ISSN (Print) 0473-453X

Discussion Paper No. 1212

ISSN (Online) 2435-0982

Non-Exponential Growth Theory

Ryo Horii

September 2023

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

Non-Exponential Growth Theory∗
Ryo Horii† (Osaka University)
September 2023

Abstract
The per capita real GDP growth rate has been remarkably stable for many decades
in most developed countries. To explain the balanced growth, however, existing endogenous growth theories typically need to assume a knife-edge degree of externality,
which is not yet confirmed by micro-level observations. We argue that this puzzle occurs because sustained growth has been commonly understood as exponential growtheither in the quantity, quality, or variety of outputs, which is hard to explain without
strong assumptions. By explicitly considering the movements of price and quantity of
individual goods after their introduction, this paper shows that the observed stabilityof the real GDP growth rate can be explained under much weaker conditions without
relying on the exponential growth of any variable. In particular, we develop an endogenous growth theory where a constant number (not exponentially many) of new goods
are introduced per unit of time. Even without externality, a constant GDP growthrate is maintained when the expenditure for older goods shrinks over time so as not to
inhibit the expenditure share given on newer goods.
Keywords: endogenous growth theory, knife-edge condition, externality, variety expansion. JEL Classification Codes: O41, O31∗

The author is grateful to Daron Acemoglu, Koske Aoki, Nick Bloom, Angus Chu, Guido Cozzi, Koichi
Futagami, Oded Galor, Peter Howitt, Chad Jones, Stelios Michalopoulos, David Weil, Hiroshi Yoshikawa,and seminar participants at Aix-Marseille School of Economics (Greqam), Brown University, GRIPS, Hitotsubashi University, University of Kent, Kobe University, Kyoto University, University of London, Universityof Macau, Nagoya City University, Osaka University, and Tokyo University for their valuable comments
and suggestions. This study is financially supported by the JSPS Grant-in-Aid for Scientific Research(20H01477, 20H05633, 20H05631), the Behavioral Economics Research Center at ISER, the Fulbright
Scholar Program (CIES), and the Abe Fellowship (SSRC) Needless to say, all remaining errors are my
own.
†

Correspondence: ISER, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan. E-mail:horii@econ.jpn.org

(a) Log per Capita Real GDP, USA(b) Per Capita Real GDP, Linear Scale

12

60,00011

50,00040,000y = 0.0158x + 7.867

30,0008

20,0007

10,000

6

-

1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
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1950
1960
1970
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1990
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9

1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010

10

Figure 1: Long-term Evolution of Real GDP Per Capita in the United States from 1830 to2018 (2011 International dollar). Source: Madison Project, Bolt and van Zanden (2020).

1

Introduction

Since around the time when the First Industrial Revolution was completed, the real GDP
per capita growth in the United States has been surprisingly stable. Figure 1(a) depicts its
time series on a log scale, where the slope of the series represents the growth rate. Although
there were short- to mid-term fluctuations, the figure clearly shows the log real GDP per
capita closely follows a linear trend, which implies the long-term rate of per capita GDP
growth is almost constant. Figure 1(b) shows the time path of the U.S. real GDP per
capita on a linear scale without taking logs. Given that the GDP growth rate is stable,
the level of the real GDP per capita is increasing exponentially in the long run.
Given these findings, it was natural for existing studies on endogenous growth to explain
the phenomenon of long-term growth by models in which the per capita output continues
to grow exponentially. Initially, this was not an easy task because it was commonly understood that reproducible inputs are subject to diminishing returns, which implies thatthe accumulation of those factors cannot explain the exponential growth by themselves.
The seminal studies in the endogenous growth theory thus overcame it by assuming strong
intertemporal knowledge spillovers.
Figure 2 graphically explains the three representative formulations in endogenous growth1(a) Variety Expansion Models: e.g. Romer (1990), Grossman and Helpman (1991a)
Number of existing varieties

New R&D= Accumulatoion of past R&D

N_ t

Nt

index of varieties(grows exponentially
if R&D labor is constant)ÁN_ t = aLRt ¢ Ntexternality (spillover) from past R&D, requires

Á=1(b) Quality Ladder models: e.g. Grossman-Helpman (1991b), Aghion and Howitt (1992)(grows exponentially if the frequency

quality of a productof improvements is constant)
When an R&D succeeds:

externality (spillover) frompast R&D, requires

improvement in quality by R&D¢qtÁ¢qt = (¸ ¡ 1) ¢ qt

qt

Á=1

quality of the existing product= due to accumulation of past R&D(c) AK-type growth models: e.g. Romer (1986), Rebelo (1991)
Existng stock of capital(of a representative ﬁrm)

Kt

= Avaerage stock of capital

investment

K_ t

Kt

amount of capital(grows exponentiallyKt
output:if labor and saving

Kt

rate are constants)savings from output°
Yt = ALt1¡®Kt® ¢ K t

unless®=1, externality from avarage stock of capital requires ° = 1 ¡ ®

Figure 2: The Exact Degree of Externality Required in Three Types of Endogenous Growth
Models2

theory and their specific assumptions. In variety-expansion models, as illustrated in panel(a), there should exist a strong knowledge externality from past R&Ds to a new R&D,
and the elasticity of this spillover ϕ need to equal exactly one. Similarly, in quality laddermodels (b), the increment in the quality by a successful R&D depends on the quality of theexisting good, which is a result of the past stock of R&Ds, and the elasticity of this relation
should again be exactly one. Finally, in AK-type growth models (c), the elasticity of production with respect to all reproducible factors and the elasticity of their externality effects
must add up exactly to one.1 In almost all existing endogenous growth models, long-term
growth can be sustained only when one of such knife-edge conditions is satisfied.2
Now, here is a puzzle. Surely, the externality and non-rivalry of knowledge play important roles in improving productivity (see, e.g., Griliches 1998). However, if we look at
the process of spillover more precisely, to my knowledge, there is no concrete evidence that
supports any of these exact assumptions. As for the elasticity of spillover ϕ in R&D-driven
growth models, Jones (1995) clearly stated, “ϕ = 1 represents a completely arbitrary degreeof increasing returns and, ... is inconsistent with a broad range of time-series data on R&D
and TFP growth.” He convincingly stated that ϕ = 0 is the most natural case, and while
ϕ can either be negative by the “fishing out effect” or positive by the “better tools effect,”
it is reasonable to assume ϕ < 1.3 Bloom et al. (2020) found that it is getting increasingly
1

When there are multiple sectors, a sector that produces a reproducible factor (typically physical or

human capital) needs to satisfy this restriction. For example, Lucas (1998) initially introduced a humancapital accumulation function ḣt = hζt G(1 − ut ) and then made an assumption of ζ = 1 following Uzawa(1965). By so doing, he wrote “the feature that recommends his formulation to us, is that it exhibits sustained per-capita income growth,” which gives a clear example of a case where such a knife-edge assumption
is justified not by micro-level observations but by the aggregate outcome. Lucas also noted that “humancapital accumulation is a social activity,” which suggests that the elasticity ζ = 1 includes the effect fromexternality.
2

Growiec (2007, 2009) formally proved that, with any generalization in the functional forms, exponential

growth cannot be explained without imposing at least one knife-edge assumption in the model.
3

By assuming ϕ < 1, Jones (1995) developed the semi-endogenous growth theory, where the long-term

rate of growth is ultimately driven by population growth. By extending the theory to include the transitional
increase in research intensity and educational attainments, Jones (2002) showed that it is possible to explain
the observed constant growth path for a certain period of time. In his theory, future growth is predicted toslow down permanently and eventually will come to an end, given that there are upper limits in population,
research intensity, and education attainments. Under a natural assumption of ϕ = 0, this paper tries topresent an alternative interpretation by developing a full-endogenous growth theory where the measured

3

harder to improve exponentially the quality of goods.4 Klenow and Rodriguez-Clare (2005,
Section 3) reviewed various AK-type models. They concluded that such models are empirically implausible based on the lack of a tight enough relationship between the investment
rates and growth rates in cross-country data.
Why have the U.S. and other successful countries been able to grow quite steadily, eventhough existing theories of endogenous growth imply sustained growth is possible only
under knife-edge conditions that seem very hard to be justified by data? This paper showsthat this inconsistency occurs because we have been interpreting the observed sustained
growth as an exponential process. If we consider a single final output (a scalar variable),sustained growth always means exponential growth. This is true even in most of theexisting R&D-based growth models with many varieties, as they measure the growth rate
after aggregating the varieties into a single final output.5
However, the real GDP growth rate in SNA statistics (e.g., the NIPA in the U.S.) ismeasured without aggregating all outputs into one variable. It is obtained by comparing
quantities of various product groups in adjacent years, using the same set of prices for both
years. Then, the aggregate level of real GDP is constructed by the chain rule. Conceptually,
the SNA statistics construct the level of real GDP at a given year t from the measured real
GDP growth rate gt as follows:
Z t

h
i h
i
Real GDP at year t = Real GDP at reference year t̄ × expgτ dτ ,t̄

where t̄ is the reference year, which can be chosen freely. Real GDP at reference year t̄
can also be set arbitrarily (because this is just an index), but for ease of interpretation, it
is usually set to the nominal GDP in year t̄. Therefore, even when we see the exponential
growth in the time series of real GDP, it only says that the gt is stationary over time.
economic growth can continue indefinitely with constant population.
4

They reported that the number of researchers required today to achieve the famous doubling of computer

chip density is more than 18 times larger than the number required in the early 1970s. This is not consistent
with the typical quality ladder specification as illustrated in 2(b).
5

A notable exception is Young (1991), who considered a learning-by-doing (LBD) model with many

goods and calculated the growth rate before aggregation. In addition, he recognized that there is an upperbound in productivity gain from LBD for individual good. To obtain a constant rate of growth, however,
he needed to assume that the upper bound itself improves exponentially through the aggregate LBD. Horii(2011) used the same disaggregated definition of growth rate, but it should be classified as a semi-endogenous
growth model because it relied on an exponential increase in population (e.g., Jones 1995).

4Focusing on this fact, this paper solves the puzzle by showing that steady state growth(constancy in the measured per capita real GDP growth rate) does not necessarily imply
an exponential increase either in the quantity, quality, or variety of products. Because
there is no need to explain the exponential increase in any variable, and we do not need tomake knife-edge assumptions to support steady-state growth.
This paper shows that the measured GDP growth rate becomes a positive constantwhen (i) a constant number of new goods are developed per unit of time, (ii) the qualityadjusted price of each new good falls after introduction, and (iii) the expenditure sharefor the very old goods is limited. Condition (i) says that it is suﬀicient for the numberof goods to increase linearly over time rather than exponentially, whereas conditions (ii)and (iii) state that the price and demand for each good should follow the well-observed
pattern of product lifecycle. In the long run, it is less counter-intuitive for us to expect thistype of economic movement to continue rather than to expect the output quantity, quality,
or variety to be expanded by astronomical orders. Of course, knowledge externalities arenevertheless important for growth as they often work behind the quality improvements andcost reductions of existing goods. Still, we show shown that the fall in quality-adjusted
prices needs not to occur at an exponential speed. A weaker externality is suﬀicient forsustaining growth.
The rest of the paper is constructed as follows. Section 2 presents a theory of sustained
growth without exponential expansion. By explicitly focusing on the product lifecycle, we
explain that the conventionally measured real GDP growth rate becomes asymptoticallyconstant even when no variable grows exponentially. Section 3 develops a prototype endogenous growth model without knife-edge assumptions and examines how the long-term
rate of economic growth is determined in equilibrium. Section 4 generalizes the theory
and the prototype model in several directions so as to demonstrate that we can obtain apositive constant real GDP growth rate in wider (even less restrictive) situations. Section
5 concludes.

2

TheoryThis section presents a theory that demonstrates the measured real GDP growth is sustained even when no underlying economic variable grows exponentially. More specifically,5

Subsection 2.1 discusses the evolution of the prices and outputs of individual goods in asetting where the number of varieties expands linearly over time rather than exponentially.
Subsection 2.2 explains the definition of the real GDP growth rate when the quantity andprice of goods change individually over their product life cycles. After deriving the real
GDP growth rate in the steady state in Subsection 2.3, Subsection 2.4 offers graphical examples that illustrate the connection between the pattern of the product lifecycle and themeasured real GDP growth rate. To retain the flexibility of the main result, this section
does not specify the full model structure. It will be shown in Section 3 that such dynamics
can be obtained as an equilibrium outcome in a general equilibrium model.

2.1

Steady-State Growth Dynamics with Product Life Cycle

Let us consider an economy with a constant population and many goods.6 We follow a
convention in the variety-expansion model by calling them goods, but it is more suitable
to think of each good in the theory as a group of products that are based on the same
technology. Each good is indexed by i ∈ [0, Nt ], where i = 0 is the oldest while i = Nt is themost recently introduced good. Suppose that the number of goods Nt increases throughR&D, and in the long run, it increases by a positive constant n per unit of time:Ṅt → n > 0

as t → ∞.

(1)

Recall that, as illustrated in Figure 2(a), existing variety-expansion models required astrong and exact degree of knowledge spillovers to maintain exponential expansion of varieties, where Ṅt /Nt is constant. In contrast, the linear increase of Nt in (1) does not
require such strong knowledge spillovers within the R&D sector, as we will see in a generalequilibrium model in Section 3.
However, because (1) implies Ṅt /Nt → 0, it is clear that the introduction of new goods
alone cannot explain the sustained growth. Therefore, we consider the changes in theprices and outputs of individual goods after their introduction explicitly. Let pet (i) andxet (i) denote the price and quantity of each good i at time t and allow them to change overtime. As in the SNA statistic, we define pet (i) and x
et (i) as quality-adjusted. For example,
if the quality of good i is doubled (so that consumers receive the same utility from half thequantity), then our measure of x
et (i) is doubled, while pet (i) is halved.
6

The assumption of constant population is relaxed in Subsection 4.2.

6

Let s(i) denote the time when good i is developed. It is convenient to label each goodby its age, τ ≡ t − s(i), i.e., time passed from its introduction. In the long run, wheren new goods are introduced per unit of time, age τ good refers to the nτ th newest good.
This means that the index of a good i and its age τ have the following relationship.
i = Nt − nτ, or equivalently, τ ≡ t − s(i) =Nt − i.
n(2)With this notation, let us say that the economy has come to a steady state if the price andquantity of every good follow the same time evolution against τ . Formally, the economy
can be said to be converging to a steady state if there exist time-invariant functions p(τ )and x(τ ) such thatpet (i) → p(t − s(i)) ≡ p(τ ),x
et (i) → x(t − s(i)) ≡ x(τ )ast → ∞.

(3)

Let T > 0 denote the age beyond which the product is never produced. In typicalvariety-expansion endogenous growth models, goods never retire from the market. In thiscase, T = ∞. However, in practice, we see many products disappear after some time. Our
theory can be applied to both cases where T is finite or infinite.
We normalize the price level at each instant so that the nominal expenditure in thesteady state is constant. We assume p(τ ) and x(τ ) satisfy the following conditions.
Assumption 1(i) Both p(τ ) and x(τ ) are non-negative and differentiable for all 0 < τ < T , where T issuch that x(τ ) = 0 for all τ ≥ T .
(ii) T can be infinite, but x(τ ) does not grow exponentially: limτ →∞ x′ (τ )/x(τ ) ≤ 0.7(iii) The newest good’s price and quantity are strictly positive: p(0) > 0 and x(0) > 0.
Assumption 1(i) will simplify the exposition in this section. In principle, the theory canaccommodate the cases where the price and output are discontinuous,8 but the present
paper focuses on the continuous setting because it is mathematically less demanding anddoes not sacrifice intuitions. Since x(τ ) represents the quality-adjusted quantity, Assumption 1(ii), combined with (1), guarantees that neither quantity, quality, or variety grows
7dNote that the time derivative of quantity in the steady state is xėt (i) = dtx(t−s(i)) = x′ (t−s(i)) = x′ (τ ).

Therefore, x′ (τ )/x(τ ) = xėt (i)/ext (i) represents the growth rate of the quantity of age τ good, or equivalently,that of index i = Nt − nτ good.
8

See the discussion in footnote 13.

7pet (i)price ofage good(stationary)

price ofgood i(changing)

N_ txet (i)
output ofgood i(changing)

N_ t

i

output ofage good(stationary)N_ tage of goods

N_ t

i

index of goods(oldest good)(newest good)

Figure 3: Evolution of Prices and Outputs of Goods in a Non-Exponential Steady-State.
exponentially in this economy. Assumption 1(iii) is an obvious one. When we say a new
good appears in the market, it should imply that the expenditure for the good, p(0)x(0),is positive.
Definition 1 A non-exponential asymptotic steady-state is the situation wherein the pathsof quality-adjusted prices and outputs of goods, pet (i) and x
et (i), satisfy condition (3) andAssumption 1 as T → ∞.
In the remainder of the paper, we simply call it a steady state unless it is confusing.
Figure 3 intuitively explains the evolution of the quality-adjusted prices and outputs
in the above definition of the steady state. The graphs can be viewed two ways: either
drawn against the i-axis (index of goods) running from left to right, or drawn against theτ -axis (age of goods) running in the opposite direction. The two variables, i and τ , arerelated according to (2), but the relationship changes over time as Nt increases. At time t,
the origin of the τ -axis coincides with the point of i = Nt on the i-axis because the newest
good i = Nt is age τ = 0 at time t. As time passes, the origin of the τ -axis moves to theright with the speed of the introduction of new goods, Ṅt = n, and so does the position of8the origin of the graph drawn against τ .
The upper panel of figure 3 illustrates a case where quality-adjusted price p(τ ) is decreasing in age τ , either because a product becomes cheaper or has higher quality as itages after its introduction. Since the newer goods have larger index i, it also means pet (i)is increasing in i at any given time t. The figure also explains the movement of the price ofeach good pet (i) over time. Even in the steady state where function p(τ ) is stationary, theprice of individual good pet (i) shifts downward to the dotted curve because the position offunction p(τ ) continues to move to the right as new goods are developed.9
Similarly, the lower panel of Figure 3 explains the evolution of quality-adjusted outputsof goods over time. The panel shows the case where x(τ ) is increasing in τ , which naturally
matches our example that the older goods have a lower quality-adjusted price. In this case,the demand for each good x
et (i) increases over time as the x(τ ) function shifts to the right.
Note that, however, Assumption 1(ii) rules out exponential growth in the quantity of any
good. Even when T = ∞, the growth rate of x(τ ) must be either zero or negative asτ → ∞. Contrary to this particular example, we can also consider the possibility for thequantity to shrink with age, even when older goods are cheaper. Such a pattern will emerge
when consumers do not like outdated goods or if newer goods replace parts of functions
that are provided by older goods.

2.2

Measuring the GDP Growth Rate

When considering the growth rate in a model with many goods, an often employed practiceis to consider an aggregation function to obtain a measure of the final output (a scalarvariable) and then measure its growth rate. A problem with this methodology is that theresulting growth rate depends on the choice of the aggregation function. Because the final
output is a virtual notion, there is no guarantee that the growth rate obtained in thisway matches the numbers in the oﬀicial statistics. Another way is to select a numeraire
good, aggregate the outputs of various goods with the observed relative prices, and thencalculate the growth rate of aggregated output. However, there is again a problem of
9

Although this is a convenient way to explain the steady-state dynamics, note that the economic envi-

ronment, such as technology, preference, and market structure, first determines the evolution of the priceof individual goods pet (i) in equilibrium. Then, the long-term pattern of movement in pet (i) shapes thestationary p(τ ) function as a result.

9

choosing the numeraire because the obtained growth rate depends on this choice since therelative price changes over time. Another substantial problem is that there is no good thatis representative enough to be the numeraire for the long time because the expenditure
shares across goods are always changing.
To avoid these problems, this paper directly looks at the changes in the prices and quantities of all goods without aggregating them into a single variable. We follow the standard
procedure to construct the GDP growth rate that is often explained in undergraduate-level
macroeconomics textbooks. In the SNA statistics, the real GDP growth rate is measured
by comparing the value of outputs between two consecutive years, say year t − 1 and year
t. Their values are measured using the common set of prices, which usually is the set ofobserved prices in a given base year. Because the base year is frequently updated in oﬀicial
statistics and also because this paper is interested in long-term dynamics, we suppose that
there is no gap between the base year and the year in which the growth rate is computed.10
Then, using the notation from the previous subsection, the real GDP growth rate between
years t − 1 and t can be written as follows.11
R Nt
gt−1,t =RNet (i)ext (i)di + 0 t−1 pet (i) (ext (i) − x
et−1 (i)) di
Nt−1 p.
R Nt
et (i)ext−1 (i)di
0 p

(4)

This formula is made up from the integrals of two functions, pet (i)ext (i) and pet (i)ext−1 (i).
Figure 4 depicts the curves of these two functions against t, for the case where the demandfor existing goods is always increasing over time (Case 1) and the case where demand forexisting goods shrinks in some part of their life cycle (Case 2). In Case 1, observe that area(a) in the figure represents the sum of the values of new goods introduced between timet − 1 and time t, evaluated by the prices at time t. Similarly, area (b) represents the value
10

In the U.S., the NIPA computes the growth rate two ways, by setting the base year to t, and also by

setting it to t − 1. Then the agency calculates their geometric average. Here, we only calculate the growthrate in which the base year is t, but the difference disappears in the limit where the period length is brought
to 0, as we do in the next subsection.
11

To ease the understanding, here we employed a slight abuse of notation and treated x
et (i) as if it is adiscrete-time variable. In the previous subsection, we defined x
et (i) as the instantaneous flow of output attime t per unit time. Since the SNA statistics use the cumulative output of good i for a given time period(e.g., a year or a quarter), we need to integrate x
et (i) for the duration of the time period to obtain the exact
real GDP growth rate. As we take the limit where the duration of one period is almost zero, we confirmedthat this exact GDP growth rate converges to the same expression as in (5).

10Case 1: When x
et (i) is always increasing in t.

pt (i) xt (i)value of production
for each good i
at time t and t-1,
measured withprice at time t

growh rate=(a+b)/c(b)(c)

pt (i) xt-1(i)

(a)

Nt-1

0

p(0) x(0)Ntiindex of varieties

Case 2: When x
et (i) becomes decreasing in t sometime after introduction.
value of production
for each good i
at time t and t-1,
measured withprice at time tpt (i) xt (i)(d)

growth rate=(a+b-d)/(c+d)

(b)
(c)

pt (i) xt-1(i)
Nt-1

0

(a)

p(0) x(0)Ntiindex of varieties

Figure 4: Calculation of the Real GDP Growth Rate: Two Casesof the increased production of goods that already existed at time t − 1. These two areas
measure how economic activity has increased from time t − 1 to time t and correspond to
the two terms in the numerator of definition (4). Area (c) represents the value of total
production in time t − 1, evaluated again by the prices at time t. This area corresponds
to the denominator of the definition (4). In this way, the real GDP growth rate can beunderstood as the ratio of area (a)+(b) to area (c), which measures the rate at which theeconomic activity at time t is increased from time t − 1.
This procedure can be generalized to the case where output x
et (i) is not monotonic
in t. Case 2 in Figure 3 illustrates an example where production of a certain range ofgoods declines between period t − 1 and t. Then, a portion of curve pet (i)ext (i) comes below
curve pet (i)ext−1 (i). In this case, the real GDP growth rate is given by the ratio of area(a)+(b)−(d) to area (c)+(d).

11

2.3

Measured GDP growth rate in the Steady StateSo far, Subsection 2.1 presented steady state dynamics with many goods where no variable
grows exponentially, and Subsection 2.2 explained how GDP growth rate can be measured
when there are many goods. Now we are ready to examine whether the non-exponential
steady state implies a positive and constant real GDP growth rate.
Note that the conventional definition of real GDP growth in (4) gives the average growthrate between two discrete time periods. To map this definition to a continuous-time growthmodel, it is convenient to consider the instantaneous growth rate gt at time t. This can beobtained by replacing t − 1 in (4) by t − ∆, and taking the limit of ∆ → 0 in gt−∆,t /∆.
RN
Ṅt · pet (Nt )ext (Nt ) + 0 t pet (i)xėt (i)di
gt−∆,t=.
gt = lim
R Nt∆→0∆pet (i)e
xt (i)di(5)0

Suppose that the economy converges to a steady state, as defined in Definition 1. The
number of goods grows linearly, and the evolution of prices and quantity in terms of age
RTbecomes stationary. Then, given that 0 p(τ )x(τ )dτ is finite, the long-term growth rate
can be obtained by substituting (1)-(3) into (5).12
RT
np(0)x(0) + n 0 p(τ )x′ (τ )dτ
gt → g ≡RTn 0 p(τ )x(τ )dτast → ∞.

(6)

The interpretation of growth rate (6) is essentially the same as in definition (4), except
the fact that now growth is represented in terms of age and also in continuous time.
In the numerator, np(0)x(0) represents the value of newly introduced goods, whereasRTn 0 p(τ )x′ (τ )dτ gives the value of changes in production given price function p(τ ). Bothterms are multiplied by n because there are n goods per unit of age. The sum of these terms
RTgives the speed of increase in economic activity. The denominator of (6), n 0 p(τ )x(τ )dτ ,gives the value of existing production, i.e., the nominal GDP of the economy given pricesp(τ ). The ratio of the two gives the real GDP growth rate.
12Equation (6) can be obtained from (5) by changing the variable of integration from i to τ ≡ t − s(i).

First, substitute p(τ ) and x(τ ) for pet (i) and x
et (i). Similarly, xėt (i) can be written as x′ (τ ) (See footnote
7). Next, we need to change the integration variable from di in (5) to dτ . By differentiating (2) for givent, we obtain di = ndτ . Finally, from (2), i = 0 and i = Nt respectively correspond to τ = Nt /n and τ = 0
RN
xt (i)di =as illustrated in Figure 3. As t → ∞, Nt /n also approaches ∞. From these, limt→∞ 0 t pet (i)eR0R∞
limt→∞ Nt /n p(τ )x(τ )(−n)dτ = n 0 p(τ )x(τ )dτ . However, since x(τ ) = 0 for τ ≥ T , the latter expression
RTRT
becomes n 0 p(τ )x(τ )dτ . Likewise, the limit of the numerator of (5) becomes np(0)x(0)+n 0 p(τ )x′ (τ )dτ .

12

In the steady state, the growth rate can also be interpreted in terms of the productlifecycle. By canceling out n and using integration by parts in (6), we obtain the following
proposition.
Proposition 1 Suppose that the economy converges to the asymptotic non-exponential
steady-state, as defined by Definition 1. Given thatZ T−
x(τ )dp(τ ) is finite and strictly positive, and0Z Tp(τ )x(τ )dτ is finite,

(7)
(8)

0

the real GDP growth rate asymptotes to a positive and finite constantRT− 0 x(τ )dp(τ )g = RT.
(9)0 p(τ )x(τ )dτRT
The expression in (7), − 0 x(τ )dp(τ ), appears in the numerator of (9).13 It represents
the cumulative reductions in the quality-adjusted price during the whole product lifecycle.
When the quality-adjusted price of goods declines, consumers have more purchasing power,
improving their utility. This income effect from price reductions is more significant whenthe quantity of the good is larger. Therefore, in (7), the price reduction −dp(τ ) is weighted
by quantity x(τ ) and then integrated. This gives the total income effect that one productRTgenerates over its product lifecycle. The expression 0 p(τ )x(τ )dτ in (8) is in the denominator of (9). It is the cumulative expenditure attracted by one product. Proposition 1says that if every product follows the same lifecycle pattern, the real GDP growth ratein the economy is given by the ratio of the two. If both values are positive and finite ina non-exponential steady state, as defined in Definition 1, then it indicates that the real
GDP growth rate can be sustained even when no variable grows exponentially. We will
first provide three examples in the following subsection and then discuss conditions (7) and(8) more in detail in Subsection 2.5.

2.4

Graphical Examples

Proposition 1 shows that the real GDP growth rate depends only on functions p(τ ) andx(τ ). We can represent the real GDP growth rate graphically using the shapes of these13

When p(τ ) is differentiable,RT0x(τ )dp(τ ) is the same as

RT0x(τ )p′ (τ )dτ . A benefit of this notation is it

allows the case where p(τ ) is not differentiable. While we do not provide the formal proof here, Proposition
1 can be extended to this case.

13

two functions. Figure 5 presents three examples.
Example 1 shows the simplest case, where the quality-adjusted price weakly falls withage throughout the product life cycle. The left panel depicts the evolution of {x(τ ), p(τ )}in the p-x diagram. T is finite in this example. The good enters the market at the pointof {x(0), p(0)} and continues to be produced until its age reaches T = τ3 . Then, thevalue of numerator (7) can be expressed by the area that is encompassed by the locus of{p(τ ), x(τ )} and the vertical axis in the p-x diagram (shown in the blue color). This can beinterpreted as follows. Whenever the quality-adjusted price falls by dp(τ ), either throughcost reductions or through quality improvements, consumers can save the purchasing powerby the amount of −x(τ )dp(τ ). The blue area shows the cumulative benefits throughout
the life of this good. The area is positive and finite as long as p(0) < p(T ).14
The right panel plots the evolution of expenditure for a (representative) good against
its age. The area below the curve gives the value of denominator (8) (shown in the yellow
color). From Assumption 1, the expenditure for the good at the time of introduction(p(0)x(0)) is strictly positive, and evolves in the non-negative region during its lifetime.
Since expenditure p(τ )x(τ ) falls to zero at finite T = τ3 , this area is positive and finite.
Proposition 1 says that the ratio of the blue area to the yellow area gives the real GDP
growth rate. Therefore, we can conclude that the real GDP growth rate in this example ispositive and finite.
Next, Example 2 considers a case where p(τ ) is not monotonic. Here, the qualityadjusted price begins to increase after τ3 years. When the price of the good (relative to thenewest good) rises in some stage of its life cycle, the area between this part of the p-x locus(from τ = τ3 to τ6 ) and the vertical axis represents the loss of the purchasing power ofconsumers. This area needs to be deducted from the benefits of the fall in quality-adjusted
prices from τ = 0 to τ3 . Therefore, the value of numerator (7) is given by area (i) minus
area (iii) because area (ii) is canceled out. It can be either positive or negative but isalways finite since T = τ6 is finite. Again the yellow area in the right panel gives the valueof denominator (8), which is positive and finite. Therefore, the real GDP growth rate is
finite, which is given by the ratio of the blue minus red area to the yellow area. Also, notethat the growth rate becomes zero only by coincidence, only when the blue and red areas
14p(0) < p(T ) requires the price to fall strictly with age in some part of the good’s life.

14

Example 1: When T is finite and p(τ ) is weakly decreasing

Example 2: When T is finite and p(τ ) is non-monotonic

Example 3: When T = ∞ and p(τ ) is decreasing

Figure 5: Graphical Representation of the Real GDP Growth Rate. The growth rate ismeasured by the ratio of the areas of the two panels.

15

are exactly the same size.15
Finally, Example 3 shows a case when the good stays in the market forever (T = ∞).
The price p(τ ) (relative to the newest good) falls throughout the life cycle, and the quantityx(τ ) remains positive as τ → ∞. For the yellow area to be finite, the expenditure for very
old goods has to shrink. More concretely, condition (8) is satisfied if expenditure for old
goods is bounded by a polynomial function of age with the power of less than −1:16
p(τ )x(τ ) ≤ [constant] · τ −ξ for all τ ≥ τ ,(10)for some ξ > 1 and τ > 0. The dotted curve in the right panel gives an example of such an
upper bound. While we need a concrete model to see whether condition (10) is satisfied,
let us note that the condition does not require an exponential fall in expenditure. The RHSof (10) falls with age at the rate of ξ/τ for τ > τ , which can be arbitrarily close to zero
when we choose a large τ . Therefore, there is no minimum rate at which the expenditureneeds to fall.
Given that the yellow area is finite, a suﬀicient condition for the finiteness of the blue
area is that the rate of change in the quality-adjusted price with respect to age is bounded,at least for older goods:17

p′ (τ )
< γ p for all τ ≥ τe,
p(τ )

(11)

for some γ p > 0 and τe > 0. These two inequality conditions, (10) and (11), jointly
guarantee that the GDP growth rate is positive and finite.

2.5

Discussion: Two Conditions for “Sustained Growth”

The previous three examples illustrated that the measured real GDP growth rate in thesteady state becomes positive and finite in various scenarios. Here, we discuss the two
15

Although Proposition 1 focuses on the case when the value of numerator (7) is positive, formula (9)holds even when (7) is negative.
16

R∞
τ

Suppose (10) is satisfied.

Then, the denominator of (9) isR∞0p(τ )x(τ )dτ ≤Rτ0p(τ )x(τ )dτ +[constant] · τ ξ dτ . The first term is finite, and the second term becomes [constant] · τ 1−ξ /(ξ − 1), whichis also finite.
R∞
Suppose that (11) is satisfied and 0 p(τ )x(τ )dτ (the yellow area) is finite. Then, the numerator of (9) is
R∞
R τe
R∞
− 0 x(τ )dp(τ ) = − 0 x(τ )dp(τ )− τe x(τ )dp(τ ). The first term is finite. The absolute value of the second
R∞
R∞
R∞
R∞
term satisfies τe x(τ )dp(τ ) = τe x(τ )p(τ ) |p′ (τ )/p(τ )| dτ ≤ γ p τe p(τ )x(τ )dτ . Here, τe p(τ )x(τ )dτ isR∞finite since we assumed 0 p(τ )x(τ )dτ is finite.
17

16

required conditions, (7) and (8), in Proposition 1 more generally.
Condition (7): quality-adjusted price falls during its product lifecycle
For this condition to be satisfied, p(τ ) needs to fall with τ at least for a portion of its productlifecycle. Recall that we normalized the price level so that the nominal expenditure in thesteady state is constant. This normalization also implies the price of the newest good isalways the same. Therefore, condition (7) just requires the quality-adjusted prices of older
goods to become lower relative to newer goods, and does not imply prices of individual
goods measured in a currency to fall.18
With this definition, the quality-adjusted price of a good may fall with age for a number of reasons. For example, the cost of production falls through learning-by-doing andknowledge spillovers. In this case, time and production experiences will contribute to pricereduction. Apart from cost reduction, changes in the form of competition may also lower
prices because older goods are typically less protected from competition by patents andtrade secrets than newer goods.
Price reductions also occur in the form of quality improvements. For example, theeffective price of computers has been declining for decades, not only because the priceof an average computer has become cheaper but also because the average performance ofcomputers has drastically improved. The SNA statistics interpret such changes as a decline
in the quality-adjusted price.
It is worth noting that our theory does not require an exponential fall in the qualityadjusted price. If the quality improvements are exponential, economic growth can easily
be maintained as in usual quality-ladder models (See panel (b) of Figure 2). The qualityof computers has been improving at a constant rate according to “Moore’s Law,” but thistrend of exponential improvement is expected to slow down. In fact, computers are aremarkable exception in terms of continued improvements in performance. Most otherproducts experience tapering in the rate of productivity improvements as they get mature.
Our theory shows that slowdowns in productivity increases in individual goods are fully
consistent with the positive rate of GDP growth, as long as a constant number of new
18

Suppose that the nominal GDP growth rate in dollars (not in our price normalization) is g $ . Then, therate of price change of age τ good in dollars is p′ (τ )/p(τ ) + g $ . If g $ is large, prices of individual goods willrise even when p(τ ) is a decreasing function of τ .

17

products are introduced per unit time.
Lastly, let us discuss the case when the quality-adjusted price of the good rises for somepart of its lifecycle, as we discussed in Example 2 of Figure 5. Although we need a concretemodel to analyze whether condition (7) is satisfied, here we present two possibilities. One
possibility is when products have antique or scarce value as they become very old. In this
scenario, p(τ ) will increase only when x(τ ) has become considerably smaller than when
they were newer. Another possibility is when it costs more to produce a good in small lots.
It happens, for example, when a particular good continues to be produced to meet a niche
demand, typically near the end of the product life cycle.
The value of numerator (7) is the weighted sum of the price changes, dp(τ ), where theweights are the quantities, x(τ ). Therefore, if the quantity x(τ ) tends to be small whenp(τ ) increases, the negative effect of such movements on the GDP growth rate is likely tobe limited. Therefore, even when the price at the end of the lifecycle p(T ) is higher than
the initial price p(0), the lifetime contribution of this representative good on the real GDP
growth rate may well be positive, as in the case of Example 2.
Condition (8): The cumulative expenditure for the representative good is finite
This condition requires the expenditure for older goods p(τ )x(τ ) to fall so that they effectively retire the market in terms of expenditure share. It is always satisfied if therepresentative good ceases to be produced at finite age T . Even when the good stays inthe market forever, the condition is satisfied if the expenditure falls with age reasonably
fast (condition 10). Notably, the speed does not need to be exponential.
The expenditure for the good can shrink with age for several reasons. One possibility
is that the price falls and the price elasticity of demand is less than one, at least for
older goods. To illustrate this possibility simply, suppose that the demand for a good isdetermined solely by its own price p(τ ), and that its price falls toward zero. Even when thegood becomes almost free, it is not plausible for a consumer to demand an infinite amountof any particular product in reality. This consideration suggests that the price demand
elasticity of a product will become small when the price has become suﬀiciently low, andthe expenditure for the good will eventually vanish as p(τ ) → 0.
More concretely, suppose that the price elasticity of demand becomes less than ε < 118

for old goods with τ ≥ τ , for some τ > 0. Then, a suﬀicient condition for (8) is19p(τ ) ≤ p(τ ) τ − ξ

1−ετ≡ p(τ ) for all τ ≥ τ ,(12)for some ξ > 1. Condition (12) implies that the positive GDP growth rate can be maintained when the price of any individual goods continues to fall at least as fast as p(τ ).
The upper bound function p(τ ) starts from p(τ ) = p(τ ) at τ = τ and falls proportional toτ −ξ/1−ε . Since this is a power function, the required rate of fall can be arbitrarily small
if we choose a large τ . Therefore, the quality-adjusted price does not need to fall exponentially to maintain positive GDP growth. If the price change is driven by quality or
productivity improvements, the improvements do not need to be exponential. In Section
3, we will present a full endogenous growth model based on this idea.
The expenditure for older goods can also fall for other reasons. The older goods may
become obsolete, and consumers no longer demand them. Sometimes, consumers are attracted by the novelty of new goods but become less interested as time passes. Changes
in the underlying economic environments may make older goods of no use. We will extend
the theory to include those possibilities in Subsection 4.1.

2.6

Implications for Endogenous Growth Theory

We have established that a constant GDP growth rate can be maintained without exponential growth either in quantity, quality, or variety. Here, we discuss how this result allows
for building a robust endogenous growth model without requiring knife-edge conditions.
In standard variety-expansion models, all goods are symmetric and receive the same
share of expenditure. Therefore, as the number of goods increases, the share of the expenditure given to a single new good dilutes. Accordingly, the contribution of each new
good to the economic growth rate also shrinks toward 0. This is why these models need toconsider exponentially accelerating R&D so as to offset the dilution effects.
This requirement restricts the way an endogenous growth model could be built. If theexpenditure share of a single new good tends to zero, profits obtained from a single suc19

Suppose that x(τ ) depends only on p(τ ), its price elasticity is less than ε for τ ≥ τ , and condition(12) is satisfied. Then, x(τ ) ≤ x(τ ) (p(τ )/p(τ ))−ε since p(τ )/p(τ ) ≤ 1 and the price elasticity is lessthan ε. Therefore, expenditure at τ satisfies p(τ )x(τ ) ≤ p(τ )x(τ ) (p(τ )/p(τ ))−ε = x(τ )p(τ )ε p(τ )1−ε ≤x(τ )p(τ )ε p(τ )1−ε = p(τ )x(τ )τ ξ τ −ξ for τ > τ . Since p(τ )x(τ )τ ξ is a positive constant, the latter inequality
implies (10). This completes the proof since (10) is a suﬀicient condition for (8).

19

cessful R&D also fall. Therefore, to give firms enough incentives to do R&D in equilibrium,
those models require a strong degree of externality in the R&D process so that the cost of
inventing new goods declines exponentially (See panel (a) in Figure 2).
By contrast, if the economy satisfies condition (8), the incentive to innovate can be
maintained without such strong externalities. Under this condition, the expenditure share R∞
for one new good, p(0)x(0) n 0 p(τ )x(τ )dτ , remains a positive constant even when more
and more goods are introduced. In such a case, a constant flow of new goods, as well as
improvements in production costs after the introduction, always constitutes a significant
addition of economic activity relative to all existing activities. This enables the measured
GDP growth rate to remain positive in the long run without accelerating R&D. In addition,
since the expenditure share for one new good does not fall, firms obtain enough profits
from R&D even when the cost of R&D does not fall as a result of externalities. In thenext section, we build a general equilibrium endogenous model without R&D externality
in creating new products.

3

A Prototype Non-Exponential Growth ModelIn this section, we present a general equilibrium model that yields non-exponential steadystate dynamics. While the theory in the previous section suggests there are many ways
to build a model that achieves non-exponential growth while capturing various aspects ofreality, here we limit ourselves to presenting the simplest prototype model so as to convey
the substance of the non-exponential growth theory as clearly as possible. Thereafter wewill discuss the generalization of the prototype model to several directions.

3.1

Consumers and Market Demand

Consider an economy with infinitely-lived representative consumers of constant population
L. At each point in time, each consumer supplies one unit of labor and obtains one unitof wage. The wage level is normalized to one.20
The lifetime utility function of the representative consumer is given byZ ∞  Z Ntu(ect (i))di e−ρt dt,
0
20

(13)

0

In the steady state where the fraction of consumption out of labor income is constant, this normalization

implies that the nominal expenditure is constant, consistent with the theory in the previous section.

20

which is separable both across time and goods. The sub-utility for individual goods takes
the same form as the standard CRRA utility function.21u(ect (i)) =e
ct (i)1−1/ε − 1,
1 − 1/ε0 < ε < 1.

(14)

Similarly to Ngai and Pissarides (2007) and Acemoglu and Guerrieri (2008), we assume
that the elasticity of substitution across goods, ε, is less than one.22 We make this assumption to better capture the behavior of demand when the price becomes low while keeping
the tractability of constant elasticity.23
The dynamic budget constraint of a representative consumer is given by
Z Nt
k̇t = rt kt + 1 −pet (i)ect (i)di,(15)0where kt is the asset holding of the consumer, and rt is the interest rate under our pricenormalization. In equilibrium, Lkt should equal the value of all firms in the economy.
Consumers maximize their lifetime utility (13) subject to (15), given interest rate rt , pricesof goods pet (i) for i ∈ [0, Nt ], the initial asset holding k0 , and the standard non-Ponzi-gamecondition.
From these, we obtain an isoelastic demand function for individual goods by a representative consumer.
ect (i) = λ−εet (i)−ε .
t p

(16)

The shadow price of the budget constraint λt evolves according to Euler equation λ̇t = (ρ−rt )λt , and its initial value is determined so that the transversality condition limt→∞ e−ρt λt kt =0 is satisfied given the evolution of kt in (15).
21Note that the sub-utility function is symmetric across goods, which means that we do not model the

obsolescence of older goods in this simplest prototype model. Subsection 4.1 introduces obsolescence andshows that a positive growth rate can be maintained under milder conditions (even when ε > 1).
22

They developed multi-sector growth models where the elasticity of substitution between the goodsof different sectors is less than one. When the productivity of a sector (e.g., agriculture) increases, theexpenditure for this sector shrinks because the demand quantity does not rise as much as the price falls.
While they examined a small and finite number of sectors, this paper considers a continuum of goods(or more precisely, product groups) and examines a steady-state shift of expenditure shares across theseproduct groups.
23

If ε > 1, the expenditure for a good will rise indefinitely as its price falls to zero. This would not be agood description of the behavior of consumers when our focus is on how goods effectively retire from themarket at the end of their product lifecycle.

21

3.2

R&D and Production Technologies

Each consumer works either as a production worker or as a researcher. A researcher
succeeds in developing a new good with a Poisson probability of a per unit of time. Let LRtdenote the number of researchers in the economy, which is to be determined in equilibrium.
Over time, the number of goods increases according toṄt = aLR
t .

(17)

Equation (17) is similar to standard variety expansion models (recall panel (a) in Figure2), except that there is no spillover term from the stock of past R&D.
Once developed, each individual good is produced with a linear production technologythat requires only labor.
xet (i) = qet (i)elt (i),(18)

where xet (i) is the output of good i, elt (i) is the labor input, and qet (i) is the marginal productof labor in producing good i. Alternatively, we can interpret x
et (i) as the quality-adjusted
output and qet (i) as the quality of good i. In this case, one unit of labor produces one unitof good i with quality qet (i). In either interpretation, we call qet (i) the productivity for goodi.
Suppose that, when any good is first developed, the productivity is 1. Then, as the
production of this good proceeds, the productivity increases according to24
qėt (i) = I(ext (i)) · β qet (i)ψ ,0 < ψ < 1.

(19)

I(e
xt (i)) is an indicator function that takes 1 when x
et (i) > 0 and 0 otherwise. It means that
productivity increases as long as production takes place. Observe that there is a similarity
between the specification in (19) and those in the quality ladder models (see panel b in
Figure 2). We assume that there are knowledge spillovers from the past productivityof the technology to today’s increments in productivity. Parameter ψ ∈ (0, 1) specifies
the degree of such spillovers. While quality ladder models need to assume ψ = 1 to24

For simplicity, here we assume that experience in terms of time matters for productivity improvement.

Alternatively, we can consider the experience in terms of cumulative production amount. Horii(2012)
analyzed a model in the latter setting and derived the GDP growth rate defined similarly to (4), although
it was a semi-endogenous growth model that required an exponentially growing population (c.f. Jones
1995).

22

achieve the exponential increase in productivity (or quality), we do not make this knifeedge assumption. For the moment, we consider the case of ψ ∈ (0, 1), and later compare
the result to the case of ψ = 1. Parameter β > 0 represents other possible factors that
affect the speed of productivity increases.
As long as x
et (i) > 0, equation (19) is an autonomous differential equation in qet (i).
Similarly to Section 2, let τ ≡ t − s(i) denote the age of the good. Then the solution to
the differential equation (19) can be written asq(τ ) = κ1 (τ + κ0 )θ ,(20)

where θ ≡ 1/(1 − ψ) > 1, κ0 ≡ θ/β > 0, and κ1 ≡ (β/θ)θ > 0. Given that ψ ∈ (0, 1), theproductivity increases less than exponentially. The rate of productivity increase is givenbygq (τ ) =θβq ′ (τ )
=
=
.
q(τ )
τ + κ0(1 − ψ)βτ + 1(21)

In this specification, gq (τ ) takes the highest value at the time of introduction, gq (0) = β,and falls to 0 as a good gets older (gq (∞) = 0). This rules out the trivial possibility that theexponential increase in the productivity of individual goods explains the sustained growth.

3.3

Behavior of Firms

Let us now turn to the behavior of production firms. While any product is protected by apatent forever, we suppose that the patent breadth is limited (e.g., O’Donoghue, Scotchmer
and Thisse 1998). This means that other producers are prohibited from using exactly thesame technology as the original inventor, but they are allowed to produce similar products
if they use a technology that is suﬀiciently different from the original. Alternatively, we
may also think that a part of technology is kept secret by the inventor, and the outsiders
need to rely on less eﬀicient technologies. In either case, outsiders face lower productivity
than the original firm.
To formalize this idea, let us assume that there are potentially many outside firms,
and they have partial access to the technology of the original inventor qet (i) to produce thesame good i, but their productivity is 1/(1 + µ) times lower, where parameter µ represents
the patent breadth or the strength of trade secret. Recall that the price elasticity of thedemand function (16) is less than unity. In this case, the profit-maximizing strategy is
to set the limit pricing, which is (1 + µ) times higher than the marginal cost. Given the23

production function (18) and the fact that wage is normalized to one, the pricing by a firmthat has τ years of experience isp(τ ) =

1+µ
.
q(τ )

(22)

From (16) and (22), we obtain the output and the maximized profit of this firm.25
x(τ ) = Lc(τ ) = Dt q(τ )ε ,(23)π(τ ) = L(p(τ ) − 1/q(τ ))c(τ ) = µDt q(τ )ε−1 ,(24)where Dt ≡ L(1 + µ)−ε λ−εt is a shifter of demand for all goods.

3.4 Steady-State Equilibrium
Now we derive the long-term growth property of the equilibrium dynamics in this prototype
model. The following defines a notion of long-term equilibrium suitable to our model.
Definition 2 An equilibrium path that satisfies the following properties as t → ∞ is called
Asymptotically Steady-State Equilibrium (ASSE).
1. The speed of the introduction of new goods converges to a positive constant: Ṅt →n∗ > 0.
2. The shifter of the demand for all goods, Dt = L(1 + µ)−ε λ−εt , converges to a positive
constant: Dt → D∗ > 0.
Later, Subsection 3.6 confirms that the real aggregate variables observed in the ASSE obey
the conventional notion of the balanced growth path.
In the following, we consider the state of the economy as t → ∞ in the ASSE, andmention it simply “in the ASSE.” The equilibrium values of n∗ and D∗ are determined by
the free entry condition for R&D and the labor market clearing condition. Let us first
focus on the R&D condition. Recall that the Euler equation is λ̇t /λt = ρ − rt . Since
λt = (1 + µ)(L/Dt )1/ε is stationary in the ASSE, the interest rate necessarily converges
to rt → ρ. Using interest rate rt = ρ and the profit function (24), we can calculate the25

Strictly speaking, we need to put subscript t for x(τ ) and π(τ ) as they depend on Dt . In the steadystate, Dt becomes constant, and then x(τ ) and π(τ ) do not depend on t.

24

present value of a new firm just after it has succeeded in developing a new good. The valueof innovation in the ASSE becomes
Z ∞
Z ∞
∗
−ρτ
∗
V =
π(τ )e dτ = µDq(τ )ε−1 e−ρτ dτ.
0

(25)

0

From the R&D function (17), the expected cost of developing a new good is 1/a.
Therefore, given that there is a positive flow of R&D, n > 0, and that the financial market
is complete, the value of the new firm (25) should be equalized to the expected cost ofdevelopment: V ∗ = 1/a. This condition gives the equilibrium value of D∗ in the ASSE.
−1
Z ∞
1
ε−1 −ρτ∗q(τ ) e dτ
.
(26)
D =
aµ
0
Substituting (20) into (26), we can obtain an explicit expression for D∗ .26
Next, let us turn to the labor market. First, equation (17) implies that the number of
research workers in the ASSE is LR∗ = n∗ /a. Second, from (18) and (23), the aggregate
demand for production workers in the ASSE is27
Z Ntelt (i)diLP ∗ = lim
t→∞ 0
Z ∞
Z ∞
→ n∗
x(τ )/q(τ )dτ = n∗ D∗q(τ )ε−1 dτ.
0

(28)

0

The labor supply is given by population L. Therefore, the labor market clearing condition
is
L=L

R∗

+L

P∗

n
= + nD∗aZ ∞q(τ )ε−1 dτ.

(29)

0

From this, we obtain the equilibrium research intensity in the ASSEn∗ =1 + aD∗aLR∞.
ε−1 dτ
0 q(τ )

(30)

From (30), LR∗ = n∗ /a and LP ∗ = L − LR∗ are also obtained. The explicit solution for n∗
can be obtained as follows. Using (26) and then (20), the ratio of two types of labor isR∞ P ∗
ε−1 dτL
0 q(τ )
R
=
(31)
∞
LR
µ 0 q(τ )ε−1 e−ρτ dτ
26

Let Γ(·, ·) denote upper incomplete Gamma function, which is defined as Γ(s, z) ≡R ∞ s−1 −tt e dt. Byzchanging the variable of integration from τ to τe = (τ + κ0 )/ρ, (26) becomes
D∗ =

κ1−ε
ρθ(1−ε)+1
1
> 0.
ρκ
0
aµe Γ(1 − θ(1 − ε), ρκ0 )(27)

Γ(s, z) is positive and finite for all s ∈ (−∞, ∞) and z ∈ (0, ∞), which implies D∗ > 0. The values ofΓ(s, z) are available in most programming platforms, such as Python, Matlab, and Mathematica.
27

The variable of integration is changed from i to τ in (28) using (2).

25The equilibrium ratio of production and research workers LP /LR∗in (32) becomes a

positive finite constant if and only if θ(1 − ε) > 1.28 Using definition θ ≡ 1/(1 − ψ),the condition reduces to ψ ∈ (ε, 1), where ψ is the degree of knowledge spillover from past
productivity to its increments. Using this ratio, the ASSE research intensity can be writtenasn∗ = aLR∗ =aL,
1 + (LP /LR )∗(33)

which becomes a positive constant if ψ ∈ (ε, 1), and 0 if ψ < ε.
The pair of D∗ and n∗ in (26) and (33) characterizes the long-term equilibrium of
this economy. These equations also show how parameters affect long-term dynamics. For
example, a larger µ means the breadth of patents is wider (or the trade secrets are betterkept). A higher a means that R&D is less costly. In these cases, innovation intensity n∗rises because of higher profitability, whereas the output of each good, proportional to D∗ ,falls because there are more production firms to which the aggregate labor needs to be
divided.29 The opposite occurs when the time preference ρ is higher because it raises theinterest rate and hence reduces the present value of profits.
When population L is larger, the research intensity n∗ is multiplied proportionally to
L. However, production of each good (proportional to D∗ ) does not change, because boththe number of products introduced each year and the number of total production workers
are multiplied by the same factor. This outcome resembles the mechanism of the secondgeneration endogenous growth models, where the horizontal number of sectors is adjusted
proportionally to the total population.30
Before closing this subsection, let us briefly compare those results against the case ofψ = 1. When ψ = 1, the solution to differential equation (19) is exponential: q(τ ) = eβτ .
28

Using (27), we obtain an explicit solution to (31) as follows. It becomes infinity if θ(1 − ε) ≤ 1.


29

LP
LR

∗

1−θ(1−ε)=κ0ρθ(1−ε)+1
if θ(1 − ε) > 1.
ρκµ(θ(1 − ε) − 1)e 0 Γ(1 − θ(1 − ε), ρκ0 )(32)

The derivative of the upper incomplete Gamma function with respect to the second argument,∂Γ(s, z)/∂z = −z s−1 e−z , is always negative. Using this, the properties in the text can be confirmed
from (27), (32) and (33).
30

However, note that the long-term growth in these models is typically maintained by the exponential

increase in the productivity (or quality) in each sector, whereas this paper focuses on the case where suchexponential improvements cannot be sustained (ψ < 1 in equation 19).

26

Then, we can calculate n∗ and D∗ in the ASSE asn∗ =µ(1 − ε)βaL,(1 + µ)(1 − ε)β + ρD∗ =(1 − ε)β + ρ
.
aµ

(34)

The comparative statistics properties with respect to µ, ρ, L and a are the same as thecase of ψ ∈ (ε, 1). Therefore, the case of exponential growth in productivity (ψ = 1) canbe viewed as a special case of our model, although we do not focus on it because it is aknife-edge case.

3.5

Long-Term Real GDP Growth Rate

Now we are ready to examine the long-term rate of economic growth in this prototype
model. The ASSE satisfies the definition of non-exponential asymptotic steady state in
Definition 1. Using (20), (22) and (23), as well as definitions θ ≡ 1/(1 − ψ) > 1 andκ0 ≡ θ/β > 0, we can confirm that p(τ ) and x(τ ) satisfies conditions (7) and (8) as long asψ ∈ (ε, 1). Therefore, we can apply Proposition 1 to calculate the real GDP growth ratein the steady-state equilibrium. Combined with the result for the case of ψ = 1, we haveR∞ ′ ψ−ε β for ε < ψ ≤ 1,
− 0 p (τ )x(τ )dτ
1−ε
∗
=
(35)
g = R∞00 p(τ )x(τ )dτfor ψ ≤ ε.
Equation (35) shows that the measured growth rate takes a positive and finite value
whenever ψ ∈ (ε, 1]. This requirement can be understood from the discussion in Subsection
2.5. There, we have seen that if the good remains in the market forever, the price of goods
need to fall fast enough so the cumulative expenditure for old goods becomes finite. A
suﬀicient condition is given by (12), which requires the power of function p(τ ) to be lessthan −1/(1 − ε). Given that ψ < 1, the price of goods in the ASSE can be written
as p(τ ) = ((1 + µ)/κ1 )(τ + κ0 )−θ from (20) and (22). The power of this price functionis −θ = −1/(1 − ψ), and it becomes lower than −1/(1 − ε) if and only if ψ > ε. In
this particular model environment, sustained economic growth requires prices to fall fastenough, and for this to occur, the degree of intertemporal knowledge spillover in production
ψ should be large enough. Still, compared to existing prototype models of endogenous
growth (See Figure 2), it is remarkable that ψ does not need to be at a knife-edge level.
Given that the markup ratio µ is constant, the growth formula (9) in the ASSE can be
represented asR∞
gq (τ )e(τ )dτ,g∗ = 0 R ∞0 e(τ )dτ27(36)

where e(τ ) = p(τ )x(τ ) is the expenditure for an age τ good and gq (τ ) is the rate ofproductivity increase for this product, defined in (21). The formula in this form clarifies
that the real GDP growth is the weighted average of the rate of productivity increases
among goods of various ages. Recall that, in our specification of the technology, the newest
goods have the fastest rate of productivity improvements, β, while the rate of improvements
is lower for the older goods because gq′ (τ ) < 0 (see equation 21). In particular, the rate ofproductivity improvement gq (τ ) is almost zero for very old goods with large τ . Therefore,
it is natural that the aggregate GDP growth rate (35) is somewhere between zero and βbecause the economy consists of goods of all ages.
Now, it is clear why growth rate g ∗ in (35) is decreasing in ε. When the elasticity of
substitution ε is high, consumers spend relatively more on old and low-priced goods andless on new and expensive goods. Since the rate of productivity increase in (21) is lower
for older goods (with high age τ ), the weighted average will also be low.
Equation (35) also shows that growth rate g ∗ in is increasing with ψ, the degree ofknowledge spillover in production. When ψ ≤ ε, (LP/LR)∗ in (31) becomes infinity, whichmeans n∗ = 0. Without the introduction of new goods, the distribution of products ages
simply moves up, and, the growth rate will fall to gq (∞) = 0. As ψ increases between ε and1, the schedule of gq (τ ) in (21) moves up, and so does the real GDP growth rate. When ψreaches 1, the long-term growth rate rises to β. This is an anticipated result; when ψ = 1,
the productivity of all goods, both the new and the old, increases with a common constant
exponential rate of β. Therefore, the case of ψ = 1 corresponds to the conventional growththeory where labor productivity increases exponentially and uniformly. However, our stressis that, even when the productivity of each product does not rise exponentially (i.e., with
ψ < 1), the economy as a whole can exhibit a constant measured growth rate, although
it is lower than β. Finally, ψ > 1 is unrealistic, because it means that the productivity of
individual goods increases more than exponentially and gq (∞) = ∞.
It is worth noting that, in this simple prototype setting, the equilibrium long-term rateof growth (35) does not depend on equilibrium values of n∗ and D∗ as long as they are
positive.31 When the research intensity n∗ is higher, there is more addition of economic
31

Of course, this property depends on the simplistic settings in this prototype model. For example,

when the aggregate R&D intensity n∗ has some positive spillovers on the rate of productivity increases inindividual goods gq (τ ), then n∗ will affect g ∗ . Also, when the amount of production has some effects on

28

activity per unit of time, but at the same time, there is also proportionally more “stock” ofexisting activities. The real GDP growth rate expresses this ratio, which is unchanged.32
Similarly, when D∗ is larger, there is more demand for each good. This means the production of new goods, as well as the increment of production of other goods per unit of time,
is higher. At the same time, however, the value of existing products is also higher, exactly
canceling out the effects on g ∗ .33 As a result, even when the changes in population L,
R&D productivity a, or patent policy µ affect n∗ and D∗ , they do not affect the real GDP
growth rate. This is a marked contrast to the implication of existing R&D-based growthmodels, where g ∗ follows directly from n∗ . Although this result depends on the simplifying
specification of the prototype model, it might provide a possible interpretation of why themeasured GDP growth rates in the U.S. and some other developed countries have beenquite stable, even though those underlying parameters seem to have changed greatly overlong time periods.

3.6

Measured Real Aggregate Variables and Balanced Growth

The ASSE in this model works very differently from the balanced growth path (BGP) in
existing growth models. Nonetheless, this section shows that when aggregate variables aremeasured in a conventional way, this model exhibits balanced growth in those measured
aggregate variables.
Note that the total labor income for production is LP ∗ since the wage rate is normalized
to one. All goods are sold at (1 + µ) times the labor cost, as shown in (22), and therefore
the aggregate value of production, which equals the aggregate value of consumption, isC ∗ = (1 + µ)LP ∗ . In our model, investments take the form of R&D, and the total value of
R&D outputs is I ∗ = n∗ V ∗ = LR∗ . The GDP in our model can be calculated as the sumof the value of production and the value of investments: Y ∗ = C ∗ + I ∗ = (1 + µ)LP ∗ + LR∗ .
gq (τ ), g ∗ will depend on D∗ .
32

Nonetheless, it is important that there is a positive flow of new innovations n∗ > 0, since otherwise, g ∗becomes 0.
33This can also be seen in Examples 3 of Figure 5. When D∗ is increased, the left panel is stretchedhorizontally (along the x(τ ) axis) while the right panel is stretched vertically (along the p(τ )x(τ ) axis) by
the same magnification ratio. As a result, the growth rate, which is given by the ratio of the two areas, isunaffected.

29

The aggregate capital is defined as the value of all firms in the economy (knowledge capital).
In Online Appendix, we derive K ∗ = (µLP ∗ − LR∗ ) > 0.
Note that those aggregate variables are measured under the price normalization of ourmodel, in which the nominal wage is set to 1. We now calculate their real values, as definedin the SNA.34 Let t be the reference year, and Yt$ be the dollar value of the GDP in yeart, which we assume is known to the researcher. Then, since the real GDP growth rate is∗constant at g ∗ in the ASSE, the real GDP level in t is Ytreal = Yt$ eg (t−t) . Since the ratios
among Y ∗ , C ∗ , I ∗ and K ∗ are constant, their dollar values grow in the same proportion:C ∗ real
1+µ∗Yt=
Yt$ eg (t−t) ,
∗
R
P
∗
Y
1 + µ + (L /L )∗I
1∗
Itreal = ∗ Ytreal =Yt$ eg (t−t) ,PR∗Y(1 + µ)(L /L ) + 1∗µ − (LR /LP )∗K∗
Yt$ eg (t−t) ,Ktreal = ∗ Ytreal =RP∗Yρ(1 + µ + (L /L ) )Ctreal =

(37)
(38)
(39)

where (LR /LP )∗ is given by (31).
Interest rate r∗ = ρ is also defined under our normalization of prices. Since the nominal
GDP growth rate in the steady state is zero, the steady-state inflation rate is −g ∗ in our
price normalization. Then, the real interest rate in the steady state is rreal = r∗ +g ∗ = ρ+g ∗ .
We can also derive other real aggregate variables in similar ways, and the growth rates of allof them are constants. Therefore, if the statistical agency measures the aggregate variables
in our model, they are observed to be exponentially growing along the BGP, even thoughneither quantity, quality, or variety of individual goods grow exponentially.

3.7

Welfare Improvements

Lastly, let us discuss the welfare of the representative consumer. From (13), the periodRN
utility is Ut = 0 t u(ect (i))di. In the ASSE, the improvements in welfare can be measured
34

The NIPA publishes two series of real GDP. One is the quantity index, which takes 100 in the reference

year (it is 2012 as of the time of writing). The values for other years are obtained by chaining the real GDP
growth rate. The other is Chained (2012) dollar series. They are calculated as the product of the quantity
index and the 2012 current-dollar value of the corresponding series, divided by 100. See U.S. Bureau of
Economic Analysis, ”Table 1.1.6. Real Gross Domestic Product, Chained Dollars.” Here, We use the latter.

30

by35
lim U̇t = n∗ lim u(c) =
c→∞

t→∞

ε
n∗ .
1−ε

(40)

Given parameter ε ∈ (0, 1), equation (40) shows that the speed of welfare improvementsin the steady state is entirely determined by the speed of innovation n∗ . This is in a stark
contrast to the real GDP growth rate g ∗ in (35), which does not depend on n∗ . At least in
this simple prototype model, the non-exponential growth theory illustrates that the GDP
growth rate is not a good measure of the speed of welfare improvements in the long run.

4

Generalizations

4.1

ObsolescenceIn the prototype model of Section 3, sustained economic growth required the price elasticity
of goods ε to be less than one, in the environment where goods stay in the market forever(T = ∞) and consumers have symmetric preference across goods (13). The condition ε < 1
was necessary to induce consumers to spend less on older (and cheaper) goods. However,even without such an assumption, consumers often spend more on new goods just because
they like new goods. They may spend less on older goods because those are obsolete. Here,
we show that condition ε < 1 can be relaxed once we include obsolescence.
Suppose now that utility function (13) is replaced by
Z ∞  Z Ntδ(t − s(i))u(ect (i))di e−ρt dt,
0

(41)

0

where t − s(i) = τ is the age of good i (time after its debut), and δ(τ ) is a decreasing
function of τ with δ(0) = 1 and δ(∞) = 0. The steepness of function δ(τ ) represents
the speed of obsolescence, or equivalently, consumers’ taste for newer goods. We keep allother settings in Section 3 except that now we allow any ε > 0. Then, the expenditure
for an age τ good in the ASSE becomes e(τ ) = p(τ )x(τ ) = (1 + µ)D∗ δ(τ )ε q(τ )ε−1 , which
illustrates that even when ε > 1, expenditure for older goods falls with age if obsolescence is
35

Equation (40) gives the difference in the period utility between time t + 1 and time t in the ASSE. Since

the economy is in the ASSE, schedules for consumption against the age of goods are the same in those 2time points, with the only difference that the economy in time t + 1 has n∗ more oldest goods than in timet. When t → ∞, the quality-adjusted amount of consumption for each of those oldest goods approaches
infinity. Therefore, the difference in the period utility is n∗ c(∞). From (14), c(∞) = ε/(1 − ε).

31fast enough.36 Proposition 1 continues to apply in an environment with obsolescence, andthe formula for the GDP growth rate (36) shows that the growth rate becomes a positiveR∞
constant if 0 e(τ )dτ is finite.
When the rate of obsolescence is constant δ > 0 per year, δ(τ ) = exp(−δτ ). In thiscase, the integration of e(τ ) = (1+µ)D∗ δ(τ )ε q(τ )ε−1 always becomes finite because δ(τ )ε is
falling exponentially and no other variable is growing exponentially. Therefore, a constantrate of obsolescence always sustains positive GDP growth regardless of ε. Growth can be
maintained with slower depreciation. Consider an example where δ(τ ) is a negative powerR∞function of τ : δ(τ ) = δ0ω (τ + δ0 )−ω where ω and δ0 are positive constants. Then, 0 e(τ )dτ
becomes finite if and only if37
ω>

ε−ψ
.
ε(1 − ψ)(42)

The condition shows that growth can be sustained even when ε > 1 and obsolesce is notexponential. In a special case of δ0 = κ0 , where κ0 is defined in (20), we obtain an explicit
expression for the long-term GDP growth rate,g∗ =ψ − ε + (1 − ψ)εω,
1 − ε + (1 − ψ)εω(43)

which is positive when (42) holds. Since ψ < 1, g ∗ is higher when the obsolescence is faster(ω is higher).
Intuitively, obsolescence skews expenditure toward newer goods. Since newer goods
have more margins for productivity increases, the overall growth rate rises with obsolescence. This result has interesting policy implications. When the government tries toprotect obsolete companies (or industries), it will reduce the GDP growth rate, not justbecause of eﬀiciency loss but also because of the way the GDP growth rate is calculated.
Conversely, advertisements and marketing practices that attract consumers from older
goods to newer goods will enhance GDP growth, even when the attractiveness of the newer
goods is illusionary.
36

−1
R∞
, which is always
Similarly to the derivation of (26), we obtain D∗ = aµ 0 δ(τ )ε q(τ )ε−1 e−ρτ dτpositive and finite because of the e−ρτ term. Using this value of D∗ , the speed of innovation is n∗ =
−1 ∗
R∞
R∞
. n is strictly positive if and only if 0 δ(τ )ε q(τ )ε−1 dτ is finite, whichaL 1 + aD∗ 0 δ(τ )ε q(τ )ε−1 dτR∞is equivalent to the finiteness of 0 e(τ )dτ .
R∞
R∞
37
Using (20), 0 e(τ )dτ = (1 + µ)δ0εω κε−1
D∗ 0 (τ + δ0 )−ωε (τ + κ0 )θ(ε−1)) dτ . The integral becomes finite
1
if and only if the sum of the powers of the integrand, −ωε + θ(ε − 1) is less than one. Using θ = 1/(1 − ψ),this condition is equivalent to (42).

32

4.2

Declining and Rising Population

The theory in Section 2 has explained that GDP growth can be maintained when the
population is constant. Here, we extend the theory to declining and rising populations.
Suppose that the population changes at the rate of gL , which can be either negative or
positive. We maintain the assumption that the creation of one new good requires a fixed
number of R&D workers and consider a steady state where a constant fraction of the
population is working in the R&D sector. Then, instead of (1), the number of goods will
increase according toṄt = n0 egL t ,(44)

for some constant n0 > 0. The relationship between the index of goods and their age in(2) changes to
i = Nt −
n0 e g L t
1 − e−gL τ ,
gL

(45)

where the second term represents the number of goods created between t − τ and t. We
also normalize prices so that the expenditure per capita is constant.38 Except these, theenvironment of the economy in the steady state is the same as defined in Definition 1,
including condition (3). The GDP growth rate in (5) still applies since it does not dependon the constant population. Therefore, using (3), (44), and (45) in (5), and then utilizing
the integration by parts, we obtain a generalized version of Proposition 1.
Proposition 2 Consider an economy where the population is changing at a constant rate
gL . Suppose that the economy converges to the asymptotic non-exponential steady-state, as
defined by Definition 1. Given thatZ T−
x(τ )e−gL τ dp(τ ) is finite and strictly positive, and0Z Tp(τ )x(τ )e−gL τ dτ is finite,

(46)
(47)

0

the growth rate of the real GDP per capita asymptotes to a positive and finite constantRT− 0 x(τ )e−gL τ dp(τ )g − gL = R T.
−gL τ dτ
p(τ
)x(τ
)e
0
38

The total expenditure in the steady state becomes n0 egL t

RT0p(τ )x(τ )e−gL τ dτ . Given that the integral

is finite, the expenditure increases proportionally to the population.

33

(48)

Note that Proposition 2 differs from Proposition 1 only in that x(τ ) is replaced by x(τ )e−gL τ .
Therefore, We can graphically calculate the growth rate as we have done in Subsection 2.4
by replacing x(τ ) by x(τ )e−gL τ .
Subsection 4.1 has shown that when there is a constant rate of obsolescence, x(τ ) will
be multiplied by δ(τ )ε = e−εδτ . Therefore, the effect of population growth (gL > 0) on
the rate of GDP growth rate is similar to having a positive rate of obsolescence, δ = gL /ε.
This similarity can be interpreted as follows. When the speed of innovation accelerates
according to (44), it means that there are relatively fewer old goods relative to new goods
because R&D were slower in the past than now. In this sense, the situation is similar
to the case where older goods disappear due to obsolescence. Therefore, a higher rate of
population growth increases the fraction of the aggregate expenditure that goes to newer
goods, and enhances the GDP growth rate, as we explained in Subsection 4.1.39
Conversely, the rate of GDP growth will be lower when the population is declining(gL < 0). It is notable that real GDP growth may still be sustained in the long run,depending on parameters. For simplicity, suppose that newer goods are relatively more
expensive than older ones, which guarantees that the expression in (46) is negative. Then,if goods retire from the market at a finite age T < ∞, both conditions (46) and (47)are satisfied, and Proposition 2 implies that the real GDP per capita will keep growing
at a positive rate. When goods stay in the market forever (T = ∞), population decline
makes positive growth more diﬀicult. In this case, positive per capita GDP growth requires
consumers to have a strong enough taste for new goods that will overcome the effect ofthe declining population. For example, when the consumers have utility function (41) andthe rate of obsolescence is constant, δ(τ ) = e−δτ , positive per capita GDP growth can besustained if εδ + gL > 0.
Note that the analysis above is about the measured per capita GDP growth, and notabout welfare improvements. With a declining population, the speed of R&D (Ṅt in equation 44) will inevitably slow down to zero (Jones 2022). Therefore, our theory predicts
stagnation of living standards despite the real GDP per capita growing at a positive rate.
This is another example of the discrepancy between GDP growth and welfare improvements, as discussed in Subsection 3.7.
39

In our theory, the rate of economic growth depends on the population growth rate, but not on the

population size. In this sense, there is no scale effect.

34

5Conclusion

The non-exponential growth theory provides an easier interpretation of the consequence oflong-term growth. We have shown that economic growth does not necessarily mean “howmany times” the economy is expanded. A constant rate of real GDP growth requires neither
quantity, quality or variety to expand to an astronomical order in the future. Rather, thereal GDP growth rate measures the new addition of the economic activity relative to theexisting economic activity in terms of the market value, which should coincide with themarginal evaluation by consumers at that time.
If the marginal evaluation of each good is unchanged, a constant rate of economic growthcannot occur unless some component of production continues to expand exponentially.
However, the marginal evaluation of each good, i.e., the relative market price of each
good, changes endogenously as the economy grows. An increase in the production of acertain good may lower the evaluation of the value of the whole production of this good
if the increased production is combined with a lower market price. Nonetheless, such amechanism will work for sustaining growth, as it keeps the evaluation of existing production
from exploding and maintains the relative evaluation of new goods high so that a constantflow of new goods can continue to constitute a positive GDP growth rate.
Our theory has several benefits when compared to conventional growth theories. First,
it provides a robust theoretical basis on which economic analysis and empirical studies canbe carried out without being constrained by knife-edge conditions. If older goods eventuallydisappear from the market, sustained growth is obtained as long as their quality-adjusted
price falls somewhere in their lifecycle. Even in the least favorable setting where no good
drops out from the market, the externality for productivity increase can be much weaker
than usually assumed and is not necessarily exactly at a certain level.
Second, the non-exponential growth theory enables growth economists to focus on factors that were previously not viewed as a determinant of economic growth. Existing R&Dbased models have shown that policies can influence the long-term growth rate only byaffecting the rate of R&D.40 The non-exponential growth theory, by contrast, shows thepossibility that various factors affect the measured long-term rate of growth through thepattern of evolution in prices and quantities of individual products.
40

See Aghion, Akcigit and Howitt (2014) for an excellent survey.

35

Third, existing endogenous growth models typically have only one engine of growththat drives the exponential growth of innovation. It is not very meaningful to consider
several growth engines at the same time because the engine that realizes the highest rateof exponential expansion will dominate all the other factors. In contrast, our theory opens
the door to combining various growth engines in one framework. All the factors that affectthe evolution of the price or quantity of individual goods, as well as those that affect theage distribution of products, will show up in the long-term growth rate. We hope that it
will eventually be used to quantify the composition of the source of economic growth.

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