ISSN (Print) 0473-453X

Discussion Paper No. 1220

ISSN (Online) 2435-0982

ENDOGENOUS CAPITAL-AUGMENTINGTECHNOLOGICAL CHANGE

Gregory Casey
Ryo Horii

November 2023

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

Endogenous Capital-Augmenting
Technological Change∗
Gregory Casey† Ryo Horii‡
November 28, 2023

Abstract
We construct a 3-factor, directed technical change growth model that exhibits capital-augmenting technical change on the balanced growth path (BGP),
circumventing the issues usually caused by the 2-factor Uzawa growth theorem.
We calibrate the model to the United States and consider a non-unitary elasticity of substitution between capital and labor. We show that the model converges to the BGP with capital-augmenting technical change from any initialcondition. Our results indicate that natural resources and directed technicalchange play a central role in explaining balanced growth.

Keywords: Balanced Growth, Uzawa Steady-State Growth Theorem, Directed Technical Change (DTC), Natural Resources, Three-Factor ModelJEL Classification Codes E13, E22, O33, O41∗

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

1

Introduction

The two-factor neoclassical growth model is central to much of macroeconomics. As
explained by Grossman et al. (2017), however, this model is inconsistent with stylized
facts observed in the United States, including balanced growth, a non-unitary elasticity of substitution (EoS) between labor and reproducible capital, and the declining
relative price of investment goods.1 We present a three-factor neoclassical growthmodel that is consistent with all of these stylized facts. Our results suggest thatnatural resources and directed technical change are essential for making neoclassicalgrowth models consistent with data.
The Uzawa (1961) steady state growth theorem explains that, on the balanced
growth path (BGP) of a two-factor neoclassical growth model, all technological changemust be labor-augmenting, unless the production function is Cobb-Douglas (Acemoglu, 2008; Jones and Scrimgeour, 2008). But, this is inconsistent with two welldocumented pieces of evidence: the EoS between capital and labor is different thanone (i.e., the production function is not Cobb-Douglas), and the relative price of investment has been falling on the balanced growth path (i.e., there is capital-augmenting
technical change) (e.g., Greenwood et al., 1997; Antras et al., 2004; DiCecio, 2009;
Oberfield and Raval, 2021).
Casey and Horii (forthcoming) argue that including three or more factors of production allows the neoclassical growth model to be consistent with data. They generalize the Uzawa growth theorem to show that a neoclassical growth model can beconsistent with balanced growth and capital-augmenting technical change as longas capital has a unitary EoS with any single other factor of production. This leaves
open the possibility that capital has a non-unitary elasticity with labor, but a unitaryelasticity with some third factor. They argue that natural resources, like land andenergy, could play the role of this third factor and discuss existing evidence consistent
with this interpretation. When the EoS condition is satisfied, the generalized Uzawa
growth theorem implies that there is a log-linear relationship between the rates offactor-augmenting technological change. The relationship guarantees stationary offactor shares in the long run. If technological change is exogenous, then this is aknife-edge condition. But, the theorem does not specify the economy mechanism
1

See Jones (2016) for a recent review of these facts.

1

through which technology growth rates are determined.
Building on their results, we formulate a directed technological change modelthat has capital-augmenting technical change on the balanced growth path, despitea non-unitary elasticity of substitution between capital and labor. The direction of
innovation is determined by profit-maximizing firms. The economy endogenously conforms to the log-linear condition identified in the generalized Uzawa growth theorem.
We also calibrate the model to U.S. data and show that the BGP is both locally andglobally stable. The transitional dynamics of our model are qualitatively different
from the standard two-factor neoclassical model. The convergence to the BGP goes
through two stages: capital and technological adjustments. The latter is much slower.
Our results contribute to the existing literature in two main ways. First, we show
how neoclassical production functions can be made consistent with data on balanced
growth and the EoS between capital and labor. Thus, our model will be useful in any
macroeconomic setting where matching these data patterns is important. Second, we
contribute to the existing literature on the Uzawa steady state theorem (e.g., Schlicht,
2006; Jones and Scrimgeour, 2008; Grossman et al., 2017). In particular, we present
the economic mechanism by which the direction of technical change endogenously
conforms to the log-linear technological condition in the generalized Uzawa growththeorem. Unlike the case of the exogenous technological change analyzed in Caseyand Horii (forthcoming), we do not need to impose a parameter restriction on technological change. The sole restriction is the unitary elasticity between capital and asingle other factor, like land or energy. In this way, our results indicate that including natural resources and directed technical change are promising ways to explainbalanced growth in neoclassical growth models.
Related literature. Our work is part of a growing literature that examines therelationship between endogenous growth and the Uzawa steady state theorem. Several
studies have used directed technical change to explain why the economy might endogenously conform to the two-factor version of the theorem. In particular, Acemoglu(2003) and Irmen and Tabaković (2017) provide models where capital-augmenting
technical change disappears in the long run, while Jones (2005) and Leon-Ledesma
and Satchi (2019) specify models that are Cobb-Douglas in the long run. To the bestof our knowledge, Grossman et al. (2017) present the only other attempt to square2balanced growth with the wider set of stylized facts observed in the United States.
They specify a specific type of capital-skill complementarity that allows growth models with schooling to match the data. Casey and Horii (forthcoming) demonstrate
how to interpret their findings in the context of the two-factor neoclassical growthmodel. We build on these works by presenting a directed technical change growthmodel that converges to a BGP that is consistent with data on both the existence ofcapital-augmenting technical change and the less-than-unitary EoS between capitaland labor.
Our work is also related to the literature on directed technical change and natural
resource use (e.g., Smulders and De Nooij, 2003; Hassler et al., 2021, 2022; Casey,
forthcoming). This literature explains how technology endogenously evolves to ensurebalanced growth when there are three or more factors of production in a neoclassicalgrowth model. These models are usually used to study resource conservation or
climate change. We show how to extend the models in this literature to be consistent
with the additional stylized facts highlighted by Grossman et al. (2017). In doing
so, we demonstrate that including directed technical change and natural resourcesin growth models is necessary to match data patterns unrelated to environmental
questions, broadening the importance of this existing literature.

2

The Uzawa Steady-State Growth Theorem

In a neoclassical growth model, continued economic growth requires technologicalchange. Theoretically, technological change in this context means time variation in
the aggregate production function, which is defined as the mapping from aggregate
production factors to aggregate output. Given that we observe sustained economic
growth in most countries, it is essential to include technological change in a macroeconomic model. However, there are limitless ways in which the mapping could evolve
over time. How should we specify technological change in the model so that the model
is consistent with data? This poses a great challenge to economists, particularly whenthe mapping is not clearly observable.
The original Uzawa growth theorem provided a simple and convenient solution.
On the balanced growth path, technological change in a 2-factor neoclassical growth3

model can always be represented as labor-augmenting technological progress.2 Thissolution is widely used in the growth literature. However, recent studies found evidence that technological change is not purely labor-augmenting. The productivity ofcapital has been increasing, as reflected in the fall in the relative price of investment.
To tackle this puzzle, Casey and Horii (forthcoming) provided a generalized versionof the Uzawa theorem that incorporates the possibility of capital-augmenting technological change on the BGP. A key difference from the original Uzawa theorem is thatthe generalized version allows for more than two aggregate factors. Below, we present
a simpler 3-factor version of their result, focusing on elements that are relevant to thedirected technical change model in this paper.
Definition 1. A 3-factor neoclassical growth model is an economic environmentthat satisfies:
1. Output, Yt , is produced from capital, Kt , labor Lt and another aggregate input
Xt :Yt = Ft (Kt , Lt , Xt ).

(1)

In any t ≥ 0, it has constant returns to scale (CRS) in each argument, and all
inputs, Kt , Lt and Xt , have positive and diminishing marginal products.
2. The amount of capital, Kt , evolves according toKt+1 = Yt − Ct − Rt + (1 − δ)Kt , K0 > 0,(2)where Ct > 0 is consumption, Rt ≥ 0 is expenditure other than capital investment or consumption (e.g., R&D inputs), and δ ∈ [0, 1] is the depreciation
rate.
The focus of the theorem is how to express the evolution of production function
Ft (·) over time. To accomplish this goal, we rely on the property that the economy
has a BGP.
2

An exception is when the production function is Cobb-Douglas. However, the estimates of theelasticity of substitution between labor and capital do not support the Cobb-Douglas specification(e.g., Oberfield and Raval, 2021).

4

Definition 2. A balanced growth path (BGP) in a 3-factor neoclassical growthmodel is a path along which all quantities, {Yt , Kt , Lt , Xt , Ct , Rt }, grow at constant
exponential rates for all t ≥ 0. We call it a non-degenerate BGP when the growthrate of Kt is larger than −δ.
On the BGP, we denote the growth factor of output by g ≡ Yt /Yt−1 , and thegrowth factors of any variable Zt by gZ ≡ Zt /Zt−1 . Unless otherwise noted, we focus
on the non-generate BGP with positive investments, i.e., gK > 1 − δ. We also need todefine the elasticity of substitution, because the definition is not obvious when there
are more than two factors.
Definition 3. The Elasticity of Substitution between capital Kt and other factors
in the 3-factor production function Ft (Kt , Lt , Xt ) is
σKL,t = −d ln(Kt /Lt ),d ln (FK,t /FL,t ) Yt ,Xt :fixed

σKX,t = −d ln(Kt /Xt ).
d ln (FK,t /FX,t ) Yt ,Lt :fixed(3)

In (3), FK,t represents ∂Ft (Kt , Xt , Lt )/∂Kt . FL,t and FX,t are similarly defined.
When evaluating σKL,t , we consider small changes in Kt and Lt while keeping Xt andYt = Ft (Kt , Xt , Lt ) constant. When two factors have a unitary EoS between them,
they can be represented with the usual Cobb-Douglas relationship. Now, we have all
the definitions necessary to state a simpler version of the generalized Uzawa theorem.
Proposition 1 (Generalized Uzawa Growth Theorem for 3 Factors). Suppose that,on a BGP of a 3-factor neoclassical growth model, σKX,t = 1 holds and σKL,t 6= 1
is constant. Also, suppose that shares of factors sK , sL , and sX are constant on theK
BGP, and let α = sKs+s. Define factor-augmenting technologies byXtt
AK,t = γK, AL,t = γLt , AX,t = γX,

(4)

and choose growth factors of technologies γK , γL , and γX so that they satisfyαγK(γX gX )1−α = g 1−α and γL = g/gL .

(5)

Then there exists a constant-returns-to-scale (CRS) function Fb(·, ·) that satisfies, on5

the BGP,
F (Kt , Lt , Xt ) = Fb (AK,t Kt )α (AX,t Xt )1−α , AL,t Lt(6)∂F (Kt , Lt , Xt )∂ Fb ((AK,t Kt )α (AX,t Xt )1−α , AL,t Lt )
=
∂Kt
∂Kt

(7)

and similarly for the first partial derivatives with respect to Xt and Lt . Moreover,similarly to definition 3, define the elasticity of substitution in the RHS of (6) asσbKL,t and σ
bKX,t . Then on the BGP,
σKL,t = σbKL,t , σKX,t = σ
bKX,t .

(8)This proposition can be obtained as a special case of the generalized Uzawa theorem in Casey and Horii (forthcoming).3 The proposition shows that technologicalchange — equivalently, the evolution of Ft — can be represented by a fixed function
Fb along with factor augmenting terms AK,t , AL,t and AX,t . We call the RHS of (6)the factor-augmenting representation of technological change. This representation
matches the level of the original production function in (6), first derivatives in (7),and the elasticity of substitution in (8) on the balanced growth path. Therefore,
when the focus of economic analysis is on or around the BGP, the factor-augmenting
representation can be used as an approximation of the true production function Ft ,even when the economist does not know precisely how Ft evolves over time.
Unlike the original Uzawa theorem, the generalized Uzawa growth theorem allows
positive capital-augmenting technological change on the BGP. Condition (5) implies
there is freedom for an economist to choose γK > 1 to match the data.4 Cancelling g
in (5) and taking log gives
log γK =1−α(log γL + log gL − log γX − log gX ) .
α

3

(9)

Proposition 1 is obtained from Propositions 4, 5 and 6 in Casey and Horii (forthcoming) when
the number of production factors is 3. They provide complete proof in the case of an arbitrary
number of factors. They also explain how Fb can be obtained when an economist has access to theshape of the true production function Ft at a point
point t = 0. Then, in our in time. Let this 
−1/α
1/α
b
M , N, X0 if F0 , the true production
3-factor setting, function F (M, N ) is obtained as F0 X0function at time 0, is known.
4
The rate of capital-augmenting technological change can be measured by the decline in therelative price of capital. By redefining the units of capital, it can also be interpreted as investmentspecific technological change.

6Thus, the BGP has positive capital-augmenting technological change (γK > 1) aslong as the effective labor is growing faster than the effective input of the third
factor (γL gL > γX gX ). Moreover, the representation preserves the elasticity of substitution of the original production function, including σKL,t 6= 1. Therefore, thegeneralized Uzawa growth theorem provides an economist with a representation of
the evolving production function that is simultaneously consistent with evidence on
the non-unitary EoS between capital and labor and on capital-augmenting technological change. This is an improvement over the existing specifications of technologicalchange, which can match only one of the two properties.
The aim of this paper is twofold. First, we demonstrate how the generalized Uzawa
growth theorem can be utilized to build an endogenous growth model that exhibits
capital-augmenting technological change in the long run. We do so starting in thenext section. The second objective is to show technology condition (5), or equivalently
its log-linear version (9), is naturally satisfied in the long-term equilibrium of theendogenous growth model. This condition means that effective inputs to function Fb
grows in a “balanced” way.5 If the values of γK , γL and γX are exogenously given,
then it is necessary to impose the log-linear relationship explained in Proposition 1.
In other words, it is necessary to place an extra restriction on the model. This paper
shows that when the rates of factor-augmenting technological change are endogenous,
the log-linear relationship is endogenously satisfied in the long run, and the economy
converges to a BGP with capital-augmenting technological progress.
Our result will contribute to a wide class of macroeconomic analysis. Even whenan economist is not interested in building an endogenous growth model, our result
provides a justification for assuming exogenous technological change that satisfies
condition (5) and, therefore, using the factor-augmenting representation (6) as agood approximation of the true evolving production function in economic analysis.6
5

The first part of Proposition 1 is directly derived from this balanced-ness. It can be shownthat g = gK holds on any balanced growth path for a neoclassical model satisfying Definition 1.
Then, technology condition (5) ensures that both arguments of Fb grow at the same constant factor
g. Since function Fb has constant returns to scale, the RHS of (6) grow at the same rate of theoutput Yt = F (Kt , Lt , Xt ). By appropriately defining the level of Fb, equation (6) holds for all t on
the BGP.
6
In this simplified version of the theorem, the assumption of σKX,t = 1 might also seem restrictive. However, in Casey and Horii (forthcoming), they showed that a similar theorem can beobtained in an environment with many input factors, and the requirement is that at least one factor

7

3

Model

In this section, we build an endogenous growth model in which firms undertake R&D
investments to improve factor-augmenting technologies. From Proposition 1, we knowthat if there is a third production factor X such that σKX = 1, then technologicalchange on the BGP can be approximated by a representation in the form of (6),which allows for capital-augmenting technological change. Since we want to build amodel that is consistent with a BGP with capital-augmenting technological change,
we provide a simple setting in which the aggregate production function takes the formof (6). In this model, we will later confirm that technology condition (5) is satisfied
endogenously as a long-rem equilibrium outcome.

3.1

Structure

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.7 The number of M-tasks, as well as that of N-tasks, determines theamount of final output. The M-tasks require effective capital AK,t Kt and effective
natural resource AX,t Xt as inputs. Xt is composed of production factors that are
not included in the conventional definition of (reproducible) capital Kt and labor Lt .
Examples are land and energy. The number of M-tasks it can complete is given by
Mt = (AK,t Kt

αAX,t Xt

1−α, α ∈ (0, 1).

(10)

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

(11)

Xj has σKXj = 1. They discuss the likely candidates for factor X, such as energy and land, referring
to evidence found in empirical studies.
7
This setting is first considered by Irmen (2017) and Irmen and Tabaković (2017), and we
expanded it to incorporate three production factors. A benefit of using the model of tasks is that it
can incorporate R&D activity within a perfectly competitive economy. Another benefit is that themodel is scale independent.

8

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

AK,t Kt

αAX,t Xt

1−α, AL,t Lt(12)units of output, where Fb(·) is a standard neoclassical production function that has
CRS and satisfies the Inada conditions.8
Now, we explain how the factor-augmenting technologies {AK,t , AX,t , AL,t } aredetermined. Technical knowledge can be kept within the firm for only one period,
after which it becomes public. Thus, the representative firm at time t can freely
use the technology of the period t − 1 firm, {AK,t−1 , AX,t−1 , AL,t−1 }. In addition,
the period t firm can improve each of factor-augmenting technologies through R&D.
Tasks are differentiated and require separate R&D investments.9 Specifically, for eachfactors Z = K, X, L, the technology follows AZ,t = (1 + aZ (iZ,t ))AZ,t−1 , where iZ isthe amount of investments in final goods. We assume that function aZ (·) satisfies
following properties: a0Z > 0, a00Z < 0, aZ (0) = 0, aZ (∞) = ∞, and a0Z (0) = ∞.10(−1)

(γZ − 1), where(−1)
aZ (·) is the inverse function of aZ (·). To improve the Z-augmenting technology in
It is convenient to define the R&D cost function by iZ (γZ ) = aZ

a task by a factor of γZ , it costs iZ (γZ ) in final goods. The properties of aZ (·) imply
i0Z > 0, i00Z > 0, iZ (1) = 0, iZ (∞) = ∞, i0Z (1) = 0 for Z = K, X, L.

(13)

Adding up the R&D costs for all tasks and technologies, the total R&D cost for the
representative firm is
 





AK,t
AX,t
AL,t
Rt = Mt · iK
+ iX
+ Nt · iL.
AK,t−1AX,t−1
AL,t−1(14)

8
There are two ways to represent the production function in an intensive form: f (M/N ) =b
F (M/N, 1) and h(N/M ) = Fb(1, N/M ). We assume that both f (·) and h(·) satisfy the Inada
conditions.
9
From the symmetry of tasks within each group (M or N) and from the convexity of the R&D
cost functions as assumed in (13), it is always optimal to choose the same levels of AK,t , AX,t , andAL,t across individual tasks. Therefore, we omit subscripts for technologies for individual tasks.
10
The declining marginal in the R&D function, a00Z < 0, 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 duplication becomes more
prominent as R&D inputs increase. See Horii and Iwaisako (2007) for a simple micro foundation.

9

The objective of the representative firm is to maximize the single period profit
net of R&D costs, because it lives only for one period and its knowledge will become
public next period. By taking the output in each period as numéraire, the period
profit is given by
πt = Fb(Mt , Nt ) − Rt − rt Kt − τt Xt − wt Lt ,(15)

where rt , τt , and wt are interest rate, payment for a unit of natural resources (e.g.,land rent), and wage rate, respectively.
The demand side of the economy is standard. There is a representative household.
The size of the representative household (i.e., population) evolves according to11
Lt = L0 gLt , L0 > 0, gL > 1 − δ : given.

(16)

As in the Ramsey-Cass-Koopman model, the period utility of the household is givenby the product of the number of household members and the per capita period felicity
function:
ut = Lt u(Ct /Lt ),(17)

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=

∞
X
t=0

Lt β t(Ct /Lt )1−θ − 1
,
1−θ

(18)

where θ > 0 is the degree of the relative risk aversion (i.e., the inverse of the intertemporal elasticity of substitution) and β > 0 is the discount factor. We later discuss theupper bound for β in Proposition 3.
The representative household owns capital, Kt , and natural resources, Xt , in addition to labor, Lt . The household also owns the representative firm and receives the
profit, πt , although, in equilibrium, profits will be zero due to perfect competition
and free entry.12 For simplicity, we assume that the supply of natural resources is
11

gL > 1 − δ is commonly assumed in neoclassical growth models to avoid a degenerate BGP.
Note that R&D cost Rt is already subtracted from profit πt . In addition, the firm can retain
the rent from R&D only for one period. Therefore, the firms are indifferent to entering the market
when πt = 0.
1210exogenous:13t, X0 , gX > 0 : given.
Xt = X0 gX(19)

As in the case of population, its available quantity can be either constant gX = 1,
shrinking gX ∈ (0, 1), or growing gX > 1. The budget constraint of the household is
Kt+1 = (rt + 1 − δ)Kt + τt Xt + wt Lt + πt − Ct ,K0 > 0 : given,(20)

where physical capital accumulates through the savings of the household.14 The
household is subject to the no-Ponzi game condition. Specifically, the present valueof its asset holding as T → ∞ should not be negative:

lim

T →∞

T
Y
(rt + 1 − δ)!−1
KT +1 ≥ 0.

(21)

t=1

This completes the description of the model economy.

3.2

R&D by Firms and the Direction of Technological Progress

Now we examine the behavior of the representative firm, focusing on the role of
R&D. The representative firm maximizes profit (15) subject to production function(12) and R&D cost (14) with respect to {Kt , Xt , Lt , AK,t , AX,t , AL,t }, taking as given
prices, {rt , τt , wt }, and lagged technology levels, {AK,t−1 , AX,t−1 , AL,t−1 }. For
convenience, we define µt ≡ Mt /Nt , which is the relative task intensity in final good
production. It also represents the ratio of effective capital composite to effective labor
13

The factor share of natural resources is around 8-9% (Caselli and Feyrer, 2007), of which about
5% is from land (Valentinyi and Herrendorf, 2008). Since the supply of land is mostly constant, we
assume Xt is exogenous in this baseline scenario. Our theory is also applicable to the case where Xt
is depleted or expanded endogenously (See robustness scenario f in Section 4.1). Note that, although
Xt is exogenous, its effective amount AX,t Xt as a production factor can be enhanced endogenously
through R&D for AX .
14
Using (15), equation (20) becomes Kt+1 = Fb(Mt , Nt ) − Ct − Rt + (1 − δ)Kt . Since Yt =Fb(Mt , Nt ), the evolution of capital in this model is exactly the same as (2) in Definition 1. The
aggregate production function (12) also conforms to Definition 1. Therefore, the model in this section
is a 3-factor neoclassical growth model. Moreover, the form of aggregate production (12) is exactly
the same as the factor-augmenting representation (6) in Proposition 1. We did this modeling choicebecause we already know that a 3-factor production function (with σKX = 1) can be represented
in the form of (6) whenever it has a BGP with capital-augmenting change, which is observed in the
data.

11

µt = AK,t Kt

αAX,t Xt

1−α/AL,t Lt . Then, because Fb(·) in (12) is a CRS function, we
can write it in intensive form, Fb(Mt , Nt )/Nt = Fb(µt , 1) ≡ f (µt ), FbM (Mt , Nt ) = f 0 (µt ),and FbN (Mt , Nt ) = f (µt ) − µt f 0 (µt ).15
Using this notation, we can conveniently express the first-order conditions forfactor demand. The firm demands capital, natural resources, and labor so as tosatisfy16,17
rt = (αMt /Kt ) (f 0 (µt ) − iK (γK,t ) − iX (γX,t )) ,(22)

τt = ((1 − α)Mt /Xt ) (f 0 (µt ) − iK (γK,t ) − iX (γX,t )) ,(23)

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

(24)

Now, we turn to R&D, starting with the condition for improving the laboraugmenting technology AL,t . The representative firm chooses AL,t , or equivalently
the speed of technological progress γL,t ≡ AL,t /AL,t−1 ≥ 1, to maximize the profit.
The first order condition is 18
γL,t i0L (γL,t ) + iL (γL,t ) = f (µt ) − µt f 0 (µt ).

(25)

15 b
FM (·) and FbN (·) represent the partial derivatives of function Fb(·) with respect to its first andsecond arguments, respectively.
16
The RHS of (22) represents the (net) marginal product of Kt in producing output Yt . It is givenby the product of two parts. The first part, αM/K, is the marginal product of Kt in increasing
the number of M-tasks performed in the firm. The second part is the net marginal product of Mt
in producing the final output. Note that, in the second part, the innovation cost for an M-task,iK (γK,t ) + iX (γX,t ), is subtracted from the “gross” marginal product of Mt , f 0 (µt ). When the firm
performs more M-tasks, it chooses to pay R&D costs to increase AK,t and AX,t in these tasks so
as to keep up with other M-tasks. Similarly, in (23), (1 − α)M/X is the marginal product of Xt in
performing more M-tasks.
17
By substituting (22), (23), and (24) into (15), it can be confirmed that the firm achieves zeroprofit, πt = 0. This is due to the CRS property of the firm’s problem.
18
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 = Fb(Mt , Nt ). The RHS of (25) shows
the marginal benefit, FbN (Mt , Nt ) = f (µt ) − µt f 0 (µt ). The LHS corresponds to the marginal costof performing a larger number of N-tasks through augmenting labor efficiency AL,t (given labor
employment Lt ). This can be broken into two components. First, by intensifying the R&D efforts
in existing N-tasks to raise labor efficiency, the representative firm can decrease labor inputs by justenough to perform one additional N-task. The cost associated with this activity is given by the firstterm γL,t i0L (γL,t ), which we call the intensive marginal R&D cost. The saved labor is then used toperform 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 termin the LHS.

12

γX

Co
m
R& binD c ed
on
d.

tio

↑

µ

Eq. with µ2(> µ 1)

1

ca
llo
a
D
R& d.
n
co

n

Eq. with µ1

γK1

Figure 1: Equilibrium innovation in K- and X-augmenting technologies.
γK and γX are determined by the intersection of the combined R&D and R&D allocation conditions.
When µt increases from µ1 to µ2 , both γK and γX decrease.

As we formally prove in Proposition 2 below, condition (25) has a unique solution
for γL,t as a function of µt = Mt /Nt , and it is strictly increasing in µt . Intuitively, ahigh value of µt ≡ Mt /Nt means that the resources to perform N-tasks (i.e., effective
labor) are relatively scarce. Then, the marginal product of an N-task is higher, andtherefore the benefit of improving AL,t to increase Nt is larger. Therefore, the firmchooses a larger γL,t when µt is higher.
Next, the first-order conditions for AK,t and AX,t yield19(γK,t i0K (γK,t ) + iK (γK,t )) + (γX,t i0X (γX,t ) + iX (γX,t )) = f 0 (µt ).

(26)

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

(27)

Condition (26) specifies the optimal combined size of R&D investments.20 Since
The first order condition for AK,t yields (γK,t /α)i0K (γK,t ) + iK (γK,t ) + iX (γX,t ) = f 0 (µt ),whereas that for AX,t gives (γX,t /1 − α)i0X (γX,t ) + iK (γK,t ) + iX (γX,t ) = f 0 (µt ). Condition (27) is
obtained by subtracting the second equation from the first. Condition (26) is from adding α times
the first equation and (1 − α) times the second equation.
20
Capital and natural resources are used in M-tasks, and therefore improving K- and Xaugmenting technologies will enable the firm to perform more M-tasks. This marginal benefit isrepresented by the RHS of (26), f 0 (µt ) = FbM (Mt , Nt ). The LHS is the marginal cost of R&D,
which has two parts, γK,t i0K (γK,t ) + iK (γK,t ) and γX,t i0X (γX,t ) + iX (γX,t ), because both K- and19

13

the LHS is increasing both in γKt and γX,t , the locus of (γK,t , γX,t ) that satisfies
this condition is depicted by the downward-sloping curve, as depicted in Figure 1.
Condition (27) gives the optimal allocation of R&D investment between K- and Xaugmenting technologies. Observe that γK,t i0K (γK,t ) and γX,t i0X (γX,t ) on the LHS arestrictly increasing in γKt and γX,t , respectively. Therefore, this condition can be
expressed as an upward-sloping curve in the (γK,t , γX,t ) space.21 The intersection of
the R&D allocation condition and the combined R&D condition gives the optimal
rates of innovation for K- and X-augmenting technologies. When µt = Mt /Nt ishigher, the combined R&D condition curve locates closer to the origin. Then thevalues of γK and γX at the intersection are smaller. Intuitively, When the resources
for M-tasks, i.e., effective capital and effective natural resources, are relatively ample,
the marginal product of an M-task is smaller. Then, the firm has less incentive toimprove γK and γX . The proposition below summarises the results.
Proposition 2. (Direction of Technological Change)
In the endogenous growth model defined in Section 3.1, the growth factors of eachof the factor augmenting technologies are uniquely determined as a function of µt =Mt /Nt ∈ (0, ∞). Let us denote by them by γbK (µt ), γbX (µt ), and γbL (µt ). Then,00(µ), γbL0 (µ)) are (−, −, +) for all µ ∈ (0, ∞).
(µ), γbX(a) The signs of (bγK(b) limµ→0 (bγK (µ), γbX (µ), γbL (µ)) = (∞, ∞, 1).
(c) limµ→∞ (bγK (µ), γbX (µ), γbL (µ)) = (1, 1, ∞).
Proof. See Appendix A.2.
Figure 2 illustrates the direction of the technological change in 3-dimensional
space. The γK -γX plane depicted at the bottom of the figure is the same as Figure
1. The equilibrium direction is obtained by extending vertically from the intersecting
X-augmenting technologies receive some R&D according to the allocation condition (27). In each of
the two parts, the first term represents the intensive marginal R&D cost, whereas the second termis the extensive marginal R&D cost, as in condition (25).
21
As the RHS of condition (27) shows, the allocation should depend on the relative contribution ofcapital and natural resources in performing M-tasks. When capital’s relative contribution is higher(i.e., when α is higher), more resources should be allocated to R&D for the capital-augmenting
technology. In addition, the slope and convexity of the R&D cost function also affect the optimal
allocation. For example, if it is relatively difficult to improve the efficiency of natural resources, i.e.,
if the marginal R&D cost i0X (γX,t ) increases more rapidly with its argument than i0K (γK,t ), then it
is optimal not to improve AX,t as fast as AK,t .

14

γL
Direction of tech.
change with µ1

Equilibrium Innovation
Possibility Frontier (EIPF)
Eq. with µ2 (> µ 1)
Eq. with µ1(1,1,1)γKµ↑Combined
R&D cond.

R&
co D al
nd lo
ca
.

γX

tio
n

Figure 2: The direction of technological change and the Equilibrium Innovation Possibility Frontier (EIPF) curve.
The direction rotates toward the vertical axis along the EIPF curve as µt increases from µ1 to µ2 .

point in the γK -γX plane by the distance of L-augmenting innovation, γbL (µt ) − 1.
As µt increases, the combined R&D locus shifts inward,22 which lowers γK,t andγX,t . At the same time γL,t increases because γ
bL0 (µt ) > 0. This way, the directionof the technological change from the solid arrow to the dashed arrow in Figure 2.
In general, property (a) in Proposition 2 says that when capital composite becomes
more abundant relative to effective labor (i.e. when µt is higher), the direction oftechnological change becomes more upright, depressing improvements in technologies
that enhance capital composite while enhancing improvements in labor-augmenting
technology. In other words, firms are ‘induced’ to do more innovation that enhances
the relatively scarce effective production factors.23
The thick downward-sloping curve in Figure 2 depicts the locus of all equilibrium
points that correspond to various values of µt . This is the equilibrium innovation
possibility frontier (EIPF).24 As µt changes, the equilibrium direction of technological22

The shift occurs because the RHS of (26) is decreasing in µt .
This notion of induced innovation was first introduced by Hicks (1932). See Acemoglu (2002)
for further discussion.
24
In this model, not only the direction within the EIPF, but also the EIPF itself is determined23

15

changes moves along the EIPF curve. Property (b) in Proposition 2 says that when
the capital composite is almost zero relative to effective labor, all the R&D efforts
are directed to enhancing technologies that are related to capital composite. In thisextreme case, the direction coincides with the vertical axis. On the contrary, property(c) implies that, when µt is almost 0, the direction rotates toward the R&D allocation
condition curve and becomes almost flat. Either way, the economy moves away fromthe corner solution.
To summarize, when the direction of technological change is chosen by the representative firm in this model, the direction is adjusted so that the ratio of two inputs
to function Fb, i.e., Mt and Nt is stabilized. This tendency provides a significant force
to achieve balanced growth in the long run. Yet, we also need to solve the full equilibrium dynamics to see how (reproducible) capital is accumulated through saving andinvestment decisions. This is the theme of the next subsection.

3.3

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 , Rt }∞t=0 , which satisfy the representative firm’s optimization problem, therepresentative 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,−1and 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 , theamount of capital per effective labor kt ≡ Kt /AL,t Lt , and consumption per effective
labor ct ≡ Ct /AL,t Lt . Below, we construct the equilibrium mapping from {µt , kt , ct }to {µt+1 , kt+1 , ct+1 } for t ≥ 0. The mapping and the initial conditions µ0 and k0 ,together with the transversality condition for ct , will pin down the equilibrium pathendogenously from the firm’s profit condition. In most models of the direction of technologicalchange, it is assumed that innovation requires a certain type of exogenously given resource (e.g.,scientists). In these cases, the innovation possibility frontier is derived from the resource constraint.
On the contrary, in our model, the total amount of R&D input (Rt ) is determined in equilibrium
through profit maximization, and hence the frontier is called the ‘equilibrium’ innovation possibility
frontier. Any innovation beyond this frontier is not profitable, although it might be feasible.

16

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 = Fb(Mt , Nt )−Rt , which means the aggregate output minus the total R&D costsin the economy. The net output per effective labor can be written as a function of µt :Vt /Nt = f (µt ) − µt (iK (bγK (µt )) + iX (bγX (µt ))) − iL (bγL (µt )) ≡ v(µt ).

(28)

Then, substituting profits (15) into the budget constraint (20), we can express thegrowth of aggregate capital supply in terms of µt , kt and ct :Kt+1
Vt + (1 − δ)Kt − Ctv(µt ) − ct==
+ 1 − δ.
Kt
Kt
kt

(29)

Dynamics for µt+1 . The growth factor of µt+1 is defined by µt+1 /µt = (Mt+1 /Mt )/(Nt+1 /Nt ).
By using (10), (11), (16), (19) and (29), its value in equilibrium can be written as
µt+1
(gX γ
bX (µt+1 ))1−α
=
µt
gL γ
bL (µt+1 )αv(µt ) − ctγbK (µt+1 )
+1−δ
,
kt

(30)

where γbK (µt ), γbX (µt ), and γbL (µt ) are the rates of technological progress defined in
Proposition 2. While equation (30) 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 bothsides of the equation depend on µt+1 .
To obtain more straightforward dynamics, let us decompose the dynamic relationship in (30) into two steps. First, we define the pre-R&D relative factor intensity
by
µpre
t+1 ≡

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

(31)

where the last equality is from (16), (19), (29) and the definition of µt . It is the valueof µt+1 before technologies are improved from their period-t state. Second, µpret+1 andthe post-R&D value of µt+1 are related by the growth of technological levels γ
bK (µt ),γbX (µt ), and γbL (µt ) as follows:
µpre
t+1 =

γ
bL (µt+1 )µt+1 ≡ Γ(µt+1 ).
γbK (µt+1 )α γbX (µt+1 )1−α
17

(32)

Note that, Proposition 2 implies that function Γ(µt+1 ) is a strictly increasing differentiable function with limµ→0 Γ(µ) = 0 and limµ→∞ Γ(µ) = ∞. Therefore, its inverseprefunction µt+1 = Γ(−1) (µpret+1 ) is well-defined for all µt+1 > 0, and is a strictly increasing

differentiable function.
Using (31) and the inverse function of (32), the dynamic relationship (30) can be
written as(−1)µt+1 = Γ 1−α α gXv(µt ) − ct
+ 1 − δ µt ≡ ψ µ (µt , kt , ct ).
gL
kt

(33)

Function ψ µ (µt , kt , ct ) gives a mapping from period-t variables µt , kt , ct to µt+1 . It provides a natural 2-step interpretation of the equivalent equation (30). The argumentof function Γ(−1) (·) in (33) represents the pre-R&D relative task intensity, which isdetermined 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 (16) and (29), the growth factor of kt ≡ Kt /AL,t Lt is
obtained as
kt+1
1
=
kt
gL γ
bL (µt+1 )
v(µt ) − ct
+1−δ .
kt(34)

While µt+1 is present in the RHS, we can replace it with (33) so that the RHS depends
only on the variables in period t.
kt+1 =1gL γ
bL (ψ µ (µt , kt , ct ))(v(µt ) − ct + (1 − δ)kt ) ≡ ψ k (µt , kt , ct ).

(35)This dynamic equation simply represents the process of capital accumulation pereffective labor. The expression (v(µt ) − ct + (1 − δ)kt ) shows the sum of the net saving
and the un-depreciated part of existing capital per effective labor in period t. It must
be divided by gL γ
bL because of the growth of effective labor between period t and t+1.
Dynamics for ct+1 . The representative household maximizes the intertemporalutility function (18) subject to the budget constraint (20) and the non-Ponzi Gamecondition (21). The Euler equation for this problem is25
25
Note that ∂U/∂Ct = β t (Ct /Lt )−θ from From (18). From this, the Euler equation is(Ct /Lt )−θ = (rt+1 + 1 − δ)β(Ct+1 /Lt+1 )−θ , which simplifies to (36).

18

−θ
.
Ct−θ = (rt+1 + 1 − δ)βgLθ Ct+1

(36)

By substituting the market interest rate (22) into the Euler equation (36) and thenapplying it to the definition ct ≡ Ct /AL,t Lt , we obtain the growth factor of consumption per effective labor:
ct+1
β 1/θ
=
ct
γ
bL (µt+1 )αµt+1 0(f (µt+1 ) − iK (bγK (µt+1 )) − iX (bγX (µt+1 ))) + 1 − δ
kt+1

1/θ
.
(37)

By replacing the period-(t + 1) variables in the RHS by (33) and (35), we can rewrite
equation (37) as
ct+1 =αψ µ (µt , kt , ct )  0 µ
β 1/θ ct
f (ψ (µt , kt , ct ))γbL (ψ µ (µt , kt , ct )) ψ k (µt , kt , ct )− iK (bγK (ψ µ (µt , kt , ct ))) − iX (bγX (ψ µ (µt , kt , ct ))) + 1 − δ!1/θ≡ ψ c (µt , kt , ct ).
(38)

Equations (33), (35) and (38) constitute the equilibrium mapping from {µt , kt , ct } to{µt+1 , kt+1 , ct+1 } for 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 (32), we have the initial value of µt :(−1)

µ0 = Γ

(AK,−1 K0 )α (AX,−1 X0 )1−α
AL,−1 L0


.

(39)

Second, using µ0 , the initial value of kt is readily obtained by
k0 =

K0
.
γ
bL (µ0 )AL,−1 L0(40)

Finally, the initial value of ct must be chosen so as to satisfy the non-Ponzi game19

condition (21) and the transversality condition
lim β

T →∞

T



CT
LT

−θ
KT +1 ≤ 0.

(41)

Combining Euler equation (36) with (21) and (41) gives the unified terminal conditionallim (βgL )T

T →∞

T
Y

!1−θ
γ
bL (µT +1 )c−θ
T kT +1 = 0.

γbL (µt )

(42)

t=0

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

3.4

The Balanced Growth Path

Now, we are ready to characterize the BGP of this economy. We will show that thedirection 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 1. Define a BGP as an equilibrium path where the growth factors of {Yt , Kt ,
Xt , Lt , Ct , Rt , Mt , Nt } are all constant.26 Then, on any BGP, the values of µt , kt
and ct must be constant.
Proof. See Appendix A.3.
We denote the BGP values of µt , kt and ct by µ∗ , k ∗ and c∗ , respectively. Theirvalues are obtained by substituting µt+1 = µt = µ∗ , kt+1 = kt = k ∗ and ct+1 = ct = c∗into (33), (35), and (38).
First, from (33) and (35), the BGP value of µt ≡ Mt /Nt will satisfy
1=

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

(43)Proposition 2 implies Φ0 (µ∗ ) < 0 with Φ(0) = ∞ and Φ(∞) = 0. Therefore, there
exists a unique value of µ∗ > 0 that satisfies Φ(µ∗ ) = 1, and hence condition (43).
26

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

20

An intuitive way to interpret (43) is to multiply both of its sides by (gL γbL (µ∗ ))α .
(gX γ
bX (µ∗ ))1−α (bγK (µ∗ )gL γ
bL (µ∗ ))α = gL γ
bL (µ∗ ) (= g ∗ ).

(44)

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 isdetermined so that Mt and Nt grow at the same speed. This condition singles out
a point on the Equilibrium Innovation Possibility Frontier (recall Figure 2), which
determines the direction of technological change on the BGP. Note that, due to the
CRS property of production function Yt = Fb(Mt , Nt ), the value of equation (44) also
represents the economic growth factor g ∗ ≡ Yt+1 /Yt .
Second, from the Euler equation (38), the BGP value of kt = Kt /(At Lt ) isk∗ =βαµ∗ (f 0 (µ∗ ) − iK (bγK (µ∗ )) − iX (bγX (µ∗ ))).
γbL (µ∗ )θ − β(1 − δ)

(45)

Intuitively, the capital-effective labor ratio on the BGP is determined from the interest
rate r∗ that yields constant consumption per effective labor on the BGP.27 Third, from(35) and gL γ
bL (µ∗ ) = g ∗ in (44), the BGP value of c∗ = Ct /AL,t Lt must satisfyc∗ = v(µ∗ ) − (g ∗ − 1 + δ)k ∗ .

(46)

These three equations describe the unique BGP in this economy. The following
proposition shows that the BGP uniquely exists when the discount factor is sufficiently
smaller than 1.28
Using (22), condition (45) is shown to be equivalent to r∗ + 1 − δ = β −1 γ
bL (µ∗ )θ . Here, theRHS is the marginal rate of intertemporal substitution given that consumption per effective laboris constant (which must be true on the BGP).
28
There are two reasons why the existence of the BGP requires an upper bound for β (or, equivalently a lower bound for ρ = (1 − β)/β). First, on the BGP, the amount of consumption for thehousehold Ct = AL,t Lt c∗ increases over time, causing the instantaneous utility to grow. Therefore,
if β is too close to one, the intertemporal utility U in (18) becomes infinity, which means that thehousehold’s problem is not well defined. Second, as effective labor AL,t Lt grows, the household accumulates 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 realinterest rate, or a too-high level of k ∗ , to the extent that preventing the dilution is impossible evenwhen all net output is invested in Kt . We rule out these extreme cases by assuming an upper boundfor β.
27

21

Proposition 3. There exists a value of β > 0 such that whenever β ∈ (0, β), there
exists a unique BGP that satisfies µ∗ > 0, k ∗ > 0, c∗ > 0, and the terminal condition(42). On this BGP, the long-term rate of capital-augmenting technological change ispositive (γK (µ∗ ) > 1).
Proof. See Appendix A.4. The exact expression for the upper bound β is given by(A.8).
An important implication from this model is that the technology condition (9)in Section 2 is now an endogenous outcome. Specifically, the BGP condition (43)is equivalent to (9), 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.
As discussed in Section 2, if the rates of innovation for the three factor-augmenting
technologies are exogenously given, then (9) becomes a knife-edge condition. In contrast, this section has shown that, once we consider endogenous technical change,
this condition is necessarily satisfied when the economy is on the BGP, which exists
if discount factor β is sufficiently less than one.
A missing link is that we have not yet shown whether the economy actually converges to this BGP. If it is shown, we can conclude that condition (9) in the generalized Uzawa growth model is naturally satisfied in the long run, greatly widening theplausibility of the theorem as an explanation of the capital-augmenting technologicalchange observed in data. We do so in the next section.

4

Numerical Analysis and Convergence to the BGP

In this section, we investigate the local and global stability of the three-factor endogenous growth model. Our primary objective is to show that the model economy
converges to a BGP with capital-augmenting technical change, where log-linear relationship (9) is endogenously satisfied. We also illustrate how having multiple technologies (including K-augmenting technology) affects the transition dynamics. To
accomplish these goals, we present a series of numerical examples for which we cancheck stability computationally. Whenever possible, we ensure that our numericalexamples are consistent with macroeconomic data characterizing the BGP of the22

United States. The data sources are provided in Appendix C. We stress, however,that this is not a complete calibration, and the results would be insufficient for a
precise quantitative analysis.

4.1

Calibration

Functional Forms
We assume that the aggregate production function takes a CES form: Fb(Mt , Nt ) =
−1

−1



(ηMt  + Nt  ) 1− , where  > 0 and η ∈ (0, 1). Output in the economy (12) can be
written as

n
o 1−
 −1
−1
α
1−α 

Yt = η (AK,t Kt ) (AX,t Xt )+ (1 − η)(AL,t Lt ).

(47)

Next, we assume that R&D costs are power functions29
λiZ (γZ ) = ζZ γZ − 1 ,ζZ > 0, λ > 1, Z = K, X, L.

(48)

We allow R&D cost parameter ζZ to differ across types of technology. We normalize
ζK to 1, and calibrate ζX and ζL . The degree of convexity, λ, is assumed to be thesame across the three types of technology.
With these functional form assumptions, our model has 11 parameters, {, η, α, λ, ζL , ζX ,β, θ, δ, gL , gX }. To calibrate the model, we also need to determine the period length,measured in years, denoted by χ. The period length in our model has an importanteconomic meaning because it represents the duration for which a firm can monopolize
the benefit from its R&D investments. Including χ, we have 12 parameters.
Exogenous Parameters
We set five parameters exogenously. Their values are given in Table 1. In the CES
production function (47), we take  = 0.7 as the baseline value. This is a common
estimate for the EoS between labor and reproducible capital (e.g., Antras et al., 2004;
Oberfield and Raval, 2014). The mapping between these estimates and a structuralparameter in our model is not exact, and we show robustness with  = 0.9 and
29

Note that function (48) satisfies condition (13).

23Parameter

λ
θ
1/χ
gL
1/χ
gX

Baseline
0.7
2.0
1.0
1.01
1.0

Alternate
0.9, 1.2
1.5, 2.25
0.5, 2
0.99

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

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

Table 1: Exogenous Parameters
Target Moment (in annual values)
Value
Capital output ratio
2.9
Labor share of income
63%
Share of R&D payments in GDP
2.7%
Consumption of fixed capital in GDP 14%
Growth rate of income per capita
1.9%
Decline in the relative price of capital 0.66%
Return on investment
4%

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

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

Table 2: Target Moments for Calibration
 = 1.2.30 In the baseline calibration, we assume the R&D cost function is quadratic(λ = 2) and also check robustness with λ = 1.5 and λ = 2.25. Quadratic cost is acommon assumption, and it is consistent with existing empirical work in endogenous
growth (Acemoglu et al., 2018; Akcigit and Kerr, 2018). As for utility function (18),
we take log preferences (θ = 1) as the baseline and also consider cases where theintertemporal EoS is higher or lower than 1 (θ = 0.5 and 2). Population growth isset to the 1960-2020 average in the U.S. (1% per year). When one period in the1/χmodel corresponds to χ years, this means gL= 1.01. We do not have good data forthe growth rate of factor X, which we interpret as natural resources, including land.
Given that land is a major factor of production, we take gX = 1 as a benchmark (i.e.,
constant X). We also consider the case where natural resources are depleted 1% per
1/χ

year (gX = 0.99).

24

Data
We calibrate the remaining parameters so that the model variables on the BGP
match data from the U.S. Table 2 reports the target moments and model variables
in annualized values (e.g,. aggregate output per year is Y /χ, where one period in
the model is χ years). For the capital-output ratio (2.9), labor share of income(63%), R&D share of income (2.7%), consumption of fixed capital as a share of GDP(14%), and real GDP per capita growth (1.9%), we use data from the Bureau of
Economic Analysis (BEA) to calculate the arithmetic averages of the annual levels in
the 1960-2020 period. To measure the growth rate of capital-augmenting technology,
we calculate the annual decline in the relative price of all capital goods from 19602020 (0.66%). Finally, we set the rate of return on investment (r∗ − δ) equal to thereturn on bonds (4%) from McGrattan and Prescott (2003).
Calibration Results
There are seven remaining parameters to calibrate, {δ, β, α, η, ζL , ζX , χ}, which we
identify with the seven moments in Table 2. We do so in two steps. First, we use
equilibrium conditions to derive four analytical relationships among these parameters.
This leaves us with three undetermined parameters, {ζL , η, χ}. In the second step,
we numerically pin them down so that the target moments in Table 2 match thecorresponding model variables on the BGP. The details of the calibration procedure
are presented in Appendix B.
Table 3 presents the results of the two-step calibration procedure with the baseline
assumptions. Period length χ is 3.94 years, which is the time until the knowledge
becomes public. At this point, someone else can freely use the knowledge to generate1

a new innovation. Discount factor β is 0.923 (β χ = 0.98/year). The depreciation
rate δ is 0.19 per period, which is about 5% per year. The share parameter α is0.76. This implies that the capital share in the GDP is 26.3%, whereas the natural
resource share (including land) is 8%. Although these shares were not targeted in thecalibration, they are consistent with the existing literature discussed in Casey andHorii (forthcoming).
30

Karabarbounis and Neiman (2014) and Piketty (2014) estimate the EoS between reproducible
capital and labor and find an elasticity that is greater than one.

25

Parameter
χ
β
δ
α
η
ζX
ζL

Calibrated
3.94
0.923
0.190
0.767
0.685
0.279
20.8

Annualized
0.980
5.21%

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

Table 3: Calibrated Parameters for Baseline Scenario
VariableκK
κX
1/χγX − 1

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

Table 4: Untargeted Variables in Calibrated Model
Robustness
Changing the free parameters, we present calibration results with λ ∈ {1.5, 2.25},
θ ∈ {0.5, 2}, and  ∈ {0.9, 1.2}. We also calibrated the model under the assumption
1/χthat natural resources X are depleted by 1% per year; i.e., gX = 0.99. In each case,we change one parameter from the baseline value and then re-calibrate the model. In
all cases, we find the set of parameters with which the model matches all the target
moments in Table 2. The results are reported in Table 5 as scenarios (a)–(g).

4.2Local Stability

Using parameters calibrated for the baseline setting and alternative scenarios, we cannow examine the local stability of the model. Recall that the dynamic system ischaracterized by three variables {µt , kt , ct }, which evolve according to equations (33),(35) and (38). Also, note that the initial values of µ0 and k0 are pre-determined,
whereas c0 should be chosen endogenously so that the system satisfies the transversality condition (42). In this system, the BGP is saddle-stable and determinate if
the Jacobian matrix evaluated at the BGP has two stable eigenvalues with absolutevalues less than one and one unstable eigenvalue with an absolute value greater than
26

(a)
(b)
(c)
(d)
(e)
(f)
(g)
1/χ
Param- Baseλ
λ


θ
θ
gX
eters
line
=1.5 =2.25 =0.9
=1.2
=0.5
=2
=0.99
χ
3.94
2.92
4.20
3.94
3.94
3.94
3.94
3.72
1/χ
β
0.98
0.98
0.98
0.98
0.98
0.97
0.998
0.98
1/χ
δ
5.21% 5.07% 5.25% 5.21% 5.21% 5.21% 5.21% 5.18%
α
0.77
0.76
0.77
0.77
0.77
0.77
0.77
0.77
η
0.69
0.38
0.79
0.41
0.27
0.69
0.69
0.70
ζL
20.82
1.88
61.74 10.05
7.64
20.82 20.82 25.69
ζX
0.28
0.29
0.28
0.28
0.28
0.28
0.28
0.11
κK
26.3% 26.1% 26.4% 26.3% 26.3% 26.3% 26.3% 26.2%
κX
8.00% 8.25% 7.93% 8.00% 8.00% 8.00% 8.00% 8.05%
1/χγX − 1 0.72% 0.80% 0.69% 0.72% 0.72% 0.72% 0.72% 1.75%
Table 5: Calibrated Parameters for the Robustness Scenarios
one.
Table 6 summarizes the results of the local stability analysis. In all cases, we findthat the BGP is saddle-stable and determinate: when state variables are near the
BGP, they will converge to the BGP along the unique saddle path. In Subsection 3.4,
we demonstrate that one of the conditions for balanced growth, (43), is equivalent to
the technology condition (9). Therefore, the saddle stability of the BGP implies thatthe technology condition is endogenously satisfied as the economy converges to the
BGP.
On this equilibrium path converging to the BGP, firms choose the intensities ofthree types of R&D, γK , γX , and γL , and hence the direction of the technologicalchange, to maximize profits. The capital-augmenting technology AK is still growing
on the BGP, because firms always benefit from improving AK . This naturally explains
the observed long-term decline in the relative price of capital, which theoreticallycorresponds to capital-augmenting technological change.
The saddle stability is confirmed in all seven alternative scenarios. It demonstrates
that our explanation of capital-augmenting technological change is robust to changes
in parameters. It is particularly interesting to note that we find stability even when
the EoS between labor and the capital-composite, , is greater than one (scenario d).

27

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

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

Table 6: Eigenvalues and Local Stability of the Calibrated Model
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., 2017).31

4.3

Transition Dynamics and Global Stability

Local stability only examines convergence within the neighborhood of the BGP. In
this subsection, we go one step further and demonstrate that convergence to the BGP
occurs even when the initial states are far away. We call this property global stability.
With three factors of production, this is not a trivial exercise, because the transitional
dynamics may take various patterns depending on the initial combination of µ0 andk0 .32 They are determined by initial stock of production factors K0 , X0 and L0 , aswell as initial technology levels AK,−1 , AX,−1 , AL,−1 . Depending on the initial stateof technology or resources, µ0 and k0 will take a wide range of combinations.
To cover various possibilities, we consider a large rectangular area in µ-k planesurrounding the BGP: namely, µt ∈ [0.2µ∗ , 2µ∗ ] and kt ∈ [0.2k ∗ , 2k ∗ ]. We choose
36 points on the border of a rectangular area and calculate the transition dynamics
from each of them. We use a forward shooting method to determine the value of c031

An exception is a model by Irmen and Tabaković (2017), which has an elasticity greater thanone. As explained above, their model has capital-augmenting technical change on the transition
path, but not on the steady state.
32
In typical macro models with two factors of production, the dynamics can be written in termsof kt and ct , where kt is the only state variable. In this case, the transition dynamics only have two
possible patterns, depending on whether k0 is higher or lower than the steady-state value. In either
case, kt typically converges monotonically to the steady state.

28

8

6

k
4

2
CapitalAdjustments

Technology
Adjustments(common arm)

BGP

0

0

50

100

6

4

6

4

c

2

2
100
50
0

c

k 4
2
0

0

8
6
0

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

Figure 3: Global Stability of the Calibrated Model (Baseline Setting).
that eventually satisfies the transversality condition (42) as t → ∞. The equilibrium
path from each starting point is depicted in Figure 3, where the parameters arefrom baseline calibration in Section 4.1. Because the graph is three-dimensional, we
depicted the same graph from two angles. We also provide the projection of the paths
to the bottom µ-k plane in darker colors.
From each of the 36 starting pairs of µ0 and k0 , we always find a unique level ofc0 such that the path from {µ0 , k0 , c0 } leads to the BGP (i.e., {µ∗ , k ∗ , c∗ }). If c0 is
higher the resource constraint is eventually violated (kt becomes negative), and if c0 is
lower the TVC is violated (ct converges to zero). This means that convergence to the
BGP is the only possible long-term outcome in equilibrium. These findings suggest
that, as long as the initial µ0 and k0 are on or within the border of the rectangle, theeconomy necessarily converges to the BGP. Since the rectangle is reasonably large,
we call it global stability.
There are a couple of properties worth observing from the figure. First, theconvergence is not monotonic. To illustrate this, let us focus on the path that starts
from the upper right corner in Figure 3(a), as indicated by a thick arrow (µ0 = 2µ∗and k0 = 0.2k ∗ ).33 Although the initial level of µ0 is double the steady-state level, µt
33

In the color PDF version of the article, the path that we now focus on is depicted in purple.

29

initially increases further, going out of the rectangular area. This phenomenon canbe interpreted as follows. At the initial state, the capital composite is abundant eventhough the reproducible capital is scarce. This happens when natural resources are soabundant that it more than offsets capital scarcity. In this setting, the consumptionof reproducible capital (i.e., the depreciation of Kt ) is small, and savings from ample
production leads to more accumulation Kt , which increases the capital composite
further. This process continues until the level of kt comes close to the steady statelevel. This is the first stage of convergence. In the second stage, the ratio of capitalcomposite to effective labor µt gradually falls to the steady state level. This is because
a high µt means that effective labor is relatively scarce, and the firms have more
incentives to improve AL through R&D, rather than AK or AX . This tendency
continues until µt reaches µ∗ . Once µt comes to µ∗ , firms have incentives to improve
all types of technologies in a ‘balanced’ way such that the ratio of capital composite
to effective labor does not change further. This illustrates how firms, in the long run,choose the direction of technological change that satisfies the BGP condition (43) or,
equivalently, the technology condition (9).
The figure also shows that, even though the stable manifold34 is two-dimensional,
the equilibrium paths first converge to a common one-dimensional arm (or curve),
and then converge to the BGP along the arm. This is because the system has two
stable eigenvalues with significantly different magnitudes. In the baseline calibration,
stable eigenvalues are 0.602 and 0.970. Given that one period χ is 3.94 years, thoseeigenvalues mean the speed of convergence is 12% and 0.7% per year, respectively.
As we discussed in the above example (the path starting from the upper-right corner,
indicated by a thick arrow), the convergence to the BGP typically goes through twostages, and each stage corresponds to a different eigenvalue. The initial adjustment
towards the common arm is driven mainly by capital accumulation. It is relatively
fast: the distance from the common arm declines by 12% every year. However, thesecond stage, along the common arm, is very slow. In the baseline example, theconvergence speed is only 0.7% per year, which means it takes about 90 years tohalve the distance. This adjustment takes much longer than capital accumulation
because it is driven by the difference in the speed of technological change among34

The stable manifold is the set of points in the (µ, k, c) space that converges to the BGP. In
Figure 3, all converging paths are on the (same) stable manifold.

30

AK , AX , and AL . Note that these numbers are just for illustration, because theeigenvalues depend on the free parameters, as shown in Table 6. Still, this result
suggests that, without considering the endogenous technological change for various
production factors (including capital-augmenting technological change), neoclassicalgrowth models may overestimate the speed of convergence to the steady state by large
margins.

5

Conclusion

We build a neoclassical growth model that has capital-augmenting technical changeon the BGP, despite a non-unitary EoS between capital and labor. As noted by
Grossman et al. (2017), standard neoclassical growth models cannot incorporate theseelements simultaneously, due to the Uzawa steady state theorem. This is a significant
limitation, because each of the elements has strong empirical support. To overcome
the restrictive nature of the theorem, we follow Casey and Horii (forthcoming) andadd natural resources to the model as a third factor of production. We then add
directed technical change and show that the model endogenously converges to the
BGP with capital-augmenting technical change. By relaxing the constraints posedby the theorem, our model should be useful in a wide range of settings.

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33

Online Appendix“Endogenous Capital-Augmenting Technological Change”
by Gregory Casey and Ryo Horii
November 28, 2023

A

Proofs of Propositions and Lemmas

A.1

Notation for derivatives

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

production function, ∂X∂j,t Fe(·) gives the marginal product of factor Xj,t .
Note that these two definitions are different when the argument of the function is
not a single variable. For example, using the chain rule, we have
∂ e
eX1 ,t X1,t , ..., AeX ,t XJ,t ). (A.1)
eX1 ,t X1,t , ..., AeX ,t XJ,t ) = A
eX ,t FeX (Kt , A
F (Kt , A
j
j
J
J
∂Xj,t

A.2

Proof of Proposition 2

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

(25)

Let us denote the LHS of (25) by ΨL (γL,t ) because it depends only on γL,t . Then,Ψ0L (γL,t ) = γL,t i00L (γL,t )+2i0L (γL,t ) > 0 for all γL,t > 1 from iL (γL,t ) > 0 and i0L (γL,t ) > 0
in (13). When γL,t = 1, the properties of iL (·) imply ΨL (1) = i0L (1) + iL (1) = 0. Also,

A-1

ΨL (∞) ≡ limγL,t →∞ ΨL (γL,t ) = ∞ from iL (∞) = ∞ and γL,t i0L (γL,t ) > 0.35 Then,since ΨL (·) is differentiable and strictly increasing, we can define its inverse function(−1)(·), which is also differentiable and strictly increasing with ΨL(−1)(∞) = ∞. Using this function, condition (25) can be solved for γL,t :

ΨL
ΨL

(−1)

(−1)

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

(0) = 1 and(A.2)

Note that f (µt )−µt f 0 (µt ) represents the marginal product of Nt in the production
function, i.e., FbN (µt , 1). We can express the production function Yt = Fb(Mt , Nt ) in
an intensive form with respect to Nt /Mt ≡ νt , instead of µt = Mt /Nt . Namely, output
per Mt can be expressed as Yt /Mt = Fb(Mt , Nt )/Mt = Fb(1, νt ) ≡ h(νt ). Since Fb(·)
is CRS, its first derivative FbN (·) is homogeneous of degree 0. Using this property,h0 (ν) = FN (1, Nt /Mt ) = FN (Mt /Nt , 1) = FN (µt , 1) = f (µt ) − µt f 0 (µt ). From thedefinition of the production function Fb(·), its alternate intensive form, h(νt ), satisfies
the Inada conditions. Therefore, limµt →0 f (µt ) − µt f 0 (µt ) = limνt →∞ h0 (νt ) = 0, andlimµt →∞ f (µt ) − µt f 0 (µt ) = limνt →0 h0 (νt ) = ∞. Substituting these into (A.2) gives(−1)

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

Finally, we show γL0 (µt ) > 0. The derivative of f (µt ) − µt f 0 (µt ) with respect toµt is −µt f 00 (µt ). It is positive for all µt > 0 since the production function satisfies(−1) 0the Inada conditions, which include f 00 (µt ) < 0. Since ΨL(·) > 0, this means

γL0 (µt ) > 0.
Properties of γbK (µK ) and γbX (µX )The representative firm chooses γK,t and γX,t according to the following two conditions:
R&D allocation:

γK,t i0K (γK,t )α=,0γX,t iX (γX,t )1−αα ∈ (0, 1),(27)

Combined R&D: (γK,t i0K (γK,t ) + iK (γK,t )) + (γX,t i0X (γX,t ) + iX (γX,t )) = f 0 (µt ).
(26)
35

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

A-2

Let us define ΩK (γK,t ) ≡ γK,t i0K (γK,t ) and similarly ΩX (γX,t ) ≡ γX,t i0K (γX,t ). Then,from properties in (13), we can confirm Ω0K (γK,t ) > 0 for γK,t > 1, ΩK (1) = 0
and ΩK (∞) = ∞. Similar conditions also hold for ΩX (·). Then, since ΩX (·) is(−1)

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

(−1)

is also differentiable and strictly increasing with ΩX (0) = 1 and ΩX (∞) = ∞.
Using this inverse function, condition (27) can be solved for γX,t as(−1)
γX,t = ΩXαΩK (γK,t ) ≡ Ω(γK,t ).
1−α

(A.3)

Now let us focus on condition (26). Let us define ΨK (γK,t ) ≡ γK,t i0K (γK,t )+iK (γK,t )and likewise ΨX (γX,t ) ≡ γX,t i0K (γX,t ) + iK (γX,t ). Using these and (A.3), the LHS ofcondition (26) can be expressed as a function only of γK,t :ΨK (γK,t ) + ΨX (Ω(γK,t )) ≡ Ψ(γK,t ).
(−1)

Note that the properties of ΩK (·) and ΩX (·) imply that Ω(γK,t ) > 0 for all γK,t > 1,Ω(0) = 0 and Ω(∞) = ∞. Also, in the same way that we derived the properties ofΨL (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 (26), f 0 (µt ) satisfies the usual
Inada conditions. The results we have obtained so far can be summarized as
γK,t

1 ···

Ψ0 (γK,t )+Ψ(γK,t )

0 %

∞

µt

∞

0

···

f 00 (µt )−f 0 (µt )

∞ &

∞
0

The tables above implies that condition (26), Ψ(γK,t ) = f 0 (µt ), gives a 1 to 1 correspondence between µt ∈ (0, ∞) and γK,t ∈ (1, ∞) that satisfies property (a):0γbK(µt ) < 0 for all µt > 0, γ
bK (0) = ∞, and γbK (∞) = 1.

Given γbK (µt ), equation (A.3) uniquely determines γX,t = Ω(bγK (µt )) ≡ γ
bX (µt ).
From the properties of Ω(·) and γbK (·) above, we can confirm that property (b) is
0satisfied: γbX(µt ) < 0 for all µt > 0, γ
bX (0) = ∞, and γbX (∞) = 1.

A-3

A.3

Proof of Lemma 1

Consider a BGP. We will show in turn that µt , kt and ct must be constant. First, fromthe definition of a BGP, Nt+1 /Nt = (AL,t+1 Lt+1 )/(AL,t Lt ) = γbL (µt+1 )gL is constant.
To keep the RHS of the latter equation constant, µt must also be constant, since γbL (·)
is a strictly increasing function from Proposition 2.
Second, since the growth factors of Ct and Nt are constant, the growth factor of
ct = Ct /At Lt = Ct /Nt is also constant. This, in turn, means that the LHS of theEuler equation (37) is constant. Then, for the RHS of (37) to be constant, kt must
be constant, since we already know that µt is constant as shown above.
Third, the growth factor of kt = Kt /At Lt = Kt /Nt is constant on the BGP, whichmeans the LHS of (34) 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.

A.4

Proof of Proposition 3

Proof of µ∗ > 0
In the text, we have already shown that there exists a unique µ∗ > 0 such thatΦ(µ∗ ) = 1 holds since Proposition 2 implies Φ0 (µ∗ ) < 0 with Φ(0) = ∞ and Φ(∞) = 0.
Therefore, there exists a unique value of µ∗ > 0.
Proof of k ∗ > 0
The value of k ∗ is explicitly given by equation (45), shown again here:k∗ =βαµ∗ (f 0 (µ∗ ) − iK (bγK (µ∗ )) − iX (bγX (µ∗ ))).
γbL (µ∗ )θ − β(1 − δ)(45)

We now show that both the numerator and the denominator of the RHS are positive.
Note that the combined R&D condition (26) is satisfied on the BGP. By rearranging
terms, it givesγX (µ∗ )) > 0,f 0 (µ∗ ) − iK (bγK (µ∗ )) − iX (bγX (µ∗ )) = γbK (µ∗ )i0K (bγK (µ∗ )) + γ
bX (µ∗ )i0X (b
(A.4)

A-4

where the inequality follows from Proposition 2 and (13). Given β ∈ (0, 1), α ∈ (0, 1),
and µ∗ > 0, this means that the numerator of (45) is strictly positive. Now, notethat γbL (µ∗ ) > 1 from Proposition 2. Combined with θ > 0, β ∈ (0, 1) and δ ∈ [0, 1],
it turns out that the denominator of (45) is also strictly positive.
Proof of c∗ > 0
The value of c∗ is given byc∗ = v(µ∗ ) − (g ∗ − 1 + δ)k ∗ .

(46)

We first show v(µ∗ ) > 0. Combining the R&D conditions (25) and (26), we have
γL,t i0L (γL,t )+iL (γL,t )+µt ((γK,t i0K (γK,t ) + iK (γK,t )) + (γX,t i0X (γX,t ) + iX (γX,t ))) = f (µt ).
Rearranging and then evaluating this condition at µt = µ∗ givesv(µ∗ ) = f (µ∗) − iL (bγL (µ∗ )) − µ∗ (iK (bγK (µ∗ )) − iX (bγX (µ∗ )))=γbL (µ∗ )i0L (bγL (µ∗ )) + µ∗ (bγK (µ∗ )i0K (bγK (µ∗ )) + γ
bX (µ∗ )i0X (bγX (µ∗ ))) > 0,
where the inequality follows from µ∗ > 0 and (13).
Note that g ∗ = γbL (µ∗ )gL is greater than 1 − δ because γbL (µ∗ ) > 1 and gL > 1 − δfrom (16). Therefore, (g ∗ − 1 + δ) in (46) is positive. From this, c∗ > 0 is equivalenttok∗ <v(µ∗ ).
g∗ − 1 + δ

Using (45), we can rewrite this condition in terms of β:∗ θβ<γbL (µ )

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

−1≡ β 1.
(A.5)∗

Note that β 1 > 0 from (A.4) and g > 1 − δ > 0. Observe also that β 1 does notdepend on β itself since µ∗ is determined entirely by the production side (see equation
43). Therefore, if β > 0 is sufficiently small, condition (A.5) holds and c∗ > 0.

A-5

Terminal Condition
On the BGP, the terminal condition (42) becomesbL (µ∗ )1−θ
lim βgL γ

T

T →∞

γ
bL (µ∗ )(c∗ )−θ k ∗ = 0.

(A.6)

Given that γbL (µ∗ ) > 1, c∗ > 0 and c∗ > 0, this condition is equivalent toβ<1≡ β 2.
gL γ
bL (µ∗ )1−θ(A.7)

Note that β 2 > 0 and that it does not depend on β since µ∗ is determined entirely by
the production side of the model. Therefore, if β > 0 is sufficiently small, condition(A.7) holds and the terminal condition (42) is satisfied.
Combining conditions (A.5) and (A.7), we have confirmed the unique existence of
BGP with µ∗ > 0, k ∗ > 0, c∗ > 0, and the terminal condition (42) wheneverβ < β ≡ min{β 1 , β 2 },(A.8)

where β > 0 is a constant that does not depend on β.

BCalibration Procedure

There are seven parameters to calibrate, {δ, β, α, η, ζL , ζX , χ}, which we identify withthe seven moments in Table 2. We do so in two steps.
Step 1: Analytical calibration. Given period length χ, exogenous parameters,
and moments, we analytically derive the values of four parameters {δ, β, α, η}. The
depreciation rate is determined by data on the consumption of fixed capital and thecapital-output ratio:
δ=

δK/Y
0.14
=
χ ≡ δ(χ).
K/Y
2.9

A-6

(B.1)

Evaluating the Euler equation (36) on the BGP gives the discount factor β:36
β=γLθ(1.019χ )1.0=
≡ β(χ).
1+r−δ
1.04χ

(B.2)

Similarly, the first-order conditions of the representative firm, (22) and (23), give theshare parameter α:α=(r − δ)(K/Y ) + δK/Y
(1.04χ − 1)(2.9χ) + 0.14
κK==
≡ α(χ), (B.3)
κK + κX
1 − κL − κR
1 − 0.63 − 0.027

where we used the identity κK + κX + κL + κR = 1.
Next, BGP relationship (43), which is equivalent to the technology condition (9),gives the growth rate for the unobserved endogenous variable γX :− α gL γL
γX = γK 1−α
=
gX

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

(B.4)

Using (B.4) and the R&D allocation condition (27), the R&D cost parameter ζX canbe derived as follows.
χζX =1 − α ζK γK (γK − 1)λ−1
1 − α(χ) (1(1.0066)(0.0066)2.0−1 )·=·≡ ζ X (χ).
αγX (γX − 1)λ−1α(χ)γ X (χ)(γ X (χ) − 1)2.0−1(B.5)

Step 2: Minimization. Among the 12 parameters of the model, five of them aregiven by Table 1, and four are given as functions of χ, in (B.1), (B.2), (B.3) and(B.5). This leaves us with three remaining parameters, {ζL , η, χ}. We calibrate themso as to minimize the squared sum of percent difference (error) between the target
moments in Table 2 and the corresponding model variables on the BGP.
36

Equation (B.2) assumes that parameter θ takes the baseline value of 1.0. When doing robustness
checks with θ = 0.8 and 1.2, the numbers in this equation are adjusted accordingly. The same appliesfor (B.3)-(B.6) when using alternative parameter values or calibration targets.

A-7

Let us define the squared sum of percent error as37

SSE =
+

K/(Y /χ) − 2.9
2.9

2



κL − 0.63
0.63

+
!2
1/χ
γL − 1 − 0.019
+
0.019

2



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

2(B.6)

In (B.6), endogenous variables K/Y , κL , κR , γL and γX represent their respective
BGP values, when the model is solved given all 12 parameters. Using exogenous
parameters and the results of analytical calibration, we determine the remaining
three parameters as the solution to the following minimization problem:{ζL , η, χ} = argmin SSE

s.t.

ζL ,η,χ{, λ, θ, gL , gX } : given by Table 1,(B.7)

δ = δ(χ), β = β(χ), α = α(χ), ζX = ζ X (χ).
We have done this minimization numerically utilizing ‘FindMinimum’ function of
Mathematica. The minimized value of SSE is virtually zero (precisely, of the orderof 10−22 ), implying that we obtained the set of parameters that fits all the moments
in Table 2.
Robustness Scenarios. To check the robustness of the result, we repeat the analytical calibration (B.1)-(B.5) with modified values for the exogenous parameters. In
all robustness scenarios, the modified version of minimization problem (B.7) yields
almost zero. This means that the model can match all the target moments in thosescenarios.

C

Data Sources

All data are originally from the Bureau of Economic Analysis (BEA) and retrieved
from FRED, Federal Reserve Bank of St. Louis, https://fred.stlouisfed.org. We
37Among seven moments in Table 2, we use five moments to define the squared sum of errors.
The other two moments, consumption of fixed capital in GDP and return on investments, always
match the data given that other moments are correct, since we impose relationships (B.1) and (B.2).
In numerical calibration, we confirmed that these two moments match the data exactly.

A-8reference series by their codes in FRED. We use annual values of Real GDP (GDPCA),
real investment (GPDICA), and real personal consumption expenditures (PCECCA).
The real capital stock is calculated as the net stock of fixed assets at current cost
(K1TTOTL1ES000) divided by the GDP price deflator (A191RD3A086NBEA). The
relative price of investment is obtained by the price deflators for gross private investment (A006RD3A086NBEA) divided by the price deflator for personal consumption
expenditures (DPCERD3A086NBEA). In addition to the variables listed above, thecalibration utilizes data on nominal consumption of fixed capital (GDICONSPA), labor compensation, and R&D expenditure (Y694RC1A027NBEA) all relative to nominal GDP (GDPA) in Table 2, as well as population (B230RC0A052NBEA) in Table
1. Labor compensation is calculated as compensation of employees (A033RC1A027NBEA)
plus proprietors’ income with inventory valuation and capital consumption adjustments (A041RC1A027NBEA).

A-9