A Multi-factor Uzawa Growth Theorem · 2020. 8. 18. · A Generalized Uzawa Growth Theorem and...

A Generalized Uzawa Growth Theorem and

Capital-Augmenting Technological Change∗

Gregory Casey† Ryo Horii‡

January 30, 2020

Abstract

We prove a generalized, multi-factor version of the Uzawa steady-state growth theorem. In

the two-factor case, the theorem implies that a neoclassical growth model cannot be simulta-

neously consistent with empirical evidence on both capital-augmenting technical change and

the elasticity of substitution between labor and reproducible capital. In the multi-factor case,

balanced growth with capital-augmenting technical change is possible as long as capital has

a unitary elasticity of substitution with any single non-reproducible factor. Since natural re-

sources are important in production, but overlooked in the standard two-factor production func-

tion, the multi-factor result increases the likelihood that neoclassical models can be consistent

with empirical findings. To illustrate the importance of this result, we also build a three-factor

growth model with endogenous and directed technical change. The model economy converges

to a saddle-path stable balanced growth path, where the long-term rate of capital-augmenting

technical change is strictly positive.

Keywords: Balanced Growth, Uzawa Steady State Growth Theorem, Endogenous Growth,

Directed Technical Change, Natural Resources

JEL Classification Codes E13, E22, O33, O41, Q55

∗An earlier version of this paper has been circulated under the title “A Multi-factor Uzawa Growth Theoremand Endogenous Capital-Augmenting Technological Change.” The authors are grateful to Been-Lon Chen, OdedGalor, Andreas Irmen, Ezra Oberfield, Cecilia Garcia-Penalosa, Alain Venditti, Ping Wang, David Weil, and seminarparticipants at Aix-Marseille School of Economics, Brown University, Chulalongkorn University, Kobe University,National Graduate Institute for Policy Studies (GRIPS), Shiga University, Society for Economic Dynamics, OsakaSchool of International Public Policy (OSIPP), Tongji University, for their helpful comments and suggestions. Thisstudy was financially supported by the JSPS Grant-in-Aid for Scientific Research (15H03329, 15H05729, 15H05728,16K13353, 17K03788). Any remaining errors are our own.†Department of Economics, Williams College & CESifo, Schapiro Hall, 24 Hopkins Hall Dr., Williamstown, MA,

02916. Email: [email protected].‡Institute of Social and Economic Research (ISER), Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,

Japan. Email: [email protected].

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1 Introduction

The neoclassical growth model was developed to explain a set of stylized macroeconomic facts that

can be classified under the umbrella of balanced growth (Solow, 1956, 1994). As conventionally

understood, the Uzawa (1961) steady state growth theorem says that on the balanced growth path

(BGP) of a neoclassical growth model, all technological change must be labor-augmenting, unless

the production function is Cobb-Douglas (Jones, 2005; Jones and Scrimgeour, 2008; Grossman

et al., 2017a). This creates a significant problem for the neoclassical growth model, because data

from the United States strongly suggest that (i) there is capital-augmenting technical change on

the BGP and (ii) the aggregate production function in not Cobb-Douglas (see, e.g., Klump et al.,

2007; Oberfield and Raval, 2014).1

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 materials. These factors do not fit well in the notion of capital in

the neoclassical growth model in that they cannot be readily accumulated (or reproduced) through

savings. In this paper, we examine whether incorporating more factors of production makes it

possible for neoclassical models to be consistent with the empirical regularities mentioned above.

We also study the usefulness of more standard neoclassical models that are only consistent with a

subset of these facts.

We start by proving a multi-factor version of the Uzawa (1961) steady state growth theo-

rem. When building macroeconomic models, researchers have incomplete knowledge of the ‘true’

aggregate production function (provided that such a production function exists) and incomplete

knowledge of how that function evolves over time due to technological change. If an economy has a

BGP, the Uzawa theorem provides guidance on how to choose a simple representation of the ever-

changing, unknown true production function. In this paper, 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 deriva-

tives and elasticity of substitution (EoS) as the true production function, suggesting that it can be

useful for some economic analyses. However, this particular representation cannot match evidence

on capital-augmenting technical change, which is observed in the data. Importantly, the multi-

factor Uzawa theorem does not rule out the existence of other representations which might have

more accurate descriptions of technological change.

Next, we prove a generalized version of the Uzawa theorem. The generalized theorem suggests

that there are a continuum of representations with capital-augmenting technical change, as long

1The evidence for capital-augmenting technical change comes from the falling relative price of capital goods(Greenwood et al., 1997; Grossman et al., 2017a).2

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as reproducible capital has a unitary EoS with any single other factor. From this broader class

of Factor-Augmenting Representations, it is possible to choose a representation that matches the

empirically observed speed of capital-augmenting technological progress. We also provide condi-

tions under which these Factor-Augmenting Representations have the same derivatives and EoS as

the true production function. In other words, the factor-augmenting representations can simulta-

neously consistent with balanced growth, a non-unitary elasticity of substitution between capital

and labor, and capital-augmenting technical change. Thus, these representations are more suitable

than standard specifications in economic analyses that are based on the neoclassical growth model.

To demonstrate the practical importance of the 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 so that the resultant aggregate production function has the

form of a Factor-Augmenting Representation that fits the Generalized Uzawa theorem. The model

economy has a balanced growth path with a positive rate of capital-augmenting technical change,

a result that is not possible with existing models.3 We also calibrate the model to U.S. data and

numerically show that the economy converges to the BGP from wide range of initial conditions.

The global stability result has a particular theoretical importance, because it demonstrates that

the direction of technological change endogenously confirms 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 neoclassical growth 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 the aggregate production function

and technical change that together are a good representation of 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 and output, first order derivatives and elasticities of

substitution between factors of production. So, our results can immediately contribute to a range

of literatures. Most directly, models with multiple types of technology allow for more accurate

descriptions of the economy in growth-, development-, and business-cycle accounting analyses (e.g.,

Caselli and Feyrer, 2007; Hsieh and Klenow, 2010; DiCecio, 2009). We also show how to embed

our findings in a general equilibrium model with endogenous directed technical change. Models

with multiple types of technology are necessary for determining how policy will affect the direction

of technical change. This is likely to be especially important in the burgeoning literature on

automation, where technological change affects the productivity of labor and capital differently, and

may have either good or bad effects for workers (e.g., Acemoglu and Restrepo, 2018). Relatedly,

3See Grossman et al. (2017a) for the sole exception.

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the direction of technical change may be important for understanding recent trends in the labor

share of income (Karabarbounis and Neiman, 2014; Grossman et al., 2017b).

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. (2017a). With the exception of Acemoglu (2008), the literature has been concerned

only with whether a particular production function can match the level of output on the BGP. We

build on this literature several ways. First, we extend the theorem to multiple factors of production.

Second, and more importantly, we prove a generalized version of the 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

elasticity of substitution as the true production function.

As noted above, the existing literature has treated the Uzawa theorem as a restrictive condition.

As a result, many studies have tried to explain why the economy might endogenously conform to

the two-factor version of the theorem. These studies frequently use models with directed technical

change. In particular, Acemoglu (2003) and Irmen and Tabakovic (2017) provide models where

capital-augmenting technical change disappears in the long run, while Jones (2005) and Leon-

Ledesma and Satchi (2019) specify models that are cobb-douglas in the long-run. We build on

these works by presenting an endogenous growth model that endogenously converges to a BGP

that is consistent with data on both the existence of capital-augmenting technical change and

the less-than-unitary elasticity of substitution between capital and labor. Our model specification

builds on the work of Irmen (2017) and Irmen and Tabakovic (2017).

To the best of our knowledge, Grossman et al. (2017a) provide the only other attempt to square

the Uzawa steady state growth theorem with data on the EoS and the capital-augmenting technical

change. In their model, schooling is both labor-augmenting and capital-dis-augmenting. In this

setting, they show that there is a scope for additional capital-augmenting technological change.

We build on their work in two ways. First, as highlighted above, we provide a generalized proof

stressing the role of representations. Second, our results indicate that there is a much wider scope

for ways in which the neoclassical growth model can be made to be consistent with the data on EoS

and capital-augmenting technical change. Indeed, their results can be understood as a particular

case of the 2-factor Uzawa theorem (see subsection 5.2).

These results stress the importance of including natural resources and directed technical change

in growth models. There is, of course, a long literature on both of these topics, but they are

generally only included in growth models to achieve specific aims. For example, energy is generally

only included in growth models when studying the depletion of finite resources (e.g., Hotelling,

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1931; Heal, 1976) or climate change (e.g., Golosov et al., 2014; Barrage, 2019). Directed technical

change is often used to understand the evolution of relative wages between different groups of

workers (e.g., Acemoglu, 1998, 2002; Acemoglu and Restrepo, 2016).4 Our results suggest a much

broader importance of directed technical change and non-accumulable factors — these factors must

be incorporated into models of economic growth in order to recreate the balanced growth facts that

originally motivated aggregate growth modeling (Solow, 1956, 1994).

Roadmap — The remainder of the paper proceeds as follows. Section 2 presents the data motivat-

ing this study. Section 3 proves a multi-factor version of the Uzawa steady state growth theorem.

In Section 4, we generalize the theorem, proving that neoclassical models can have a positive rate

of capital-augmenting on the BGP. Section 5 explains the application of these results, focusing on

simple cases and existing literature. Section 6 presents a full economic model with endogenous

direction of technological change. Section 7 demonstrates that the the model endogenously con-

verges to a BGP with positive K-augmenting technical change. Section 8 discusses the broader

implications of our findings, and Section 9 concludes.

2 Empirical Motivation

In this section, we discuss the empirical facts that motivate this study. In particular, we quickly

review facts on balanced growth and evidence for the existence of capital-augmenting technical

change. The Uzawa (1961) steady state growth theorem suggests that these two sets of facts cannot

be reconciled in a standard two-factor growth model. So, we conclude by presenting evidence that

other factors, such as land and energy are important in the production process.

The neoclassical growth model, first developed by Solow (1956) and Swan (1956), is the central

building block of much contemporary research in economic growth. Such models are designed to

explain a set of stylized facts, known as ‘balanced’ or ‘steady’ growth (Jones, 2016). The main

stylized is that income per capita has grown at a constant rate over long periods of time. Panel

(a) in figure 1 presents U.S. data from 1950-2012, which clearly demonstrates this pattern.5 It also

demonstrates that other macroeconomics aggregates have grown at similar rates to GDP, capturing

the notion of balance.6

To explain these facts, the neoclassical growth model focuses on an aggregate production func-

4There is also a growing literature that combines directed technical change and natural resources to ask resource-related questions (e.g., Smulders and De Nooij, 2003; Acemoglu et al., 2012; Hassler et al., 2016).

5See Papell and Prodan (2014), Jones (2016), and others for longer time series and data on other countries.6As shown in Section 3, the formal definition of balanced growth in the steady state growth theorem only relies

on the notion that various macroeconomic aggregates grow at constant (and possibly different) rates. In this sense,the constant labor share and capital-output ratio need not be part of our formal definition of balanced growth. Beingable to recreate these facts, however, was an important part of the motivation for the original neoclassical model. Aswe will discuss, they are also important for understanding the implications of theorem.

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1

10

100

1000

10000

100000

1950 1960 1970 1980 1990 2000 2010

Billi

ons o

f $20

12 (l

og sc

ale)

I GDP C K

(a) BGP

0

20

40

60

80

100

120

1950 1960 1970 1980 1990 2000 2010

Inde

x (1

950=

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All Non-Residential Equipment

(b) K-Augmenting Technology

Figure 1: Balanced Growth with capital-augmenting technical change. This figures presents some of the mainfeatures of balanced growth in the United States. Panel (a) demonstrates the real output, investment, consumptionand the capital stock have grown at roughly constant rates over long periods of time. These empirical patternssummarize the notion of balanced growth. Panel (b) demonstrates that the price of investment goods, and equipmentin particular, having been falling relative to the price of consumption goods in the United States. This indicatescapital-augmenting technical change has been occurring along the BGP. See appendix section C for details on datasources.

tion that has constant returns to scale in two factors, reproducible capital and labor.7 The abil-

ity of the neoclassical growth model to provide a simple explanation for these facts 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 theorem. As conven-

tionally stated, the Uzawa steady state growth theorem is as follows: on a balanced growth path,

all technological 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 (Chirinko, 2008). For example,

Oberfield and Raval (2014) estimate the macro elasticity of around 0.7 using firm-level micro data,

and Antras et al. (2004) estimates an elasticity of 0.6 directly from macro time-series data.8

Panel (b) of Figure 1 demonstrates that the relative price of investment goods has been 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

7Intuitively, the key to explaining balanced growth is that capital is reproducible (i.e., it is accumulated fromsaved output). Thus, capital ‘inherits’ the constant growth rate of output, implying that the capital-output ratio willbe constant in the long-run (Jones and Scrimgeour, 2008). Their joint growth rate is then determined by populationgrowth and technological progress.

8See also, Klump et al. (2007) and Chirinko et al. (2011). Using cross-country data, Karabarbounis and Neiman(2014) find an elasticity greater than one.

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Factor Share Source

Natural Resources 8% Caselli and Feyrer (2007)Land 5% Valentinyi and Herrendorf (2008)Energy 8.5% Energy Information Administration

Table 1: This table presents some estimates of U.S. factor shares for inputs other than reproducible capital andlabor. Definitions and methodologies vary. See Appendix Section C for details.

efficiency of the investment goods sector. Depending on how we measure the amount of capital,

the declining relative price of capital goods can be interpreted as investment-specific technological

change (IST) or capital-augmenting technical change. Lemma 1 in Section 3 formalizes the notion

that these two types of technological change are equivalent to each other. This result is also

discussed by Grossman et al. (2017a), who demonstrate this equivalence in a proof of the Uzawa

steady-state growth theorem.

A long literature demonstrates that declining investment prices are a quantitatively important

source of growth in the United States (e.g., Greenwood et al., 1997; Krusell, 1998; Krusell et al.,

2000).9 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 exhibited

signs of balanced growth.

These findings create a puzzle. Given that the elasticity of substitution 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 ex-

plain 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 materials,

exist in the production process. Table 1 collects some evidence on the importance of these factors

in the United States.10 Broadly speaking, estimates suggest that non-reproducible factor other

than labor account for about 10% of total factor payments.

3 A Multi-factor Uzawa Theorem

This section shows 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 Solow (1956) has shown, sustained economic growth requires the shape

9See, He et al. (2008) and Maliar and Maliar (2011) for discussions of the Uzawa steady state growth theorem inthis context.

10Estimating factor shares for inputs other than labor is notoriously difficult and often requires structural assump-tions. Our intention is not to endorse any particular estimate. Instead, we simply note that there is ample evidencethat factors other than labor and reproducible capital play a non-negligible role in production.

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of the production function to change over time, which we usually call technological change. Given

the existence of a BGP, the Uzawa theorem provides a convenient representation of the evolution

of the production function. We stress the importance of making a clear distinction between this

representation and the true production function, for which we often have limited information.

In particular, we prove a new set of propositions that clarifies the conditions under which the

representation implied by the Uzawa theorem matches important properties of the true production

function. In this situation, 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. We start with a 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 settings, 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 environment that

satisfies:

1. Output, Yt, is produced from capital, Kt, and J ≥ 1 kinds of other inputs, Xj,tJj=1 :

Yt = F (Kt, X1,t, ..., XJ,t; t). (1)

In any t ≥ 0, it has constant returns to scale (CRS) in all inputs, Kt, X1,t, ..., XJ,t, and each

input has positive and diminishing marginal products.11

2. The amount of capital, Kt, evolves according to

Kt+1 = Yt − Ct −Rt + (1− δ)Kt, K0 > 0, (2)

where Ct > 0 is consumption, Rt ≥ 0 is expenditure other than capital investment or con-

sumption (e.g., R&D inputs), and δ ∈ [0, 1] is the depreciation rate. Yt−Ct−Rt in the RHS

represents physical capital investment.

There are a number of points to note regarding Definition 1. First, production function F (·; t)in (1) captures technological progress, i.e., changes in the shape of the production function, through

11If we allow J = 0, the only constant-returns-to-scale production function is in the form of Yt = AK,tKt. Althoughit cannot satisfy decreasing marginal products of its input (Kt), this AK functional form is also subject to the Uzawatheorem, in the sense that AK,t must be constant on the BGP (i.e., there may not be any technological change).

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t. Importantly, we place no restrictions on how the F (·) changes over time. As discussed below,

the Uzawa theorem provides insight about how to approximate the time dependence of F (·) with

standard factor-augmenting terms.

Second, if J equals 1 and X1,t is interpreted as labor, Lt, then equation (1) reduces to a familiar

two-factor neoclassical production function, Yt = F (Kt, Lt; t). In addition, if we assume Lt grows

exogenously, Definition 1 essentially coincides with the definition of a neoclassical growth model in

Schlicht (2006) and Jones and Scrimgeour (2008), who provide a simple statement and proof of the

two-factor Uzawa theorem.

Third, we allow for the term Rt term in (2). If we set Rt = 0, equation (2) is in line with the

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 is captured 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 factorsX1,t, ..., XJ,t

is that we explicitly specify its accumulation process as in (2), which guarantees that Kt can be ac-

cumulated linearly with the output. From the theoretical viewpoint, Kt needs not to be limited to

physical capital.12 We will show that the Uzawa theorem holds regardless of the evolution process

for other inputs. 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 is merely a choice

of units, and therefore should not pose substantial limitation in the applicability of our results.

For example, Greenwood et al. (1997) and Grossman et al. (2017a) considered investment-specific

technological change, which enables more capital to be produced from a unit of final output. The

following Lemma shows that, by change of valuables, Definition 1 can accommodate such a case.

Lemma 1. Consider an economic environment where physical capital Kt accumulates according to

Kt+1 = (Yt − Ct −Rt)qt+1 + (1− δ)Kt, (3)

where qt > 0 is the investment-specific technology. The production function is given by Yt =

F (Kt, X1,t, ..., XJ,t; t), which has CRS and positive and diminishing returns to all inputs.

12It can be any combination of factors that can be accumulated linearly with the output. For example, in thepre-industrial Malthusian economy where population was proportional to output (e.g., Galor and Weil, 2000; Galor,2011; Ashraf and Galor, 2011; Li et al., 2016), labor could be included in Kt, not in Xj,t.

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If the growth factor of investment-specific technological change gq = qt+1/qt is constant,13 this

environment fits Definition 1 through a change of variables of Kt ≡ Kt/qt and δ = (δ+ gq − 1)/gq.

Proof. See Appendix A.2.

The change of variables effectively normalizes the unit of capital so that capital Kt ≡ Kt/qt is

always measured in terms of the previous period’s final goods. The depreciation rate, δ, after this

normalization should be higher than δ, because positive investment specific technological change

decreases the value of older capital. Thus, the new value of δ must accommodate changes in both

the numerator (physical depreciation) and denominator (falling investment price) of Kt ≡ Kt/qt.

In the rest of the paper, we describe the economy in terms of Definition 1, where any investment-

specific technological change is included in the change of the shape of the production function

F (·; t).

Balanced Growth Path

Now, we turn to the second requirement of the Uzawa theorem, the balanced growth path.

Definition 2. A balanced growth path (BGP) in a multi-factor neoclassical growth model 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 by g ≡ Yt/Yt−1, and the growth

factors of any variable Zt ∈ Kt, X1,t, ..., XJ,t, Ct, Rt by gZ ≡ Zt/Zt−1. A non-degenerate

balanced growth path is a BGP with gK > 1− δ.

From (2), condition gK > 1− δ means that physical capital investment Yt − Ct −Rt is strictly

positive along the balanced growth path. The rest of the paper focuses on this non-trivial case. We

call it a non-degenerate BGP and simply mention it as a BGP when there is no risk of confusion.

Note that, while a BGP requires variables to grow at constant rates, it does not require them to

grow at the same rate. Still, the following lemma confirms that capital and consumption need to

grow at the same speed as output to maintain a BGP.

Lemma 2. On any non-degenerate BGP in a multi-factor neoclassical growth model, the capital-

output ratio Kt/Yt and the consumption-output ratio Ct/Yt are constant and strictly positive.


The proof utilizes the assumption of C0 > 0 from Definition 1. If R0 > 0, we can similarly show

that Rt/Yt is constant.

13We assume gq to be constant for simplicity. This condition is not necessary if we extend Definition 1 to allowdepreciation rate to change over time. As long as we focus on the BGP, the results will not be affected.

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Uzawa Representation and Its Properties

Having defined the neoclassical growth model and the BGP, we are ready to present a multi-factor

version of the Uzawa Theorem.

Proposition 1. (a Multi-Factor Uzawa Theorem) Consider a non-degenerate BGP in a multi-

factor neoclassical growth model, and define AXj ,t ≡ (g/gXj )t where j = 1, ..., J . Then, on the BGP,

Yt = F (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t) holds for all t ≥ 0, (4)

where F (·) ≡ F (·; 0).

Proof. From the definition of AXj ,t ≡ (g/gXj )t, the growth factor of AXj ,tXj,t is g for all j. The

growth factor of Kt is also g from Lemma 2. Therefore, all the arguments in function F (·) are

multiplied by g each period. This means that the RHS of (4) is multiplied by g each period since

F (·) ≡ F (·; 0) has constant returns to scale. Note that in period 0, equation (4) holds true because

it is identical with (1). Therefore, (4) holds for all t ≥ 0, where the both sides are multiplied by g

in every period.

It is important to understand what the theorem does and does not imply. Recall that the

neoclassical production function F (·; t) in (1) is a time-varying function that potentially depends

on t in complex ways. If a BGP exists, the Uzawa theorem says that there should be a simple

representation of this dependence of function F (·; t) on t, which holds 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 F (·) and exponentially growing AXj ,t terms.

Caution is needed when interpreting F (·) as a production function, because (4) is not a func-

tional relationship. Proposition 1 only guarantees that the value of F (·) coincides with that of the

original production function F (·; t) on a particular BGP. As is clear from the proof of the propo-

sition, function F (·) contains no information about what will happen when inputs deviate even

slightly from the BGP. When the amount of one of the factors is changed from the BGP value,

equation (4) does not hold in general. In this sense, the Uzawa Theorem does not say that F (·) in

(4) is a production function.

As a result, there is no guarantee that the derivatives of function F (·), 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 the theorem by focusing on the condition under which the Uzawa

representation has the ‘correct’ marginal properties. We start by looking at first-order derivatives.

Proposition 2. (Derivatives of the Uzawa representation) Let FZ(·; t) denote the partial

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derivative of function F (·; t) with respect to its argument Z ∈ Kt, X1,t, ..., XJ,t.14 If the share

of factor Z, i.e., sZ,t = FZ(·; t)Zt/Yt, is constant on a non-degenerate BGP of a multi-factor

neoclassical growth model, then the following holds on the BGP:

∂

∂ZtF (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t) = FZ(Kt, X1,t, ..., XJ,t; t) for all t ≥ 0. (5)


If the factor shares are constant on the BGP, equation (5) says that F (·) has the same derivatives

as the original production function F (·; t) on the BGP. We can also show that the elasticity of

substitution (EoS) between capital and other production factors in the Uzawa representation F (·)coincides with the EoS in the original production function F (·; t) on the BGP, if the latter does not

change over time.15 Let us first define the EoS when there are more than two inputs.16

Definition 3. The Elasticity of Substitution between capital Kt and input Xj in multi-factor

neoclassical production function F (K,X1, ..., XJ ; t) in (1) is defined by

σKXj ,t = − d ln(Kt/Xj,t)

d ln(FK(Kt, X1,t, ..., XJ,t; t)/FXj (Kt, X1,t, ..., XJ,t; t)

)∣∣∣∣∣Yt,X−j,t:const

, (6)

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 σKXj ,t denote

the EoS in the Uzawa representation, as in Definition 3. If the EoS of the original production

function, σKXj ,t for some j ∈ 1, ..., J, is constant over time on the BGP, then σKXj ,t = σKXj ,t

holds for all t ≥ 0 on the BGP.


Benefits and Limitations of the Uzawa Theorem

When developing a dynamic macroeconomic model, researchers need to take a stand on how to

represent technical change. In other words, they need to decide how the shape of the production

14Appendix A.1 discusses the details regarding notation for derivatives.15To the best of our knowledge, Acemoglu (2008) is the only example of previous work considering first-order

properties implied by the Uzawa Theorem. He looks at first-order conditions in the two-factor case, providing aspecial case of Proposition 2.

16When there are more than two production factors, there are various ways to define the elasticity of technicalsubstitution. See Stern (2011) for a concise taxonomy. The elasticity in (6) is calculated using the inverse of thesymmetric elasticity of complementarity (SEC), defined in Stern (2010), which has a desirable property of symmetrybetween the two variables.

12

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 pro-

vides guidance in choosing an effective representation of the evolution of the production function. As

shown in equation (1), the definition of the neoclassical growth model allows the aggregate produc-

tion 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 constant on the BGP. Relatedly, Proposition 3 says that the Uzawa rep-

resentation has the correct elasticity of substitution between capital and other variables, as long as

that elasticity is constant. Thus, the Uzawa representation has desirable properties which suggest

that it can be useful as an approximation of the true function.

Correctly modeling the level of output, factor prices, factor shares, and elasticity of substitution

between inputs is likely to be sufficient in answering many research questions in macroeconomics,

especially those related to business cycles. In this sense, the Uzawa theorem is a positive result,

highlighting the usefulness — rather than the limitations — of neoclassical growth models. At

the same time, the Uzawa representation is inconsistent with evidence on the existence of capital-

augmenting technical change. Thus, it will not be appropriate for analyzing questions where the

direction of technological change is important. In the next section, we generalize the Uzawa theorem

to include representations with capital-augmenting technical change.

4 A Generalized Steady-State Growth Theorem

The Uzawa representation invites us to interpret the time dependence of production function

in terms of factor-augmenting technological change. By viewing the AXj ,t’s as the factor Xj,t-

augmenting technology terms, Proposition 1 implies that it is always possible to interpret the time

variation of the original production function F (·; t) on the BGP in terms of exponential augmenta-

tion of production factors. It is tempting to conclude that there should be no technological change

that enhances the productivity of capital on the BGP, because there is no AK,t term in (4).

This reasoning is insufficient, because Proposition 1 does not establish uniqueness. As a result,

it does not rule out the existence of other convenient representations of the original production

function. In this section, we explore the possibility that the original production function has more

than one factor-augmenting representations. Under a certain condition, there will be one among

those representations that matches the data on the capital-augmenting technological change shown

in Section 2. We also discuss that, for the required condition to be satisfied, it is essential to include

factors of production beyond labor and reproducible capital.

13

Factor Substitution and Capital-Augmenting Technological Change

Let us start by defining a factor-augmenting representation. Note that we still do not know the

functional form of FAUG(·) below. Rather, the following definition sets out the goal of this section.

Definition 4. A Factor-Augmenting Representation of the original production function (1)

is a combination of a time-invariant constant-returns-to-scale function FAUG(·) and the growth

factors of factor-augmenting technologies γK > 0 and γXj > 0, j ∈ 1, ..., J, such that the paths

of output and inputs on a BGP satisfy

Yt = FAUG(AK,tKt, AX1,tX1,t, ..., AXJ ,tXJ,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 factor-augmenting

representation. As we explain below, (4) focuses only on a certain special case where 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 the reality.

To see this, suppose that every effective input, including effective capital, grows at the same

speed as the output. Recall that the physical capital is already growing at the same speed as

output on the BGP (from Lemma 2). Then, there is no room for additional capital-augmenting

technological progress to further augment its effectiveness. As discussed in Section 2, however, there

is clear evidence that the productivity of capital, measured in terms of output as in our model, has

steadily been increasing on the BGP. Thus, the interpretation of the BGP as being a homothetic

expansion of every input is at odds with a well-established stylized fact.

Motivated by this contradiction, we now consider a broader range possibilities in which effective

inputs grow at different constant rates. To have balanced growth with non-homothetic expansion of

production factors, it is necessary to further restrict the possible functional forms of the the factor-

augmenting representation. However, before moving to formal propositions, it help to start from a

heuristic discussion that highlights the key intuition. Suppose that the original production function

can be represented in the factor-augmenting way (7) on the BGP with the correct derivatives and

14

EoS. Then, the growth rate of output can approximately be written as follows:17

g ≡ Yt+1/Yt ≈ sk,tγKgK +

J∑j=1

sXj ,tγXjgXj , (8)

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, γKgK and γXjgXj ’s are different. Specifically, let us assume that effective capital

grows faster than output due to K-augmenting technological change (γKgK > 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. γX1gX1 < 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 = FAUG(AK,tKt

Yt,AX1,tX1,t

Yt, constants

). (9)

In this form, it is evident the growing effective capital-output ratio AK,tKt/Yt permits produc-

tion of unit output with the shrinking effective X1-output ratio AX1,tX1,t/Yt. This exactly means

that factor substitution is happening.

Now, let us check if this on-going factor substitution is consistent with the definition of the

BGP. On the BGP, output grows at a constant rate, g, which means that the RHS of (8) must

also be constant. Given γKgK > γX1gX1 , RHS of (8) remains constant only when their factor

shares, sk,t and sX1,t, does not change over time. This happens if and only if the EoS of FAUG(·)between K and X1, defined similarly to Definition 3, is one. To summarize, for K-augmenting

technological change to happen on the BGP in a factor-augmenting representation, the functional

form of FAUG(·) needs to have a unitary EoS between capital and some other factor.18 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 approximation of

the original production function. In particular, as in Proposition 3, the representation is useful if

the EoS of FAUG(·) match that of the original the original production function. This is possible

only when the original production function F (·; t) has a unitary EoS between capital and some

other factor, because we already know that FAUG(·) 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

17This 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 divide the result by the RHS of (7) for t. The Taylor-expansion isexact when the variables in t and t+ 1 are sufficiently close, or equivalently, in continuous time.

18In the Uzawa representation, γKgK = γXjgXj holds for all j. Because the production function is assumed to

have constant returns to scale (which guarantees sK,t +∑Jj=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 the cost thatit cannot accommodate the possibility of K-augmenting technological change.

15

different than one. However, our definition of neoclassical growth model allows 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 satisfies has a unitary EoS with capital. We focus on this case

for the remainder of the paper.19

A Generalized Steady-State Growth Theorem

Here, we formally construct a function that can be used as a basis for a factor-augmenting rep-

resentation. 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 so that the first j∗ ∈ 1, ..., Jof them can be substituted with capital with the unitary elasticity of substitution.

If capital is substitutable with other j∗ factors with unit elasticity, we can interpret them as

if they are combined together in the Cobb-Douglas fashion to form an intermediate input. The

intermediate input, which we call the capital composite, will then be one argument in the final

production function. Using the share of factors in period 0, sK,0 ≡ FK(K0, X1,0, ..., XJ,0; 0)Kt/Yt

and sXj ,0 ≡ FK(K0, X1,0, ..., XJ,0; 0)Xj,t/Yt, we define period-0 relative shares within the capital

composite:

α = sK,0/(sK,0 +

j∗∑j=1

sXj ,0), ξj = sXj ,0/(sK,0 +

j∗∑j=1

sXj ,0). (10)

Using these relative shares, we can represent the production function in a nested form:20

F (k, x1, ..., xJ) ≡ F(kα∏j∗

j=1xξjj , xj∗+1, ..., xJ

). (11)

The first argument of the RHS, m = kα∏j∗

j=1 xξjj , 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 F (·), along with other factors xj∗+1, ..., xJ . The shape of the

outside function F (·) is defined using the period-0 production function F (·; 0):

F (m,xj∗+1, ..., xJ) ≡ F

((∏j∗

j=1Xξjj,0

)−1/αm1/α, X1,0, ..., Xj∗,0, xj∗+1, ..., xJ ; 0

). (12)

19As discussed in Section 8, we hope that our results will motivate further empirical work to examine the elasticityof substitution between reproducible capital and a wide range on non-reproducible factors. In the exiting literature,there is some suggestive evidence that certain forms of land may have a unitary elasticity of substitution with capital.For example, Epple et al. (2010) and Ahlfeldt and McMillen (2014) suggest that the elasticity of substitution betweenland and structures is likely close to one for residential housing. These papers, however, do not estimate aggregateelasticities and, therefore, do not measure the structural parameters of interest in our model. Further work focusingexplicitly on a macroeconomic context is necessary before meaningful conclusions can be drawn.

20In this section, we use lowercase letters k, x1, ..., xJ to denote variables, while uppercase letters Kt, X1,t, ..., XJ,tare the BGP values, unless otherwise noted.

16

The first argument of F (·), m, collects the j∗ relevant inputs and combines them with capital in the

first argument. As a result, function F (·) has j∗ fewer arguments than F (·; 0). Note that the RHS

of (12) includes the BGP values Xj,0, J = 1, . . . , j∗, which are treated as constants. The impact of

changes in the xj,0’s operates only through m.

The following lemma establishes that the nested representation, F (·) with F (·), approximates

the original production function around the BGP in period 0 almost as good as the Uzawa repre-

sentation.

Lemma 3. (Nested representation of 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, FZ(K0, X1,0, ..., XJ,0) = FZ(K0, X1,0, ..., XJ,0; 0).

c. For any j = 1, ..., j∗, σKXj ,0 = σKXj ,0, where σKXj ,0 is the EoS of function F (k, x1, ..., xJ)

between k and xj, evaluated at the period-0 BGP.

d. Functions F (m,xj∗+1, ..., xJ) and F (k, x1, ..., xJ) have constant returns to scale.


Properties a, b and c respectively confirm that the nested representation F (·) matches with the

period-0 original production function F (·; 0) in terms of the level, first derivatives, and the EoS,

when the function is evaluated around the period-0 BGP: k, x1, ..., xJ = K0, X1,0, ..., XJ,0.Property d confirms the CRS property.

Thanks to the CRS property, the nested representation can be used not only for period 0, but

for all t 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 Steady-State 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 satisfy the technology condition

(γKg)α∏j∗

j=1(γXjgXj )

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. Then, on the BGP,

Yt = F (AK,tKt, AX1,tXj,t, ..., AXJ ,tXJ,t) for all t ≥ 0. (14)


Note that (14) constitutes a factor augmenting representation, as defined by Definition 4, be-

cause function F (·) has constant returns to scale from Lemma 3. Thus, proposition 4 characterizes

17

the set of factor-augmenting representations of the true production function along the BGP. When

there is no factor that is substitutable with capital with unit elasticity at time 0 (i.e., j∗ = 0),

then Proposition 4 becomes identical with Proposition 1.21 Given that there are many factors of

production in reality, we cannot rule out the possibility that at least one of them is substitutable

with capital with unit elasticity (j∗ ≥ 1). There are several aspects of the proposition that warrant

further discussion.

First, condition (13) requires the amount of effective capital composite, Mt = (AK,tKt)α∏j∗

j=1(AXj ,tXj,t)ξj ,

to grow by factor of g every period on the BGP. By taking logs, condition (13) can be expressed in

a log-linear form:

α log γK +∑j∗

j=1ξj log γXj = (1− α) log g −

∑j∗

j=1ξj log gXj . (15)

Proposition 4 states that, whenever γK and γXj ’s satisfy this log-linear condition, there exist a

representation of the BGP with factor-augmenting technologies AK,t = (γK)t and AXj ,t = (γXj )t.

When the growth rates of the factor-augmenting technologies are exogenous, the log-linear condition

is restrictive. In sections 6 and 7, we build an endogenous growth model and show that the model

has a locally and globally stable BGP with positive capital-augmenting technical change, implying

that this condition is endogenously satisfied.

Second, unlike Proposition 1, the generalized theorem implies that the are a continuum of

representations, each of which has different growth rates for factor-augmenting terms. Thus applied

researchers can pick the representation that is consistent with data. The Uzawa representation is

a special case of the representation with γK = 0,

Third, similar to the original Uzawa Theorem (Proposition 1), equation (14) is not a functional

relationship. It only states that the level of this representation matches with the original produc-

tion 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 and other factors around the BGP.

Proposition 5. (Derivatives of the Factor-Augmenting Representation) Suppose that

σKXj ,0 = 1 for j = 1, ..., j∗. If the share of factor Zt ∈ Kt, X1,t, ..., XJ,t, i.e., sZ,t = FZ(·; t)Zt/Yt,is constant on a non-degenerate BGP of a multi-factor neoclassical growth model, the following holds

on the BGP:

∂

∂ZtF (AK,tKt, AX1,tX1,t, ..., AXJ ,tXJ,t) = FZ(Kt, X1,t, ..., XJ,t; t) for all t ≥ 0. (16)


21If j∗ = 0, condition α+∑j∗

j=1 ξj = 1 in Lemma 3 implies α = 1. Then, condition (13) reduces to γK = 1, whichmeans AK,t = 1 for all t. Then, (14) becomes identical to (4).

18

Proposition 6. (The EoS of the Factor-Augmenting Representation) Suppose that σKXj ,0 =

1 for j = 1, ..., j∗ and let σKXj ,t denote the the EoS in the factor-augmenting representation

F (AK,tKt, AX1,tX1,t, ..., AXJ ,tXJ,t). If the EoS of the original production function, σKXj ,t for some

j ∈ 1, ..., J, is constant over time on the BGP, then σKXj ,t = σKXj ,t holds for all t ≥ 0 on the

BGP.


Propositions 4 – 6 have important implications for applied macroeconomic research. Even when

the true form of technological progress is unknown, researchers can choose a factor augmenting

representation that has the correct level of output, first derivatives, elasticities of substitution, and

growth rate of capital productivity in the vicinity of the BGP. These results imply that the factor-

augmenting representation (14) can serve as an effective approximation of the true production

function.

Implications and Comparison to Uzawa Theorem

The results obtained in this section can be contrasted with those in Section 3. The Uzawa

thoerem (Proposition 1) has shown that if the economy exhibits balanced growth, as observed

in many countries, there always exists a representation of the evolution of production function,

F (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t). This simple representation, called the Uzawa Representation, ex-

plains the balanced growth by homothetic expansion of every effective production factor. In other

words, the Uzawa representation hypothesizes that no factor substitution is taking place along the

BGP.

While the Uzawa representation matches the behavior of the actual, often unknown, production

function around the BGP (Propositions 2 and 3), it fails to explain one critical aspect of growth.

In the Uzawa representation, the productivity of capital does not improve, because there is no AK

term. Instead, our generalized theorem in Proposition 4 clarifies that the Uzawa theorem is only a

single possibility out of a continuum of possible factor-augmenting representations, as long as the

production function allows factor substitution on the BGP (which requires at least one factor of

production that have a unitary EoS with capital).22 Every candidate representation can explain

the observed quantities on the BGP, but they differ 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 of capital-augmenting technological progress observed in data. Given the

evidence of positive capital-augmenting technological change, the Uzawa representation will be

ruled out as an appropriate representation of technological change.

22We can confirm that the Uzawa representation, where γK = 1 and γXj = g/gXj for all j, satisfies condition (15).

19

Propositions 5 and 6 guarantee that, if factor shares and the elasticity of substitution are sta-

tionary, the chosen factor-augmenting representation will have correct derivatives and EoS. Thus,

the representation constitutes a local approximation of the actual production function along the

BGP, implying that it should be at least as useful as the Uzawa representation in any economic

analysis. Since many questions in macroeconomics require understanding the evolution of pro-

ductivity, the factor-augmenting representation will be more useful in many applications. In the

reminder of the paper, we explain the use 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 these results

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 data on the relative price of

capital and the elasticity of substitution 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. (2017a) as a special case of the

2-factor neoclassical environment. Finally, subsection 5.3 shows that the 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 will not specify the source of technological change. We will develop a full macroeconomic model

in Section 6.

5.1 Standard 2-Factor Neoclassical Growth Model

Suppose that the original production function uses only two kinds of inputs, capital, Kt, and

labor, Lt, i.e., Yt = F (Kt, Lt; t). The production function F (·; t) depends on time due to the

technological change. Then, Proposition 1 says that, on any BGP with positive investment, the

technological change can always be represented as Yt = F (Kt, AL,tLt). However, if these two factors

are substitutable with unit elasticity (σKL = 1), Proposition 4 shows there are other possible factor-

augmenting representations of the same BGP:23

Yt = A(AK,tKt)α(AL,tLt)

1−α,where A > 0 is a constant, (17)

which includes an Uzawa representation Yt = AKαt (AL,tLt)

1−α as a special case. Given the

growth factors of output and labor on the BGP, condition (13) implies that any combination

23When there are two factors (J = 1) and they are substitutable with unit elasticity (j∗ = 1), equation (14) in

Proposition 4 implies that Yt = F((AK,tKt)

α(AL,tLt)1−α). Because function F (·) has constant returns to scale and

has only one argument, we can write F (x) = Ax for some A > 0, which gives (17).

20

of γK = AK,t+1/AK,t and γL = AL,t+1/AL,t is consistent with the BGP as long as they satisfy

γαK(γLgL)1−α = g1−α. By rewriting (17) as Yt = AtKαt L

1−αt , where the TFP At is given by

At ≡ AAαK,tA1−αL,t , it is clear that various combinations of capital- and labor-augmenting technolog-

ical changes give the same rate of growth for the TFP and, therefore, output.

This result confirms the widely understood version of the Uzawa theorem: on a BGP, all

technological progress must be labor-augmenting, unless the production function is Cobb-Douglas.

As we have seen in Section 2, this theoretical result is in contradiction with the two stylized facts:

(i) the productivity of capital has been steadily increasing, and (ii) the elasticity of substitution

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.

5.2 Inclusion of Schooling in a Two-Factor model

Grossman et al. (2017a) 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 directly started their analysis from a factor-

augmenting representation, it is worthwhile to consider an underlying time-varying production

function in the form of (1):24

Yt = F (Kt, Lt; t) = F s(D(st)aKt, D(st)

−bLt; t), (18)

where a > 0, b > 0, D(·) ∈ [0, 1], and D′(·) < 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. (2017a) specified a certain type of complementarity between

schooling and capital.

Note that st is not a production factor in the neoclassical sense, because the production function

has constant returns to scale only in capital and labor. Still, as the D(st) term changes over time,

it affects the amount of output produced from given Kt and Lt. This is a particular form of

technological change, and we can consider D(st) as being included in the t term of F (Kt, Lt; t),

as in the middle part of (18). Therefore, it falls within the definition of a two-factor neoclassical

growth model (i.e., Definition 1 with J = 1).

From Proposition 1, this production function has an Uzawa representation Yt = F (Kt, AL,tLt)

with AL = (g/gL)t on a BGP, where both effective factors Kt and AL,tLt grow at the same speed

as output. Grossman et al. (2017a) interpret the production function in the following way, keeping

24They considered not only factor-augmenting technological progress, but also investment-specific technologicalchange. Definition 1 can include both cases as we have shown in Lemma 1.

21

the multiplier D(st) term in the expression:

Yt = F (Kt, AL,tLt) = F (AK,tD(st)aKt, AL,tD(st)

−bLt). (19)

Comparing the arguments in the right hand side and those in the middle, we immediately obtain

AK,t = D(st)−a and AL,t = AL,tD(st)

b on the BGP. Because the multiplier D(st)a shrinks as st

increases, the capital-augmenting technology AK,t must grow so as to exactly offset the shrinking

D(st)a term. Conversely, the labor-augmenting term AL,t should grow slower than that in the

Uzawa representation AL,t because the multiplier D(st)−b is also augmenting labor.25 In this sense,

there is no overall growth in capital productivity in the Grossman et al. (2017a) formulation.

Within the limits of the two-factor Uzawa theorem, Grossman et al. (2017a) propose a new

interpretation of the production function, which provides the first possible solution to the contra-

diction raised by the Uzawa theorem. In their formulation, it is important that schooling enters

the production function precisely in the form of (19), where the same function 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 expo-

nentially 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 discusses

a particularly simple example.

5.3 A Simple Three-Factor Model with Land

As shown in Section 2, a significant portion of GDP is paid to production factors that do not fit well

in the notion of Kt or Lt. Thus, it is natural to consider production functions with more than two

factors. Adding these additional factors makes it possible to reconcile neoclassical models with the

data. While labor cannot be substituted by capital with unitary elasticity (σKL 6= 1), Proposition

4 only requires that there 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,

Yt = Ft(Kt, Lt, Xt; t), where Xt = X0gtX for all t,X0 > 0, gX > 0. (20)

25From these observations, the main result of (Grossman et al., 2017a, proposition 2) can easily be obtained asfollows. Taking the growth factor of the both sides of AK = D(st)

−a gives γK = g−aD . From this we obtain a discrete

time equivalent of their proposition 2(ii): gD = γ−1/aK . Note that Grossman et al. (2017a) assumed Lt = D(st)Nt,

which means gL = gDgN . Because effective labor AL,tD(st)−bLt in (19) must grow at the same rate as output,

g = γLg−bD gL = γLg

1−bD gN = γLγ

(b−1)/aK gN , which is a discrete time equivalent of their proposition 2(i).

22

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. Another example is natural resources.

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 have a unitary

elasticity of substitution with capital: σKX = 1. For concreteness, let us call this factor land.

Then, Proposition 4 implies that, along a non-degenerate BGP, the technological change can be

represented in a factor-augmenting fashion:

Yt = F

((AK,tKt

)α(AX,tXt

)1−α, AL,tLt

), α ∈ (0, 1), (21)

where the growth factor of technology variables must satisfy γL = g/gL and γαK(γXgX)1−α = g1−α.

As in (15), the latter condition can be written in a log-linear form:

log γK =1− αα

(log γL + log gL − log γX − log gX) . (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 = γLgL > γXgX).

This finding raises an important question: even if there is a factor with σXjK = 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 long-term 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 regardless of the initial state of technologies, condition (22)

is always satisfied in the long run.

6 A Full Model with Endogenous Directed Technological Change

So far, we have discussed the implications of the generalized Uzawa theorem focusing on the pro-

duction sector.In this section, we develop a complete endogenous growth model, where the direction

of technological progress is determined by profit-maximizing firms. We will show that the log-linear

technology condition (22) is endogenously satisfied on the BGP. We base this section on a stream-

23

lined version of the model of tasks developed by Irmen (2017) and Irmen and Tabakovic (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 perfectly competitive 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,tKt and effective land AX,tXt as inputs, where AK,t and AX,t are the representa-

tive firm’s capital-augmenting and labor-augmenting technologies (these will be explained in detail

below). Effective capital and effective land are substitutable with each other with unit elasticity.

Specifically, if an M-task uses AX,tx units of effective land, it requires at least AK,tk ≥ (AX,tx)−ω

units of effective capital, where ω > 0. Then, when the representative firm uses Kt units of capital

and Xt units of land in total, the maximum number of M-tasks it can complete is given by26

Mt = (AK,tKt

)α(AX,tXt

)1−α, α = 1/(1 + ω) ∈ (0, 1). (23)

N-task uses only effective labor, AL,tLt, where AL,t is the labor-augmenting technology of the

representative firm. Assuming that an N-task requires at least one unit of effective labor, the

maximum number of N-tasks that the representative firm can perform using Lt is simply

Nt = AL,tLt. (24)

By performing Mt and Nt tasks, the representative firm produces

Yt = F (Mt, Nt) = F

((AK,tKt

)α(AX,tXt

)1−α, AL,tLt

)(25)

units of output, where = F (·) is a standard neoclassical production function that has constant

returns to scale and an intensive form that satisfies the Inada conditions when it is expressed either

in terms of Mt/Nt or Nt/Mt.

Now, we explain how the factor-specific technologies AK,t, AX,t, AL,t are determined. Tech-

26With unit elasticity of substitution between capital and land, it is optimal to allocate capital and landto individual M-tasks with equal quantities. If the representative firm is to operate Mt kinds of M-tasks, itmeans k = Kt/Mt and x = Xt/Mt. Substituting these into the input requirement AK,tk ≥ (AX,tx)−ω gives

Mt ≤ (AK,tKt

)α(AX,tXt

)1−α, where α = 1/(1 + ω).

24

nical knowledge can be kept within the firm for only one period, after which there are knowledge

spillovers. Thus, the representative firm at time t can freely use the previous firm’s average factor-

augmenting technologies. In addition, the firm can improve each of factor-augmenting technologies

through R&D in each period. Specifically, when ik units of final goods are used for capital aug-

menting R&D in a M-task, AK for that task is enhanced to

AK,t = AK,t−1 (1 + eK (ik)) , (26)

where we omit the subscript for each task because all M-tasks are symmetric. Here, eK(ik) repre-

sents the enhancement to technological knowledge that results from investing ik units of final goods

in R&D. Function eK(·) satisfies Inada-like conditions: it is an increasing and concave function,

e′K(ik) > 0 and e′′K(ik) < 0 for ik > 0, with eK(0) = 0, eK(∞) =∞, e′K(0) =∞, and e′K(∞) = 0.27

It is convenient to rewrite (26) in terms of the investment cost function. Let γK,t ≡ AK,t/AK,t−1denote the growth factor of AK,t. Then, we can define the R&D cost function as28

ik(γK,t) ≡ e(−1)K (γK,t − 1) , defined for γK,t ≥ 1, (27)

where e(−1)K (·) denotes the inverse function of eK(·). From properties of eK(·), it can be seen that

R&D cost function iK(γK) satisfies iK(1) = 0, i′K(1) = 0, iK(∞) = ∞ and i′K(∞) = ∞. Also,

iK(γK) > 0, i′K(γK) > 0, and i′′K(γK) > 0 hold for all γK > 1. 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.29

We assume that the firm faces similar constraints when improving AX,t and AL,t. R&D cost

functions for land and labor-augmenting technologies are defined accordingly as iX(·) and iL(·).Recall that factor augmenting technologies are all task-specific and the R&D costs must be incurred

for each of Mt and Nt tasks. This means the total R&D costs for all M- and N-tasks are:

RK,t = Mt · iK(AK,t/AK,t−1),

RX,t = Mt · iX(AX,t/AX,t−1),

RL,t = Nt · iL(AL,t/AL,t−1).

(28)

The objective of the representative firm is to maximize the single period profit net of the R&D

costs, because it lives only for one period and its knowledge will become public next period. By

27To minimize notations, limik→∞ e′K(ik) is written as e′K(∞). We use similar conventions hereafter when they

cause no ambiguities.28With a slight abuse of notation, hereafter iK(·) represents a function, not a variable.29This 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. The risk of duplicationbecome more prominent as R&D inputs increase, which makes the R&D cost function iK(·) convex (or, equivalently,the R&D output function eK(·) concave). See Horii and Iwaisako (2007) for a simple micro foundation.

25

taking the output in each period as numeraire, the period profit is given by

πt = F (Mt, Nt)−RK,t −RX,t −RL,t − rtKt − τtXt − wtLt, (29)

where rt, τt, and wt are interest rate, land rent, 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 to30

Lt = L0gtL, L0 > 0, gL > 1− δ : given. (30)

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 = Ltu(Ct/Lt), (31)

where Ct/Lt > 0 is per capita consumption. We assume the felicity function u(·) takes the CRRA

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

U =

∞∑t=0

Ltβt (Ct/Lt)

1−θ − 1

1− θ, (32)

where θ > 0 is the degree of the relative risk aversion and β > 0 is the discount factor. The value

of β must be significantly smaller than 1, since otherwise U will become infinity when the number

of household members and its per capita consumption are increasing over time. In Proposition 8,

we derive the upper limit for β below which the household’s problem has a finite solution on the

BGP.

The representative household owns capital, Kt, and land, 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 land

is exogenous:

Xt = X0gtX , X0, gX > 0 : given. (33)

As in the case of population, the available quantity of land can be either constant gX = 1, shrink-

ing gX ∈ (0, 1), or growing gX > 1.31 Physical capital accumulates through the savings of the

household:

Kt+1 = (rt + 1− δ)Kt + τtXt + wtLt + πt − Ct, K0 > 0 : given, (34)

30We assume gL > 1 − δ so as to avoid the possibility of a degenerate BGP, where physical capital investmentbecomes zero or even negative in the long run. (See Definition 2). Note that, as long as δ > 0, condition gL > 1− δallows declining population. However, population should not fall faster than the speed of capital depreciation.

31Recall that we just call the third production factor Xt land for convenience. If Xt literally means the acreage ofland, then gX > 1 should be ruled out unless we want to consider space migration.

26

where (rt+1−δ)Kt+τtXt+wtLt+πt represents the household’s income. The household is subject

to the non-Ponzi game condition. Specifically, the present value of its asset holding as T → ∞should not be negative:

limT→∞

(T∏t=1

(rt + 1− δ)

)−1KT+1 ≥ 0 (35)

This completes the description of the model economy.

Before proceeding to the analysis of the model, we demonstrate that it conforms to our defi-

nition 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 belongs to the

definition of the multi-factor neoclassical production function (1). In fact, Proposition 4 guarantees

that, if the elasticity of substitution between Kt and Xt is unity and the economy has a BGP in

equilibrium, then the aggregate production function can always be written in the form of (25) at

least along the BGP. Our microeconomic setting gives an example of such an economy. Second, by

substituting (29) into (34), we obtain the evolution of capital in the same 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 is 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 strictly positive rate of 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 described above,

focusing on the role of R&D. The representative firm maximizes profit (29) subject to the production

and R&D functions (23)–(28) 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, define µt ≡Mt/Nt, which is the relative task intensity in final good production. Then, because F (·) in (25)

is a CRS function, we can write F (Mt, Nt) = NtF (µt, 1) ≡ Ntf(µt), FM (Mt, Nt) = f ′(µt), and

FN (Mt, Nt) = f(µt)− µtf ′(µt).32

Using this notation, we can conveniently express the first order conditions for factor demand.

The firm demands capital so as to satisfy

rt = (αMt/Kt)(f ′(µt)− iK(γK,t)− iX(γX,t)

). (36)

The RHS of (36) 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

32FM (·) and FN (·) represent the partial derivative of function F (·) with respect its first and second arguments,respectively.

27

the number of M-tasks performed in the firm, while 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′(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.33 The demand for land X is determined in a similar way,

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

), (37)

where (1 − α)M/X is the marginal product of Xt in performing more M-tasks. Lastly, the firm

employs labor according to

wt = AL,t(f(µt)− µtf ′(µt)− iL(γL,t)

), (38)

where the first part says that an additional unit of Lt can perform AL N-tasks. By substituting

(36), (37), and (38) into (29), it can be confirmed that the firm achieves zero profit, πt = 0. This

is due to the constant-returns-to-scale property of the firm’s problem.

Now, let us turn to R&D. We first explain the R&D 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:

R&D for N-tasks: γL,ti′L(γL,t) + iL(γL,t) = f(µt)− µtf ′(µt). (39)

The firm’s private benefit from improving technology AL,t is the ability to perform a larger number

of N-tasks, which increases the final output Yt = F (Mt, Nt). The RHS of (39) shows the marginal

benefit, FN (Mt, Nt) = f(µt) − µtf ′(µt). The LHS corresponds to the marginal cost of performing

a larger number of N-tasks through augmenting labor efficiency AL,t (given labor employment Lt).

This can be achieved by two steps. First, by intensifying the R&D efforts in the existing N-tasks

to raise labor efficiency, the representative firm can save a certain amount of labor, which is just

enough to perform one additional N-task. The cost associated with this activity is given by the

first term γL,ti′L(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 to invest 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.

As we formally prove in Proposition 7 below, condition (39) has a unique solution for γL,t as a

33Recall that, although we omit subscripts for individual tasks, AK,t and AX,t are task-specific and thereforethe R&D cost will increase with the number of tasks, given the target rate of technological improvement. From thesymmetry of tasks within each group (M or N) and from the convexity of the R&D cost functions, it is always optimalto spend equal amounts of R&D costs for individual tasks.

28

γX

γK

µ↑µ

( 1

Eq. with 2> µ )

Eq. with µ1

CombinedR&D cond.

11

R&D allocation

cond.

(a) Equilibrium innovation in (γK , γX)space. When µt increases, the combinedR&D curve shifts towards the origin

γL

γX

γKµ↑

Equilibrium Innovation

Eq. with µ2 (> µ1)

Eq. with µ1

R&D allocationCombinedR&D cond.

Possibility Frontier (EIPF)Direction of tech.change with µ1

(1,1,1)

cond.

(b) The direction of innovation and the EIPF curvein (γK , γX , γL) space. As µt changes, the equilibriuminnovation combination moves along the EIPF.

Figure 2: Determination of the direction of the technological change and the Equilibrium InnovationPossibility Frontier (EIPF) Curve

function of µt = Mt/Nt, and it is strictly increasing in µt. Intuitively, when the firm is performing

relatively few N-tasks (i.e., when µt ≡ Mt/Nt is higher), the benefit of increasing 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 capital- and land-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

to satisfy 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 capital- and land-augmenting technologies, as well as the condition that specifies the

optimal combined amount of R&D for M-tasks:34

R&D allocation:γK,ti

′K(γK,t)

γX,ti′X(γX,t)=

α

1− α, (40)

Combined R&D:(γK,ti

′K(γK,t) + iK(γK,t)

)+(γX,ti

′X(γX,t) + iX(γX,t)

)= f ′(µt). (41)

Observe that γK,ti′K(γK,t) and γX,ti

′X(γX,t) in condition (40) are similar to the intensive marginal

34The first order condition for AK,t yields (γK,t/α)i′K(γK,t) + iK(γK,t) + iX(γX,t) = f ′(µt), whereas that for AX,tgives (γX,t/1− α)i′X(γX,t) + iK(γK,t) + iX(γX,t) = f ′(µt). Condition (40) is obtained by subtracting the secondequation from the first. Condition (41) is from adding α times the first equation and (1 − α) times the secondequation.

29

R&D cost in (39). Since they are strictly increasing in γKt and γX,t, respectively, this condition

can be expressed as an upward sloping curve in the (γK,t, γX,t) space, as depicted in Figure 2(a).

As the RHS of condition (40) shows, the allocation should depend on the relative contribution of

capital and land 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 land productivity, i.e., if the marginal R&D cost

i′X(γX,t) increases more rapidly with its argument than i′K(γK,t), then it is optimal not to improve

AX,t as fast as AK,t.

Condition (41) specifies the optimal combined size of R&D investments. Capital and land are

used in M-tasks, and therefore improving capital- and land-augmenting technologies will enable the

firm to perform more M-tasks. This marginal benefit is represented by the RHS of (41), f ′(µt) =

FM (Mt, Nt). The LHS is the marginal cost of R&D, which has two parts, γK,ti′K(γK,t) + iK(γK,t)

and γX,ti′X(γX,t) + iX(γX,t), because both capital- and land-augmenting technologies receive some

R&D according to the allocation condition (40). 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 (39). The locus of (γK,t, γX,t) that satisfies the combined R&D condition (41) is

depicted by the downward-sloping curve in Figure 2(a).35

The intersection of the R&D allocation condition and the combined R&D condition gives the

optimal rates of innovation for capital- and land-augmenting technologies. When µt is increased

(e.g., from µ1 to µ2 in the figure), the RHS of (41) declines, which pushes the combined R&D

condition towards the origin at (1, 1). This means that both γK,t and γX,t are decreasing in

µt. Intuitively, when Mt/Nt is higher, there is an abundance of M-tasks relative to N-tasks, which

reduces the incentives to further improve the M-task related technologies. The following Proposition

provides a formal statement regarding the direction of technological change.

Proposition 7. (Direction of Technological Change)

In the endogenous growth model defined in Section 6.1, the growth factors of each of the factor

augmenting technologies are function of µt = Mt/Nt, denoted by γK(µt), γX(µt), and γL(µt). These

functions satisfy:

(a) γ′K(µt) < 0 for all µt > 0, γK(0) =∞, and γK(∞) = 1.

(b) γ′X(µt) < 0 for all µt > 0, γX(0) =∞, and γX(∞) = 1.

(c) γ′L(µt) > 0 for all µt > 0, γL(0) = 1, and γL(∞) =∞.

Proof. In Appendix A.10

35The curve is downward-sloping because the first part of the LHS of (41) is increasing in γK,t, whereas the secondterm is in γX,t. See the proof of Proposition 7 for the formal proof.

30

Figure 2(b) illustrates the direction of the technological change in the 3-dimensional space.

The (γK , γX)-plane depicted at the bottom of figure is the same as in panel (a). For a given

value of µt, the intersection gives the value of γK(µ) and γX(µ). In addition, the vertical distance

between this point and equilibrium point shows the size of L-augmenting innovation, γL(µ) − 1.

As µt increases (e.g., from µ1 to µ2), the combined R&D locus shifts inward, which lowers γK,t

and γX,t, but at the same time γL,t increases because γ′L(µt) > 0. The thick downward-sloping

curve depicts the locus of such equilibrium points that correspond to various values of µt. This

is the equilibrium innovation possibility frontier (EIPF), and the technological change occurs to

the direction of either point in this curve. Any innovation beyond this frontier is not profitable,

although it might be materialistically feasible.36

The dependence of the direction on µt can be interpreted in terms of relative scarcity of effective

factors. Note that the relative task intensity µt = Mt/Nt also represents the ratio of effective

capital composite to effective labor µt =(AK,tKt

)α(AX,tXt

)1−α/AL,tLt. Proposition 7 says that

the direction of technological progress is chosen so that it enhances effective factors which are in

relatively short supply. In other words, firms are “induced” to do more innovation that enhance

the relatively more scarce effective production factors.37

Another point to note is that µt is endogenous and depends on the amount of R&D activities

in period t. Therefore, Proposition 7 should be understood as an equilibrium relationship between

the task intensity and the direction of technological change, rather than causality. To see how they

are determined in each period, we need to 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 capital per effective

labor kt ≡ Kt/AL,tLt, and consumption per effective labor ct ≡ Ct/AL,tLt. Below, we construct

the equilibrium mapping from µt, kt, ct to µt+1, kt+1, ct+1 for t ≥ 0. The mapping and the

36In most models of direction of technological change, it is assumed that innovation requires a certain type ofresource (e.g., scientists), and its amount is given exogenously. In such a case, the innovation possibility frontieris derived from the resource constraint. To the contrary, in our model, the total amount of R&D input (Rt) isdetermined in equilibrium through profit maximization, and hence the frontier is called the “equilibrium” innovationpossibility frontier.

37This notion of induced innovation was first introduced by ?. See Acemoglu (1998) for more discussion

31

initial conditions µ0 and k0, together with the transversality condition for ct, will pin down the

equilibrium path of µt, kt, ct, from which the path of all variables in the model can be recovered.

Before so doing, it is convenient to define the net aggregate output in the economy as Vt =

F (Mt, Nt)−RK,t −RX,t −RL,t, which means the aggregate output minus the total R&D costs in

the economy. The net output per effective labor can be written as a function of µt:

Vt/Nt = f(µt)− µt(iK(γK(µt)) + iX(γX(µt)))− iL(γL(µt)) ≡ v(µt). (42)

Then, substituting profits (29) into the budget constraint (34), we can express the growth of

aggregate capital supply in terms of µt, kt and ct:

Kt+1

Kt=Vt + (1− δ)Kt − Ct

Kt=v(µt)− ct

kt+ 1− δ. (43)

Dynamics for µt+1. The growth factor of µt+1 is defined by µt+1/µt = (Mt+1/Mt)/(Nt+1/Nt).

By using (23), (24), (30), (33) and (43), its value in equilibrium can be written as

µt+1

µt=

(gX γX(µt+1))1−α

gLγL(µt+1)

(γK(µt+1)

(v(µt)− ct

kt+ 1− δ

))α, (44)

where γK(µt), γX(µt), and γL(µt) are the rates of technological progress defined in Proposition 7.

While equation (44) gives a relationship between the period-t variables µt, kt, ct 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 (44) into two steps.

First, we define the pre-R&D relative factor intensity by

µpret+1 ≡(AK,tKt+1

)α(AX,tXt+1

)1−αAL,tLt+1

=g1−αX

gL

(v(µt)− ct

kt+ 1− δ

)αµt, (45)

where the last equality is from (30), (33), (43) and the definition of µt. It is the value of µt+1 before

technologies are improved from their period-t state. Second, µpret+1 and the post-R&D value of µt+1

are related by the growth of technological levels γK(µt), γX(µt), and γL(µt) as follows:

µpret+1 =γL(µt+1)

γK(µt+1)αγX(µt+1)1−αµt+1 ≡ Γ(µt+1). (46)

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 µt+1 = Γ(−1)(µpret+1) is

well-defined for all µpret+1 > 0, and is a strictly increasing differentiable function.

32

Using this inverse function and (45), the dynamic relationship (44) can be written as

µt+1 = Γ(−1)

(g1−αX

gL

(v(µt)− ct

kt+ 1− δ

)αµt

)≡ ψµ(µt, kt, ct). (47)

Equation (47) explains how µt+1 is determined given the period-t variables µt, kt, ct. We write

this mapping as µt+1 = ψµ(µt, kt, ct). It provides a natural 2-step interpretation of the equivalent

equation (44). The argument of function Γ(−1)(·) in (47) represents the pre-R&D relative task

intensity, which is determined by the relative supply of production factors, as well as the period-t

technology levels. Then, function Γ(−1)(·) describes how R&D in period t+1 transforms the relative

task intensity.

Dynamics for kt+1. From (30) and (43), the growth factor of kt ≡ Kt/AL,tLt is obtained as

kt+1

kt=

1

gLγL(µt+1)

(v(µt)− ct

kt+ 1− δ

). (48)

While µt+1 is present in the RHS, we can replace it with (47) so that the RHS depend only on the

variables in period t.

kt+1 =1

gLγL(ψµ(µt, kt, ct))(v(µt)− ct + (1− δ)kt) ≡ ψk(µt, kt, ct). (49)

This dynamic equation simply represents the process of capital accumulation per effective labor.

The expression (v(µt)− ct + (1− δ)kt) shows the sum of the net saving and the undepreciated part

of existing capital, per effective labor in period t. It must be divided by gLγL because of the growth

of effective labor between period t and t+ 1.

Dynamics for ct+1. The representative household maximizes the intertemporal utility function

(32) subject to the budget constraint (34) and the non-Ponzi Game condition (35). From (32),

∂U/∂Ct = βt(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

C−θt = (rt+1 + 1− δ)βgθLC−θt+1. (50)

By substituting the market interest rate (36) into the Euler equation (50) and then applying it to

the definition ct ≡ Ct/AL,tLt, we obtain the growth factor of consumption per effective labor:

ct+1

ct=

β1/θ

γL(µt+1)

(αµt+1

kt+1

(f ′(µt+1)− iK(γK(µt+1))− iX(γX(µt+1))

)+ 1− δ

)1/θ

. (51)

By replacing the period-(t+ 1) variables in the RHS by (47) and (49), we can rewrite equation (51)

33

as

ct+1 =β1/θct

γL(ψµ(µt, kt, ct))

(αψµ(µt, kt, ct)

ψk(µt, kt, ct)

(f ′(ψµ(µt, kt, ct))− iK(γK(ψµ(µt, kt, ct)))

−iX(γX(ψµ(µt, kt, ct))))

+ 1− δ

)1/θ

≡ ψc(µt, kt, ct).

(52)

Equations (47), (49) and (52) constitute the equilibrium mapping from µt, kt, ct to µt+1, kt+1, ct+1for all t ≥ 0.

Boundary Conditions. To obtain the equilibrium path of µt, kt, ct∞t=0, we need three boundary

conditions. First, since K0, X0, L0, AK,−1,AX,−1 and AL,−1 are given, we can construct µpre0 , the

pre-R&D relative task intensity for period 0. Using it with the inverse function of Γ from (46), we

have the initial value of µt:

µ0 = Γ(−1)(

(AK,−1K0)α(AX,−1X0)

1−α

AL,−1L0

). (53)

Second, using µ0, the initial value of kt is readily obtained by

k0 =K0

γL(µ0)AL,−1L0. (54)

Finally, the initial value of ct must be chosen so as to satisfy the non-Ponzi game condition (35)

and the transversality condition

limT→∞

βT(CTLT

)−θKT+1 ≤ 0. (55)

By using the Euler equation (50), we show in Appendix A.11 that conditions (35) and (55) jointly

mean

limT→∞

(βgL)T

(T∏t=0

γL(µt)

)1−θ

γL(µT+1)c−θT kT+1 = 0. (56)

The next subsection will show that the economy has a BGP that satisfies this terminal condition.

6.4 The Balanced Growth Path

Now, we are ready to derive the balanced growth path (BGP) of this economy. We will show

that the direction of 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, Lt, Ct,

34

Rt, Mt, Nt are all constant.38 Then, on any BGP, the values of µt, kt and ct must be constant.


We denote the BGP values of µt, kt and ct by µ∗, k∗ and c∗, respectively. Note that the RHS of

(44), (48) and (51) represent the growth factors of µt, kt and ct, and therefore they take the value

of 1 on the BGP. Rearranging them, we obtain the values of µ∗, k∗ and c∗ as follows.

First, from (44) and (48), the BGP value of µt ≡Mt/Nt will satisfy

1 =(gX γX(µ∗))1−α (γK(µ∗))α

(gLγL(µ∗))1−α.≡ Φ(µ∗). (57)

Note that Proposition 7 implies Φ′(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 (57). An intuitive way to

interpret (57) is to multiply the both of its sides by (gLγL(µ∗))α.

(gX γX(µ∗))1−α (γK(µ∗)gLγL(µ∗))α = gLγL(µ∗) (58)

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 determined so that Mt and

Nt grow at the same speed on the Equilibrium Innovation Possibility Frontier (EIPF), as depicted

in Figure 2. Because of the CRS property of production function Yt = F (Mt, Nt), the value of

equation (57) also represents the the economic growth factor g∗ ≡ Yt+1/Yt.

Second, from the Euler equation (51), the BGP value of kt = Kt/(AtLt) is

k∗ =βαµ∗(f ′(µ∗)− iK(γK(µ∗))− iX(γX(µ∗)))

γL(µ∗)θ − β(1− δ), (59)

where µ∗ is obtained from (57). Intuitively, (59) means that the capital-effective labor ratio on the

BGP is determined by the fact that the interest rate r∗ yields constant consumption per effective

labor on the BGP.39 Third, from the resource constraint (48), the BGP value of c∗ = Ct/AL,tLt

must satisfy

c∗ = v(µ∗)− (g∗ − 1 + δ)k∗, (60)

where µ∗ and k∗ are given by (57) and (59), and g∗ = gLγL(µ∗). These three equations describe

the unique BGP in this economy.

38Here, we slightly extend Definition 2 by requiring constancy of the growth factors of Mt and Nt, i.e., the numbersof tasks performed in the economy.

39Using (36), condition (59) is shown to be equivalent to r∗ + 1− δ = β−1γL(µ∗)θ. Here, the RHS is the marginalrate of intertemporal substitution given that consumption per effective labor is constant (which must be true on theBGP).

35

Proposition 8. There exists a value of β > 0 such that whenever β ∈ (0, β), there exists a unique

BGP that satisfy µ∗ > 0, k∗ > 0, c∗ > 0, and the terminal condition (56).40

Proof. See Appendix A.13. The exact expression for the upper bound β is given by (A.38).

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 (57) is equivalent to (22), except

that the speed of technological progress is endogenously determined by profit-maximizing producers.

This difference has important implications for the plausibility of capital-augmenting technological

progress on the BGP. In the example discussed in Section 5.3, the rates of innovation for the three

factor-augmenting technologies are exogenously given, and they must satisfy a knife-edge condition

(22). We have shown that, once we consider endogenous technical change, this condition is naturally

satisfied when the economy is on the BGP.

So far, we have shown the existence and uniqueness of the BGP. The remaining task is to

demonstrate that the economy necessarily converges to it. For this aim, section 7 explores the model

numerically. We calibrate the model to standard macroeconomic facts describing the BGP of the

United States. We then provide evidence that the model is both locally and globally stable. Once

we establish the stability and convergence, it means that condition (22) is endogenously satisfied

in the long run—i.e., balanced growth with capital-augmenting progress is possible without this

knife-edge condition.

7 Numerical Analysis and Stability

The multi-factor Uzawa steady state growth theorem presented in Proposition 1 and 4 requires that

the growth rates of factor-augmenting technologies conform to a particular log-linear relationship

as in equation (22). Section 6 demonstrates that this log-linear relationship can be an endogenous

outcome of a model with directed technical change, suggesting that this requirement is not actually

restrictive. The analytic results, however, did not demonstrate that the economy will necessarily

converge to the BGP with capital-augmenting technical change.

In this section, therefore, we investigate the local and global stability of the three factor endoge-

nous growth model. To do so, we present a series of numerical examples for which we can check

both local and global stability computationally. Whenever possible, we ensure that our numerical

40There are two reasons why β should not be too large (in other words, the discount rate ρ = (1 − β)/β shouldnot be too small). First, on the BGP, the amount of consumption for the household Ct = AL,tLtc

∗ increases overtime, causing the instantaneous utility to grow. Therefore, if β is too close to one, the intertemporal utility U in (32)becomes infinity, which means that the household’s problem is not well defined. Second, as effective labor AL,tLtgrows, the household need to accumulate more capital Kt so as to prevent the dilution of capital per effective labor,k∗. However, when β is too large (i.e., when the discount rate ρ is too small), the BGP requires a too low real interestrate, or a too high level of k∗, to the extent that preventing the dilution is impossible even when all net output isinvested in Kt. We rule out these extreme cases by assuming an upper bound for β.

36

examples are consistent with macroeconomic data characterizing the balanced growth path of the

United States. We stress, however, that this is not a complete calibration, and the results would be

insufficient for a complete quantitative analysis. Our contribution is theoretical, and the numerical

results are meant to be informative about stability. As discussed in section 8, future empirical

and quantitative work to fully pin down the parameters of such a growth model is a particularly

promising area for future work.

7.1 Calibration

7.1.1 Preliminaries

In this section, we describe the partial calibration procedure that we use to discipline our numerical

examples. We start by making some important assumptions. In line with Section 6, we assume

that there is some aggregate variable X that incorporates all factors of production other than labor

and capital.41 For ease of exposition, we call this factor land.

Next, we need to make two important functional form assumptions. We start by assuming that

the aggregate production function (25) is CES,

Yt =

η((AKK)α (AXX)1−α

) ε−1ε + (1− η)(ALL)

ε−1ε

ε1−ε

. (61)

Next, we assume that the cost function for R&D (27) takes the form42

iZ(γZ) = ζZ(γZ − 1

)λ, ζZ > 0, λ > 1, (62)

for Z = K,X,L. The cost of R&D investment differs across types of technology due only to the ζZ

parameters. The degree of convexity, λ is the same across the three types of technology.

7.1.2 Free Parameters

Our calibration procedure will not be able to pin down all of parameters in the model. Thus, we set

some parameters exogenously. Again, we stress that our goal is to show that the model is stable,

not to conduct exact quantitative exercises. In the appendix, we show that the stability results are

robust to a wide range of choices for the free parameters.

To start, we assume log preferences (θ = 1). In the appendix, we show robustness with θ = .5

and θ = 1.5. Next, we assume that the R&D cost function is quadratic (λ = 2). This is a common

assumption, and it is consistent with existing empirical work in endogenous growth (Acemoglu et al.,

41This is done for transparency and tractability. The theorem only requires that any one factor non-reproduciblefactor is combined with capital with a unit-elastic elasticity of substitution.

42As in (26), R&D cost function (62) can also be represented in terms of R&D production function: e(iZ) =(iZ/ζZ)1/λ. This e(iZ) function satisfies the required conditions: e′Z(0) = ∞, e′Z(iZ) > 0 and e′′Z(iZ) < 0 for iZ > 0and eK(∞) =∞.

37

2013; Akcigit and Kerr, 2018). We show robustness with λ = 1.5 and λ = 2.5. Finally, we take

ε = .8. This is a common estimate for the elasticity of substitution between labor and reproducible

capital (e.g., Oberfield and Raval, 2014; Antras et al., 2004; Klump et al., 2007; Herrendorf et al.,

2015; Alvarez-Cuadrado et al., 2018). We show robustness with ε = .5 and ε = 1.5.43 Unlike many

other endogenous growth models, we find that our model has a globally stable balanced growth

path with an elasticity of substitution that is greater than one (Acemoglu, 2003; Grossman et al.,

2017a).44 Interpreting X as land, we take gX = 1, implying that there is no change in natural

resource availability over time.

7.1.3 Data

Our goal is to calibrate the BGP endogenous growth model to data from the U.S. We report annual

values. We discuss choice of period length later in this section. To start, We take the capital output

ratio (K/Y = 3), the labor share of income (κL ≡ wtLtYt

= 65%), the growth rate of income per

capita (γL = 1.02) from Jones (2016).45 These are all fairly standard parameters. We also set

the growth rate rate of capital-augmenting technology to match the decline in the relative price

of capital goods from 1980-2015. We use 1980 rather than 1960 because there was a significant

revision of the measurement method in 1978, with a string of subsequent updates.46 The 1980-

2015 data yield γK − 1 = 1.02%. Compared to the 1960 data, this is much closer the growth rate

measured by DiCecio (2009), which is used in (e.g., Grossman et al., 2017a).47 In the appendix, we

show that our results are robust to using γK − 1 = 0.63%, which is the average growth rate from

1960-2015. Next, we set the rate of return on investment equal to the return on bonds using data

from McGrattan and Prescott (2003) (r∗−δ = 4%). We also take consumption of fixed capital from

Rognlie (2016) (δK/Y = 11%). In a standard two-sector model, the income share of reproducible

capital would be equal to one minus the labor share. In our model, however, there are two other

factors, land and R&D, that also receive payments income. We take the share of R&D payments

in total income from NIPA (R/Y = 3%). We calibrate the model using these seven moments.

7.1.4 Calibration Procedure

To calibrate the model, we use the Euler equation (51), the law of motion for capital (48), the law

of motion for µ = M/N , (44), and the six first order conditions for the profit maximization problem

43Karabarbounis and Neiman (2014) and Piketty (2014) estimate the elasticity of substitution between reproduciblecapital and labor and find elasticities that are greater than one.

44See, Irmen and Tabakovic (2017) for a growth model that also has a stable BGP with an elasticity of substitutionthat is greater than one.

45Our choice of round numbers is meant to highlight the fact that we do not claim to have a precise and accuratecalibration. Instead, we calibrate the model to demonstrate stability.

46See BLS Handbook of Measurement, chapter 14, p.11, available here: https://www.bls.gov/opub/hom/pdf/

ppi-20111028.pdf.47The procedure used by DiCecio (2009) is an update of the older measurement approach developed by Gordon

(1990) and Krusell et al. (2000).

38

https://www.bls.gov/opub/hom/pdf/ppi-20111028.pdf

https://www.bls.gov/opub/hom/pdf/ppi-20111028.pdf

(three technology conditions and three for factor demand), which are captured by equations (36)

– (41). This yields nine equations, but since the production function is CRS, one of the first order

conditions is redundant, implying that we have eight degrees of freedom.

Taking λ, θ, and ε as exogenous parameters, we have seven remaining parameters to calibrate,

(β, δ, α, η, ζK , ζL, ζX). In addition, we calibrate the period length, which we will denote with χ

below. While this is somewhat unusual, the period length in our model has a specific economic

meaning, as it is the length of time that firms can benefit from private investment. As a result, we

have eight unknown parameters. Together with the eight equations, this implies that our calibration

procedure is exactly identified.48

We break the calibration process into two steps. First, we use the data and analytic results

from section 6 to calibrate as many parameters as possible. Then, we perform a minimization

procedure to pick the remaining parameters in a way that is consistent with both the data and

the parameters that were calibrated analytically. In practice, we could iterate between the analytic

and minimization steps to ensure that they are internally consistent, but in practice we find that

this is not necessary as we are able to exactly the data.

Analytic Calibration Steps. – We start the calibration by deriving as many parameters as

possible analytically. To start, note that δK/Y = .11 and K/Y = 3. This directly implies that

δannual = 3.7%. This is the annual value. In the model, we use δ = 3.7%χ. We also know that

r∗ − δ = 4%, which implies r∗ = 7.7%. Now, κK ≡ rK/Y = 21%. From the data, we have

κL = 65% and κR = 3%. This leaves κX = 9%, which is an untargeted moment in our calibration

procedure. Comparing this results to Table 1, this is a reasonable estimate for the factor share of

natural resources. Finally, from equations (36) and (37), we have α =κ∗K

κ∗K+κ∗X= .74. Next, we have

βannual = γθL/(1 + r∗ − δ) = .98. In the model, therefore, we use β = 0.98χ.

While we cannot derive any further parameters analytically, we can still place an extra restriction

on the analytic outcomes. To start, we can use the log-linear technology restriction to derive

γX−1 = gLγL ·γ−α1−αK −1 = 0.15%. Then, using equations (40) and the functional form assumptions

(62), we have

ζKζX

=α

1− α·(γX − 1

γK − 1

)λ−1. (63)

Plugging in the values calibrated above yields ζK/ζX = 0.41. We impose this restriction in the

minimization procedure.

Minimization. – We are not able to make any further progress analytically. The remaining

48We confirm this fact with the data. We are able to exactly the moments in the data given above for a single setof parameters only.

39

parameters are (η, ζK , ζL). To calibrate these remaining parameters, we simulate the model after

imposing our parameter values and restrictions found above. Let Γ = (K/Y κL κR γL γK) be a a

1x5 vector of the estimated moments from the model and Γ = (3 0.65 0.037 1.02 1.01) be a 1x5

vector of the moments in the data. We do not include the other two moments, δK/Y = 0.11 and

r∗ − δ = 4%, because they are redundant given analytic steps taken above.49 Then, subject to the

analytic results derived above, we choose

(η, ζL, ζK) = arg min(Γ− Γ)W (Γ− Γ)′, (64)

where W is a 5x5 weighting matrix. We minimize the squared errors in percentage terms, i.e.,

Wii = 1/Γi, and Wij = 0, i 6= j. As noted above, our calibration procedure is exactly identified

and we are able to exactly mach the moments from the data.50

7.2 Calibration Results

Table 2 presents the results of the two part calibration procedure, with the baseline assumptions.

Appendix section B presented results with alternate assumptions. Changing the free parameters,

we also present results with λ ∈ 1.5, 2.5, θ ∈ .5, 1.5, and ε ∈ .5, 1.1. In each case, we change

one parameter from the baseline value and then re-calibrate the model. As discussed in the next

subsection, the model is stable under all of these alternate parameter choices. Finally, we also test

results with γK = 0.6%/yr, which is the average value from 1960-2015.

Parameter Baseline Description Source

λ 2.0 R&D Convexity Free ParametergX 1.0 Growth of X Definition of Xε 0.8 Elast. Sub. Oberfield and Raval (2014)

θ 1.00 IES Log PreferencesgL 1.01 Population growth BEA

χ 4.04 Period Length (years) Calibratedβ 0.98 Discount factor Calibratedδ 3.7% Depreciation Calibratedα 0.74 Cap. Distribution Calibratedη 0.34 CES Distribution CalibratedζK 2.17 Research efficiency CalibratedζL 4.04 Research efficiency CalibratedζX 5.27 Research efficiency Calibrated

Table 2: CALIBRATION. This table presents the baseline calibration. Growth rates and rates ofreturn (gL, gX , β, δ) are presented in terms of annualized values.

49We already calibrated δ assuming that K/Y = 3. Similarly, κL, κR, and α uniquely identify κK . Together withK/Y , this uniquely identifies r. So, δ and r∗− δ are redundant if we are able to exactly match the other parameters.

50We check that numerically that minimizing arguments are unique.

40

7.3 Stability

7.3.1 Local Stability

Table 3 examines the stability of the parameterized model. In particular, we calculate the eigenval-

ues of the three-variable system, (µt, kt, ct), at the steady state. The system is saddle path stable

when two eigevalues have absolute less than one (i.e., are stable) and one eigenvalue has absolute

value greater than one (i.e., is unstable).

Our baseline case is described in Table 2, but we also show robustness with a wide range of

alternate parameters. In particular, we vary our three free parameters, (λ, ε, θ), along with our

estimated value of growth in capital-augmenting technology, γK . When changing one of these

four variables, we re-do the entire calibration procedure. The calibrated parameters for robustness

exercises are presented in appendix section B.

Scenario Eigenvalues Stability

Baseline 0.97, 0.63, 1.60 Saddleλ = 1.5 0.95, 0.69, 1.43 Saddleλ = 2.5 0.97, 0.57, 1.78 Saddleε = .5 0.96, 0.57, 1.72 Saddleε = 1.2 0.97, 0.67, 1.52 Saddleθ = 0.5 0.97, 0.53, 1.92 Saddleθ = 1.5 0.97, 0.68, 1.48 Saddle

γK − 1 = 0.63% 0.95, 0.62, 1.81 Saddle

Table 3: STABILITY. This table shows the eigenvalues and local stability for different calibrationsof the endogenous growth model.

In all cases, we find that that the steady state is locally saddle path stable. It is particularly

interesting to note that we find stability even when the elasticity of substitution between labor and

the capital-composite (ε) is greater than one. Most directed technical change growth models require

a low elasticity to be stable, especially when allowing for the possibility of capital-augmenting

technical change (e.g., Acemoglu, 2003; Grossman et al., 2017a). The model of Irmen and Tabakovic

(2017) is stable with 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.

These results demonstrate that the log linear technology restriction is an endogenous outcome

of the model. The unit-elastic substitution between K and X is the key exogenous assumption

that must hold to have a BGP with a positive rate of capital-augmenting technical change. In our

model, there is a single X that captures all factors of production that do not fall under the headings

of labor or reproducible capital. The broader theory only requires that a single production factor

has this unit-elastic property.

41

7.3.2 Global Stability

The preceding section shows that the technology restriction from Proposition (4) does not pose

extra constraints on the neoclassical model when technology is endogenous. This was done by

showing that the model with endogenous technology has a locally stable BGP with positive K-

augmenting technical change. In this section, we go one step further and show that the model is

also globally stable.

0.10.2

0.3k

0.0 0.5 1.0 1.5

0.2

0.4

0.6

0.8c

0.0

1.0

0.00.10.20.3k

0.0

0.2

0.4

0.6

0.8c

Figure 3: Global Stability. This figure shows the dynamic paths of the model starting from a wide range of valuesof the two state variables µt and kt. In particular, the we test starting values for the state variables that are up totwo times the steady state values. The fact the model converges to the steady state from such a wide range of ofstarting values strongly suggests that the steady state with positive capital-augmenting technical change is globallysaddle-path stable. The figure shows the three-dimensional space from two angles in order to give a clear picture ofthe saddle paths.

To solve for the global dynamics, we use a forward shooting algorithm. There is only one control

variable, ct in the system. Since we already demonstrated that the model is locally saddle-path

stable, we know that there is only one value of ct that leads to convergence to the BGP, for any

given starting values of kt and µt. Once this value is known, it is straightforward to solve for all of

the subsequent values of all three variables in the normalized system. We can solve for the global

dynamics by iterating over guesses for ct and updating our guesses via a bisection algorithm.

We test for global stability from a wide range of starting values for the state variables. We

set up a two-dimensional grid for the starting values of kt and µt, testing values up to twice the

steady state value. Then, we find the global dynamics starting from each point on the outer edge

42

of this grid. Figure 3 presents the results. We show the three-dimensional figure from two angles

to better illuminate the transition paths. The figures show that the economy converges to the

BGP, even when they have started from initial states that are far away from the BGP. This means

that the economy endogenously converges to the state where BGP condition (57) is satisfied. This

result demonstrates that the log-linear relationship between technological growth rates (22), which

is equivalent to (57) in our endogenous growth model, can be viewed as an endogenous outcome

and does not pose an extra restriction on the conditions needed for balanced growth in neoclassical

models with improvements in the productivity of capital.51

7.4 Implications

The endogenous growth model demonstrates the practical importance of our theoretical results. We

view aggregate neoclassical growth models as using representations of some unknown production

function. Our results demonstrate how to choose a representation that is simultaneously consistent

with balanced growth, non-unit-elastic substitution between capital and labor, and the growth rate

of capital-augmenting technical change.

These results stress the importance of including natural resources and directed technical change

in growth models. There is, of course, a long literature on both of these topics, but they are

generally only included in growth models to achieve specific aims. 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; Barrage, 2019). Directed technical

change is often used to understand the evolution of relative wages between different groups of

workers (e.g., Acemoglu, 1998, 2002; Acemoglu and Restrepo, 2016). Our results suggest a much

broader importance of directed technical change and non-accumulable factors — these factors must

be incorporated into models of economic growth in order to recreate the balanced growth facts that

originally motivated aggregate growth modeling (Solow, 1956, 1994).

8 Discussion and Future Research

8.1 Estimating σKXj

The obvious next step for future research would be to determine whether there is some factor,

Xj , that has a unit-elastic elasticity of substitution with reproducible capital. In our illustrative

growth model, we assumed that this third factor was an amalgamation of all factors not included

in reproducible capital or labor. Proposition 4, however, only requires that a single factor of

51The simulation results also illustrate interesting properties of the 3-factor endogenous directed technologicalchange model. For example, the figure demonstrates that µt converges much slower than kt, a result that is consistentwith the eigenvalues reported in Table 3. This result suggests that ignoring land and endogenous technical change canlead neoclassical models to overestimate the speed of convergence to the steady state. Of course, these quantitativeresults are preliminary and warrant further investigation.

43

production has this property. Thus, it is necessary to test many different factors to see if they

satisfy this property.52 The obvious candidates would be land, energy, and materials, but it could

also be any subcategory of these types of inputs. Given the vast array of choices and the limited

restrictions created by the theory, there is reasonable possibility that such a factor exists.

8.2 Applications for a multi-factor model

The Uzawa steady state theorem is not a mere theoretical curiosity. It has important implications

for understanding the economy. If it is possible to identify a factors that has a unit-elastic substitu-

tion with reproducible capital, then models with aggregate productions can include both labor- and

capital-augmenting technical change on the balanced growth path, allowing for more informative

analysis of data and policy interventions.

Most obviously, the counter-factual notion that there is a single type of technology places severe

restrictions on our ability to describe the economy, even in an aggregate sense. In growth accounting,

there has been a great deal of work focusing on the role of investment specific technical change.

Without utilizing our results, such analyses must focus on the Cobb-Douglas case (e.g., Greenwood

et al., 1997; Krusell, 1998; Cummins and Violante, 2002) or be inconsistent with balanced growth

(e.g., Krusell et al., 2000).53 The same would is true of business-cycle- and development accounting

analyses (e.g., Fisher, 2006; DiCecio, 2009; Hsieh and Klenow, 2010; Schoellman, 2011). Work

by Caselli and Feyrer (2007) also demonstrates the importance of including natural resources in

development accounting exercises.

Separate from the accounting literate, models of economic growth with capital-augmenting

technical change would be useful in the study of automation, artificial intelligence, and labor-

saving technical change, topics that have recently received a great deal of attention in the growth

literature (e.g., Benzell et al., 2015; Hemous and Olsen, 2016; Brynjolfsson and McAfee, 2012, 2014).

An outstanding question in this literature is how to best model the impact of technology on labor

market outcomes (Acemoglu and Autor, 2011; Acemoglu and Restrepo, 2016, 2018). Models with

only one type of technology in the long run – e.g., those constrained the Uzawa (1961) theorem –

are naturally of limited use when trying to determine when technological change is worker-friendly

and when it is not. In standard neoclassical production functions, capital- and labor-augmenting

technical change have different impacts on the labor share of income. With low elasticities of

substitution, positive shocks to labor- and capital-augmenting technologies also have opposite effects

on the levels of wages and rental rates (Acemoglu, 2010; Acemoglu and Restrepo, 2018). Thus,

including both capital- and labor-augmenting technical greatly increases the usefulness of these

52While, of course, avoiding the pitfalls of multiple hypothesis testing.53See, He et al. (2008) and Maliar and Maliar (2011) for discussions of how to make these models consistent with

balanced growth.

44

models in studying impacts of new technologies.54

8.3 What if there is no Xj with σKXj = 1?

Despite the wide range of factors used in production, it is possible that no factor has unit-elastic

substitution with reproducible capital. In this case, our analysis suggests that it is not possible

to write down a multi-factor growth neoclassical growth model with capital-augmenting technical

change.

If the result of the Uzawa theorem does not fit the data, we need 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 assumption: (1) the economy can be expressed by a neoclassical growth

model, and (2) there is a balanced growth path. The NIPA data strongly suggests the existence of

the balanced growth path. Therefore, we can narrow down the concern 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 Uzawa theorem, the latter is only 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 trend 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 the there is a mapping from aggregated factor inputs to the aggregate output.

While vast majority of all macroeconomic models use some form of aggregate production function,

it is neither a weak assumption nor a fully proven property. For example, Figure 1 shows that the

movement of the relative price of capital depends the type of capital (e.g., equipment and structure).

Even within equipment category, the relative price of different capital goods changes dramatically

over time. The same can be said for different kinds of output. The aggregate production function

implicitly assumes that capital and output can be aggregated and there are stable relationship

between these aggregates. If there is no Xj with σKXj = 1, it might suggest that the aggregation

54On a related note, much of the literature on automation has argued that factor-augmenting technologies are anineffective way of modelling labor-saving technical change. Instead, such papers argue for using a task-based modelalong the lines of Zeira (1998) (e.g., Acemoglu and Autor, 2011; Acemoglu and Restrepo, 2016, 2018). However, asdiscussed in Acemoglu and Restrepo (2016), task-based growth models are still subject to the Uzawa theorem, aslong as they can be represented by an aggregate production function. Indeed, Propositions 1 and 4 imply that thesemodels have Uzawa and Factor-Augmenting representations on the BGP. To the best of our knowledge, no task-basedmodel of endogenous growth has capital-augmenting technical change on the BGP.

55Another concern with the resource constraint might be that it assumes a one-to-one conversion between Ct andKt+1. However, as discussed in Section 3, if the conversion rate changes over time, we can redefine the units of capitalso Definition 1 still applies (see equation 3). In this sense, it does not pose a real restriction.

45

created the problem. Exploring disaggregated models seems important in this respect.

9 Conclusion

The Uzawa (1961) steady state theorem has long posed significant problems for neoclassical models

of balanced growth (Jones and Scrimgeour, 2008). As conventionally understood, the theorem

states that all technological change must be labor-augmenting on a balanced growth path, unless

the production function is Cobb-Douglas. This constraint makes it impossible for the neoclassical

model to be consistent with data indicating that (i) the two-factor aggregate production function is

not Cobb-Douglas and (ii) the relative price of capital goods has been declining. This is a significant

limitation. Substitution between capital and labor and technological progress are the key forces

that generate balanced growth in the standard model (see, e.g., Solow, 1956, 1994).

To provide insight into this apparent contradiction, we prove a multi-factor version of the Uzawa

(1961) steady state theorem. The generalized version of the theorem says that, on a balanced

growth path, production can be represented by a neoclassical production function without capital-

augmenting technical change. As in the standard Uzawa theorem, this does not imply that the

true production function does not have capital-augmenting technical change. More precisely, the

theorem does not rule out the existence of multiple different representations of aggregate production

along the balanced growth path. Thus, we also examine the circumstances under which there is

a representation of the aggregate production function with capital-augmenting technical change.

We prove that, with more than two factors of production, neoclassical production functions can

accommodate capital-augmenting technical change, as long as there is a single factor than has a

unit-elastic elasticity of substitution with reproducible capital.

This suggests a natural way to resolve the contradiction affecting standard models: add more

factors of production. In reality, there are many factors of production besides labor and reproducible

capital. If any of these has a Cobb-Douglas relationship with capital, then multi-factor neoclassical

models can be consistent with the data. We also build a three-factor endogenous growth model

and show that is converges to a globally-stable BGP with a positive rate of capital-augmenting

technical change.

We strongly suggest that future empirical work investigate whether there is a factor of pro-

duction that is Cobb-Douglas with reproducible capital. If such a factor can be identified, then

tractable aggregate models can be made more consistent with data and will be better able to study

a wide range of economic phenomena.

46

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51

Appendix

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 F (·).Following the convention in economics, ∂

∂Ktand ∂

∂Xj,trepresent the partial derivatives with

respect to variable Kt and Xj,t, respectively. For example, if F (·) is the production function,∂

∂Xj,tF (·) 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

∂

∂Xj,tF (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t) = AXj ,tFXj (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t). (A.1)

A.2 Proof of Lemma 1

Note that δ = (δ+gq−1)/gq means 1−δ = (1−δ)qt+1/qt. By dividing equation (3) by qt+1 and using

the above, we have Kt+1 = Kt+1/qt+1 = (Yt−Ct−Rt)+(1− δ)Kt/qt+1 = (Yt−Ct−Rt)+(1−δ)Kt,

which coincides with (2).

Production function F (Kt, X1,t, ..., XJ,t; t) = F (qt−1Kt, X1,t, ..., XJ,t; t) is a CRS function of

Kt, X1,t, ..., XJ,t and depends on time both through the shape of F (·; t) and through the growth

of qt. Therefore we can define function F (Kt, X1,t, ..., XJ,t; t) ≡ F (qt−1Kt, X1,t, ..., XJ,t; t), where

dependence of F (·; t) on t includes the effect from qt−1. From the assumptions on F (·), function

F (·; t) obviously satisfies the required marginal product properties in (1).


Using the notation in Definition 2, equation (2) can be written as K0gt+1K = Y0g

t−C0gtC −R0g

tR +

(1− δ)K0gtK . Dividing all terms by gt and rearranging them gives

Y0 = C0(gC/g)t +R0(gR/g)t +K0(gK + δ − 1)(gK/g)t. (A.2)

Because all three terms on the right hand side (RHS) of (A.2) are non-negative exponential functions

of t, every one of them needs to be constant for the sum of all the terms to become constant (Y0).

For the first term C0(gC/g)t to be constant, gC = g must hold since C0 > 0 from Definition 1. This

means Ct/Yt = C0/Y0 > 0. For the third term (gK + δ − 1)(gK/g)t to be constant, gK = g must

52

hold since K0 > 0 and gK > 1− δ. This implies 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) ≡ F (·), we can write the share of factor Z in

period 0 as

sZ,0 = FZ(K0, X1,0, ..., XJ,0)Z0

Y0, (A.3)

where FZ(·) represents the derivative of function F (·) with respect to its argument (see Appendix

A.1). Note that, since function F (·) has constant returns to scale, its partial derivative function

FZ(·) must be homogeneous of degree 0 (See Theorem M.B.1 in Mas-Colell et al., 1995). Therefore,

the value of FZ(·) will be unchanged when all of its arguments are multiplied by the same factor

gt = Yt/Y0 = Kt/K0 = AXj ,tXj,t/Xj,0 (Here we used gK = g from Lemma 2.) Applying this for

(A.3) gives

sZ,0 = FZ(Kt, AX1,tX1,t, ..., AXJ ,tXJ,t)Z0

Y0.

In addition, because the effective amount of production factors and the output grow at the same

speed, Z0/Y0 = AZ,tZt/Yt holds on the BGP. (In the case of Zt = Kt, we define AK,t ≡ 1.)

Therefore,

sZ,0 = FZ(Kt, AX1,tX1,t, ..., AXJ ,tXJ,t)AZ,tZtYt

=∂F (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t)

∂Zt

ZtYt, (A.4)

where the validity of the second equality is guaranteed by the chain rule.56 Recall that we assumed

that the share is constant over time, which means

sZ,0 = sZ,t = FZ(Kt, X1,t, ..., XJ,t; t)ZtYt. (A.5)

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


As in Definition 3, the EoS between Kt and Xj , j ∈ 1, ..., J, in the Uzawa Representation

F (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t) is defined as

σKXj ,t = − d ln(Kt/Xj,t)

d ln

(FK(Kt,AX1,t

X1,t,...,AXJ,tXJ,t)

AXj,tFXj (Kt,AX1,tX1,t,...,AXJ,tXJ,t)

)∣∣∣∣∣∣∣∣Yt,X−j,t,:const

(A.6)

56See Appendix A.1.

53

We used (A.1) for calculating the marginal product of Xj in the denominator. Note that, in addition

to output Yt and other production factors X−j,t, we keep technologies AX1,t, ..., AXJ ,t 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 the specific BGP values,

and k and xj the variables to be perturbed. Then, (A.6) can be written as57

σKXj ,t = − d ln(k/xj)

d ln

(FK(k,AX1,t

X1,t,...,AXj,txj ,...,AXJ,tXJ,t)

AXj,tFXj (k,AX1,tX1,t,...,AXj,txj ,...,AXJ,tXJ,t)

)∣∣∣∣∣∣∣∣F (k,AX1,t

X1,t,...,AXj,txj ,...,AXJ,tXJ,t)=Yt

k=Kt,xj=Xj,t

(A.7)

Condition F (k, AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t) = Yt says that k and xj need to move 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.

Now, consider a change of variables: k′ = g−tk and x′j = g−tAXj ,txj . Then, k in (A.7) is replaced

by k = gtk′ and xj is by (gt/AXj ,t)x′j . Specifically, k/xj in the numerator becomes AXj ,tk

′/x′j . In

the denominator,

FK(k, AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t) = FK(gtk′, gtX1,0, ..., gtx′j , ..., g

tXJ,0)

= FK(k′, X1,0, ..., x′j , ..., XJ,0),

where we used the definition of AXj ,t ≡ gtXj,0/Xj,t and the homogeneity of degree 0 property of

the FK(·) function.58 Similarly,

FXj (k, AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t) = FXj (k′, X1,0, ..., x

′j , ..., XJ,0).

Note that, using the CRS property of F (·) condition F (k, AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t) =

Yt can be simplified as

F (gtk′, gtX1,0, ..., gtx′j , ..., g

tXJ,0) = gtF (k′, X1,0, ..., x′j , ..., XJ,0) = Yt.

Since Yt = gtY0, the condition reduces to F (k′, X1,0, ..., x′j , ..., XJ,0) = Y0. The point of evaluation,

k = Kt, becomes gtk′ = Kt, or k′ = g−tKt = K0. Similarly, xj = Xj,t becomes x′j = g−tAXj ,tXj,t =

Xj,0. Therefore, (A.7) can be expressed in terms of k′ and x′j as follows:

57We omit condition “X−j,t: const” because we already made it clear that Xj,t’s are the BGP values, not variables.58For the homogeneity of degree 0 property, see the proof of proposition 2 in appendix A.4.

54

σKXj ,t = −d ln(AXj ,tk

′/x′j)

d ln

(FK(k′,X1,0,...,x′j ,...,XJ,0)

AXj,tFXj (k′,X1,0,...,x′j ,...,XJ,0)

)∣∣∣∣∣∣∣∣F (k′,X1,0,...,x′j ,...,XJ,0)=Y0k′=K0,x′j=Xj,0

(A.8)

Recall that we keep technology AXj ,t fixed when calculating the EoS. We can eliminate AXj ,t

from the numerator from d ln(AXj ,tk′/x′j) = d(ln(k′/x′j) + ln AXj ,t) = d ln(k′/x′j). In the same

way, AXj ,t in the denominator can also be eliminated (or replaced by AXj ,0 ≡ 1). Finally, using

F (·) ≡ F (·; 0), (A.8) can be written as

σKXj ,t = −d ln(k′/x′j)

d ln

(FK(k′,X1,0,...,x′j ,...,XJ,0;0)

FXj (k′,X1,0,...,x′j ,...,XJ,0;0)

)∣∣∣∣∣∣∣∣F (k′,X1,0,...,x′j ,...,XJ,0;0)=Y0k′=K0,x′j=Xj,0

(A.9)

Then, comparing with Definition 3, it turns out that the RHS of (A.9) exactly matches the

definition of σKXj ,0, evaluated at the period-0 BGP. Since it is assumed that σKXj ,t does not

change over time, we have σKXj ,t = σKXj ,0 = σKXj ,t.


Proof of part a

By substituting K0, X1,0, ..., XJ,0 into (11) and then using (12),

F (K0, X1,0, ..., XJ,0) = F

(Kα

0

∏j∗

j=1Xξjj,0, Xj∗+1,0, ..., XJ,0

)= F

((∏j∗

j=1Xξjj,0

)−1/α(Kα

0

∏j∗

j=1Xξjj,0

)1/α

, X1,0, ..., XJ,0; 0

)= F (K0, X1,0, ..., XJ,0; 0) .

Proof of part b

Let M0 = Kα0

∏j∗

j=1Xξjj,0 denote the amount of capital composite m in period 0, and FM (·) denote

the derivative of function F (·) with respect to its first argument. By differentiating both sides of

(12) by m with chain rule and substituting the period-0 BGP values for k, xj∗+1, ..., xJ ,

FM (M0, Xj∗+1,0, ..., XJ,0) = FK(K0, X1,0, ..., XJ,0; 0)

(∏j∗

j=1Xξjj,0

)−1/α 1

αM

(1−α)/α0

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

αM0.

(A.10)

55

Now consider the case of Z = K. By differentiating the both sides of (11) by k with chain rule and

substituting the period-0 BGP values,

FK(K0, X1,0, ..., XJ,0) = FM (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∗,

FXj (K0, X1,0, ..., XJ,0) = FM (M0, Xj∗+1,0, ..., XJ,0) ξjM0/Xj,0

= FK(K0, X1,0, ..., XJ,0; 0)ξjα

K0

Xj,0.

(A.11)

Note that, from the definitions of α and ξj in (10), ξj/α = sXj ,0/sK,0. Therefore, (A.11) becomes

FK(K0, X1,0, ..., XJ,0; 0)FXj (K0, X1,0, ..., XJ,0; 0)Xj,0

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

K0

Xj,0= FXj (K0, X1,0, ..., XJ,0; 0).

Finally, consider the case of Z = Xj , where j ∈ j∗ + 1, ..., J. similarly to the proof of part

a, we can confirm that F (K0, X1,0, ..., Xj∗,0, xj∗+1, ...xJ) = F (K0, X1,0, ..., Xj∗,0, xj∗+1, ..., xJ ; 0)

for any xj∗+1, ..., xJ . This means that they are identical functions of xj∗+1, ..., xJ , and have

the same derivative with respect to these variables. Therefore, for j ∈ j∗ + 1, ..., J, we have

FXj (K0, X1,0, ..., XJ,0) = FXj (K0, X1,0, ..., XJ,0; 0).

Proof of part c

The EoS for function F (·) between capital and factor j, evaluated at the period-0 BGP, is defined

as

σKXj ,0 = − d ln(k/xj)

d ln

(FK(k,X1,0,...,xj ,...,XJ,0)

FXj (k,X1,0,...,xj ,...,XJ,0)

)∣∣∣∣∣∣∣∣F (k,X1,0,...,xj ,...,XJ,0)=Y0k=K0,xj=Xj,0

, (A.12)

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 σKXj ,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

F (m,Xj∗+1,0, ..., XJ,0), wherem is the amount of capital composite, defined asm = kαxξjj

∏j′∈1,...,j∗\j X

ξj′

j′,0.

Using the chain rule, its derivative with respect to k becomes

FK(k,X1,0, ..., xj , ..., XJ,0) =∂

∂kF (m,Xj∗+1,0, ..., XJ,0)

= FM (m,Xj∗+1,0, ..., XJ,0)∂m

∂k

= FM (m,Xj∗+1,0, ..., XJ,0)αm

k.

56

Similarly, FXj (k,X1,0, ..., xj , ..., XJ,0) = FM (m,Xj∗+1,0, ..., XJ,0)ξjmxj

. Substituting these into (A.12)

gives

σKXj ,0 = − d ln(k/xj)

d ln(αξj

xjk

)∣∣∣∣∣∣F (k,X1,0,...,xj ,...,XJ,0)=Y0k=K0,xj=Xj,0

. (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 σKXj ,0 = 1. Recall that σKXj ,0 = 1

because j ∈ 1, ..., j∗. Therefore, σKXj ,0 = σKXj ,0 holds.

Next, we examine σKXj ,0 for the case of j ∈ j∗ + 1, ..., J. Using (11) and (12), function

F (k,X1,0, ..., xj , ..., XJ,0) is identical with F (k,X1,0, ..., xj , ..., XJ,0; 0) as a function of k and xj .

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 σKXj ,0 = σKXj ,0.

Proof of part d

Let us first consider the CRS property of function F (m,xj∗+1, ..., xJ). We multiply every argument

by an arbitrary factor of λ > 0. From (12),

F (λm, λxj∗+1, ..., λxJ) = F

(λ1/α

(∏j∗

j=1Xξjj,0

)−1/αm1/α, X1,0, ..., Xj∗,0, λxj∗+1, ..., λxJ ; 0

).

(A.14)

Recall that the period-0 original production function F (·; 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 when the output and other inputs are kept constant.59 In

other words, this differential equation is satisfied on the isoquant curve in the k-x1 space. Inte-

grating equation d ln(FK/FX1)/d ln(k/x1) = −1 gives ln(FK/FX1) = − ln(k/x1) + ξ1, where ξ1 is a

constant of integration. Taking the exponential of the both sides gives

FK/FX1 = (exp ξ1)(x1/k). (A.15)

From the definition of the isoquant curve, the amount of output must be constant: dY =

FKdk + FX1dx1 = 0. Rearranging and using (A.15), we have the slope of the isoquant curve

as dx1/dk = −FK/FX1 = −(exp ξ1)(x1/k). Integrating this differential equation gives ln k =

−(1/ exp ξ1) lnx1 + y1, where y1 is another constant of integration. By taking the exponential, the

isoquant curve is written as

k = (exp y1)x−1/ exp ξ11 . (A.16)

Different values of y1 gives isoquant curves corresponding to different output values. However,

59To minimize notation we omit the arguments of the functions FK(k, x1, ..., xJ ; 0) and FX1(k, x1, ..., xJ ; 0).

57

the value of ξ1 must be the same across all isoquant curves, since otherwise they intersect with

each other, which is impossible by definition of the isoquant curve. Now, consider the isoquant

curve that runs through the period-0 BGP value. Then, at the period-0 BGP, (A.15) implies that

FK/FX1 = (exp ξ1)(X1,0/K0) must hold. Note that, from (10), sK,0/sX1,0 = FKK0/FX1X1,0 =

α/ξ1. Comparing these gives (exp ξ1) = α/ξ1. Using this, the isoquant curve (A.16) becomes

k = (exp y1)x−ξ1/α1 .

Note that this isoquant curve means that the amount of output will not change when we

multiply x1 (i.e., the second argument of F (·, 0)) by λ > 0 and simultaneously multiply k (i.e., the

first argument of of F (·, 0)) by λ−ξ1/α. Applying this operation to the RHS of (A.14) gives

F

(λ

1−ξ1α

(∏j∗

j=1Xξjj,0

)−1/αm1/α, λX1,0, X2,0, ..., Xj∗,0, λxj∗+1, ..., λxJ ; 0

).

By repeating this operation for j = 2, ..., j∗ and using 1−∑j∗

j=1 ξj = α from (10), (A.14) becomes

F (λm, λxj∗+1, ..., λxJ) = F

(λ

(∏j∗

j=1Xξjj,0

)−1/αm1/α, λX1,0, ..., λXj∗,0, λxj∗+1, ..., λxJ ; 0

)

= λF

((∏j∗

j=1Xξjj,0

)−1/αm1/α, X1,0, ..., Xj∗,0, xj∗+1, ..., xJ ; 0

)= λF (m,xj∗+1, ..., xJ),

where the second equality is from the CRS property of F (·; 0).

Next, we prove the CRS property of function F (k, x1, ..., xJ). From (11),

F (λk, λx1, ..., λxJ) = F

((λk)α

∏j∗

j=1(λxj)

ξj , λxj∗+1,0, ..., λxJ,0

)= F

(λkα

∏j∗

j=1xξjj , λxj∗+1,0, ..., λxJ,0

)= λF

(kα∏j∗

j=1xξjj , xj∗+1,0, ..., xJ,0

)= λF (k, x1, ..., xJ).

The second equality utilizes α+∑j∗

j=1 ξj = 1 from (10), whereas the third equality is from the CRS

property of function F (·).


Using (11), the RHS of equation (14) can be written as

F

((AK,tKt)

α∏j∗

j=1(AXj ,tXj,t)

ξj , AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t

). (A.17)

58

The first argument of in function F (·) represent the effective amount of capital composite on the

BGP. It is multiplied by g each period from condition (13). Also, all the other arguments of F (·)are multiplied by g each period because it is assumed γXj = g/gXj for j ∈ j∗+1, ..., J. Since F (·)has constant returns to scale from property d of Lemma 3, (A.17) 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.


Let us first consider the case of Zt = Kt. Using property b of Lemma 3 and (11), the share of factor

K in period 0 can be written as (11),

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

Y0

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

Y0

=∂

∂K0F

(Kα

0

∏j∗

j=1Xξjj,0, Xj∗+1,0, ..., XJ,0

)K0

Y0.

(A.18)

Let FM (·) be the derivative of function F (·) with respect to its first argument. Note that, in (A.18),

the first argument is the capital composite in period 0, M0 = Kα0

∏j∗

j=1Xξjj,0. Using the chain rule,

the RHS of (A.18) becomes

sK,0 = FM (M0, Xj∗+1,0, ..., XJ,0)dM0

dK0

K0

Y0= FM (M0, Xj∗+1,0, ..., XJ,0)

αM0

Y0, (A.19)

where the second equality follows from dM0/dK0 = αM0/K0.

Recall that F (·) has constant returns to scale from Lemma 3, and therefore its derivative FM (·)is a homogeneous function of degree 0. Let Mt = (AK,tKt)

α∏j∗

j=1(AXj ,tXj,t) denote the effective

amount of capital composite in period t. From condition (13), Mt grows by factor of g every period.

The same applies to the effective amounts of factors not in the capital composite: AXJ ,tXj,t for

j = j∗ + 1, ..., J. Therefore, when we consider function FM

(Mt, AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t

), its

every argument is multiplied by g every period, which does not change the value of FM (·) over time

due to homogeneity of degree 0. Therefore (A.19) can be written as

sK,0 = FM


) αM0

Y0. (A.20)

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 repre-

sents dMt/dKt, which can be confirmed by differentiating Mt = (AK,tKt)α∏j∗

j=1(AXj ,tXj,t) by Kt.

59

Therefore, (A.20) becomes

sK,0 = FM


) dMt

dKt

Kt

Yt

=∂

∂KtF

((AK,tKt)

α∏j∗

j=1(AXj ,tXj,t), AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t

)Kt

Yt

=∂

∂KtFK (AK,tKt, AX1,tX1,t, ..., AXJ ,tXJ,t)

Kt

Yt,

(A.21)

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.22)

By comparing (A.21) with (A.22), 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∗, ..., J, can be confirmed in a similar way as in Proposition

2, because the value of AXj ,t is the same as AXj ,t in the Uzawa theorem. In particular, we use

F (Mt, AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t) instead of F (Kt, AX1,tX1,t, ..., AXJ ,tXJ,t), and define FXj (·),j ∈ j∗, ..., J, as the derivative of function F (·) with respect to its (j − j∗ + 1)th argument.60

Except for these, the proof proceeds exactly as in Appendix A.4.


Similarly to Definition 3, the EoS σKXj ,t on the BGP is defined as


d ln

(AK,tFK(AK,tk,AX1,t

X1,t,...,AXj,txj ,...,AXJ,tXJ,t)

AXj,tFXj (AK,tk,AX1,tX1,t,...,AXj,txj ,...,AXJ,tXJ,t)

)∣∣∣∣∣∣∣∣F (AK,tk,AX1,t


k=Kt,xj=Xj,t

(A.23)

where Yt,Kt, X1,t, ..., XJ,t indicate the BGP values, and k and xj are the variables to be perturbed.

Let us first consider the case of j ∈ 1, ..., j∗. In this case, factors Xj∗+1,t, ..., XJ,t are fixed at

the BGP values. Using (11), function F (AK,tk,AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t) can be written

as F (m,AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t), where m is the effective amount of capital composite, m =

(AK,tk)α(AXj ,txj)ξj∏j′∈1,...,j∗\j(AXj′ ,tXj′,0)

ξj′ . Note that dm/dk = αm/k. Using the chain rule,

60FXj (·) needs to be defined this way because j∗ arguments are eliminated from function F (·) in definition (12).

Also, note that similarly to function F (·), function F (·) has a CRS property from Lemma 3.

60

the derivative of this function with respect to k can be written as

AK,tFK(AK,tk,AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)

=∂

∂kF (AK,tk,AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)

=∂

∂kF (m,AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t)

= FM (m,AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t)αm

k.

(A.24)

Similarly, the derivative of F (m,AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t) with respect to xj is

AXj ,tFXj (AK,tk,AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)

= FM (m,AXj∗+1,tXj∗+1,t, ..., AXJ ,tXJ,t)ξjm

xj.

(A.25)

Substituting (A.24) and (A.25) into (A.23) gives


d ln(αξj

xjk

)∣∣∣∣∣∣F (AK,tk,AX1,t


k=Kt,xj=Xj,t

. (A.26)

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.26) gives σKXj ,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 = σKXj ,t holds.

Next, we examine σKXj ,t for the case of j ∈ j∗+1, ..., J. Similarly to the proof of Proposition

3, consider a change of variables: k′ = g−tk and x′j = g−tAXj ,txj . Then, k in (A.23) is replaced by

k = gtk′ and xj is replaced by (gt/AXj ,t)x′j . In the numerator, k/xj becomes AXj ,tk

′/x′j . In the

denominator, by the same operations as in (A.24), AK,tFK(·) can be written as

FM (m,AXj∗+1,tXj∗+1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)αm

k, (A.27)

where m = (AK,tk)α(AXj ,txj)ξj∏j∗

j′=1(AXj′ ,tXj′,0)ξj′ . The definition of m does not include xj

because j ∈ j∗ + 1, ..., J means that xj is not a part of capital composite. Instead, AXj ,txj

appear in (A.27) as the (j − j∗+ 1)th argument of F (·) function. Using xj = (gt/AXj ,t)x′j , AXj ,txj

can be written as gtx′j . Since Mt = (AK,tKt)α∏j∗

j=1(AXj ,tXj,t) grows by a factor of g every period,

the capital composite m can also be written as

m =(gtk′/Kt

)αMt =

(k′/K0

)αgtM0 = gtm′,

61

where m′ = (k′)α∏j∗

j′=1(AX′j ,0Xj′,0). Other effective factors also grow by factor of g: AXj′ ,tXj′,t =

gtXj′,0 for j′ ∈ j∗ + 1, ..., J\j. Using these, (A.27) becomes

FM (gtm′, gtXj∗+1,0, ..., gtx′j , ..., g

tXJ,0)αgtm′

gtk′

= FM (m′, Xj∗+1,0, ..., x′j , ..., XJ,0)α

m′

k′

=∂

∂k′F (m′, Xj∗+1,0, ..., x

′j , ..., XJ,0)

= FK(k′, X1,0, ..., x′j , ..., XJ,0),

where the first equality is from the homogeneity of degree 0 property of the FM (·) function, the

second equality is from the chain rule and dm′/dk′ = αm′/k′, and the last equality is from the

definition of F (·) in (11).

Likewise, AXj ,tFXj (·) in the denominator of (A.23) can be expressed in terms of k′ and x′j as

∂

∂xjF (AK,tk,AX1,t,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)

=∂

∂xjF (m,AXj∗+1,tXj∗+1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)

= FXj (m,AXj∗+1,tXj∗+1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)dAXj ,txj

dxj

= FXj (gtm′, gtXj∗+1,0, ..., g

tx′j , ..., gtXJ,0)AXj ,t

= FXj (m′, Xj∗+1,0, ..., x

′j , ..., XJ,0)AXj ,t

=∂

∂x′jF (m′, Xj∗+1,0, ..., x

′j , ..., XJ,0)AXj ,t

= AXj ,tFXj (k′, X1,0, ..., x

′j , ..., XJ,0).

We also need to rewrite the conditions in (A.23). Note that

F (AK,tk,AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)

= F (m,AXj∗+1,tXj∗+1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t)

= F (gtm′, gtXj∗+1,0, ..., gtx′j , ..., g

tXJ,0)AXj ,t

= gtF (m′, Xj∗+1,0, ..., x′j , ..., XJ,0)AXj ,t

= gtF (k′, X1,0, ..., x′j , ..., XJ,0).

Therefore, condition F (AK,tk,AX1,tX1,t, ..., AXj ,txj , ..., AXJ ,tXJ,t) = Yt can be substituted by

F (k′, X1,0, ..., x′j , ..., XJ,0) = Y0.

62

The point of evaluation, k = Kt, becomes gtk′ = Kt, or k′ = g−tKt = K0. Similarly, xj = Xj,t

becomes x′j = g−tAXj ,tXj,t = Xj,0. Using all these results, (A.23) can be expressed in terms of k′

and x′j as follows:

σKXj ,t = −d ln(AXj ,tk

′/x′j)

d ln

(FK(k′,X1,0,...,x′j ,...,XJ,0)

AXj,tFXj (k′,X1,0,...,x′j ,...,XJ,0)

)∣∣∣∣∣∣∣∣F (k′,X1,0,...,x′j ,...,XJ,0)=Y0k′=K0,x′j=Xj,0

(A.28)

We can eliminate AXj ,t from the numerator of d ln(AXj ,tk′/x′j) = d(ln(k′/x′j) + lnAXj ,t) =

d ln(k′/x′j). In the same way, AXj ,t in the denominator can also be eliminated. Also, recall that

AK,0 = AXj ,0 = 1. Then, comparing (A.28) with (A.23), it turns out that the RHS of (A.28)

coincides with σKXj ,0. From Lemma 3, σKXj ,0 = σKXj ,0 holds in period 0. In addition, it is

assumed that σKXj ,0 does not change over time. Therefore, σKXj ,t = σKXj ,0 = σKXj ,0 = σKXj ,t.


Recall that the R&D cost functions, defined by (27), have the following properties:

iZ(γZ) > 0, i′Z(γZ) > 0, i′′Z(γZ) > 0 for all γZ ≥ 1,

iZ(1) = 0, i′Z(1) = 0, iZ(∞) =∞ for Z = K,X,L.(A.29)

Proof existence and uniqueness of γL(µt) and property (c)

As explained in the main text, the representative firm chooses γL,t so as to satisfy

R&D for N-tasks: γL,ti′L(γL,t) + iL(γL,t) = f(µt)− µtf ′(µt). (39)

Let us denote the LHS of (39) by ΨL(γL,t) because it depends only on γL,t. Then, Ψ′L(γL,t) =

γL,ti′′L(γL,t)+2i′L(γL,t) > 0 for all γL,t > 1 from iL(γL,t) > 0 and i′L(γL,t) > 0 in (A.29). When γL,t =

1, the properties of iL(·) implies ΨL(1) = i′L(1)+iL(1) = 0. Also, ΨL(∞) ≡ limγL,t→∞ΨL(γL,t) =∞from iL(∞) =∞ and γL,ti

′L(γL,t) > 0.61 Then, since ΨL(·) is differentiable and strictly increasing,

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

Ψ(−1)L (0) = 1 and Ψ

(−1)L (∞) =∞. Using this function, condition (39) can be solved for γL,t:

γL,t = Ψ(−1)L (f(µt)− µtf ′(µt)) ≡ γL(µt). (A.30)

Note that f(µt) − µtf ′(µt) represents the marginal product of Nt in the production function,

i.e., FN (µt, 1). Since F (·) is CRS, its first derivative FN (·) is homogeneous of degree 0. Therefore,

61Similarly to the main text, we employ an abuse of notation by writing iL(∞) to represent limγL→∞ iL(γL). Wewill do similar abbreviations as long as they cause no confusion.

63

dividing the both arguments of FN (µt, 1) by µt, FN (µt, 1) becomes FN (1, 1/µt) = FN (1, νt) ≡h(νt). This is an intensive form representation of the partial derivative. From the definition of the

production function F (·), h(νt) satisfies the Inada conditions. Therefore, limµt→0 f(µt)−µtf ′(µt) =

limνt→∞ h(νt) = 0, and limµt→∞ f(µt) − µtf ′(µt) = limνt→0 h(νt) = ∞. Substituting these into

(A.30) gives γL(0) = Ψ(−1)L (0) = 1 and γL(∞) = Ψ

(−1)L (∞) =∞.

Finally, we show γ′L(µt) > 0. The derivative of f(µt)−µtf ′(µt) with respect to µt is −µtf ′′(µt).It is positive for all µt > 0 since the production function satisfies the Inada conditions, which

include f ′′(µt) < 0. Since Ψ(−1)L

′(·) > 0, this means γ′L(µt) > 0.

Proof of existence and uniqueness of γK(µK) and γX(µX), as well as properties (a) and

(b)

The representative firm chooses γK,t and γX,t according to the following two conditions:

R&D allocation:γK,ti

′K(γK,t)

γX,ti′X(γX,t)=

α

1− α, α ∈ (0, 1), (40)

Combined R&D:(γK,ti


)+(γX,ti


)= f ′(µt). (41)

Let us define ΩK(γK,t) ≡ γK,ti′K(γK,t) and similarly ΩX(γX,t) ≡ γX,ti

′K(γX,t). Then, from proper-

ties in (A.29), we can confirm Ω′K(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 increasing, we can de-

fine its inverse function Ω(−1)X (·), which is also differentiable and strictly increasing with Ω

(−1)X (0) = 1

and Ω(−1)X (∞) =∞. Using this inverse function, condition (40) can be solved for γX,t as

γX,t = Ω(−1)X

(α

1− αΩK(γK,t)

)≡ Ω(γK,t). (A.31)

Now let us focus on condition (41). Let us define ΨK(γK,t) ≡ γK,ti′K(γK,t) + iK(γK,t) and

likewise ΨX(γX,t) ≡ γX,ti′K(γX,t) + iK(γX,t). Using these and (A.31), the LHS of condition (41)

can be expressed as a function only of γK,t:

ΨK(γK,t) + ΨX(Ω(γK,t)) ≡ Ψ(γK,t).

Note that the properties of ΩK(·) and Ω(−1)X (·) implies 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 (41), f ′(µt) satisfies the usual Inada conditions. The results we have obtained so

64

far can be summarized as

γK,t 1 · · · ∞Ψ′(γK,t) +

Ψ(γK,t) 0 ∞

µt 0 · · · ∞f ′′(µt) −f ′(µt) ∞ 0

The tables above implies that condition (41), Ψ(γK,t) = f ′(µt), gives a 1 to 1 correspondence

between µt ∈ (0,∞) and γK,t ∈ (1,∞) that satisfies property (a): γ′K(µt) < 0 for all µt > 0,

γK(0) =∞, and γK(∞) = 1.

Given γK(µt), equation (A.31) uniquely determines γX,t = Ω(γK(µt)) ≡ γX(µt). From the

properties of Ω(·) and γK(·) above, we can confirm that property (b) is satisfied: γ′X(µt) < 0 for

all µt > 0, γX(0) =∞, and γX(∞) = 1.

A.11 Derivation of the terminal condition (56)

The Euler equation (50) implies that rt + 1 − δ = (Ct−1/Lt−1)−θ/β(Ct/Lt)

−θ. Through repeated

multiplication,T∏t=1

(rt + 1− δ) =(C0/L0)

−θ

βT (CT /LT )−θ.

Using this, the Non Ponzi Game condition (35) becomes

(C0

L0

)θlimT→∞

βT(CTLT

)−θKT+1 ≥ 0 (A.32)

Since C0/L0 > 0, we can divide the both sides of (A.32) by (C0/L0)θ to eliminate this term. Then,

it turns out that (A.32) has exactly the same form as the transversality condition (55), except that

the direction of the inequality is opposite. By combining (55) and (A.32), therefore, we have a

unified terminal condition

limT→∞

βT(CTLT

)−θKT+1 = 0. (A.33)

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

βT(CTLT

)−θKT+1 = βT (AL,T cT )−θ AL,T+1LT+1kT+1

= βT (AL,T cT )−θ AL,T γL(µT+1)L0gT+1L kT+1

= L0gL(βgL)TA1−θL,T c

−θT γL(µT+1)kT+1

= L0gLAL,−1(βgL)T

(T∏t=0

γL(µt)

)1−θ

c−θT γL(µT+1)kT+1.

65

By substituting the last expression into (A.33) and then dividing both sides by L0gLAL,−1 > 0, we

obtain (56).


Consider a BGP. We will show that µt, kt and ct must be constant in turn. First, from the definition

of a BGP, Nt+1/Nt = (AL,t+1Lt+1)/(AL,tLt) = γL(µt+1)gL is constant. To keep the RHS of the

latter equation constant, µt must also be constant, since γL(·) is a strictly increasing function

from Proposition 7. Second, since growth factors of Ct and Nt are constant, the growth factor of

ct = Ct/AtLt = Ct/Nt is also constant. This, in turn, means that the LHS of the Euler equation

(51) is constant. Then, for the RHS of (51) to be constant, kt must be constant, since we already

know that µt is constant as shown above. Similarly, the growth factor of kt = Kt/AtLt = Kt/Nt is

constant on the BGP, which means the LHS of (48) is constant. For its RHS to be constant, given

that µt and kt are already shown to be constant, ct also needs to be constant.


Proof of µ∗ > 0

In the text, we have already shown that there exist a unique µ∗ > 0 such that Φ(µ∗) = 1 holds,

since Proposition 7 implies Φ′(µ∗) < 0 with Φ(0) = ∞ and Φ(∞) = 0. Therefore, there exist a

unique value of µ∗ > 0.

Proof of k∗ > 0

The value of k∗ is explicitly given by equation (59), shown again here:

k∗ =βαµ∗(f ′(µ∗)− iK(γK(µ∗))− iX(γX(µ∗)))

γL(µ∗)θ − β(1− δ). (59)

We now show that both the numerator and the denominator of the RHS are positive. Note that

the combined R&D condition (41) is satisfied on the BGP. By rearranging terms, it gives

f ′(µ∗)− iK(γK(µ∗))− iX(γX(µ∗)) = γK(µ∗)i′K(γK(µ∗)) + γX(µ∗)i′X(γX(µ∗)) > 0, (A.34)

where the inequality follows from Proposition 7 and (A.29). Given β ∈ (0, 1), α ∈ (0, 1), and

µ∗ > 0, this means that the numerator of (59) is strictly positive. Now, note that γL(µ∗) > 1 from

Proposition 7. Combined with θ > 0, β ∈ (0, 1) and δ ∈ [0, 1], it turns out that the denominator of

(59) is also strictly positive.

66

Proof of c∗ > 0

The value of c∗ is given by

c∗ = v(µ∗)− (g∗ − 1 + δ)k∗. (60)

We first show v(µ∗) > 0. Combining the R&D conditions (39) and (41), we have

γL,ti′L(γL,t) + iL(γL,t) + µt

((γK,ti


)+(γX,ti


))= f(µt).

Rearranging and then evaluating this condition at µt = µ∗ gives

v(µ∗) = f(µ∗)− iL(γL(µ∗))− µ∗ (iK(γK(µ∗))− iX(γX(µ∗)))

= γL(µ∗)i′L(γL(µ∗)) + µ∗(γK(µ∗)i′K(γK(µ∗)) + γX(µ∗)i′X(γX(µ∗))

)> 0,

where the inequality follows from µ∗ > 0 and (A.29).

Note that g∗ = γL(µ∗)gL is greater than 1 − δ because γL(µ∗) > 1 and gL > 1 − δ from (30).

Therefore, (g∗ − 1 + δ) in (60) is positive. From this, c∗ > 0 is equivalent to

k∗ <v(µ∗)

g∗ − 1 + δ.

Using (59), we can rewrite this condition in terms of β:

β < γL(µ∗)θ(αµ∗

v(µ∗)

(f ′(µ∗)− iK(γK(µ∗))− iX(γX(µ∗))

)(g∗ − 1 + δ) + 1− δ

)≡ β1. (A.35)

Note that β1 > 0 from (A.34) and g∗ > 1 − δ > 0. Observe also that β1 does not depend on β

itself since µ∗ is determined entirely by the production side (see equation 57). Therefore, if β > 0

is sufficiently small, condition (A.35) holds and hence c∗ > 0.

Terminal Condition

On the BGP, the terminal condition (56) becomes

limT→∞

(βgLγL(µ∗)1−θ

)TγL(µ∗)(c∗)−θk∗ = 0. (A.36)

Given that γL(µ∗) > 1, c∗ > 0 and c∗ > 0, this condition is equivalent to

β <1

gLγL(µ∗)1−θ≡ β2. (A.37)

Note that β2 > 0 and that it does not depend on β since µ∗ is determined entirely by the production

side. Therefore, if β > 0 is sufficiently small, condition (A.37) holds and hence the terminal

67

condition (56) is satisfied.

Combining conditions (A.35) and (A.37), we have confirmed the unique existence of BGP with

µ∗ > 0, k∗ > 0, c∗ > 0, and the terminal condition (56) whenever

β < β ≡ minβ1, β2, (A.38)

where β > 0 is a constant that does not depend on β.

B Calibration Details

In this section, we present the full calibration results for our robustness exercises. The stability

results are presented in table 3 in the main text.





Table 4: CALIBRATION. Robustness check with λ = 1.5. Growth rates and rates of return(gL, gX , β, δ) are presented in terms of annualized values.

68




χ 5.13 Period Length (years) Calibratedβ 0.98 Discount factor Calibratedδ 3.7% Depreciation Calibratedα 0.74 Cap. Distribution Calibratedη 0.21 CES Distribution CalibratedζK 44.79 Research efficiency CalibratedζL 3.53 Research efficiency CalibratedζX 11,162.1 Research efficiency Calibrated

Table 5: CALIBRATION. Robustness check with λ = 2.5. Growth rates and rates of return(gL, gX , β, δ) are presented in terms of annualized values.





Table 6: CALIBRATION. Robustness check with ε = 0.5. Growth rates and rates of return(gL, gX , β, δ) are presented in terms of annualized values.

69





Table 7: CALIBRATION. Robustness check with ε = 1.2. Growth rates and rates of return(gL, gX , β, δ) are presented in terms of annualized values.





Table 8: CALIBRATION. Robustness check with θ = 0.5. Growth rates and rates of return(gL, gX , β, δ) are presented in terms of annualized values.

70





Table 9: CALIBRATION. Robustness check with θ = 1.5. Growth rates and rates of return(gL, gX , β, δ) are presented in terms of annualized values.

71

C Data Sources

C.1 Figure 1

Real GDP. U.S. Bureau of Economic Analysis, Real Gross Domestic Product [GDPCA], retrieved

from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GDPCA, Febru-

ary 15, 2019. Shown in Panel (a).

Investment. U.S. Bureau of Economic Analysis, Real Gross Private Domestic Investment [GPDIC1],

retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/GPDIC1,

February 15, 2019. Shown in Panel (a).

Consumption. U.S. Bureau of Economic Analysis, Real Personal Consumption Expenditures

[PCECC96], retrieved from FRED, Federal Reserve Bank of St. Louis;

https://fred.stlouisfed.org/series/PCECC96, February 15, 2019. Shown in Panel (a).

Nominal Capital Stock. U.S. Bureau of Economic Analysis, Current-Cost Net Stock of Fixed

Assets, Fixed Asset Table 1.1. Not shown in Table.

GDP Deflator. Bureau of Economic Analysis, Implicit Price Deflator for Gross Domestic Product,

NIPA Table 1.1.9. Not shown in Table.

Real Capital Stock (K). Nominal Capital Stock/GDP Deflator. Shown in Panel (a).

Investment Deflator. Bureau of Economic Analysis, Implicit Price Deflator for Gross Private

Domestic Investment, NIPA Table 1.1.9. Not shown in Table.

Non-Residential Investment Deflator. Bureau of Economic Analysis, Implicit Price Deflator

for Fixed, Non-Residental Investment, NIPA Table 1.1.9. Not shown in Table.

Equipment Investment Deflator. Bureau of Economic Analysis, Implicit Price Deflator for

Equipment, NIPA Table 1.1.9. Not shown in Table.

Personal Consumption Expenditures Deflator. Bureau of Economic Analysis, Implicit Price

Deflator for Personal Consumption Expenditures, NIPA Table 1.1.9. Not shown in Table.

Relative Price of Investment (various). Investment Deflator/Consumption Deflator. Applied

to all investment, non-residential, and equipment deflators. Shown in panel (b).

72

C.2 Table 1

Energy Expenditure Share. Energy expenditure as a share of GDP (%). Data available from

1970–2014. Source: ‘Table 1.5: Energy consumption, expenditures, and emissions indicators esti-

mates, 1949–2011’ at https://www.eia.gov/totalenergy/data/annual/.

73

https://www.eia.gov/totalenergy/data/annual/

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