2007) for a simple micro foundation.
29
We assume gL > 1 − δ so as to avoid the possibility of a degenerate BGP, where physical capital
investment becomes zero or even negative in the long run. (See Definition 2). Note that, as long as δ > 0,condition gL > 1 − δ allows declining population. However, population should not fall faster than the speedof capital depreciation.

29

CRRA form. Then, the intertemporal objective function of the household can be written as
U=

∞
X
t=0

Lt β t(Ct /Lt )1−θ − 1
,
1−θ

(31)

where θ > 0 is the degree of the relative risk aversion (i.e., the inverse of the intertemporalelasticity of substitution) and β ∈ (0, 1) is the discount factor.30
The representative household owns capital, Kt , and natural resources, Xt , in addition
to labor, Lt . The household also owns the representative firm and receives the profit, πt ,
although in equilibrium profits will be zero due to perfect competition. For simplicity, we
assume that the supply of natural resources is exogenous:31t
Xt = X0 gX, X0 , gX > 0 : given.

(32)

As in the case of population, its available quantity can be either constant gX = 1, shrinking
gX ∈ (0, 1), or growing gX > 1. Physical capital accumulates through the savings of thehousehold:Kt+1 = (rt + 1 − δ)Kt + τt Xt + wt Lt + πt − Ct ,K0 > 0 : given,(33)

where (rt + 1 − δ)Kt + τt Xt + wt Lt + πt represents the household’s income. The household
is subject to the no-Ponzi game condition. Specifically, the present value of its asset holding
as T → ∞ should not be negative:

lim

T →∞

!−1
T
Y
(rt + 1 − δ)KT +1 ≥ 0.

(34)

t=1

This completes the description of the model economy.
Before proceeding to the analysis of the model, we demonstrate that it conforms to ourdefinition of the multi-factor neoclassical growth model, given by (1) and (2) in Definition
1. First, the aggregate production function (25) has exactly the same form as (21), which
30

While we tentatively assume β ∈ (0, 1), it needs to be significantly smaller than 1 since otherwise U will
become infinite in a representative household model with growing population and per capita consumption.
The actual upper bound for β will be derived in Proposition 8.
31
As we have shown in Table 1, among the estimated 8% factor share of natural resources, a majority(5%) is from land. Since the supply of land is mostly constant, we assume Xt is exogenous in this baseline
scenario. Our theory is also applicable to the case where Xt is depleted or expanded endogenously (Seerobustness scenario f in Section 7.1). Note that, although Xt is exogenous, its effective amount AX,t Xt as a
production factor can be enhanced endogenously, through R&D for AX .

30belongs to the definition of the multi-factor neoclassical production function (1). In fact,
Proposition 4 guarantees that, if the EoS between Kt and Xt is unity and the economy hasa BGP in equilibrium, then the aggregate production function can always be represented in
the form of (25) at least along the BGP. Our microeconomic setting gives an example of suchan economy. Second, by substituting (28) into (33), we obtain the evolution of capital in thesame form as (2), where the total R&D expenditure is defined as Rt = RK,t + RX,t + RL,t .
The difference between Definition 1 and the current model is that we now have a complete
description of the economy, including how the speed and direction of technological change
are determined. We are now ready to explore whether this economy can generate a BGP in
equilibrium, paying special attention to whether there is a BGP with a strictly positive rateof capital-augmenting technological progress.

6.2

R&D by Firms and the Direction of Technological Progress

We start by examining the behavior of the representative firm in the economy describedabove, focusing on the role of R&D. The representative firm maximizes profit (28) subject
to the production and R&D functions (23)–(27) with respect to {Kt , Xt , Lt , AK,t , AX,t , AL,t },taking as given prices, {rt , τt , wt }, and lagged technology levels, {AK,t−1 , AX,t−1 , AL,t−1 }.
For convenience, we define µt ≡ Mt /Nt , which is the relative task intensity in final good
production. It also represents the ratio of effective capital composite to effective laborα1−αµt = AK,t Kt AX,t Xt/AL,t Lt . Then, because Fb(·) in (25) is a CRS function, we canwrite it in intensive form, Fb(Mt , Nt )/Nt = Fb(µt , 1) ≡ f (µt ), FbM (Mt , Nt ) = f 0 (µt ), andFbN (Mt , Nt ) = f (µt ) − µt f 0 (µt ).32
Using this notation, we can conveniently express the first order conditions for factor
demand. The firm demands capital, natural resources, and labor so as to satisfy33,34
rt = (αMt /Kt ) (f 0 (µt ) − iK (γK,t ) − iX (γX,t )) ,

(35)

32 b
FM (·) and FbN (·) represent the partial derivatives of function Fb(·) with respect to its first and second

arguments, respectively.
33
The RHS of (35) represents the (net) marginal product of Kt in producing output Yt . It is given by
the product of two parts. The first part, αM/K, is the marginal product of Kt in increasing the number ofM-tasks performed in the firm. The second part is the net marginal product of Mt in producing the final
output. Note that, in the second part, the innovation cost for an M-task, iK (γK,t ) + iX (γX,t ), is subtracted
from the “gross” marginal product of Mt , f 0 (µt ). When the firm performs more M-tasks, it chooses to pay
R&D costs to increase AK,t and AX,t in these tasks so as to keep up with other M-tasks. Similarly, in (36),(1 − α)M/X is the marginal product of Xt in performing more M-tasks.
34
By substituting (35), (36), and (37) into (28), it can be confirmed that the firm achieves zero profit,
πt = 0. This is due to the CRS property of the firm’s problem.

31τt = ((1 − α)Mt /Xt ) (f 0 (µt ) − iK (γK,t ) − iX (γX,t )) ,(36)

wt = AL,t (f (µt ) − µt f 0 (µt ) − iL (γL,t )).

(37)

Now, we turn to R&D, starting with the condition for improving the labor-augmenting
technology AL,t . The representative firm chooses AL,t , or equivalently the speed of technological progress γL,t ≡ AL,t /AL,t−1 ≥ 1, according to first order condition ∂πt /∂AL,t = 0.
Simplifying this condition yields:35
R&D for N-tasks: γL,t i0L (γL,t ) + iL (γL,t ) = f (µt ) − µt f 0 (µt ).

(38)

As we formally prove in Proposition 7 below, condition (38) has a unique solution for
γL,t as a function of µt = Mt /Nt , and it is strictly increasing in µt . Intuitively, when thefirm is performing relatively few N-tasks (i.e., when µt ≡ Mt /Nt is higher), the benefit ofincreasing AL,t to perform another N-task is larger, and therefore it is optimal to improve
the labor-augmenting technology AL,t at a faster pace (i.e., γL,t should be higher).
Next, we examine the R&D investments for K- and X-augmenting technologies. As in
the case of labor-augmenting technology, γK,t ≡ AK,t /AK,t−1 and γX,t = AX,t /AX,t−1 need tosatisfy the first order conditions, ∂πt /∂AK,t = 0 and ∂πt /∂AX,t = 0. Combining these two
equations, we obtain two intuitive conditions that determine the allocation of relative R&D
effort between K- and X-augmenting technologies, as well as the condition that specifies the
optimal combined amount of R&D for M-tasks:36
R&D allocation:

γK,t i0K (γK,t )α=,0γX,t iX (γX,t )
1−α

(39)

Combined R&D: (γK,t i0K (γK,t ) + iK (γK,t )) + (γX,t i0X (γX,t ) + iX (γX,t )) = f 0 (µt ).
35

(40)

The firm’s private benefit from improving technology AL,t is the ability to perform a larger numberof N-tasks, which increases the final output Yt = Fb(Mt , Nt ). The RHS of (38) shows the marginal benefit,
FbN (Mt , Nt ) = f (µt ) − µt f 0 (µt ). The LHS corresponds to the marginal cost of performing a larger numberof N-tasks through augmenting labor efficiency AL,t (given labor employment Lt ). This can be broken into
two components. First, by intensifying the R&D efforts in existing N-tasks to raise labor efficiency, therepresentative firm can decrease labor inputs by just enough to perform one additional N-task. The costassociated with this activity is given by the first term γL,t i0L (γL,t ), which we call the intensive marginal R&D
cost. The saved labor is then used to perform a new N-task, which means the representative firm needs toinvest in R&D for one more N-task, which costs iL (γL,t ). This extensive marginal R&D cost is represented
by the second term in the LHS.
36
The first order condition for AK,t yields (γK,t /α)i0K (γK,t ) + iK (γK,t ) + iX (γX,t ) = f 0 (µt ), whereas thatfor AX,t gives (γX,t /1 − α)i0X (γX,t )+iK (γK,t )+iX (γX,t ) = f 0 (µt ). Condition (39) is obtained by subtracting
the second equation from the first. Condition (40) is from adding α times the first equation and (1 − α)times the second equation.

32

γL

γX

Combined
R&D cond.

Direction of tech.
change with µ1

Equilibrium Innovation
Possibility Frontier (EIPF)
Eq. with µ2 (> µ 1)

R&D allocation
cond.

Eq. with µ1(1,1,1)
µ↑

γK
1

1

Combined
R&D cond.

γK(a) Equilibrium innovation in K- andX-augmenting technologies.
γK ≡AK,t /AK,t−1 and γX ≡ AX,t /AX,t−1
are determined by the intersection of
the R&D allocation and combined R&D
conditions.

R&
co D al
nd lo
ca
.

γX

tio

n

(b) When µt increases from µ1 to µ2 , the equilibrium direction of technological change moves fromthe solid arrow to the dashed arrow, closer to theγL axis. The set of all these equilibrium points
gives the EIPF curve. The bottom (γK , γX ) plane
corresponds to panel (a).

Figure 2: Determination of the direction of the technological change and the Equilibrium
Innovation Possibility Frontier (EIPF) Curve
Condition (39) gives the optimal allocation of R&D investment between K- and Xaugmenting technologies. Observe that γK,t i0K (γK,t ) and γX,t i0X (γX,t ) on the LHS are strictlyincreasing in γKt and γX,t , respectively. Therefore, this condition can be expressed as an
upward sloping curve in the (γK,t , γX,t ) space, as depicted in Figure 2(a).37 Condition (40)specifies the optimal combined size of R&D investments.38 Since the LHS is increasing
increasing both in γKt and γX,t , the locus of (γK,t , γX,t ) that satisfies this condition is depicted
37
As the RHS of condition (39) shows, the allocation should depend on the relative contribution of capitaland natural resources in performing M-tasks. When capital’s relative contribution is higher (i.e., when α
is higher), more resources should be allocated to R&D for the capital-augmenting technology. In addition,
the slope and convexity of the R&D cost function also affects the optimal allocation. For example, if it
is relatively difficult to improve the efficiency of natural resources, i.e., if the marginal R&D cost i0X (γX,t )increases more rapidly with its argument than i0K (γK,t ), then it is optimal not to improve AX,t as fast as
AK,t .
38
Capital and natural resources are used in M-tasks, and therefore improving K- and X-augmenting
technologies will enable the firm to perform more M-tasks. This marginal benefit is represented by the RHSof (40), f 0 (µt ) = FbM (Mt , Nt ). The LHS is the marginal cost of R&D, which has two parts, γK,t i0K (γK,t ) +iK (γK,t ) and γX,t i0X (γX,t ) + iX (γX,t ), because both K- and X-augmenting technologies receive some R&D
according to the allocation condition (39). In each of the two parts, the first term represents the intensive
marginal R&D cost, whereas the second term is the extensive marginal R&D cost, as in condition (38).

33

by the downward-sloping curve. The intersection of the R&D allocation condition and the
combined R&D condition gives the optimal rates of innovation for K- and X-augmenting
technologies. Combined with γL,t given by (38), we have the direction of technological change
in equilibrium.
We can also study how the direction of technological change moves endogenously, focusing
on µt , the ratio of effective capital composite to effective labor. The following propositionsummarizes the result.
Proposition 7. (Direction of Technological Change)
In the endogenous growth model defined in Section 6.1, the growth factors of each of thefactor augmenting technologies are function of µt = Mt /Nt , denoted by γbK (µt ), γbX (µt ), andγbL (µt ). These functions satisfy:
0
(a) γ
bK
(µt ) < 0 for all µt > 0, γ
bK (0) = ∞, and γbK (∞) = 1.
0(µt ) < 0 for all µt > 0, γ
bX (0) = ∞, and γbX (∞) = 1.
(b) γ
bX

(c) γ
bL0 (µt ) > 0 for all µt > 0, γ
bL (0) = 1, and γbL (∞) = ∞.
Proof. See Appendix A.10.
Figure 2(b) illustrates the direction of the technological change in the 3-dimensional
space. The γK -γX plane depicted at the bottom of the figure is the same as in panel (a).
For a given value of µt , the intersection gives the value of γbK (µ) and γbX (µ). In addition, thevertical distance between this point and equilibrium point shows the size of L-augmenting
innovation, γbL (µ)−1. As µt increases, the combined R&D locus shifts inward,39 which lowers
γK,t and γX,t . At the same time γL,t increases because γ
bL0 (µt ) > 0. This way, the directionof the technological change moves from the solid arrow to the dashed arrow in Figure 2(b).
The dependence of the direction on µt can be interpreted in terms of relative scarcity ofeffective factors. Proposition 7 says that the direction of technological progress is chosen sothat it enhances effective factors which are in relatively short supply. In other words, firms
are ‘induced’ to do more innovation that enhances the relatively scarce effective production
factors.40
The thick downward-sloping curve in Figure 2(b) depicts the locus of all equilibrium
points that correspond to various values of µt . This is the equilibrium innovation possibility
frontier (EIPF). Depending on the equilibrium value of µt , the direction of technological39

The shift occurs because the RHS of (40) is decreasing in µt .
This notion of induced innovation was first introduced by Hicks (1932). See Acemoglu (2002) for morediscussion.
40

34

change is captured by some point on this curve.41 To see how µt is determined in eachperiod, we now consider the equilibrium dynamics.

6.3

Equilibrium Dynamics

The equilibrium path of this economy is given by the sequence of output, consumption,
production factors, technologies, and R&D investments, {Yt ,Ct ,Kt ,Xt ,Lt ,AK,t ,AX,t ,AL,t ,RK,t ,RX,t ,RX,t }∞t=0 , which satisfy the representative firm’s optimization problem, the representative consumer’s utility maximization problem, and the market clearing conditions for output
and production factors. The economy is endowed with K0 , X0 and L0 at time 0, as well as
the initial levels of publicly available technologies, AK,−1 , AX,−1 , and AL,−1 .
While the equilibrium involves many variables, we can analytically characterize its dynamic path in terms of only three: relative task intensity µt = Mt /Nt , the amount of capitalper effective labor kt ≡ Kt /AL,t Lt , and consumption per effective worker ct ≡ Ct /AL,t Lt .
In Appendix A.11, we derive the dynamics these three variables in detail with intuitive
explanations. The following summarizes the result.
First, the dynamics of relative task intensity µt can be written as 1−α α gXv(µt ) − ct
µt+1 = G+ 1 − δ µt ≡ ψ µ (µt , kt , ct ),
gL
kt

(41)

where v(µt ) ≡ f (µt )−µt (iK (bγK (µt ))+iX (bγX (µt )))−iL (bγL (µt )) is the net output per effective
labor, and G(·) is the inverse function of Γ(µ) = γbL (µ)µ/b
γK (µ)α γbX (µ)1−α .42 Intuitively,
the relative task intensity Mt+1 /Nt+1 is determined in two steps. The relative supply of
production factors determines the relative numbers of task that the representative firm canperform, given the previous period’s technology levels. This is the argument of function G(·)
in (41). The firm can also increase Mt and Nt by factor-augmenting R&D, which affects theequilibrium µt+1 . The latter effect is captured by G(·).
41

In this model, not only the direction within the EIPF, but also the EIPF itself is determined endogenously form the firm’s profit condition. In most models of direction of technological change, it is assumed
that innovation requires a certain type of exogenously given resource (e.g., scientists). In these cases, theinnovation possibility frontier is derived from the resource constraint. To the contrary, in our model, thetotal amount of R&D input (Rt ) is determined in equilibrium through profit maximization, and hence thefrontier is called the ‘equilibrium’ innovation possibility frontier. Any innovation beyond this frontier is notprofitable, although it might be materialistically feasible.
42
In appendix A.11, we confirm G(·) is well defined and strictly increasing.

35

Second, the dynamics of capital per effective worker, kt , are
kt+1 =1gL γ
bL (ψ µ (µt , kt , ct ))(v(µt ) − ct + (1 − δ)kt ) ≡ ψ k (µt , kt , ct ),(42)

where ψ µ (µt , kt , ct ) is µt+1 from (41).43 The expression (v(µt ) − ct + (1 − δ)kt ) is the sum of
the net saving and the un-depreciated part of existing capital, per effective labor in periodt. It must be divided by gL γ
bL (µt+1 ), because of the growth of effective labor between periodt and t + 1.
Third, the Euler equation for the consumer optimization gives the dynamics for ct :ct+1 =

β 1/θ ctαψ µ (µt , kt , ct )  0 µf (ψ (µt , kt , ct )) − iK (bγK (ψ µ (µt , kt , ct )))γbL (ψ µ (µt , kt , ct )) ψ k (µt , kt , ct )!1/θ−iX (bγX (ψ µ (µt , kt , ct ))) + 1 − δ≡ ψ c (µt , kt , ct ).

(43)

The seemingly complex expression inside (· · · )1/θ is simply rt+1 + 1 − δ, expressed in terms of
period t variables. Equations (41), (42), and (43) constitute the equilibrium mapping from{µt , kt , ct } to {µt+1 , kt+1 , ct+1 } for all t ≥ 0.
In appendix A.11, we explain starting levels of µ0 and k0 are given by initial factor
endowments and initial technology levels. In addition, transversality condition and the nonPonzi game condition jointly requirelim (βgL )T

T →∞

T
Y

!1−θ
γ
bL (µT +1 )c−θ
T kT +1 = 0.

γbL (µt )

(44)

t=0

These three boundary conditions, µ0 , k0 , and (44), pin down the equilibrium path of {µt , kt , ct }.
The next subsection will examine the property of the equilibrium path, focusing on the BGP.

6.4

The Balanced Growth Path

Now, we are ready to characterize the BGP of this economy. We will show that the directionof technological progress is endogenously chosen so that in equilibrium there is a unique
BGP with a positive rate of capital-augmenting technical change.
Lemma 4. Define a BGP as an equilibrium path where the growth factors of {Yt , Kt , Xt ,43

This expressions shows that the RHS is a function of only period t variables.

36

Lt , Ct , Rt , Mt , Nt } are all constant.44 Then, on any BGP, the values of µt , kt and ct must
be constant.
Proof. See Appendix A.12.
We denote the BGP values of µt , kt and ct by µ∗ , k ∗ and c∗ , respectively. Their values
are obtained by substituting µt+1 = µt = µ∗ , kt+1 = kt = k ∗ and ct+1 = ct = c∗ into (41),(42), and (43).
First, from (41) and (42), the BGP value of µt ≡ Mt /Nt will satisfy
1=

(gX γ
bX (µ∗ ))1−α (bγK (µ∗ ))α≡ Φ(µ∗ ).
(gL γ
bL (µ∗ ))1−α .

(45)Proposition 7 implies Φ0 (µ∗ ) < 0 with Φ(0) = ∞ and Φ(∞) = 0. Therefore, there exists a
unique value of µ∗ > 0 that satisfies Φ(µ∗ ) = 1, and hence condition (45). An intuitive way
to interpret (45) is to multiply the both of its sides by (gL γbL (µ∗ ))α .
(gX γ
bX (µ∗ ))1−α (bγK (µ∗ )gL γ
bL (µ∗ ))α = gL γ
bL (µ∗ ) (= g ∗ ).

(46)

The LHS represents the growth factor of Mt on the BGP, while the RHS is that for Nt .
Therefore, this condition means that the relative factor intensity µ∗ = Mt /Nt is determinedso that Mt and Nt grow at the same speed. This condition singles out a point on the
Equilibrium Innovation Possibility Frontier (recall Figure 2), which determines the directionof technological change on the BGP. Note that, due to the CRS property of production
function Yt = Fb(Mt , Nt ), the value of equation (46) also represents the economic growth
factor g ∗ ≡ Yt+1 /Yt .
Second, from the Euler equation (43), the BGP value of kt = Kt /(At Lt ) isk∗ =βαµ∗ (f 0 (µ∗ ) − iK (bγK (µ∗ )) − iX (bγX (µ∗ ))).
γbL (µ∗ )θ − β(1 − δ)(47)

Intuitively, the capital-effective labor ratio on the BGP is determined from the interest rater∗ that yields constant consumption per effective labor on the BGP.45 Third, from (42) and44

Here, we slightly extend Definition 2 by requiring constancy of the growth factors of Mt and Nt , i.e.,
the numbers of tasks performed in the economy.
45
Using (35), condition (47) is shown to be equivalent to r∗ + 1 − δ = β −1 γ
bL (µ∗ )θ . Here, the RHS is themarginal rate of intertemporal substitution given that consumption per effective labor is constant (whichmust be true on the BGP).

37

gL γ
bL (µ∗ ) = g ∗ in (46), the BGP value of c∗ = Ct /AL,t Lt must satisfyc∗ = v(µ∗ ) − (g ∗ − 1 + δ)k ∗ .

(48)

These three equations describe the unique BGP in this economy. The following propositionshows that the BGP uniquely exists when the discount factor is sufficiently smaller than 1.
Proposition 8. There exists a value of β > 0 such that whenever β ∈ (0, β), there exists a
unique BGP that satisfies µ∗ > 0, k ∗ > 0, c∗ > 0, and the terminal condition (44).46
Proof. See Appendix A.13. The exact expression for the upper bound β is given by (A.51).

The most important implication from this model is that the technology condition (22) in
Section 5.3 is now an endogenous outcome. Specifically, BGP condition (45) is equivalent to(22), except that the speed of technological progress is endogenously determined by profitmaximizing producers. This difference has important implications for the plausibility ofcapital-augmenting technological progress on the BGP. As discussed in Section 5.3, if thethe rates of innovation for the three factor-augmenting technologies are exogenously given,and then (22) becomes a knife-edge condition. In contrast, this section has shown that,once we consider endogenous technical change, this condition is necessarily satisfied whenthe economy is on the BGP, which exists if discount factor β is sufficiently less than one.
Thus, if we can show that the model economy converges to the BGP, then condition (22) isnaturally satisfied in the long run. We do so in the next section.

7

Numerical Analysis and Stability

In this section, we investigate the local and global stability of the three-factor endogenous
growth model. Our primary objective is to show that the model economy converges to a BGP
with capital-augmenting technical change, where log-linear relationship (22) is endogenously
46

There are two reasons why the existence of the BGP requires an upper bound for β (or, equivalently
a lower bound for ρ = (1 − β)/β). First, on the BGP, the amount of consumption for the household
Ct = AL,t Lt c∗ increases over time, causing the instantaneous utility to grow. Therefore, if β is too close toone, the intertemporal utility U in (31) becomes infinity, which means that the household’s problem is notwell defined. Second, as effective labor AL,t Lt grows, the household accumulates more capital Kt so as toprevent the dilution of capital per effective labor, k ∗ . However, when β is too large (i.e., when the discountrate ρ is too small), the BGP requires a too low real interest rate, or a too high level of k ∗ , to the extent thatpreventing the dilution is impossible even when all net output is invested in Kt . We rule out these extreme
cases by assuming an upper bound for β.

38satisfied. We also illustrate how having multiple technologies (including K-augmenting technology) affects the transition dynamics. To accomplish these goals, we present a series ofnumerical examples for which we can check stability computationally. Whenever possible, we
ensure that our numerical examples are consistent with macroeconomic data characterizing
the BGP of the United States. We stress, however, that this is not a complete calibration,and the results would be insufficient for a precise quantitative analysis.

7.1

Calibration

Functional Forms
−1

We assume that the aggregate production function takes a CES form: Fb(Mt , Nt ) = (ηMt  +−1Nt  ) 1− , where  > 0 and η ∈ (0, 1). Output in the economy (25) can be written as

o 1−
n
 −1
−1
α
1−α 
.
Yt = η (AK,t Kt ) (AX,t Xt )+ (1 − η)(AL,t Lt ) (49)

Next, we assume power function for R&D costs47
λiZ (γZ ) = ζZ γZ − 1 ,ζZ > 0, λ > 1, Z = K, X, L.

(50)

We allow R&D cost parameter ζZ to differ across types of technology. We normalize ζK to
1, and calibrate ζX and ζL . The degree of convexity, λ, is assumed to be the same across thethree types of technology.
With these functional form assumptions, our model has 11 parameters, {, η, α, λ, ζL , ζX ,β, θ, δ, gL , gX }. To calibrate the model, we also need to determine the period length, measured
in years, denoted by χ. The period length in our model has important economic meaning,
because it represents the duration for which a firm can monopolize the benefit from its R&D
investments. Including χ, we have 12 parameters.
Exogenous Parameters
We set five parameters exogenously. Their values are given in Table 2. In the CES production
function (49), we take  = 0.7 as the baseline value. This is a common estimate for the
EoS between labor and reproducible capital (e.g., Antras et al., 2004; Oberfield and Raval,2014). The mapping between these estimates and a structural parameter in our model is not
47

Note that function (50) satisfies condition (26).

39

Parameter

λ
θ
1/χ
gL
1/χ
gX

Baseline
0.7
2.0
1.0
1.01
1.0

Alternate
0.9, 1.2
1.5, 2.25
0.5, 2
0.99

Description
EoS b/w M and N
R&D Cost convexity
Inverse of IES
Population growth
Growth of X

Explanation / Source
Oberfield and Raval (2014)
Quadratic
Log Preferences
BEA 1960-2020 average
Fixed Supply of Land

Table 2: Exogenous Parameters
Target Moment (in annual values)
Value
Capital output ratio
2.9
Labor share of income
63%
Share of R&D payments in GDP
2.7%
Consumption of fixed capital in GDP 14%
Growth rate of income per capita
1.9%
Decline in the relative price of capital 0.66%
Return on investment
4%

Model Variable
K/(Y /χ)κL ≡ wL/YκR ≡ RK +RYX +RL
δK/Y1/χ
γL − 11/χγK − 1(1 + r − δ)1/χ − 1

Source
BEA 1960-2020 average
BEA 1960-2020 average
BEA 1960-2020 average
BEA 1960-2020 average
BEA 1960-2020 average
BEA 1960-2020 average
McGrattan et al. (2003)

Table 3: Target Moments for Calibration
exact and we show robustness with  = 0.9 and  = 1.2.48 In the baseline calibration, we
assume the R&D cost function is quadratic (λ = 2) and also check robustness with λ = 1.5
and λ = 2.25. Quadratic cost is a common assumption, and it is consistent with existing
empirical work in endogenous growth (Acemoglu et al., 2018; Akcigit and Kerr, 2018). As
for utility function (31), we take log preferences (θ = 1) as the baseline and also consider
cases where the intertemporal EoS is higher or lower than 1 (θ = 0.5 and 2). Population
growth is set to the 1960-2020 average in the U.S. (1% per year). When one period in the1/χmodel corresponds to χ years, this means gL= 1.01. We do not have a good data for thegrowth rate of factor X, which we interpret as natural resources, including land. Given thatland is a major factor of production,49 we take gX = 1 as a benchmark (i.e., constant X).
1/χWe also consider the case where natural resources are depleted 1% per year (gX = 0.99).
Data
We calibrate the remaining parameters so that the model variables on the BGP match datafrom the U.S. Table 3 reports the target moments and model variables in annualized values
48
Karabarbounis and Neiman (2014) and Piketty (2014) estimate the EoS between reproducible capitaland labor and find an elasticity that is greater than one.
49
As shown in Table 1, the factor share of land is estimated to be around 5%, while the share of all natural
resources (including land) is 8%.

40

Parameter
χ
β
δ
α
η
ζX
ζL

Calibrated
3.94
0.923
0.190
0.767
0.685
0.279
20.8

Annualized
0.980
5.21%

Description
Period Length (years)
Discount Factor
Depreciation Rate
Capital Share within K-X composite
CES Distribution parameter
Cost parameter for AX R&D
Cost parameter for AL R&D

Table 4: Calibrated Parameters for Baseline Scenario(e.g,. aggregate output per year is Y /χ, where one period in the model is χ years). For
the capital-output ratio (2.9), labor share of income (63%), R&D share of income (2.7%),
consumption of fixed capital as a share of GDP (14%), and real GDP per capita growth(1.9%), we use data from the Bureau of Economic Analysis (BEA) to calculate the arithmetic
averages of the annual levels in the 1960-2020 period. To measure the growth rate of capitalaugmenting technology, we calculate the annual decline in the relative price of all capital
goods from 1960-2020 (0.66%). Finally, we set the rate of return on investment (r∗ −δ) equal
to the return on bonds (4%) from McGrattan and Prescott (2003).
Calibration Results
There are seven remaining parameters to calibrate, {δ, β, α, η, ζL , ζX , χ}, which we identify
with the seven moments in Table 3. We do so in two steps. First, we use equilibrium
conditions to derive four analytical relationships among these parameters. This leaves us
with three undetermined parameters, {ζL , η, χ}. In the second step, we numerically pin themdown so that the target moments in Table 3 match the corresponding model variables on
the BGP. The details of the calibration procedure are presented in Appendix A.14.
Table 4 presents the results of the two-step calibration procedure with the baseline assumptions. Period length χ is 3.94 years, which is time it takes for a firm’s R&D outcome
1

to be overtaken by a new innovation. Discount factor β is 0.923 (β χ = 0.98/year). The
depreciation rate δ is 0.19 per period, which is about 5% per year. The share parameter α
is 0.76. As reported in Table 5, this number implies that capital share in the GDP is 26.3%,
whereas the natural resource share (including land) is 8%. Although these shares were nottargeted in the calibration, they are consistent with the values reported in Table 1.

41

Variable
κK
κX
1/χγX − 1

Value
Description
26.3%
Capital Share
8.0% Natural Resource Share (incl. Land)
0.72%
Tech. Change in AX per year

Table 5: Untargeted Variables in Calibrated Model

(a)
(b)
(c)
(d)
(e)
(f)
(g)
1/χ
Param- Baseλ
λ


θ
θ
gX
eters
line
=1.5 =2.25 =0.9
=1.2
=0.5
=2
=0.99
χ
3.94
2.92
4.20
3.94
3.94
3.94
3.94
3.72
β 1/χ
0.98
0.98
0.98
0.98
0.98
0.97
0.998
0.98
1/χ
5.21% 5.07% 5.25% 5.21% 5.21% 5.21% 5.21% 5.18%
δ
α
0.77
0.76
0.77
0.77
0.77
0.77
0.77
0.77
η
0.69
0.38
0.79
0.41
0.27
0.69
0.69
0.70
ζL
20.82
1.88
61.74 10.05
7.64
20.82 20.82 25.69
ζX
0.28
0.29
0.28
0.28
0.28
0.28
0.28
0.11
κK
26.3% 26.1% 26.4% 26.3% 26.3% 26.3% 26.3% 26.2%
κX
8.00% 8.25% 7.93% 8.00% 8.00% 8.00% 8.00% 8.05%
1/χγX − 1 0.72% 0.80% 0.69% 0.72% 0.72% 0.72% 0.72% 1.75%
Table 6: Calibrated Parameters for the Robustness Scenarios
Robustness
Changing the free parameters, we present calibration results with λ ∈ {1.5, 2.25}, θ ∈{0.5, 2}, and  ∈ {0.9, 1.2}. We also calibrated the model under the assumption that natural
1/χresources X are depleted by 1% per year; i.e., gX= 0.99. In each case, we change one

parameter from the baseline value and then re-calibrate the model. In all cases, we find theset of parameters with which the model matches all the target moments in Table 3. The
results are reported in Table 6 as scenarios (a)–(g).

7.2

Local Stability

Using parameters calibrated for the baseline setting and alternative scenarios, we can now
examine the local stability of the model. Recall that the dynamic system is characterized
by three variables {µt , kt , ct }, which evolve according to equations (41), (42) and (43). Also,
note that the initial values of µ0 and k0 are pre-determined, whereas c0 should be chosen
endogenously so that the system satisfies the transversality condition (44). In this system,
42

Eigenvalues
Scenario
StableUnstable
Baseline
0.602 0.970
1.672(a) λ = 1.5
0.667 0.957
1.479(b) λ = 2.25
0.587 0.971
1.722(c)  = .9
0.633 0.971
1.610(d)  = 1.2
0.664 0.974
1.550(e) θ = 0.5
0.496 0.969
2.034(f) θ = 2
0.692 0.971
1.454(g) gX = 0.99 0.620 0.964
1.629

BGPStability
Saddle/Determinate
Saddle/Determinate
Saddle/Determinate
Saddle/Determinate
Saddle/Determinate
Saddle/Determinate
Saddle/Determinate
Saddle/Determinate

Table 7: Eigenvalues and Local Stability of the Calibrated Model
the BGP is saddle-stable and determinate if the Jacobian matrix evaluated at the BGP has
two stable eigenvalues with absolute values less than one and one unstable eigenvalue withabsolute value greater than one.
Table 7 summarizes the results of the local stability analysis. In all cases, we find thatthe BGP is saddle-stable and determinate: when state variables are near the BGP, they will
converge to the BGP along the unique saddle path. In Subsection 6.4, we demonstrate that
one of the conditions for balanced growth, (45), is equivalent to the technology condition(22). Therefore, the saddle stability of the BGP implies that the technology condition is
endogenously satisfied as the economy converges to the BGP.
On this equilibrium path converging to the BGP, firms choose the intensities of three typesof R&D, γK , γX , and γL , and hence the direction of the technological change, to maximize
profits. The capital-augmenting technology AK is still growing on the BGP, because firms
always benefit from improving AK . This naturally explains the observed long-term decline
in the relative price of capital, which theoretically corresponds to the capital-augmenting
technological change.
The saddle stability is confirmed in all seven alternative scenarios. It demonstrates
that our explanation of capital-augmenting technological change is robust to changes inparameters, although we still need to maintain the assumption that there is a production
factor X (e.g. natural resources or land) that can be substituted with conventional capital
K with unit elasticity. It is particularly interesting to note that we find stability even when
the EoS between labor and the capital-composite, , is greater than one (scenario d). Most
directed technical change growth models require a low elasticity to be stable, especiallywhen allowing for the possibility of capital-augmenting technical change (e.g., Acemoglu,

43

6

8

k
4

2
CapitalAdjustments

Technology
Adjustments(common arm)

BGP

0

0

50

100

6

4

6

4

c

2

2
100
50
0

0

c

k 4
2
0

8
6
0

(a) Dynamic equilibrium paths from various (b) The same figure as (a), viewed from µ-c
starting points, viewed from k-c plane
plane

Figure 3: Global Stability of the Calibrated Model (Baseline Setting).
2003; Grossman et al., 2017).50

7.3

Transition Dynamics and Global Stability

Local stability only examines convergence within the neighborhood of the BGP. In thissubsection, we go one step further and demonstrate that convergence to the BGP occurs
even when the initial states are far away. We call this property global stability. With
three factors of production, this is not a trivial exercise, because the transitional dynamics
may take various patterns depending on the initial combination of µ0 and k0 .51 They are
determined by initial stock of production factors K0 , X0 and L0 , as well as initial technology
levels AK,−1 , AX,−1 , AL,−1 . Depending on the initial state of technology or resources, µ0 and
k0 will take a wide range of combinations.
To cover various possibilities, we consider a large rectangular area in µ-k plane surrounding the BGP: namely, µt ∈ [0.2µ∗ , 2µ∗ ] and kt ∈ [0.2k ∗ , 2k ∗ ]. We choose 36 points on the50

An exception is the model by Irmen and Tabaković (2017), which has an elasticity greater than one.
As explained above, their model has capital-augmenting technical change on the transition path, but not on
the steady state.
51
Values of µ0 and k0 are respectively defined in equations (A.42) and (A.43) in Appendix A.11. In
typical macro models with two factors of production, the dynamics can be written in terms of kt and ct ,where kt is the only state variable. In this case, the transition dynamics only have two possible patterns,
depending on whether k0 is higher or lower than the steady-state value. In either case, kt typically converges
monotonically to the steady state.

44

border of rectangular area and calculate the transition dynamics from each of them. We use
a forward shooting method to determine the value of c0 that eventually satisfies the transversality condition (44) as t → ∞. The equilibrium path from each starting point is depicted
in Figure 3, where the parameters are from baseline calibration in Section 7.1. Because thegraph is three-dimensional, we depicted the same graph from two angles. We also provide
the projection of the paths to the bottom µ-k plane in darker colors.
From each of 36 starting pair of µ0 and k0 , we always find a unique level of c0 such thatthe path from {µ0 , k0 , c0 } leads to the BGP (i.e., {µ∗ , k ∗ , c∗ }). If c0 is higher the resource
constraint is eventually violated (kt becomes negative), and if c0 is lower the TVC is violated(ct converges to zero). This means that convergence to the BGP is the only possible longterm outcome in equilibrium. These findings suggest that, as long as the initial µ0 and k0
is on or within the border of the rectangle, the economy necessarily converges to the BGP.
Since the rectangle is reasonably large, we call it global stability.
There are couple of properties worth observing from the figure. First, the convergence is
not monotonic. To illustrate this, let us focus on the path that starts form the upper rightcorner in Figure 3(a), as indicated by a thick arrow (µ0 = 2µ∗ and k0 = 0.2k ∗ ).52 Althoughinitial level of µ0 is double the steady-state level, µt initially increases further, going out of
the rectangular area. This phenomenon can be interpreted as follows. At the initial state, thecapital composite is abundant even though the reproducible capital is scarce. This happens
when natural resources are so abundant that it more than offsets the capital scarcity. In thissetting, the consumption of reproducible capital (i.e., the depreciation of Kt ) is small, andsavings from ample production leads to more accumulation Kt , which increases the capitalcomposite further. This process continues until the level of kt come close to the steadystate level. This is the first stage of convergence. In the second stage, the ratio of capitalcomposite to effective labor µt gradually falls to the steady state level. This is because ahigh µt means that effective labor is relatively scarce, and the firms have more incentives toimprove AL through R&D, rather than AK or AX . This tendency continues until µt reachesµ∗ . Once µt comes to µ∗ , firms have incentives to improve all types of technologies in a‘balanced’ way such that the ratio of capital composite to effective labor does not change
further. This illustrates how firms, in the long run, choose the direction of technologicalchange that satisfies the BGP condition (45), or equivalently the technology condition (22).
The figure also shows that, even though the stable manifold53 is two-dimensional, the52

In the color PDF version of the article, the path that we now focus is depicted in purple.
The stable manifold is the set of points in the (µ, k, c) space that converges to the BGP. In Figure 3,
all converging paths are on the (same) stable manifold.
53

45

equilibrium paths first converges to a common one-dimensional arm (or curve), and thenconverges to the BGP along the arm. This is because the system has two stable eigenvalues
with significantly different magnitudes. In the baseline calibration, stable eigenvalues are
0.602 and 0.970. Given that one period χ is 3.94 years, those eigenvalues means the speed ofconvergence is 12% and 0.7% per year, respectively. As we discussed in the above example(the path starting from the upper-right corner, indicated by a thick arrow), the convergenceto the BGP typically goes through two stages, and each stage corresponds to a different
eigenvalue. The initial adjustment towards the common arm is driven mainly by capital
accumulation. It is relatively fast: the distance from the common arm declines 12% every
year. However, the second stage, along the common arm, is very slow. In the baseline
example, the convergence speed is only 0.7% per year, which means it takes about 90 years
to halve the distance. This adjustment takes much longer than capital accumulation because
it is driven by the difference in the speed of technological change among AK , AX , and AL .
Note that these numbers are just for illustration, because the eigenvalues depend on thefree parameters, as shown in Table 7. Still, this result suggests that, without considering
endogenous technical change for various production factors (including capital-augmenting
technological change), neoclassical growth models may overestimate the speed of convergenceto the steady state by large margins.

8

Conclusions and Future Research

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 had to be ignored in macroeconomic models that incorporate empirically relevant value for the elasticity of substitution
between labor and capital.
This paper shows that this limitation can be overcome once we take the realistic stepof adding more factors of production. For example, Caselli and Feyrer (2007) estimate
that, among capital share of 26%, only 18% is reproducible capital (i.e., the factor that canbe accumulated by savings). The other 8% cannot fall within the category of L or K in
neoclassical growth models. We have shown that once we explicitly introduce other factors
than L and K into the model, such as natural resources and land, we do not need to ignore
capital-augmenting technological change either, as long as there is at least one production
factor that has a unitary elasticity of substitution with reproducible capital.

46

For a model with multiple factor-augmenting technologies to have a BGP, the growth ratesof these technologies need to satisfy a log-linear condition. In an endogenous growth model
where technologies are improved by profit-maximizing R&D, we show that the condition isnaturally satisfied as an long-term outcome of the equilibrium path, regardless of the initialconditions.
Implications for macroeconomic research
The generalized Uzawa growth theorem is not a mere theoretical curiosity. In standard
macroeconomic models, the counter-factual notion that there is a single type of technology
has implicitly placed severe restrictions on our ability to describe the economy. The literatureon growth accounting, for example, emphasizes the role of investment specific technicalchange. Without utilizing our results, such analyses must focus on the Cobb-Douglas case(e.g., Greenwood et al., 1997) or be inconsistent with balanced growth (e.g., Krusell et al.,2000).54 The same is true of business-cycle- and development accounting analyses (e.g.,
Fisher, 2006; DiCecio, 2009; Hsieh and Klenow, 2010; Schoellman, 2011).
In addition to improving descriptions of the economy, our results may also be useful formodeling medium-run transition dynamics. Standard models capture transition dynamics
purely through capital accumulation, but directed technical change introduces another dimension of adjustment. Our numerical example (Subsection 7.3) suggests that adjustments
among technologies are significantly slower than capital accumulation. This result implies
that shocks have highly persistent effects when they have disproportionate effects on different effective production factors, because technological adjustment will be required for the
economy to be back to the BGP (see also, Leon-Ledesma and Satchi, 2019). Existing studies suggest that the transition dynamics in standard neoclassical models are too rapid tomatch data (e.g., King and Rebelo, 1993; Banerjee and Moll, 2010). Our results indicatethat adding natural resources and directed technical change can allow researchers to build
models that match BGP data, while generating slower transition dynamics.
Models of economic growth with capital-augmenting technical change are also useful forunderstanding the relationship between technological progress and inequality (e.g., Karabarbounis and Neiman, 2014; Acemoglu and Restrepo, 2016; Hémous and Olsen, forthcoming).
An important question in this literature is how to best model the impact of technology onlabor market outcomes (Acemoglu and Autor, 2011; Acemoglu and Restrepo, 2018). There
54

See, He et al. (2008) and Maliar and Maliar (2011) for discussions of how to make these models consistent
with balanced growth.

47

is a consensus in the literature that capital- and labor-augmenting technical change have different impacts on the labor share of income. Thus, including both types of technical change
greatly increases the usefulness of these models in studying impacts of new technologies on
workers.
Estimating σKXj
To utilize the generalized Uzawa Growth theorem, we need some factor, Xj , that has aunit-elastic EoS with reproducible capital. An obvious next step for future research would
be to determine whether such a factor exists. In our illustrative growth model, we assumed
that this third factor was an amalgamation of all factors not included in reproducible capitalor labor. Proposition 4, however, only requires that at least one factor of production has
this property. Thus, it is necessary to test many different factors to see if they satisfy thisproperty. The obvious candidates would be land, energy, and natural materials, but it could
also be any subcategory of these types of inputs. Given the vast array of choices and thelimited restrictions created by theory, there is a reasonable possibility that such a factor
exists.
What if there is no Xj with σKXj = 1?
Despite the wide range of factors used in production, it is still a open question whether
there exist a production factor that has unit-elastic substitution with reproducible capital.
If future research determines there is no such factor, the generalized Uzawa Growth theorem
again implies that the speed of capital-augmenting technological change must be exactly
zero on any BGP.
This result does not fit the data on the falling relative price of investment. In thiscase, it is necessary to question whether the assumptions of the theorem hold in reality. A
remarkable property of the Uzawa theorem is that it depends on very few assumptions: (1)the economy can be expressed by a neoclassical growth model, and (2) there is a BGP. The
NIPA data strongly suggests the existence of the BGP. Therefore, we can narrow down theconcern to the assumption in the neoclassical growth model, as given in Definition 1.
The definition consists of two parts, aggregate production function Yt = F (Kt , X1,t , ..., XJ,t ; t)and resource constraint Kt+1 = Yt − Ct − Rt + (1 − δ)Kt . In the Uzawa theorem, the latter isonly utilized in Lemma 2, which showed that the K/Y ratio must be constant in the BGP.
The result of this lemma is clearly visible in the NIPA data depicted by Figure 1, where
Y and K grows at the same rate, confirming the statement that “capital inherits the trend48

of output” (Jones and Scrimgeour, 2008). Therefore, as long as an economic analysis uses
aggregate output Y and aggregated capital K, as defined in NIPA, this resource constraint
seems to do no additional harm.55
The remaining suspect, then, is the aggregate production function Yt = F (Kt , X1,t , ..., XJ,t ; t).
It assumes that there is a mapping from aggregated factor inputs to the aggregate output.
While the vast majority of all macroeconomic models use some form of aggregate production
function, it is not a weak assumption. For example, Figure 1 shows that the movement of
the relative price of capital depends on the type of capital (e.g., equipment and structure).
Even within the equipment category, the relative prices of different capital goods change
dramatically over time. The same can be said for the left hand side of the production function, i.e., aggregated output Yt . The aggregate production function implicitly assumes that
capital and output can be aggregated and that there are stable relationships between theseaggregates. If there is no Xj with σKXj = 1, it might suggest that aggregation created the
problem. Exploring disaggregated models seems important in this respect.

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52

Online Appendix“A Generalized Uzawa Growth Theorem and Capital-Augmenting Technological Change”
by Gregory Casey and Ryo Horii
January 2022

A

Proofs of Propositions and Lemmas

A.1

Notation for derivatives

Throughout this paper, 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 + jth argument. The same applies to other functions, such as Fe(·).
∂
and ∂X∂j,t represent the partial derivatives
Following the convention in economics, ∂Ktwith 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 not a single
variable. For example, using the chain rule, we have
∂ e
eX1 ,t X1,t , ..., AeX ,t XJ,t ) = A
eX ,t FeX (Kt , A
eX1 ,t X1,t , ..., AeX ,t XJ,t ).
F (Kt , A
j
j
J
J
∂Xj,t

A.2

(A.1)

Proof of Lemma 1

Note that δ = (δ̆ + gq − 1)/gq means 1 − δ̆ = (1 − δ)qt+1 /qt . Dividing equation (3) by qt+1
and using the above result, we have Kt+1 = K̆t+1 /qt+1 = (Yt − Ct − Rt ) + (1 − δ̆)K̆t /qt+1 =(Yt − Ct − Rt ) + (1 − δ)Kt , which coincides with (2).
Production function F̆ (K̆t , X1,t , ..., XJ,t ; t) = F̆ (qt−1 Kt , X1,t , ..., XJ,t ; t) is a CRS functionof Kt , X1,t , ..., XJ,t and depends on time both through the shape of F̆ (·; t) and through thegrowth of qt . Therefore, we can define function F (Kt , X1,t , ..., XJ,t ; t) ≡ F̆ (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 F̆ (·),
function F (·; t) obviously satisfies the required marginal product properties in (1).

A-1

A.3

Proof of Lemma 2

t+1
Using the notation in Definition 2, equation (2) can be written as K0 gK= Y0 g t − C0 gCt −t
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 exponentialfunctions of t, every one of them needs to be constant for the sum of all the terms tobecome constant (Y0 ). For the first term C0 (gC /g)t to be constant, gC = g must holdsince 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 and gK > 1 − δ. Thisimplies Kt /Yt = K0 /Y0 > 0. If R0 > 0, gR = R must hold since otherwise the second term
cannot be constant.

A.4

Proof of Proposition 2

Because the production function in period 0 is F (·; 0) ≡ Fe(·), we can write the share offactor 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 (seeAppendix Section A.1). Note that, since function Fe(·) has constant returns to scale, itspartial derivative function FeZ (·) must be homogeneous of degree 0 (See Theorem M.B.1 in
Mas-Colell et al., 1995). Therefore, the value of FeZ (·) will be unchanged when all of its
eX ,t Xj,t /Xj,0 . (Herearguments are multiplied by the same factor g t = Yt /Y0 = Kt /K0 = Aj
we used gK = g from Lemma 2.) Applying this for (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 at theeZ,t Zt /Yt holds on the BGP. (In the case of Zt = Kt , we definesame speed, Z0 /Y0 = A
eK,t ≡ 1.) Therefore,AeeeeX1 ,t X1,t , ..., AeX ,t XJ,t )AeZ,t Zt = ∂ F (Kt , AX1 ,t X1,t , ..., AXJ ,t XJ,t ) Zt ,sZ,0 = FeZ (Kt , A
J
Yt
∂Zt
Yt

A-2

(A.4)

where the validity of the second equality is guaranteed by the chain rule.56 Recall that weassumed that the share is constant over time, which means
sZ,0 = sZ,t = FZ (Kt , X1,t , ..., XJ,t ; t)

Zt
.
Yt

(A.5)

By comparing (A.4) and (A.5), we obtain (5).

A.5

Proof of Proposition 3

As in Definition 3, the EoS between Kt and Xj , j ∈ {1, ..., J}, in the Uzawa Representation
eX1 ,t X1,t , ..., AeX ,t XJ,t ) is defined as
Fe(Kt , AJ

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

j

1

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
eX1 ,t , ..., AeX ,taddition to output Yt and other production factors X−j,t , we keep technologies A
J

fixed when calculating the EoS.
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 thespecific BGP values, and k and xj the variables to be perturbed. Then, (A.6) can be written
as57
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 move
Condition Fe(k, AjJ
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, k in (A.7) isNow, consider a change of variables: k 0 = g −t k and x0j = g −t A
j56

See Appendix A.1.
We omit condition “X−j,t : const” because we already made it clear that Xj,t ’s are the BGP values, notvariables.
57

A-3

eX ,t )x0 . Specifically, k/xj in the numerator becomes
replaced by k = g t k 0 and xj is by (g t /A
j
j
0
0
e
AX ,t k /xj . In the denominator,jeX1 ,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 ),eX ,t ≡ g t Xj,0 /Xj,t and the homogeneity of degree 0 property
where we used the definition of Aj58
of the FeK (·) 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 point ofevaluation, k = Kt , becomes g t k 0 = Kt , or k 0 = g −t Kt = K0 . Similarly, xj = Xj,t becomeseX ,t Xj,t = Xj,0 . Therefore, (A.7) can be expressed in terms of k 0 and x0 as follows:
x0 = g −t AjjjeX ,t k 0 /x0 )d ln(Ajj

σeKXj ,t = −d ln

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

j

j

.



(A.8)

Fe(k0 ,X1,0 ,...,x0j ,...,XJ,0 )=Y0
k0 =K0 ,x0j =Xj,0eX ,t fixed when calculating the EoS. We can eliminate
Recall that we keep technology AjeeeX ,t ) = d ln(k 0 /x0 ). In
AX ,t from the numerator from d ln(AX ,t k 0 /x0 ) = d(ln(k 0 /x0 ) + ln A
j

j

j

j

j

j

eX ,t in the denominator can also be eliminated (or replaced by AX ,0 ≡ 1).
the same way, Ajje
Finally, using F (·) ≡ 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 does58

For the homogeneity of degree 0 property, see the proof of Proposition 2 in appendix A.4.

A-4

not change over time, we have σ
eKXj ,t = σKXj ,0 = σKXj ,t .

A.6

Proof of Lemma 3

Proof of part a
By substituting K0 , X1,0 , ..., XJ,0 into (11) and then using (12),


Yj ∗
ξj
α
b
F (K0 , X1,0 , ..., XJ,0 ) = F K0
Xj,0 , Xj ∗ +1,0 , ..., XJ,0
j=1

=F

Y ∗
j
j=1

ξ
Xj,0j

!
−1/α 
1/α
Yj ∗
ξ
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, and FbM (·)
denote the derivative of function Fb(·) with respect to its first argument. By differentiating
both sides of (12) by m with the chain rule and substituting the period-0 BGP values fork, xj ∗ +1 , ..., xJ ,FbM (M0 , Xj ∗ +1,0 , ..., XJ,0 ) = FK (K0 , X1,0 , ..., XJ,0 ; 0)

Y ∗
j
j=1

ξ
Xj,0j

−1/α

1 (1−α)/αM
α 0

K0= FK (K0 , X1,0 , ..., XJ,0 ; 0),αM0where the last equality follows from the definition of M0 = K0α(A.10)Qj ∗ξj
j=1 Xj,0 . Now, consider thecase of Z = K. By differentiating both sides of (11) 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),
where the last equality is from (A.10). Similarly, for the case of Z = Xj , where j ∈ {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,0A-5(A.11)

Note that, from the definitions of α and ξj in (10), 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 the proof ofpart a, we can confirm that F (K0 , X1,0 , ..., Xj ∗ ,0 , xj ∗ +1 , ...xJ ) = F (K0 , X1,0 , ..., Xj ∗ ,0 , xj ∗ +1 , ..., xJ ; 0)for any xj ∗ +1 , ..., xJ . This means that they are identical functions of xj ∗ +1 , ..., xJ , and havethe same derivatives with respect to these variables. Therefore, for j ∈ {j ∗ + 1, ..., J}, wehave 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, isdefined 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 (11), 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, defined as m =ξ 0
ξ Q
k α xjj j 0 ∈{1,...,j ∗ }\j Xj 0j,0 . Using the chain rule, its derivative with respect to k becomes∂ b
F (m, Xj ∗ +1,0 , ..., XJ,0 )∂k∂m= FbM (m, Xj ∗ +1,0 , ..., XJ,0 )∂kmb= FM (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
k=K0 ,xj =Xj,0

A-6(A.13)

Since 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 (11)and (12) 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). This means σ KXj ,0 =σKXj ,0 .
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 (12),Fb(λm, λxj ∗ +1 , ..., λxJ )
!
Y ∗
−1/α
j
ξ
j
m1/α , X1,0 , ..., Xj ∗ ,0 , λxj ∗ +1 , ..., λxJ ; 0= F λ1/α
Xj,0(A.14)

j=1= λF(1−α)/α

λ

Y ∗
j

j=1

ξ
Xj,0j

−1/α

!

m1/α ,

X1,0
Xj ∗ ,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 that the last line of(A.14) coincides with λFb(m, xj ∗ +1 , ..., xJ ). Using (12), this desired condition 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 and x1 .
From Definition 3, σKX1 = 1 means that equation d ln(FK /FX1 )/d ln(k/x1 ) = −1 holds
A-7when the output and other inputs are kept constant.59

In other words, this differen-tial equation is satisfied on the isoquant curve in the k-x1 space. Integrating equationd ln(FK /FX )/d ln(k/x1 ) = −1 gives ln(FK /FX ) = − ln(k/x1 ) + ξe1 , where ξe1 is a constant1

1of integration. Taking the exponential of the both sides gives
FK /FX1 = (exp ξe1 )(x1 /k).

(A.16)

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 the isoquantcurve as dx1 /dk = −FK /FX = −(exp ξe1 )(x1 /k). Integrating this differential equation by1

separation of variables gives ln k = −(1/ exp ξe1 ) ln x1 + ye1 , where ye1 is another constant ofintegration.60 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 share between k and x1
is written as kFK /x1 FX = exp ξe1 . The result does not depend on k or x1 , which means that1

the relative share is constant on the isoquant curve. Also, notice that the value of ξe1 must
be the same across all isoquant curves, since otherwise they intersect with each other, which
is impossible by the definition of the isoquant curve. From (10), 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, consider
Q ∗
−1/α
ξj
j
a particular isoquant curve that goes through k =X
m1/α and x1 = X1,0 ,
j,0
j=1
Q ∗
−1/α
ξj
ξ1 /α
j
. From (A.18), we can confirm that thiswhich means exp ye1 =X
m1/α X1,0
j,0
j=1
Q ∗
−1/α
ξj
j
0
ξ1 /α
isoquant curve also goes through k = λm1/α and x01 = X1,0 /λ.61 Since
j=1 Xj,059

To minimize notation we omit the arguments of the functions FK (k, x1 , ..., xJ ; 0) andFX1 (k, x1 , ..., xJ ; 0).
60This 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 equation sepR
R
R
R
arately gives (1/k)dk = −(1/ exp ξe1 ) (1/x1 )dx1 . Since (1/k)dk = ln k + constant and (1/x1 )dx =ln x1 + constant, we obtain the result in the text.
61
This can be confirmed by substituting k 0 and x01 into (A.18). It yields the same exp ye1 as in the previous
sentence.

A-8

the output is the same on an isoquant curve, we have
F

Y ∗
j

=F

j=1

λξ1 /α

ξ

−1/α

Xj,0j
Y ∗
j

j=1

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

−1/α

Xj,0j

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

Pj ∗j=1 ξj = 1 − α from (10), 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 (11),


Yj ∗
ξj
α
b
(λxj ) , λxj ∗ +1,0 , ..., λxJ,0F (λk, λx1 , ..., λxJ ) = F (λk)
j=1Yj ∗ ξjαb= F λkxj , λxj ∗ +1,0 , ..., λxJ,0
j=1
 Y∗

j
ξj
α
b
= λF kxj , xj ∗ +1,0 , ..., xJ,0j=1

= λF (k, x1 , ..., xJ ).
P
The second equality utilizes α + j∗
j=1 ξj = 1 from (10), whereas the third equality is fromthe CRS property of function Fb(·).

A.7

Proof of Proposition 4

Using (11), the RHS of equation (14) 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 .

(A.20)

j=1

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 (13). Also, all the otherarguments of Fb(·) are multiplied by g each period because it is assumed that γX = g/gXjj∗

for j ∈ {j + 1, ..., J}. Since Fb(·) has CRS from property d of Lemma 3, (A.20) is multiplied
by g each period.
Also, the LHS of (14), Yt , is multiplied by g every period by the definition of the BGP.
In period 0, (14) holds from property a of Lemma 3. Therefore, (14) holds for all t ≥ 0.

A-9

A.8

Proof of Proposition 5

The proof relies on Lemma 3, 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 3 and (11), theshare of factor K in period 0 can be written as (11),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, inQ∗ξ(A.21), the first argument is the capital composite in period 0, M0 = 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 ),dK0 Y0Y0(A.22)

where the second equality follows from dM0 /dK0 = αM0 /K0 .
Recall that Fb(·) has CRS from Lemma 3, and therefore its derivative FbM (·) is a hoQ∗
mogeneous function of degree 0. Let Mt = (AK,t Kt )α jj=1 (AXj ,t Xj,t ) denote the effective amount of capital composite in period t. From condition (13), Mt grows by a factor 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, whichJ

j +1

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

 αM0
.
Y0

(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 the latter expression repQ∗
resents dMt /dKt , which can be confirmed by differentiating Mt = (AK,t Kt )α jj=1 (AXj ,t Xj,t )A-10

by Kt . Therefore, (A.23) becomes dMt Kt
sK,0 = FbM Mt , AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,tdKt Yt


Yj ∗
Kt
∂ b
α
(AXj ,t Xj,t ), AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,tF (AK,t Kt )
=
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 of function
F (·) in (11). 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 (16) for the case of Zt = Kt . The proof of 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 similar way as ineX ,t in the Uzawa theorem. In parProposition 2, because the value of AX ,t is the same as Ajj

eX1 ,t X1,t , ..., AeX ,t XJ,t ),ticular, we use Fb(Mt , AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t ) instead of Fe(Kt , AJ∗bband define FX (·), j ∈ {j , ..., J}, as the derivative of function F (·) with respect to its
j∗(j − j + 1)th argument.62 Except for these slight modifications, the proof proceeds exactly
as in Appendix A.4.

A.9

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 to be
perturbed.63
62 b
FXj (·) needs to be defined this way because j ∗ arguments are eliminated from function Fb(·) in definition(12). Also, note that similarly to function Fe(·), function Fb(·) has a CRS property from Lemma 3.
63
In definition (A.26), condition F (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t ) = Yt means that k and xj
are perturbed so that output Yt is unchanged from the BGP value. Condition k = Kt , xj = Xj,t says that

A-11

Let us first consider the case of j ∈ {1, ..., j ∗ }. In this case, factors Xj ∗ +1,t , ..., XJ,t are fixedat the BGP values. Now, we simplify the denominator of (A.26), particularly focusing on
the fraction inside ln(·). Using (11), function F (AK,t k, AX1 ,t X1,t , ..., AXj ,t xj , ..., AXJ ,t XJ,t ) canbe written as Fb(m, AXj∗ +1 ,t Xj ∗ +1,t , ..., AXJ ,t XJ,t ), where m is the effective amount of capitalQ
composite, m = (AK,t k)α (AXj ,t xj )ξj j 0 ∈{1,...,j ∗ }\j (AXj0 ,t Xj 0 ,0 )ξj0 . Note that dm/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,

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 proof of
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 as in (A.27), AK,t F K (·)
the EoS is evaluated at the BGP values. It is also important to keep in mind 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, whichcorresponds to FK (·; t) in Definition 3.

A-12

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)αQj ∗j 0 =1 (AXj 0 ,t Xj 0 ,0 )

ξj 0(A.30). The definition of m does not include xj becausej ∈ {j ∗ + 1, ..., J} means that xj is not a part of capital composite. Instead, AXj ,t xj appearsin (A.30) as the (j − j ∗ + 1)th argument of the Fb(·) function. Using xj = (g t /AXj ,t )x0j ,Q∗
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 )αQj ∗j 0 =1 (AXj ,0 Xj 0 ,0 ).
0
∗
0

α

α

Mt = (k 0 /K0 ) g t M0 = g t m0 ,Other effective factors also grow by a factor of g:

AXj0 ,t Xj 0 ,t = g t Xj 0 ,0 for j ∈ {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
gk0m= 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 the last equality isfrom the definition of F (·) in (11).
Likewise, AXj ,t F Xj (·) in the denominator of (A.26) can be expressed in terms of k 0 and

A-13

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.64 We alsoneed 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 expressedin terms of k 0 and x0j as follows:d ln(AXj ,t k 0 /x0j )σ KXj ,t = −


d ln

64

F K (k0 ,X1,0 ,...,x0j ,...,XJ,0 )AXj ,t F Xj (k0 ,X1,0 ,...,x0j ,...,XJ,0 )I.e., the conditions that are written to the right of “|”.

A-14

.


F (k0 ,X1,0 ,...,x0j ,...,XJ,0 )=Y0
k0 =K0 ,x0j =Xj,0(A.31)

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 can also 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 3, σ KXj ,0 = σKXj ,0 holds
in period 0. In addition, it is assumed that σKXj ,0 does not change over time. Therefore,σ KXj ,t = σ KXj ,0 = σKXj ,0 = σKXj ,t .

A.10

Proof of Proposition 7

Proof of (c), as well as existence and uniqueness of γbL (µt )As explained in the main text, the representative firm chooses γL,t so as to satisfy
R&D for N-tasks: γL,t i0L (γL,t ) + iL (γL,t ) = f (µt ) − µt f 0 (µt ).

(38)

Let us denote the LHS of (38) by ΨL (γL,t ) because it depends only on γL,t . Then, Ψ0L (γL,t ) =γL,t i00L (γL,t ) + 2i0L (γL,t ) > 0 for all γL,t > 1 from iL (γL,t ) > 0 and i0L (γL,t ) > 0 in (26).
When γL,t = 1, the properties of iL (·) imply ΨL (1) = i0L (1) + iL (1) = 0. Also, ΨL (∞) ≡limγL,t →∞ ΨL (γL,t ) = ∞ from iL (∞) = ∞ and γL,t i0L (γL,t ) > 0.65 Then, since ΨL (·) is(−1)

differentiable and strictly increasing, we can define its inverse function ΨL(·), which is

(−1)
(−1)
also differentiable and strictly increasing with ΨL (0) = 1 and ΨL (∞) = ∞.

Using thisfunction, condition (38) can be solved for γL,t :(−1)

γL,t = ΨL(f (µt ) − µt f 0 (µt )) ≡ γ
bL (µt ).

(A.32)

Note that f (µt ) − µt f 0 (µt ) represents the marginal product of Nt in the production
function, i.e., FbN (µt , 1). We can express the production function Yt = Fb(Mt , Nt ) in
an intensive form with respect to Nt /Mt ≡ νt , instead of µt = Mt /Nt . Namely, output per Mt can be expressed as Yt /Mt = Fb(Mt , Nt )/Mt = Fb(1, νt ) ≡ h(νt ). Since
Fb(·) is CRS, its first derivative FbN (·) is homogeneous of degree 0.

Using this prop-

erty, h0 (ν) = FN (1, Nt /Mt ) = FN (Mt /Nt , 1) = FN (µt , 1) = f (µt ) − µt f 0 (µt ). From
the definition of the production function Fb(·), its alternate intensive form, h(νt ), satisfies the Inada conditions.

Therefore, limµt →0 f (µt ) − µt f 0 (µt ) = limνt →∞ h0 (νt ) = 0,

and limµt →∞ f (µt ) − µt f 0 (µt ) = limνt →0 h0 (νt ) = ∞. Substituting these into (A.32) gives
65

Similarly to the main text, we employ an abuse of notation by writing iL (∞) to represent
limγL →∞ iL (γL ). We will employ similar abbreviations as long as they cause no confusion.

A-15

(−1)

γL (0) = ΨL(−1)(0) = 1 and γL (∞) = ΨL(∞) = ∞.

Finally, we show γL0 (µt ) > 0. The derivative of f (µt ) − µt f 0 (µt ) with respect to µt is−µt f 00 (µt ). It is positive for all µt > 0 since the production function satisfies the Inada(−1) 0conditions, which include f 00 (µt ) < 0. Since ΨL (·) > 0, this means γL0 (µt ) > 0.
Proof of (a) and (b), as well as existence and uniqueness of γbK (µK ) and γbX (µX )The representative firm chooses γK,t and γX,t according to the following two conditions:
R&D allocation:

αγK,t i0K (γK,t )=,0
γX,t iX (γX,t )1−αα ∈ (0, 1),

Combined R&D: (γK,t i0K (γK,t ) + iK (γK,t )) + (γX,t i0X (γX,t ) + iX (γX,t )) = f 0 (µt ).

(39)
(40)

Let us define ΩK (γK,t ) ≡ γK,t i0K (γK,t ) and similarly ΩX (γX,t ) ≡ γX,t i0K (γX,t ). Then, fromproperties in (26), we can confirm Ω0K (γK,t ) > 0 for γK,t > 1, ΩK (1) = 0 and ΩK (∞) =∞. Similar conditions hold also for ΩX (·). Then, since ΩX (·) is differentiable and strictly(−1)

increasing, we can define its inverse function ΩX (·), which is also differentiable and strictly
(−1)

(−1)

increasing with ΩX (0) = 1 and ΩX (∞) = ∞. Using this inverse function, condition (39)can be solved for γX,t as(−1)
γX,t = ΩXαΩK (γK,t ) ≡ Ω(γK,t ).
1−α(A.33)

Now let us focus on condition (40). Let us define ΨK (γK,t ) ≡ γK,t i0K (γK,t ) + iK (γK,t ) andlikewise ΨX (γX,t ) ≡ γX,t i0K (γX,t ) + iK (γX,t ). Using these and (A.33), the LHS of condition(40) can be expressed as a function only of γK,t :ΨK (γK,t ) + ΨX (Ω(γK,t )) ≡ Ψ(γK,t ).
(−1)

Note that the properties of ΩK (·) and ΩX (·) imply that Ω(γK,t ) > 0 for all γK,t > 1,Ω(0) = 0 and Ω(∞) = ∞. Also, in the same way that we derived the properties of ΨL (γL,t )earlier in this proof, we can confirm ΨK (γK,t ) > 0 for all γK,t > 1, ΨK (1) = 0, ΨK (∞) = ∞,and similar properties for ΨX (γX,t ). From these, we have Ψ(γK,t ) > 0 for all γK,t > 1,
Ψ(1) = 0, Ψ(∞) = ∞. On the RHS of (40), f 0 (µt ) satisfies the usual Inada conditions. The

A-16

results we have obtained so far can be summarized as
1 ···

γK,t
Ψ0 (γK,t )Ψ(γK,t )∞µt
f 00 (µt )

+
0 %

0

0

∞

f (µt )

···

∞

−
∞ &

0

The tables above implies that condition (40), Ψ(γK,t ) = f 0 (µt ), gives a 1 to 1 correspondence0
between µt ∈ (0, ∞) and γK,t ∈ (1, ∞) that satisfies property (a): γ
bK(µt ) < 0 for all µt > 0,γbK (0) = ∞, and γbK (∞) = 1.
Given γbK (µt ), equation (A.33) uniquely determines γX,t = Ω(bγK (µt )) ≡ γ
bX (µt ). From the0properties of Ω(·) and γbK (·) above, we can confirm that property (b) is satisfied: γbX(µt ) < 0for all µt > 0, γ
bX (0) = ∞, and γbX (∞) = 1.

A.11

Derivation of Equilibrium Dynamics

In this section, we derive the equilibrium dynamics of the endogenous growth model in
Section 6. To do so, it is convenient to define the net aggregate output in the economy as
Vt = Fb(Mt , Nt ) − RK,t − RX,t − RL,t , which is aggregate output minus total R&D costs inthe economy.
The net output per effective labor can be written as a function of µt :Vt /Nt = f (µt ) − µt (iK (bγK (µt )) + iX (bγX (µt ))) − iL (bγL (µt )) ≡ v(µt ).

(A.34)

Then, substituting profits (28) into the budget constraint (33), we can express the growthof aggregate capital supply in terms of µt , kt and ct :Vt + (1 − δ)Kt − Ctv(µt ) − ct
Kt+1==
+ 1 − δ.
Kt
Kt
kt
Dynamicsforµt+1 .

(A.35)The growth factor of µt+1 is defined by µt+1 /µt=(Mt+1 /Mt )/(Nt+1 /Nt ). By using (23), (24), (29), (32) and (A.35), its value in equilibrium
can be written as
µt+1
(gX γ
bX (µt+1 ))1−α
=
µt
gL γ
bL (µt+1 )αv(µt ) − ctγbK (µt+1 )
+1−δ
,
kt

(A.36)

where γbK (µt ), γbX (µt ), and γbL (µt ) are the rates of technological progress defined in Proposition 7. While equation (A.36) gives a relationship between the period-t variables {µt , kt , ct }A-17

and µt+1 , it is not easy to understand how µt+1 is determined since both sides of the equation
depend on µt+1 .
To interpret it intuitively, let us decompose the dynamic relationship in (A.36) into two
steps. First, we define the pre-R&D relative factor intensity by
µpre
t+1 ≡

α
1−α
α
1−α 
AK,t Kt+1 AX,t Xt+1
gX
v(µt ) − ct=
+ 1 − δ µt ,AL,t Lt+1
gL
kt(A.37)

where the last equality is from (29), (32), (A.35) and the definition of µt . It is the valueof µt+1 before technologies are improved from their period-t state. Second, µpret+1 and thepost-R&D value of µt+1 are related by the growth of technological levels γ
bK (µt ), γbX (µt ), andγbL (µt ) as follows:
µpre
t+1 =

γ
bL (µt+1 )µt+1 ≡ Γ(µt+1 ).
γbK (µt+1 )α γbX (µt+1 )1−α(A.38)

Note that Proposition 7 implies that function Γ(µt+1 ) is a strictly increasing differentiable
function with limµ→0 Γ(µ) = 0 and limµ→∞ Γ(µ) = ∞. Therefore, its inverse function iswell-defined for all µpret+1 > 0, and is a strictly increasing differentiable function. We write
this inverse function by µt+1 = G(µpret+1 ). Combined with (A.37), the dynamic relationship(A.36) can be written as (41).
Dynamics for kt+1 . From (29) and (A.35), the growth factor of kt ≡ Kt /AL,t Lt is given by
kt+1
1
=
kt
gL γ
bL (µt+1 )
v(µt ) − ct
+1−δ .
kt(A.39)

While µt+1 is present in the RHS, we can replace it with (41) so that the RHS depends only
on the variables in period t, which gives (42).
Dynamics for ct+1 . The representative household maximizes the intertemporal utility
function (31) subject to the budget constraint (33) and the no-Ponzi game condition (34).
From (31), ∂U/∂Ct = β t (Ct /Lt )−θ . Therefore, the Euler equation for this problem is(Ct /Lt )−θ = (rt+1 + 1 − δ)β(Ct+1 /Lt+1 )−θ , which simplifies to−θCt−θ = (rt+1 + 1 − δ)βgLθ Ct+1.

(A.40)

By substituting the market interest rate (35) into the Euler equation (A.40) and then applying it to the definition ct ≡ Ct /AL,t Lt , we obtain the growth factor of consumption per

A-18

effective labor:
ct+1
β 1/θ
=
ct
γ
bL (µt+1 )αµt+1 0(f (µt+1 ) − iK (bγK (µt+1 )) − iX (bγX (µt+1 ))) + 1 − δ
kt+1

1/θ
.

(A.41)

By replacing the period-(t + 1) variables in the RHS by (41) and (42), we can rewrite equation(A.41) as (43).
Boundary Conditions. To obtain the equilibrium path of {µt , kt , ct }∞t=0 , we need threeboundary conditions. First, since K0 , X0 , L0 , AK,−1 ,AX,−1 , and AL,−1 are given, we canconstruct µpre0 , the pre-R&D relative task intensity for period 0. Using it with the inverse
function of Γ from (A.38), we have the initial value of µt :
µ0 = G(AK,−1 K0 )α (AX,−1 X0 )1−α
AL,−1 L0

.

(A.42)

Second, using µ0 , the initial value of kt is readily obtained by
K0.
γ
bL (µ0 )AL,−1 L0

k0 =(A.43)

Finally, the initial value of ct must be chosen so as to satisfy the no-Ponzi game condition(34) and the transversality condition
lim β

T



T →∞

CT
LT

−θ
KT +1 ≤ 0.

(A.44)In the remainder of this section, we establish that conditions (34) and (A.44) jointly
mean (44). The Euler equation (A.40) implies that rt + 1 − δ = (Ct−1 /Lt−1 )−θ /β(Ct /Lt )−θ .
Through repeated multiplication,TY(rt + 1 − δ) =t=1

(C0 /L0 )−θ
.
β T (CT /LT )−θUsing this, the no-Ponzi game condition (34) becomes


C0
L0

θ
lim β

T →∞

T



CT
LT

−θ
KT +1 ≥ 0.

(A.45)

Since C0 /L0 > 0, we can divide the both sides of (A.45) by (C0 /L0 )θ to eliminate this term.
Then, it turns out that (A.45) has exactly the same form as the transversality condition
A-19

(A.44), except that the direction of the inequality is opposite. By combining (A.44) and(A.45), therefore, we have a unified terminal condition
lim β

T

T →∞



CT
LT

−θ
KT +1 = 0.

(A.46)

Using ct ≡ Ct / (AL,t Lt ) and kt ≡ Kt / (AL,t Lt ), the expression on the LHS can be written as
β

T



CT
LT

−θ

KT +1 = β T (AL,T cT )−θ AL,T +1 LT +1 kT +1= β T (AL,T cT )−θ AL,T γ
bL (µT +1 )L0 gLT +1 kT +1−θ= L0 gL (βgL )T A1−θbL (µT +1 )kT +1
L,T cT γ
!1−θ
T
Y
c−θ
= L0 gL AL,−1 (βgL )Tγ
bL (µt )bL (µT +1 )kT +1 .
T γ
t=0By substituting the last expression into (A.46) and then dividing both sides by L0 gL AL,−1 >0, we obtain (44).66

A.12

Proof of Lemma 4

Consider a BGP. We will show in turn that µt , kt and ct must be constant. First, from thedefinition of a BGP, Nt+1 /Nt = (AL,t+1 Lt+1 )/(AL,t Lt ) = γbL (µt+1 )gL is constant. To keep
the RHS of the latter equation constant, µt must also be constant, since γbL (·) is a strictlyincreasing function from Proposition 7.
Second, since growth factors of Ct and Nt are constant, the growth factor of ct =Ct /At Lt = Ct /Nt is also constant. This, in turn, means that the LHS of the Euler equation(A.41) is constant. Then, for the RHS of (A.41) to be constant, kt must be constant, since
we already know that µt is constant as shown above.
Third, the growth factor of kt = Kt /At Lt = Kt /Nt is constant on the BGP, which means
the LHS of (A.39) is constant. For its RHS to be constant, given that µt and kt are alreadyshown to be constant, ct also needs to be constant.
66

Initial population L0 and initial level of technology AL,−1 > 0 are exogenously given (see Subsection
6.3). Population growth factor gL > 0 is also exogenous.

A-20

A.13

Proof of Proposition 8

Proof of µ∗ > 0
In the text, we have already shown that there exists a unique µ∗ > 0 such that Φ(µ∗ ) = 1
holds, since Proposition 7 implies Φ0 (µ∗ ) < 0 with Φ(0) = ∞ and Φ(∞) = 0. Therefore,
there exists a unique value of µ∗ > 0.
Proof of k ∗ > 0
The value of k ∗ is explicitly given by equation (47), shown again here:k∗ =βαµ∗ (f 0 (µ∗ ) − iK (bγK (µ∗ )) − iX (bγX (µ∗ ))).
γbL (µ∗ )θ − β(1 − δ)(47)

We now show that both the numerator and the denominator of the RHS are positive. Notethat the combined R&D condition (40) is satisfied on the BGP. By rearranging terms, itgives
f 0 (µ∗ ) − iK (bγK (µ∗ )) − iX (bγX (µ∗ )) = γbK (µ∗ )i0K (bγK (µ∗ )) + γ
bX (µ∗ )i0X (bγX (µ∗ )) > 0,

(A.47)

where the inequality follows from Proposition 7 and (26). Given β ∈ (0, 1), α ∈ (0, 1), andµ∗ > 0, this means that the numerator of (47) is strictly positive. Now, note that γbL (µ∗ ) > 1
from Proposition 7. Combined with θ > 0, β ∈ (0, 1) and δ ∈ [0, 1], it turns out that thedenominator of (47) is also strictly positive.
Proof of c∗ > 0
The value of c∗ is given byc∗ = v(µ∗ ) − (g ∗ − 1 + δ)k ∗ .

(48)

We first show v(µ∗ ) > 0. Combining the R&D conditions (38) and (40), we have
γL,t i0L (γL,t ) + iL (γL,t ) + µt ((γK,t i0K (γK,t ) + iK (γK,t )) + (γX,t i0X (γX,t ) + iX (γX,t ))) = f (µt ).
Rearranging and then evaluating this condition at µt = µ∗ givesv(µ∗ ) = f (µ∗) − iL (bγL (µ∗ )) − µ∗ (iK (bγK (µ∗ )) − iX (bγX (µ∗ )))γK (µ∗ )) + γ
bX (µ∗ )i0X (bγX (µ∗ ))) > 0,=γ
bL (µ∗ )i0L (bγL (µ∗ )) + µ∗ (bγK (µ∗ )i0K (bA-21

where the inequality follows from µ∗ > 0 and (26).
Note that g ∗ = γbL (µ∗ )gL is greater than 1 − δ because γbL (µ∗ ) > 1 and gL > 1 − δ from
(29). Therefore, (g ∗ − 1 + δ) in (48) is positive. From this, c∗ > 0 is equivalent tok∗ <v(µ∗ ).
g∗ − 1 + δ

Using (47), we can rewrite this condition in terms of β:∗ θβ<γbL (µ )

αµ∗(f 0 (µ∗ ) − iK (bγK (µ∗ )) − iX (bγX (µ∗ ))) (g ∗ − 1 + δ) + 1 − δv(µ∗ )

−1≡ β 1.
(A.48)

Note that β 1 > 0 from (A.47) and g ∗ > 1 − δ > 0. Observe also that β 1 does not depend on
β itself since µ∗ is determined entirely by the production side (see equation 45). Therefore,
if β > 0 is sufficiently small, condition (A.48) holds and c∗ > 0.
Terminal Condition
On the BGP, the terminal condition (44) becomeslim βgL γ
bL (µ∗ )1−θ

T →∞

T

γ
bL (µ∗ )(c∗ )−θ k ∗ = 0.

(A.49)

Given that γbL (µ∗ ) > 1, c∗ > 0 and c∗ > 0, this condition is equivalent toβ<1≡ β 2.
gL γ
bL (µ∗ )1−θ(A.50)

Note that β 2 > 0 and that it does not depend on β since µ∗ is determined entirely by the
production side of the model. Therefore, if β > 0 is sufficiently small, condition (A.50) holds
and the terminal condition (44) is satisfied.
Combining conditions (A.48) and (A.50), we have confirmed the unique existence of BGP
with µ∗ > 0, k ∗ > 0, c∗ > 0, and the terminal condition (44) wheneverβ < β ≡ min{β 1 , β 2 },where β > 0 is a constant that does not depend on β.

A-22(A.51)

A.14

Calibration Procedure

There are seven parameters to calibrate, {δ, β, α, η, ζL , ζX , χ}, which we identify with theseven moments in Table 3. We do so in two steps.
Step 1: Analytical calibration. Given period length χ, exogenous parameters, and moments, we analytically derive the values of four parameters {δ, β, α, η}. The depreciation
rate is determined by data on the consumption of fixed capital and the capital-output ratio:
δ=

δK/Y
0.14
=
χ ≡ δ(χ).
K/Y
2.9(A.52)

Evaluating the Euler equation (A.40) on the BGP gives the discount factor β:67
β=(1.019χ )1.0γLθ=≡ β(χ).
1+r−δ
1.04χ

(A.53)

Similarly, the first order conditions of the representative firm, (35) and (36), gives the shareparameter α:α=κK(r − δ)(K/Y ) + δK/Y
(1.04χ − 1)(2.9χ) + 0.14==
≡ α(χ),
κK + κX
1 − κL − κR
1 − 0.63 − 0.027(A.54)

where we used the identity κK + κX + κL + κR = 1.
Next, BGP relationship (45), which is equivalent to the technology condition (22), gives
the growth rate for the unobserved endogenous variable γX :− α gL γL
γX = γK 1−α
=
gX


χ
α(χ)
− 1−α(χ) (1.01)(1.019)(1.066)≡ γ X (χ).
1.0

(A.55)

Using (A.55) and the R&D allocation condition (39), the R&D cost parameter ζX can bederived as follows.
χ

1 − α ζK γK (γK − 1)λ−1
1 − α(χ) (1(1.0066)(0.0066)2.0−1 )
ζX =
·
=
·
≡ ζ X (χ).
αγX (γX − 1)λ−1α(χ)γ X (χ)(γ X (χ) − 1)2.0−1(A.56)

Step 2: Minimization. Among the 12 parameters of the model, five of them are given by
Table 2, and four are given as functions of χ, in (A.52), (A.53), (A.54) and (A.56). This
leaves us with three remaining parameters, {ζL , η, χ}. We calibrate them so as to minimize
67

Equation (A.53) assumes that parameter θ takes the baseline value of 1.0. When doing robustness
checks with θ = 0.8 and 1.2, the numbers in this equation are adjusted accordingly. The same applies for(A.54)-(A.57) when using alternative parameter values or calibration targets.

A-23

the squared sum of percent difference (error) between the target moments in Table 3 andthe corresponding model variables on the BGP.
Let us define the squared sum of percent error as68

SSE =
+

K/(Y /χ) − 2.9
2.9

2



κL − 0.63
0.63

+
!2
1/χ
γL − 1 − 0.019
+
0.019

2



κR − 0.027
+
0.027
!2
1/χ
γK − 1 − 0.0066.
0.0066

2(A.57)In (A.57), endogenous variables K/Y , κL , κR , γL and γX represent their respective BGP
values, when the model is solved given all 12 parameters. Using exogenous parameters andthe results of analytical calibration, we determine the remaining three parameters as thesolution to the following minimization problem:{ζL , η, χ} = argmin SSE

s.t.

ζL ,η,χ{, λ, θ, gL , gX } : given by Table 2,(A.58)

δ = δ(χ), β = β(χ), α = α(χ), ζX = ζ X (χ).
We have done this minimization numerically utilizing ‘FindMinimum’ function of Mathematica. The minimized value of SSE is virtually zero (precisely, of order of 10−22 ), implyingthat we obtained the set of parameters that fits all the moments in Table 3.
Robustness Scenarios. To check the robustness of the result, we repeat the analyticalcalibration (A.52)-(A.56) with modified values for the exogenous parameters. In all robustness scenarios, the modified version of minimization problem (A.58) yields almost zero. Thismeans that the model can match all the target moments in those scenarios.

B

Data Sources

All 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 series by their codes in FRED. Panel a of Figure 1 shows annual values of Real GDP (GDPCA), real investment (GPDICA), and real personal consump68

Among seven moments in Table 3, we use five moments to define the squared sum of errors. The othertwo moments, consumption of fixed capital in GDP and return on investments, always match the data giventhat other moments are correct, since we impose relationship (A.52) and (A.53). In numerical calibration,
we confirmed that these two moments matches the data exactly.

A-24

tion expenditures (PCECCA). The real capital stock is calculated as the net stockof fixed assets at current cost (K1TTOTL1ES000) divided by the GDP price deflator(A191RD3A086NBEA). Panel b of Figure 1 shows three different price deflators for grossprivate investment– all (A006RD3A086NBEA), non-residential (A008RD3A086NBEA), andequipment (Y033RD3A086NBEA) – relative to the price deflator for personal consumption expenditures (DPCERD3A086NBEA). 1950 values are normalized to 100.

In ad-

dition to the variables listed above, the calibration utilizes data on nominal consumption of fixed capital (GDICONSPA), labor compensation, and R&D expenditure(Y694RC1A027NBEA) all relative to nominal GDP (GDPA) in Table 3, as well as population (B230RC0A052NBEA) in Table 2. Labor compensation is calculated as compensationof employees (A033RC1A027NBEA) plus proprietors’ income with inventory valuation andcapital consumption adjustments (A041RC1A027NBEA).

A-25