A Schumpeterian Microfoundation of the Geometric
Brownian Motion of Firm Size and Zipf’s Law
Tetsugen HARUYAMAMay 2024
Discussion Paper No. 2403

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANA Schumpeterian Microfoundation of the Geometric
Brownian Motion of Firm Size and Zipf’s Law
Tetsugen HARUYAMA∗May 2024

Abstract
A geometric Brownian motion is often used in dynamic economic analysis when variables of interest grow stochastically. What economic mechanisms are working behind?
What economic forces contribute to shaping such stochastic processes? The existing studies leave those questions unanswered. The present paper represents an effort to answer
them, focusing upon the firm size distribution. Using the otherwise standard Schumpeterian growth model, Poisson-distributed innovations in “many” sectors give rise to thegeometric Brownian motion of a firm size via the Lindberg-Feller Central Limit Theorem.
The resulting distribution of firm sizes is Pareto, and the Pareto exponent can take a low
or high value. Local stability analysis reveals that the lower Pareto exponent, close to 1,
is locally stable.

JEL Code: C00, E13, O40, O30, D39

Keywords: Geometric Brownian Motion, Firm Size, Pareto Distribution, Innovation,
Growth∗Graduate School of Economics, Kobe University.

11

Introduction

A Brownian motion and a geometric Brownian motion are often used in dynamic economic
analysis. The former is applied to, e.g. short-run macroeconomic fluctuations (Ahn, Kaplan,Moll, Winberry, and Wolf (2018)), while the latter has become popular in analysis where avariable of interest grows stochastically and its distribution is examined. For example, thewealth distribution (e.g. Benhabib, Bisin, and Zhu (2011)), the income distribution (e.g.
Gabaix, Lasry, Lions, and Moll (2016)), firm dynamics (e.g. Luttmer (2007)), and the city
size distribution (e.g. Gabaix (1999)) in addition to finance (e.g. Duffie (2003)). This factproved that Brownian and geometric Brownian motions are instrumental in generating valuable
insights on those topics.
Having said this, however, a (geometric) Brownian motion is merely a mathematical apparatus, and hence the following questions remain. What mechanisms are working behind?
What economic forces contribute to shaping such stochastic processes? The existing studies
leave those questions unanswered. An attempt to answer them is important, as Steindl (1987)
wrote “The stochastic models have often been criticised for their lack of economic content.
Perhaps it has been overlooked that they only represent the first steps in a new and exceedingly difficult terrain.” (p.810) The present paper represents an effort in the direction Steindl
perceived, focusing upon the firm size distribution.
Our starting point is the Schumpeterian model of economic growth, pioneered by Aghion
and Howitt (1992) and Grossman and Helpman (1991). Typically, innovation follows a Poissonprocess in those models. As in other studies (see below), we introduce incumbent R&D in“many” manufacturing sectors. Then, invoking the Lindberg-Feller Central Limit Theorem,
we will show that the rate of firm growth in terms of the number of products is normally distributed, and hence, the size of a firm exactly follows a geometric Brownian motion (hereafter,
GBM). That is, Poisson distributed innovations in many sectors give rise to the GBM of a firmsize. Note that this mechanism fundamentally differs from a simple normal approximation of
the sum of independent Poisson random variables. What is required for the firm size GBM isa normally distributed rate of growth in the number of products, not the level of the numberof goods. A driving force for this mechanism is a positive externality which makes it possiblefor incumbents to generate more innovations as their size expands.
Given the GBM of an individual firm size, it is easy to establish that the firm sizes is
Pareto-distributed. This is a standard result. What differentiates our analysis from othersis the existence of multiple values for the Pareto exponents because of the afore-mentioned
externality. A mechanism of this result can be understood by distinguishing three types ofcreative destruction of innovation. The first type is by entrant firms. Their new products
replace the existing goods, thinning the tail distribution. Incumbent innovation without theexternality constitutes the second type of creative destruction. Incumbents forge ahead withnew products on average, thickening the tail distribution. The third creative destruction isrealised via incumbent innovation amplified by the positive externality. The third type works
against the first two types of creative destruction in terms of how they affect the Pareto2

exponent. Multiple values for the exponent arises due to those opposing creative destruction
effects.
While this is an interesting result, stability analysis is required to determine what value
the Pareto exponent takes in equilibrium. For this, we develop a full-fledged growth model,
endogenising entrant and incumbent Poisson rates. Firms have to be successful in R&D first
before entry, and incumbent firms invest in R&D to expand its portfolio of products. They
make R&D decisions by choosing the optimal number of workers. In this framework, we
conduct local stability analysis based on the Kolmogorov forward equation and establish thata lower value for the Pareto exponent is stable, and a higher value is unstable. In addition,
we show that the Pareto exponent is arbitrarily close 1, i.e. Zipf’s law, if the degree of thepositive externality is small enough.
Our study is related to the literature of firm dynamics, pioneered by Lucas (1978) andHopenhayn (1992) who consider a competitive industry. In the sense of monopoly profit and
R&D activity, the present paper is closer to Klette and Kortum (2004) and Luttmer (2011).
In particular, we borrow a valuable insight of the former study that products are countable ina continuum of industries. The latter study extends the former to show that the firm sizes are
Pareto-distributed. Our study is more closely related to Luttmer (2007) who uses a GBM andestablishes that the stationary distribution of firm sizes closely approximates Zipf’s law, i.e.
a Pareto exponent is one. A difference is that we obtain a Pareto exponent close to 1 using
the fairly standard Schumpeterian endogenous growth model. In this sense, our model has anadvantage in that it can be easily extended to, for example, international trade and analysisof policy instruments like patents and taxes. Di Giovanni, Levchenko, and Ranciere (2011)
consider power laws in firm sizes in the context of international trade. Acemoglu and Cao(2015) is also related to our study in that entrant and incumbent innovations drive economic
growth. Clementi and Palazzo (2016), Arellano, Bai, and Kehoe (2019) and Bilal, Engbom,
Mongey, and Violante (2022) are more recent studies which apply firm dynamics models toshed light on business cycles and labour market issues. Luttmer (2010) gives an overview of theliterature on firm dynamics, and Gabaix (2009) reviews studies on power laws in economics. In
a more general setting based on Markov processes, Beare and Toda (2022) explore a rigorous
mechanism of generating a Pareto distribution of sizes.
The structure of the paper is as follows. Section 2 takes Poisson arrival rates of entrant
and incumbent innovations as given. It is shown that (i) the growth rate of a firm size isnormally distributed, (ii) a firm size follows the GBM, (iii) the stationary distribution of firm
sizes is Pareto, and (iv) there exists multiplicity of Pareto exponents. Section 3 endogenises
entrant and incumbent Poisson rates in the otherwise standard Schumpeterian model. Local
stability is examined and a lower Pareto exponent, closer to 1, is shown to be locally stable.
Section 4 concludes.

3

2

Exogenous Arrival Rates of Innovation

The purpose here is to demonstrate that the GBM of a firm size arises in a model of Poissonprocess innovations in a familiar framework of the Schumpeterian model of technological innovation. For this, Poisson arrival rates of innovation are taken as given in this section.

2.1

Assumptions

Time t is continuous, and we initially partition t into small time intervals ∆t such that T ∆t = t,
T > 0. All variables are taken to be constant during ∆t. We will ∆t → 0 in a later analysis.
The production function of final output Y (t) is given byK

1 X
ln [Yk (t)]
K
k=1
Z 1
ln [qk (j, t) xk (j, t)] dj
ln [Yk (t)] =ln [Y (t)] =

(1)
(2)

0

There are K number of what we call manufacturing sectors, denoted by k = 1, 2, · · · K. (2)shows that in each sector there are a unit continuum of industries, indexed by j ∈ [0, 1].
xk (j, t) denotes an intermediate product in an industry j in a sector k at time t. Its quality
level is given by qk (j, t) ≡ λℓk (j,t) , λ > 1, ℓk (j, t) = 0, 1, 2, · · · . In a later analysis, we will let
K → ∞ for a continuum of sectors.
In developing our model, we borrow an insight of Klette and Kortum (2004) that the
number of goods n produced by firms is countable in the product space. That is, firms startwith n = 1 upon successful entry, and n = 1, 2, 3, · · · for incumbent firms. Using π for
profit per product, which is taken as given in this section, firms with n products earn profitsnπ. Countability of n has an important implication. It allows us to calculate the mean andvariance of a proportionate rate of change in n in discrete numbers, which we take advantage
of in invoking a Central Limit Theorem in an environment of a continuum of products.
The quality levels qk (j, t) improve through R&D activities, entrant and incumbent. Free
entry prevails in R&D. Suppose that an entrant firm decides to invest in a sector k. A
research success occurs with a Poisson arrival rate gE . If successful, the entrant innovation is
implemented in a randomly selected industry in the entering sector. An entrant becomes asole producer of a single product, i.e. n (t) = 1, and starts R&D as an incumbent firm.
To describe incumbent R&D, consider a firm producing n (t) ≥ 1 number of goods. It
conducts R&D in all sectors, and gI is used to denote a common Poisson arrival rate of a
research success. If the firm succeeds in a sector k, it can generate n (t) number of potentiallylaunchable goods due to the positive externality of its past R&D successes.1 Each of thoseinnovations is implemented with the probability 0 < ϕ ≤ 1 in industries which are randomly
selected out of those in a sector k. In this sense, ϕ is the measure of the positive externality
captured by n.
1

A similar assumption is used in Garcia-Macia, Hsieh, and Klenow (2019) to quantify the extent of thecreative destruction effects of innovation in the US.

4

We assume that gE is the same in all sectors and so is gI in all sectors for all incumbent
firms. This symmetric feature greatly simplifies exposition. The time argument t is dropped
in what follows unless ambiguity arises.

2.2

A Normal Distribution of Firm Growth Rate

Consider a firm producing n =PKknk products where nk denotes the number of goods thatthe firm produces in a sector k. Using mIk for the random number of R&D successes in asector k during a time interval ∆t, the number of new products that the firm creates via
incumbent R&D in a sector k, denoted by ∆ngain
k , is given by= ϕnmIk .
∆ngaink(3)Note that the presence of n captures the positive externality, mentioned above.
On the other hand, the firm suffers losses due to rival innovations, entrant and incumbent.
The number of goods that it loses in a sector k, denoted by ∆nloss
k , during ∆t is given by=∆nlosskZ
mEk + ϕnkmIk (s) ds × nk .

(4)

s∈Mk

Let us consider the terms inside the parentheses. mEk is the number of innovations by entrants
in the sector. Turning to the second term, nk denotes the average number of products produced
per firm in a sector k. Given that there is a unit continuum of industries in the sector, wehave nk = 1/Mk where Mk is the number of firms operating. Next, define Mk is a set of firms
R
investing in R&D in a sector k, and s∈M mIk (s) ds is the total number of R&D successes
in a sector k. Because each research success generates n number of new products for firms
with n products with the probability ϕ, the mean of implemented innovation is given by ϕnk ,Rand hence ϕnk s∈Mk mIk (s) ds is equivalent to the number of implemented innovations by
incumbents in a sector k. All told, the terms inside the parentheses is the total number of
newly implemented products in the sector.
Given that those new products are implemented in randomly selected industries, theyrandomly hit products produced by the incumbent firm we are considering, and its probability
is given by nk , the last term outside the parentheses, because there is a unit continuum of
industries in the sector.
Now recall that all sectors are symmetric in the sense that innovation is random, andindustries are randomly selected for implementation. Indeed, equilibrium will be symmetric,
as shown below, in the sense that Mk = M and nk = n ∀k. Taking advantage of this featureand using (3) and (4), a proportionate rate of change in nk relative to n during ∆t is given by
the following random variable
gain

∆nk
∆nk
=
n



Z
− ∆nloss
1
nkk= ϕmIk − mEk + ϕmIk (s) dsn
M s∈Mkn

5(5)

We are ready to state the following lemma:∆nκ
are given by
n




∆nk
∆nk
= µk ,
V
= σk2
E
n
n

Lemma 1. The mean and variance of

(6)

where
nk
µk = ϕgI ∆t − [gE ∆t + ϕgI ∆t] ,n n 2
2
ϕ
k
gI ∆t
.
σk2 = ϕ2 gI ∆t + gE ∆t +
M
n

(7)
(8)

Proof. Taking expectations of (5) gives

E
using

R


∆nk
nk
= ϕE [mIk ] − [E [mEk ] + ϕE [mIk (s)] ds]
n
n

(9)

s∈M ds = M . The first equation of (6) is obvious from E [mIk ] = gI ∆t and E [mEk ] =gE ∆t. Similarly, the variance of (5) is given by


 
∆nk
ϕ2
nk 2
2
V
= ϕ V [mIk ] + V [mEk ] +V [mIk ]
.
n
M
n


(10)

V [mIk ] = gI ∆t and V [mEk ] = gE ∆t enable us to rewrite it as the second equation of (6).
∆nkdepend on the share of productsnnk. This comes from the fact that the number of products lost due to rival innovations is
nproportional to nk . Another aspect worth mentioning is that the variance is affected by the
The lemma shows that the mean and variance of

number of firms M . The effect is due to the average product per firm n. The larger thenumber of firms M , the lower the average number of product n. Therefore, a higher M means
less number of implemented incumbent innovations on average, hence a lower variance of the
proportionate change in nk . This channel is the source of multiple equilibria, as shown below∆nk
LetXk =. Having identified the first two moments of the distribution of Xk , we turnnto the distribution of the random growth rate of n. For this, define
K

1 X
SK =
Xk
K

(11)

k=1

ZK =

SK − E [SK ]p
V [SK ]

(12)

SK is the average of Xk and is random because Xk is random. ZK is the standardised randomvariable of SK . Though we do not know the distribution of SK , hence ZK , we can identify
it by invoking the Lingberg-Feller Central Limit Theorem. The result is summarised in thefollowing proposition:

6

Proposition 1. As K → ∞, ZK approaches the standard normal distribution, i.e.
lim ZK = Z ∼ N (0, 1)

(13)

K→∞

Proof. Define χk ≡ Xk − µk . The Lindberg condition in our case is
 
K
i
h χ
p
1 X
χk 2
k
E
> ϵ V [SK ] = 0·1
K→∞ V [SK ]K
Klim∀ϵ > 0(14)

k=1

where 1 [.] is an indicator function. (14) is the necessary and sufficient condition for (13).2
Given that ∆t is a short time interval such that mEκ = {0, 1} and mIκ = {0, 1} and using(9), we havemax [χk ] = ϕ + gE ∆t ≡ χ+
k > 0,min [χk ] = − (1 + ϕ) − ϕgI ∆t ≡ χ−
k < 0.
It means that |χk | is bounded such that−|χk | ≤ max χ+
k , χk

≡χ<∞

∀k.

(15)

gI ∆t < ∞(16)

Moreover, we have the following from (10)max {V [Xk ]} = ∆t + ϕ

2



1
1+
M



which is independent of K. Using these, we rewrite (14) as
 
K
h χ
i
p
1 X
χk 2
k
·1
E
> ϵ V [SK ]K→∞ V [SK ]
K
K
k=1
(
)
 2
K
h χ
i
p
χ
1 X
k
≤ lim
·1
> ϵ V [SK ]K→∞ V [SK ]
K
K
k=1
( 

)
K
p
χ 2
χ
1 X
·1
> ϵ V [SK ]≤ lim
K→∞ V [SK ]
K
K
lim

k=1

=0
 
The first line is due to E χ2k ≤ χ2 and the second line uses (15). The last line holds becausepχ> ϵ V [SK ] = 0 as K → ∞. The Lindberg(1) V [SK ] < ∞ due to (16), and (2) 1
K
condition (14) is met.
In general, the Lindberg condition implies that values in both tails have little influenceon the variance. This holds in our case because the maximum value of Xk is finite and eachrandom variable has less and less impacts as K goes to infinity.
2

For example, see Spanos (1986).

7

Having established Proposition 1, we need to accommodate K → ∞ in our model. For
this, we assume K∆k = 1. This assumption reduces the production function (1) and (2) toZ 1 Z 1ln [q (j, k) x (j, k)] dj dk.

ln (Y ) =
0

(17)

0

K → ∞ allows us to introduce an infinitely many manufacturing sectors in a unit continuumk ∈ [0, 1]. In line with K → ∞, therefore, the number of products, n, produced by anincumbent firm products is now defined asn = lim

K
X

K→∞

Z 1
nk ∆k =n (k) dk

(18)

0

k=1

where n (k) is the number of goods in a sector k. Note that n is equivalent to the aver∆nk∆n (k)
age of n (k). Accordingly, the random variablein (5) is replaced with
in an
n
n
infinitesimally small sector interval dk.
Based on those results, we state the next lemma.
Lemma 2. The following holds:
lim V [SK ] = σ 2 ∆tlim E [SK ] = µ∆t,

K→∞

K→∞

where
µ = −gE

(19)



1
1
gI
σ 2 = gE + ϕ2 1 +
3
3M

(20)

Proof. Using (11),lim E [SK ] = lim

K→∞

K
X

K→∞

Z 1
E [Xk ] ∆k =

µk dk
0

k=1

Z 1
= ϕgI ∆t − (gE ∆t + ϕgI ∆t)
0

n (k)
dk
n

= −gE ∆twhere the third equality uses (7). Similarly,
lim V [SK ] = lim

K→∞

K→∞

K
X

Z 1
V [Xk ] ∆k =σk2 dk

0

k=1


Z 1

ϕ2
n (k) 2
= ϕ gI ∆t + gE ∆t +
gI ∆t
dk
M
n
0

Z 1
ϕ2
2
= ϕ gI ∆t + gE ∆t +gI ∆t
k 2 dk
M
0
2

8


 
1
1
2
=
gI ∆t
gE + ϕ 1 +
3
3M


n (k)
= k is used innthe third line because n (k) is randomly distributed over a unit continuum of sectors [0, 1].

where the third equality makes use of (8). The linear change of variable∆n
is negative, implying that on average the firmnsuffers the loss of all products in all sectors, resulting in an exit from the market on average.
The lemma shows that the mean ofThis is required for a stationary distribution of firm sizes because it will be modelled as adiffusion process in a GBM. Conveniently µ and σ 2 are both constant, and its significanceshould be clear in the following proposition:
Proposition 2. The following holds:
lim SK = S ∼ N µ∆t, σ 2 ∆t .

K→∞

(21)

Proof. The result is clear from Proposition 1 and Lemma 2.
We have established that the growth rate of a firm size follows a normal distribution. This
result is crucial in deriving the GBM of n.

2.3

The Geometric Brownian Motion of Firm Size

Proposition 2 concerns what happens during a small time interval ∆t. We turn to the wholetime period up to t starting from the initial time t0 = 0. Let us define
GT (t) =TX∆n (τ ∆t)
τ =0n (τ ∆t),n (0) = 1

(22)

where ∆n (t) = n (t + ∆t) − n (t). Recalling T ∆t = t, GT (t) is a cumulative rate of growth ofn up to time t. We are ready state the next lemma:
Lemma 3. The following holds:
lim GT (t) = G (t) ∼ N µt, σ 2 t .

T →∞

(23)

Proof. It is obvious using the fact that the sum of normally distributed random variables arenormally distributed.
The cumulative growth rate of firm size also follows a normal distribution, and this is anindispensable ingredient for the GBM of n. Given the definition of G (t), the size of a firm n
is given byn (t) = eG(t)(24)

where n (0) = 1 requires G (0) = 0. It implies that the size of a firm is defined by a log-normal
distribution. Using (23) and (24), the next proposition follows:9Proposition 3. The number of products produced by an incumbent firm n obeys
dn (t) = θn (t) dt + σn (t) dW (t)
where

(25)

σ2
2

(26)

G (t) − µt
∼ N (0, t)
σ

(27)

θ =µ+
and W (t) is the Wiener process.
Proof. First define
W (t) =
Using (27), rewrite (24) asn (t) = eµt+σW (t) ≡ Ψ (t, W (t)) .
Ψt (t, W (t)) = µn (t), ΨW (t, W (t)) = σn (t) and ΨW W (t, W (t)) = σ 2 n (t). Using Ito’s
Formula3
Z t1
Ψ (t, W (t)) − Ψ (0, W (0)) =Ψt (t, W (t)) + ΨW W (t, W (t)) dt
2
0
Z t
+
ΨW (t, W (t)) dW (t)0which is expressed as a differential form (25).
Note that θ is the sum of the mean of ṅ/n and a half of its variance. This result is dueto the fact that the cumulative growth rate of n or G (t) in (23) is normally distributed, andhence n itself is defined by a log-normal distribution (see (24)). The term σ 2 /2 is called theIto correction term and arises because of the convexity of the exponential function.
Proposition 3 is one of the key results of the paper. We started from the otherwise standard quality-ladder framework, and an additional assumption regarding “many” manufacturingsectors enables the model itself to serve as a microfoundation of the GBM of the size of a firm.

2.4

A Pareto Distribution of Firm Size

It is well known that if a random variable follows the GBM, its distribution is Pareto. To
show this, we follow Gabaix (2009). Let the distribution of n be denoted by f (n, t), which isthe density function of firms producing n products at time t. Its change is governed by the
Kolmogorov forward equation∂f (n, t)∂ [θnf (n, t)] 1 ∂ 2 σ 2 n2 f (n, t)
=−
+ ·
.
∂t
∂n
2
∂n2
3

For example, see Theorem 7.5 of Brzezniak and Zastawniak (2000)

10

(28)

High
Low
<latexit sha1_base64="CYSjZsLtwqh2CP0WXRCoNo3hOGE=">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</latexit><latexit sha1_base64="EAtdI5txssP3lnckDmZ/vqWC5xU=">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</latexit>⌅(⇣)<latexit sha1_base64="Fu5NT7u+VobfLDqK1MyDMZm0qMk=">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</latexit><latexit sha1_base64="y7Z3miTEvZmotL8DK5uQnD/K8Gw=">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</latexit>⇠
2⇣<latexit sha1_base64="PBsx0D/bUMr5qkhSaC1487MV6S4=">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</latexit>

1<latexit sha1_base64="59RiNhV1+m+EtfOAHWmcGCGqyz8=">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</latexit>

⇣L

1.409

<latexit sha1_base64="YE9Ya1ECqEVngB3uGIx7/eKKv8k=">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</latexit><latexit sha1_base64="Dn9dvcQIer7RDI4DhTnbCKxBip4=">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</latexit>⇣H<latexit sha1_base64="R/hARMtEXLAIw3KtVg7nL8GI/zk=">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</latexit>

6<latexit sha1_base64="jZ+8koCITJ6UZBOkbZ99c6U+AEo=">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</latexit>

Figure 1: Determination of the Pareto exponent ζ.
∂f (n, t)= 0 in steady state. Using the candidate solution f (n) = Cn−ζ−1 , it is∂teasy to show
σ2(29)
0=µ+ ζ
2

We havewhich defines the Pareto exponent ζ. Note that the number of incumbent firms M is given byR∞
M = 1 f (n) dn, which allows us to pin down the value C = M ζ. Therefore, the distributionof n is given by
f (n) = M ζn−ζ−1 .

(30)

Recall that all sectors are symmetric, and hence (30) holds in each of all sectors k ∈ [0, 1].
R∞
To express the number of firms M in terms of the Pareto exponent, note that 1 nf (n) dn =1 where nf (n) is equivalent to the number of goods produced by firms with n products. The
LHS is equated to the total number of intermediate goods in each sector , 1 on the RHS. One
can easily confirm that1
M =1− .
ζ(31)

To understand (31), recall that the number of goods n produced by a firm is countable in acontinuum of (j, k) ∈ [0, 1] × [0, 1], and that firms produce one or more products. Therefore,
we must have M < 1, and M = 1 holds only when each of all firms produces a single product.
(31) shows that the number of firms is increasing in the Pareto exponent ζ. An intuition goesas follows. A higher ζ implies that a tail is thinner with less of larger firms, and that more
firms are smaller than otherwise. Note that monopoly profits are also Pareto-distributed withthe exponent ζ because they are proportional to the number of products, i.e. nπ.
Now, making use of (19), (20), (26), (29) and (31), one obtains

ξ
=
ϕ2


ζ
3 + ζ−1
ζ
6−ζ

where
ξ≡

gE
.
gI

11

≡ Ξ (ζ)

(32)

(33)

This equation should be interpreted to determine the Pareto exponent ζ, taking ξ and ϕ asgiven. That is, (32) captures the mechanism which translates the Poisson arrival rates gE andgI into the distribution of firm size. Figure 1 shows (32). It takes a U shape with asymptotes
at 1 and 6. (32) also shows that (i) two values of ζ, denoted by ζL and ζH , are consistent
with a given value of ξ, as long as the condition in the next lemma is met (ζL = ζH is also
possible), and (ii) a given value of ζ is consistent with “many” combinations of gE and gI as
long as their ratio is constant.
Lemma 4. For a given value of ξ, the existence of real-valued ζ requires
 v

r ! 
 u

u
56
ϕ ∈ 0, min 1, t 3 −
ξ .


9


(34)

Proof. Rewrite (32) as a (ϕ) ζ 2 + b (ϕ) ζ + c = 0 where a (ϕ) = ξ + 4ϕ2 , b (ϕ) = − 7ξ + 3ϕ2

and c = 6ξ. Then, real-valued ζ exists forh (ϕ) = b (ϕ)2 − 4a (ϕ) c ≥ 0(35)which is a fourth-degree polynomial equation in ϕ. The coefficient of ϕ4 is positive, hence∂h (ϕ)
limϕ→−∞ h (ϕ) = limϕ→∞ h (ϕ) = ∞. We have= 6 7ξ + 3ϕ2 − 96ξ 2ϕ, which∂ϕmeans that the local maximum is achieved at ϕ = 0 and h (0) = 25ξ 2 > 0. To find roots for2
h (ϕ) = 0, rewrite the equality part of (35) as h̃ (Φ) = AΦ2 + BΦ + C whereq Φ≡ ϕ , A = 9,

B = −54ξ and C = 25ξ 2 . Then, the roots for h̃ (Φ) = 0 are Φ1 , Φ2 = 3 ±56
9

ξ. Using thedefinition of Φ ≡ ϕ2 , the roots for h (ϕ) = 0 are given by
v
r !
u
u
56
t
ϕ1 , ϕ2 = ±
3+
ξ,
9

v
r !
u
u
56
t
ϕ3 , ϕ4 = ±3−ξ9Recalling ϕ ∈ (0, 1], the condition (34) follows.
This lemma shows a meaningful value of ζ is guaranteed, as long as the strength of thepositive externality ϕ is sufficiently small. An interesting implication is that the mere existenceof the externality, however small it is, is a necessary and sufficient condition for an equilibrium.
Given Lemma 4, the next result is obvious.
Proposition 4. (1) Multiple equilibria exist such that ζL < ζH if (35) holds with a strict
inequality, and (2) ζL = ζH if the inequality is replaced with an equality in (35).
Let us develop an intuition for multiple values for the Pareto exponent. This is basically
due to the positive externality in incumbent R&D, which is captured in the form of n in(3), and in particular nk in (4). To explain its role, let us distinguish three types of creative
destruction of innovation, which affect the Pareto exponent ζ in (32). Consider entrant firms.
As they enter the market, their higher-quality goods replace the existing goods. Incumbents
that suffer losses become smaller in size and even exit the market. It is easy to understand12

that a higher gE bring more of younger and smaller firms, tending to increase ζ. This is thefirst creative destruction effect.
The second and third effects are due to incumbent innovation. To isolate them, let us
consider first the situation where there was no positive externality, i.e. each incumbent R&D
success generates a single innovative product rather than multiple goods for n ≥ 2. Then, ahigher gI tends to reduce the Pareto exponent ζ because incumbents grow faster and forcing“marginal” firms with a few products to exit. As a result, there are less of smaller firms andmore of larger firms in the right tail. Conversely, a lower gI raises ζ. This is the second
creative destruction effect. Note that a higher gE and a lower gI work in the same way in that
the Pareto exponent ζ and the number of firms M both increase.
Introducing the positive externality into the picture makes it possible to identify the third
creative destruction effect. Now each incumbent R&D success generates ϕn number of new
goods on average. Such externality seems to reinforce the second creative destruction effect,
but it turns out that it affects the Pareto exponent in a different way. Remember the following
relationship:
n=

1
ζ
=
M
ζ −1

∀k.

(36)

n is the number of products per firm which is equivalent to the inverse of the number of firms
1/M , which in turn negatively related to ζ (see (31)). Consider a higher gI . The Pareto
exponent ζ tends to fall due to the second creative destruction effect, as explained above.
This reduces the number of firms M falls, and the resulting increase in n causes the following
ramifications. Each incumbent R&D success creates a greater number of innovative goods on
average, and that all incumbents become more likely to lose their own goods due to rival R&D
successes than otherwise. As a result, ceteris paribus, firms grow slower, and even shrink insize, falling leftward in the distribution of firm size, i.e. less of larger firms. This is the third
creative destruction, induced by the externality, and tends to raise the Pareto exponent ζ. A
similar story holds for entrant innovation. A lower gE reduces the pool of firms M via the firstcreative destruction effect, increasing the average number of products n. Less entry intensifies
the third creative destruction effect. Conversely, a higher gE and/or a lower gI weakens thethird creative destruction effect and helps incumbents consolidate their monopoly positions
on average, reducing ζ. Note that this mechanism exists due to the externality captured by nin (3), however small its strength ϕ is. Using ξ ≡ gE /gI , let us summarise the three effects:
1. In the first creative destruction effect, a higher gE raises ξ and tends to increase ζ.
2. In the second creative destruction effect without the externality effect, a lower gI raises
ξ and tends to increase ζ.
3. In the third creative destruction effect with the externality effect, a lower gI raises ξ andtends to decrease ζ.
In Figure 1, the third creative destruction effect dominates the other two effects at ζL , and thereverse holds at ζH . Now, let us consider the impact of a lower ϕ. It shifts up the horizontal
line. If we focus on ζL , a sufficiently small ϕ can make the Pareto exponent ζL arbitrarily
13

close to 1, which is the empirically relevant value (more on this point later). This statement,
however, does not take into account how ξ responds to ϕ. In addition, the deviation of ζHfrom 1 gets larger. This highlights the importance of stability analysis of an equilibrium. To
examine those issues, we next endogenise gE and gI to develop a full-fledged Schumpeterianmodel.

3

Endogenising gE and gI

3.1

Consumers

The number of consumers is normalised to one, and they maximise the intertemporal utility
with a logarithmic felicity function. This assumption gives the Euler condition
Ė
=r−ρ
E

(37)

where E is equivalent to aggregate consumption expenditure, r is the interest rate and ρ is therate of time preference. Consumers are identical except that L of them are unskilled and H are
skilled. Both L and H are constant, and L+H = 1. The assumption of heterogeneous workers
enables us to show our key results in the simplest possible setup. The appendix develops themodel with homogeneous workers and establishes that the key results remain intact.

3.2

Intermediate Goods

Final output Y is competitively produced. Normalising its price PY = 1, a demand functionof intermediate product is given by
x (j, k) =E
p (j, k)(38)

where E = Y and p (j, k) is the price of x (j, k).
One unit of intermediate goods is produced with an unskilled worker. Like other Schumpeterian models, firms charge the price with the quality step λ as a constant markup over themarginal cost, i.e. p (j, k) = λwL where wL is unskilled wage. Profit per product is
π = ΛE
where Λ ≡ 1 −

3.3

(39)

1
. This is equivalent to π in Section 2.1.
λIncumbent R&D

After entry, a firm starts incumbent R&D in all sectors, employing skilled workers, in order toincrease profits further. Consider a firm producing n ≥ 1 products. An R&D success in each14

sector follows a Poisson process with an arrival rate ofgI (k) = δI

RI (k)nγ,

δI > 0,

0<γ<1

(40)

where RI (k) is the number of skilled workers used in a sector k. The presence of n in the
denominator on the RHS is due to the negative congestion externality. As the portfolio ofgoods expands, it gets harder to generate an additional research success. As explained above,
if successful in a given sector, n number of new goods are created and ϕn are implemented
on average in randomly selected industries in the sector, raising quality of those goods by afactor λ.
Let us consider an R&D decision facing an incumbent firm with n products. Let Vn denotethe value of the firm, which is defined as
Z 1
rVn = nπ − wHdVn
RI (k) dk + E
dt
0


(41)


dVn
is an Ito calculus term and represents the capital gain/loss
where wH is skilled wage. E
dt
due to the GBM of n. The next lemma shows what is involved in the term.
Lemma 5. The Ito calculus term is given by





Z
∂Vn
gE
ϕ2 g I
ϕ2 1
RI (k) γ∂Vn
dVn=+ −gE +
+
+
δI
dk
n
E
dt
∂t n fixed
6
6M
2 0
n
∂n



 2
Z
gE
ϕ2 g I
ϕ2 1
RI (k) γ∂ Vn 2
+
+
+
δI
dk
n
6
6
2 0
n
∂n2


(42)

where g I is the arrival rate for rival incumbent firms.
Proof. By Ito’s Lemma, we have
dVn =

∂Vn
∂Vn
1 ∂ 2 Vn 2dt +[θn (t) dt + σn (t) dW (t)] +σ n (t)2 dt
∂t
∂n
2 ∂n2

where we used (25), (dt)2 = dt · dW (t) = 0 and (dW (t))2 = dt. Dividing both sides by dt andtaking expectations, we have

dVn
∂Vn
∂Vn
1 ∂ 2 Vn 2E=+
θn (t) +σ n (t)2
dt∂t n fixed
∂n
2 ∂n2


(43)

where E [dW (t)] = 0 is used. The first term on the RHS takes n as given. Making use of (19),(20) and (26), (43) can be rearranged into (42).
(41) and (42) show that the incumbent firm can increase the capital gain term by investingin R&D, incurring skilled wage costs. The optimal number of skilled workers can be found by15

maximising the RHS of (41) by choosing RI (k). The first-order condition is

wH =

ϕ2
γδI RIγ−1
2

where
RI ≡



∂Vn ∂ 2 Vn
n
+
∂n
∂n2

RI (k)
n



∀k.

(44)

(45)

(45) implies that R&D workers per product is the same in all sectors and for all incumbent
firms. This result allows us to rewrite (40) as
gI = δI RIγ

∀k.

(46)

This is indeed equivalent to gI in Section 2.1. Because each of n products equally contributes tothe value of the firm Vn by generating a stream of profit π, equilibrium must be characterised
by Vn = nv where v is the contribution of a single product. Using this, it is obvious that∂Vn
∂ 2 Vn= v and= 0. Now, the FOC (44) is reduced to
∂n
∂n2
1

wH =

ϕ2 γδIγ
1−γ
γ

v.

(47)

2gI

The RHS is the expected marginal benefit of employing an additional skilled worker, and the
LHS is the associated marginal cost. The RHS negatively depends on the Poisson arrival rate
gI because of the diminishing marginal product of skilled workers. This optimal condition
holds for all incumbent firms.

∂Vn= nv̇ in (42). Using this result, (26), (29), (41)∂t n fixedand (47), the value of a single product to the the firm, v, is determined by the following asset
Note also that Vn = nv means

equation
r=
where

π wH
v̇
Σ (ζ, gE , gI )−RI + − gE +
v
v
v
2



1
ζ
2
σ = gE + ϕ 1 +gI ≡ Σ (ζ, gE , gI )3
3 (ζ − 1)
2

(48)

(49)

is derived from (20) and (29). The RHS of (48) is the overall return from producing a single
product to the firm. The first term depends on profit per product and the second term captures
R&D costs. The remaining terms represent capital gains/losses. v̇/v reflects the cost side of
R&D in equilibrium. −gE captures a risk of losing profit because of entrant innovation. The
last term is due to the GBM of n, without which the model collapses to a standard qualityladder model. Lemma 5 shows that the expected change in Vn follows a stochastic process.
Given Vn = nv, v goes through similar stochastic changes, and such changes appear in theform of Σ (ζ, gE , gI ). Its positive impact is realised via the convexity of exponentiating G (t)in (23), i.e. Jensen’s inequality. Intuitively, such convexity amplifies the effect of an additional

16

product gained through R&D.

3.4

Entrant R&D

To enter a sector k, a firm has to be successful in R&D first. It employs skilled workers, andeach of them brings about an R&D success with a Poisson arrival rate of δE > 0. Let RE (k)be the total number of skilled worker employed in entrant R&D in a sector k. The Poisson
rate in the sector as a whole isgE (k) = δE RE (k) .

(50)

Once successful in R&D, innovation is implemented in an industry randomly selected from[0, 1] in a sector k.4 The production of a higher quality intermediate good begins with profit
π accruing to the firm, rendering the existing goods obsolete.
i (k) skilled workers. It generates a Poisson arrival rate ofConsider a firm i employing RE
i (k), incurring costs of w Ri (k). Free entry leads to
δE RE
H E

δ E v = wH ,

∀k.

(51)

Its LHS is the expected benefit of entry and its associated cost is on the RHS. This condition
holds in all sections, which means gE (k) = gE ∀k. gE is equivalent to gE introduced in Section
2.1.

3.5

Labour Market

Intermediate goods are produced using unskilled workers only. Their full employment requires
L=

E
.
λwL

(52)

Its RHS is the unskilled labour demand. Skilled workers are used in R&D activities. Those
R1
R1
who engage in entrant R&D is 0 RE (k) dk = 0 (gE /δE ) dk = gE /δE using (50).
Turning to incumbent R&D workers, consider a sector k. An incumbent firm with n 1γ
products employs RI (k) = gδIIn from (45) and (46). Summing all workers across all firms
 1

R∞
R ∞ gI  γ1R∞
gI γ
gives 0 RI (k) f (n) dn = 0
nf
(n)
dn
=
because 0 nf (n) dn = 1, i.e. theδIδItotal number of products in a sector k is one. Finally, summing over all sectors, the total 1
R 1   γ1γnumber of skilled workers employed in incumbent R&D is given by 0 gδIIdk = gδII .
Therefore, skilled workers are fully employed for
gE
H=
+
δE
4



gI
δI

1
γ

.

(53)

We could assume free entry in each and all industries, but there is no significant change in key results.

17

gE<latexit sha1_base64="N8dNQ0Jv919ctBOVbmQ29o7euOs=">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</latexit>

The R&D Arbitrage
Condition<latexit sha1_base64="6vCb3n5mXr69bMfmwCpJid+jkBo=">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</latexit><latexit sha1_base64="zwKL9UWWqDweNWHJrJWJg7FRYsg=">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</latexit>⇠<latexit sha1_base64="0LeUCiwv/e52sbOgRPeZxFoIWFw=">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</latexit>

The Skilled LabourMarket Condition<latexit sha1_base64="5lGp4fZ9tUFGjSj6XyPEjqtGg8k=">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</latexit><latexit sha1_base64="3XsC6xJGgTLH4whmlkCNw9GOKBc=">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</latexit>

gI

0<latexit sha1_base64="vYe8HYUXoKupc+OMdpjZzZDzLi4=">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</latexit><latexit sha1_base64="tIfZNkJLDNubinZ2+kocXMuvL20=">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</latexit>

Figure 2: Determination of the Poisson arrival rates gE and gI .

3.6

Equilibrium

The model can be solved with four equilibrium conditions for four endogenous variables, gE ,
gI , ζ and ω ≡ wL /wH . Use (47) and (51) to derive what we call the R&D arbitrage condition
1−γ
γ

2
1
γ



gI

ϕ2 γδI

v
=
wH


=

1
.
δE

(54)

This condition equalises returns from incumbent and entrant R&D, making incumbent firms
indifferent between incumbent and entrant R&D. We assume that incumbent firms engage
in incumbent R&D only in what follows. The condition (54) defines the equilibrium valueof gE and gI along with the skilled market labour condition (53). Figure 2 shows a uniqueequilibrium. The slope of the ray from the origin is equivalent to ξ ≡ gE /gI . Once ξ is found,(32) determines ζ, i.e. ζL and ζH in Figure 1.
Next we derive the condition which determines ω. Use (37), (39), (47), (51) and (48) toobtainω̇ϕ2
Σ (gE , gI , ζ)= δE ΛLλω − γgI − gE +
− ρ.
ω
2
2

(55)

ω follows this condition, taking gE , gI and ζ as given.
To derive the growth rate of output, rewrite the production function (17) as ln (Y ) =R1R1
0 0 {ℓ (j, k) ln [q (j, k)] + ln (L)} djdk, using (38) and (52). Considering a small time intervalẎdt, the rate of growth in Y is given by gY ≡= (λ − 1) (meE + meI ) where meE and meIY
are the expected number of innovative products created via entrant and incumbent R&D,
respectively. The presence of λ is due to the fact that each innovation improves the qualityof goods by a factor λ.5 It should be clear that meE = gE . Regarding meI , it is given byR
meI = ϕn s∈M mI (s) ds, which is equivalent to the second term inside the parenthesis in (4)
R1R1
R1R1
To be more precise, we have meE ≡ 0 0 mE (j, k) djdk and m̂I ≡ 0 0 mI (j, k) djdk where mE (j, k)
and mI (j, k) are the number of implemented innovations in an industry j in a sector k due to entrant andincumbent R&D, respectively.
518

if Mk , a set of incumbents in a sector k, is replaced with M, a set of firms in the economy asa whole. Using n = 1/M and (31), the growth rate isζ
gY = (λ − 1) gE + ϕ
gI
ζ −1


.

(56)

It should be intuitively clear that gY increases in gE , gI and ϕ. The average number ofζproducts produced by incumbents n =is due to the positive externality. An intuition
ζ −1for gY falling in ζ goes as follows. A higher ζ means a thinner Pareto tail, i.e. more of smallerfirms and less of larger firms, which in turn implies a fall in the average number of innovation,
decelerating growth.

3.7

Local Stability

In the above analysis, the Pareto exponent can take either ζL or ζH , ignoring the case ofζL = ζH . This section explores their stability property to investigate which Pareto exponent
is likely to prevail in equilibrium. (32) shows that ζ depends on gE and gI only, which in turnare pinned down by (53) and (54). An important property is that gE and gI are constant in as
well as off steady state. Taking advantage of this, we follow a heuristic approach to examine
stability. Assume that the economy is not in steady state, but sufficiently close to it such thatthe distribution of n is given by
f (n, t) = [ζ (t) − 1] n−ζ(t)−1(57)

where ζ (t) is a function of time and differs from ζL and ζH . This function collapses to asteady state Pareto distribution, defined by (30) and (31). In a nutshell, we are considering
the economy in the neighbourhood of a steady state where n is Pareto-distributed but withan exponent different from ζL and ζH . Note that f (n, t) in (57) still follows the Kolmogorov
forward equation (28). Given this observation, we state the next lemma:
Lemma 6. The following differential equation dictates the evolution of ζ (t):Σ (ζ (t) , gE , gI )ζ̇ (t) = −gE +ζ (t) [ζ (t) − 1] ζ (t)2 ≡ Ψ (ζ (t)) .
2(58)

Proof. Evaluating both sides of (28) with the use of (57), we haveζ̇ (t)
1
Σ (ζ (t) , gE , gI )− log (n) f (n, t) = −gE +ζ (t) ζ (t) f (n, t) .
ζ (t) − 12

where (19) and (49) are used. Integrating both sides w.r.t. n from 1 to infinity gives (58) afterR∞R∞
using 1 f (n, t) dn = M (t) = 1 − 1/ζ (t) and 1 log (n) n−ζ(t)−1 dn = 1/ζ (t)2 .
A convenient feature of (58) is that it alone determines the nature of local stability of ζ (t),given gE and gI . To explain (58) further, consider what if σ 2 or Σ (.) in (49) was independent of
ζ. In this hypothetical case, ζ would be fixed by ζ = 1 − 2θ/σ 2 (where the RHS is independent
19

˙
⇣(t)
<latexit sha1_base64="8dAKahESZUILP+widGhHeKvWI8g=">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</latexit><latexit sha1_base64="91WyXbhxSTD4LmtHIHOBqmINwCU=">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</latexit>(⇣(t))

1<latexit sha1_base64="zw+CPPSBXT4xERfYAaF5qNkZ9GM=">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</latexit>

0<latexit sha1_base64="QXklKmTP6rY8U9C6Yy22TgrijKM=">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</latexit>⇣(t)<latexit sha1_base64="vYe8HYUXoKupc+OMdpjZzZDzLi4=">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</latexit>⇣L⇣H<latexit sha1_base64="cVsh7gqpM4BhCxnx3O8uhvExSAA=">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</latexit><latexit sha1_base64="VMbA0PEdbWvq49YJVAwQ3JGq868=">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</latexit>

Figure 3: The local dynamics of ζ (t).
of ζ), which would replace (32). Obviously the “gradual” dynamics of ζ does not arise. In thissense, the differential equation (58) makes sense due to the dependence of σ 2 or Σ (.) on ζ inthe (49).
We are ready to state the next proposition.
Proposition 5. ζL is locally stable, and ζH is locally unstable.
To explain this proposition, consider ζ̇ (t) = 0 or
0 = Ψ (ζ (t)) .

(59)

Basically it is a fourth-degree polynomial equation. It is easy to confirm the following properties:
1. the coefficient of ζ 2 is positive, i.e. limζ→−∞ Ψ (ζ) = limζ→∞ Ψ (ζ) = ∞.
2. there are two repeated roots of 0, corresponding to 0 = ζ 2 .
ih
3. The remaining two roots are defined by 0 = −gE + Σ(ζ,g2E ,gI ) ζ (ζ − 1), which is in fact
equivalent to (32).6 It means that those roots are ζL and ζH .
These observations allow us to draw Figure 3. The shaded area is irrelevant for our analysis.
The figure confirms Proposition 5

3.8

Key Comparative Statics in Steady State

Table 1 summarises key comparative static results in steady state with ζ̇ = ω̇ = 0. An increasein ϕ encourages incumbent R&D, and hence, skilled workers are diverted from entrant R&D
with a drop in gE . In Figure 2, the R&D arbitrage condition shifts rightward with a lower
ξ. Consequently, ζL rises, as can be confirmed in Figure 1. The effect of changing δE and δI
involves general equilibrium effects. Consider a higher δI . It makes incumbent R&D relatively6(ζ − 1) does not give a root because of the presence of (ζ − 1) in the denominator in σ 2 or Σ (ζ, gE , gI ).

20

gI
gE
ξ
ζL
gY

ϕ
+
−
−
+
±

δI
+
−
−
+
±

δE
−
+
+
−
±

H
0
+
+
−
+

L
0
0
0
0
0

ρ
0
0
0
0
0

λ
0
0
0
0
0

Table 1: Comparative statics in steady state. The sign ± indicates an ambiguous change, and0 means no change.
more attractive compared with entrant R&D, shifting the R&D arbitrage condition rightward
in Figure 2. At the same time, incumbents become more productive in the sense that agiven gI can be achieved with less workers. It makes more skilled worker available to both
entrant and incumbent R&D. This is represented by an anti-clockwise pivot of the skilled
labour market condition around the vertical intercept in Figure 2. gI increases more than in
the case of an increase in ϕ, while gE falls because the general equilibrium effects on gE is
not sufficiently large. As a result, ξ unambiguously decreases. Its effect on ζL is the sameas in ϕ. Similar mechanisms work for a higher δE , but reversing the effect on gI , gE , ξ andζL . Regarding gY , the impacts of ϕ, δI and δE on it are ambiguous because gI and gE movein opposite directions. On the other hand, a greater H unambiguously raises output growth
partly because gI is independent of H. Table 1 shows that L, ρ and λ have no effects on
the Poisson rates, the Pareto exponent and output growth. This result is an artefact of theassumption that skilled workers only are used in R&D activities.

3.9

Zipf ’s Law

A firm size distribution is often found to follow a Pareto distribution with an exponent beingslightly above 1. Measuring firm sizes by employment, Axtell (2001) shows that the Pareto
exponent is 1.059 for the US data of 5.5 million firms in 1997. Toda (2017) also reports 1.0967
using the 2011 US data and slightly higher values (e.g. around 1.1) over the period 1992–2011
7 Let us consider how our model can be used to explain such data regularity.

Suppose that the externality weakens, i.e. ϕ decreases. Table 1 shows that ζL falls, andthis result is represented by an upward shift of the horizontal line in Figure 1 because it ξ/ϕincreases. If we reduce ϕ further, ζL can be made arbitrarily close to 1 as long as ϕ > 0. In
this sense, Zipf’s law emerges with a sufficiently weak positive externality.
Then, how small ϕ should be? To answer this question using a numerical example basedon estimates, we slightly generalises the assumption regarding the quality improvement caused
by innovation. Let us assume that the quality of a product increases by a factor λE > 1 if
innovation is generated by entrants, and by a factor λI in the case of incumbents. Following
Acemoglu and Cao (2015), assume λE > λI to capture the “radical” nature of entrant innovation. To keep the equilibrium conditions in tact for simplicity, let us further assume that
7

There are studies which propose non-Zipf distributions. For example, see Kondo, Lewis, and Stella (2023)
for a recent study.

21

part of entrants’ innovative ideas is available in public, so that it can be used by competitive
copycat firms. To be more precise, if q−1 (j, k) is the quality of the previous product, andλE
the fringe firms can produce the goods with quality
q−1 (j, k) whenever entrant innovation
λI
occurs. Entrants’ monopoly position is eroded and they charge the price λI wL , as incumbents
do. If we let λ = λI , all equations derived above unchange except the growth rate (56), which
is now given by
gY = (λE − 1) gE + (λI − 1) ϕ

ζ
gI .
ζ −1

(60)

We take advantage of this equation to calibrate the values of ϕ and ξ. For this, define(λE − 1) gE
the contribution of entrant and incumbent innovation to growth as SE ≡andgY
ζ(λI − 1) ϕ
gI
ζ −1
SI ≡
, which in turn givegYξ≡λ I − 1 SE
ζ
gE
=
·
·
ϕ.
gI
λ E − 1 SI ζ − 1(61)This equation relates ξ and ϕ. In particular, (32) and (61) constitute the system of two
equations for (ϕ, ξ), taking the values of ζ, λE , λI , SI and SE as given. We use three values(1.01, 1.059, 1.1) for ζ as the target Pareto exponent (1.059 is the value reported in Axtell(2001)). In setting the quality steps, we adopt λI = 1.2 and λE = 2 used in Acemoglu andCao (2015). They also consider λE = 3, and we briefly mention that case later. Regarding
SE and SI , we take values from Garcia-Macia, Hsieh, and Klenow (2019) who decompose thecontribution of entrant and incumbent innovation to aggregate TFP growth. In the period of
1983-1993, for example, 32.3% of TFP growth is attributed to entrants, and the complementary
fraction 67.7% comes from incumbents.
Table 2 reports the result. We take ζ = 1.059 as the benchmark case. First note thatthe contribution of entrants to growth becomes smaller in more recent years, while that ofincumbents gets larger, as SE and SI show. This is interpreted as the declining businessdynamism (see Akcigit and Ates (2021) for example). This fact is reflected in a decline inboth ξ and ϕ over time. Second, all values of ϕ are less than 0.5. Note that ϕ does not need
to be extremely low for a reasonable value of the Pareto exponent. Third, in the benchmark
case, the average of ϕ is approximately 0.27, meaning that only about a quarter of incumbent
innovation succeeds to be implemented. Fourth, the table is based on λI = 1.2 and λE = 2.
λE − 1
It means= 5, i.e. productivity gains from entrant innovation is five times as large as
λI − 1
λE − 1incumbent innovation. Acemoglu and Cao (2015) use λE = 3, i.e.
= 10. In that case,
λI − 1the value of ϕ halves, and ξ also moderately gets lower.
Let us compare our mechanism to generate Zipf’s law with the existing explanations using
the GBM in the literature. A typical account goes as follows. Consider a GBM dm (t) =ηm (t) dt + νm (t) W (t) where m (t) is the size of units, e.g. firms or cities, with the minimum22

λI = 1.2,

λE = 2

ζ = 1.010
ζ = 1.059
ζ = 1.100

Periods(SE , SI )
ϕ
ξ
ϕ
ξ
ϕ
ξ

1983 − 1993
(0.323, 0.677)
0.458
4.412
0.381
0.653
0.334
0.351

1993 − 2003
(0.224, 0.776)
0.277
1.615
0.231
0.239
0.202
0.128

2003 − 2013(0.198, 0.802)
0.237
1.181
0.197
0.175
0.173
0.094

Table 2: Calibrated values of (ϕ, ξ) required for different values of ζ, λE , SE and SI for λI .
The values of λE and λI follow Acemoglu and Cao (2015). The values of SE and SI for thethree periods are taken from Table V of Garcia-Macia, Hsieh, and Klenow (2019).
−ζm2η
where ζ = 1 − 2 .
mminν
Then, if the mean growth rate η is sufficiently close to 0, we obtain ζ ≈ 1. The result
ζmmin . This gives
can also be expressed in terms of the mean of m, denoted by m =
ζ −1
1
ζ=
. If the mmin is sufficiently small relative to the mean, we have again ζ ≈ 1.
1 − mmin /m
Alternatively, introducing Poisson birth/death of a unit at a given size without a minimum
being mmin . The stationary distribution is given by f˜ (m) =size gives rise to a double-Pareto distribution (see (Gabaix (2009))). No externality plays arole in these explanations.

4

Conclusion

The paper is an attempt to provide a microfoundation of the GBM of firm size in the sense that
we have derived it in the standard Schumpeterian endogenous growth model where Poissondistributed innovation drives growth. The key driving force in our analysis is the positive
externality which makes larger firms more productive. It was established that the distributionof firm size is Pareto, and the Pareto exponent can have a low or high value because of theexternality. In a locally stable equilibrium, the Pareto exponent takes a low value where theincumbent creative destruction effect dominates the entrant counterpart. In particular, thestable value of the Pareto exponent can be arbitrarily close 1 if the degree of the externality issmall. Our model is based on the fairly familiar endogenous growth model, and in this sense,
it can be easily extended to analyse, for example, international trade and the effect of policy
such as patents and taxes on long-run growth and firm size distribution.

References
Acemoglu, D., and D. Cao (2015): “Innovation by Entrants and Incumbents,” Journal of
Economic Theory, 157, 255–294.
Aghion, P., and P. Howitt (1992): “A Model of Growth Through Creative Desctruction,”
Econometrica, 60(2), 323–351.

23

Ahn, S., G. Kaplan, B. Moll, T. Winberry, and C. Wolf (2018): “When inequality
matters for macro and macro matters for inequality,” NBER macroeconomics annual, 32(1),
1–75.
Akcigit, U., and S. T. Ates (2021): “Ten facts on declining business dynamism and lessons
from endogenous growth theory,” American Economic Journal: Macroeconomics, 13(1),257–298.
Arellano, C., Y. Bai, and P. J. Kehoe (2019): “Financial frictions and fluctuations involatility,” Journal of Political Economy, 127(5), 2049–2103.
Axtell, R. L. (2001): “Zipf distribution of US firm sizes,” science, 293(5536), 1818–1820.
Beare, B. K., and A. A. Toda (2022): “Determination of Pareto exponents in economic
models driven by Markov multiplicative processes,” Econometrica, 90(4), 1811–1833.
Benhabib, J., A. Bisin, and S. Zhu (2011): “The distribution of wealth and fiscal policy ineconomies with finitely lived agents,” Econometrica, 79(1), 123–157.
Bilal, A., N. Engbom, S. Mongey, and G. L. Violante (2022): “Firm and worker
dynamics in a frictional labor market,” Econometrica, 90(4), 1425–1462.
Brzezniak, Z., and T. Zastawniak (2000): Basic stochastic processes: a course throughexercises. Springer Science & Business Media.
Clementi, G. L., and B. Palazzo (2016): “Entry, exit, firm dynamics, and aggregate
fluctuations,” American Economic Journal: Macroeconomics, 8(3), 1–41.
Di Giovanni, J., A. A. Levchenko, and R. Ranciere (2011): “Power laws in firm sizeand openness to trade: Measurement and implications,” Journal of International Economics,
85(1), 42–52.
Duffie, D. (2003): “Intertemporal Asset Pricing Theory,” in Handbook of the Economics ofFinance, ed. by G. Constantinides, M. Harris, and R. Stulz, vol. 1B, chap. 11, pp. 639–742.
Elsevier, Amsterdam.
Gabaix, X. (1999): “Zipf’s law for cities: an explanation,” The Quarterly journal of economics, 114(3), 739–767.
(2009): “Power laws in economics and finance,” Annual Review of Economics, 1(1),255–294.
Gabaix, X., J.-M. Lasry, P.-L. Lions, and B. Moll (2016): “The Dynamics of Inequality,”
Econometrica, 84(6), 2071–2111.
Garcia-Macia, D., C.-T. Hsieh, and P. J. Klenow (2019): “How destructive is innovation?,” Econometrica, 87(5), 1507–1541.

24

Grossman, G., and E. Helpman (1991): “Quality Ladders in the Theory of Growth,” Review
of Economic Studies, 58(1), 43–61.
Hopenhayn, H. A. (1992): “Entry, exit, and firm dynamics in long run equilibrium,” Econometrica, pp. 1127–1150.
Klette, T. J., and S. Kortum (2004): “Innovating Firms and Aggregate Innovation,”
Journal of Political Economy, 112(5), 986–1018.
Kondo, I. O., L. T. Lewis, and A. Stella (2023): “Heavy tailed but not Zipf: Firm andestablishment size in the United States,” Journal of Applied Econometrics, 38(5), 767–785.
Lucas, R. E. (1978): “On the size distribution of business firms,” The Bell Journal of Economics, pp. 508–523.
Luttmer, E. G. (2007): “Selection, growth, and the size distribution of firms,” The Quarterly
Journal of Economics, 122(3), 1103–1144.
(2010): “Models of growth and firm heterogeneity,” Annu. Rev. Econ., 2(1), 547–576.
Luttmer, E. G. J. (2011): “On the Mechanics of Firm Growth,” The Review of Economic
Studies, 78(3), 1042–1068.
Spanos, A. (1986): Statistical Foundations of Econometric Modelling. Cambridge University
Press.
Steindl, J. (1987): “Pareto Distribution,” in New Palgrave: A Dictionary of Economics, ed.
by J. Eatwell, M. Milgate, and P. Newman, vol. 3, pp. 809–811. Macmillan, London andBasingstoke.
Toda, A. A. (2017): “Zipf’s Law: A Microfoundation,” Available at SSRN 2808237.

A

The Case of Homogenous Labour

A.1

Equilibrium Conditions

In this appendix, we develop the model assuming no distinction between skilled and unskilledworkers, i.e. there are homogeneous workers who are employed to produce intermediate goods
and conduct entrant and incumbent R&D. The full employment of workers is now given by
Z
gE
L= +
+
λ
δE



gI
δI

1
γ

(62)

which replaces (52) and (53). L is the total number of workers, and w is used to denote wage.
Regarding R&D activities, the incumbent FOC (47) and the free entry condition (51) still
hold with wH being replaced with w. The R&D arbitrage condition (54) unchanged, while the25asset equation (55) is now
ϕ2Ż
Σ (gE , gI , ζ)= δE ΛZ − I γgI − gE +
−ρ
Z
2
2

(63)

E
. The evolution of ζ still follows (58).
w
Using those equations, the model can be reduced to the system of two differential equations

where Z ≡with two unknowns (Z, ζ):
 2


ϕ2I
ϕ gI
5
ζ
Ż = Z δE ΛZ − γgI − Γ (Z) + 1 +−ρ
2
6
3 (ζ − 1)
2


 
 
ϕ2 gI
ζ
ζ
Γ (Z) +1+ζ (ζ − 1) ζ 2
ζ̇ = − 1 −
62
3 (ζ − 1)


(64)
(65)



 1
Z
gI γ
where Γ (Z) ≡ δE L̃ −and gI is determined in (47).
, L̃ (gI ) ≡ L −
λ
δI

A.2

Steady State: 3 Cases

Consider steady state. (65) with ζ̇ = 0 gives


ϕ2 gI
Z = λ L̃ (gI ) − I Ξ (ζ) ≡ Θ (ζ) .
δE

(66)

ϕ2I gIΞ (ζ). There is Ξ (ζ) with the minus sign on the RHS. Therefore, Θ (ζ)δE
takes an inverted U shape, as in Figure 4. Two horizontal intercepts are defined bywhere L̃ (gI ) >δE L̃ (gI )= Ξ (ζ) ,ϕ2I gIand hence, we can reuse Lemma 4 to define the two roots, replacing ξ with(67)
δE L̃ (gI )in (34).
gI

Let us use ζ < ζ, to denote those roots. (64) with Ż = 0 gives

  

 2 
1 δE −1
5
ζ
ϕ gI
Z=
λ−
ρ + δE L̃ (gI ) − 1 − γ +≡ Ω (ζ)6 λ
63 (ζ − 1)2(68)

It is upward sloping with the horizontal intercepthi
3 ρ + 56 δE L̃ (gI ) ϕ22gI − (1 − γ)iζ̂ = h>13 ρ + 56 δE L̃ (gI ) ϕ22gI − (1 − γ) − 1(69)

We requireΩ (ζ) > 0

⇐⇒



5
2
ρ + δE L̃ (gI )> (1 − γ) .
6
ϕ2 gI

(70)

n
o
There are three cases, as illustrated in Figure 4 where three values of ζ̂ = ζ̂1 , ζ̂2 , ζ̂3 are26

<latexit sha1_base64="nYn+sItAkePnzVgRvBx9k3+rSrI=">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</latexit>

0 = ⇣˙ = ⇥(⇣)
<latexit sha1_base64="htiR0EucxvX+IUV0kMy/REG24+E=">AAACgnichVE9SwNBEH2eXzF+RW0EEcSoqEWYU4giCkEbS7+igpFwd1mTw8vdcbcJaLCy8w9YWCmIiJ3+BBv/gEV+glgq2Fg4uRyIijrL7sy8nTf7dld3LdOXRNUGpbGpuaU10hZt7+js6o719G76TskzRNpwLMfb1jVfWKYt0tKUlth2PaEVdUts6ftLtf2tsvB807E35IErdota3jb3TEOTDGVjgyO0kMk5spI5FFI7WshsFNiPB9nESDYWpwQFNvQzUMMgjtBWnNgVMsjBgYESihCwITm2oMHnsQMVBJexXVQY8zgyg32BI0SZW+IqwRUao/u85jnbCVGb81pPP2AbfIrF02PmEEbpka7phR7ohp7o/ddelaBHTcsBe73OFW62+6R//e1fVpG9ROGT9admiT3MBlpN1u4GSO0WRp1fPjx9WZ9bG62M0QU9s/5zqtI938AuvxqXq2LtDFH+APX7c/8MNqcSajKRXJ2KpxbDr4hgAMMY5/eeQQrLWEGazz3GNW5xpzQpk4qqTNdLlYaQ04cvpsx/ALa2lCM=</latexit>

Z<latexit sha1_base64="ozclheYkEPWITNzZoRpVl9Ig05M=">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</latexit>

0 = Ż = ⌦(⇣)⇣
<latexit sha1_base64="AAAalY1YHzLK3n3p8wcOzeJrlck=">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</latexit>

0<latexit sha1_base64="vYe8HYUXoKupc+OMdpjZzZDzLi4=">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</latexit>

1 ⇣ˆ1 ⇣
<latexit sha1_base64="RwBGzOGlQOtUUDW5mKWkxJsjcco=">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</latexit><latexit sha1_base64="QguiYjltZRRgxXBwZvTWbyy1rhk=">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</latexit><latexit sha1_base64="zw+CPPSBXT4xERfYAaF5qNkZ9GM=">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</latexit>⇣L ⇣ˆ2<latexit sha1_base64="p+m1YQCtlCUe7gLNC0TM28swRNA=">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</latexit>⇣ ⇣ˆ3<latexit sha1_base64="FWlL7qQ3IjWFPGQjMh3o0uOkxtc=">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</latexit><latexit sha1_base64="PWnemkhevNhrgczPtAEVKVuukeM=">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</latexit><latexit sha1_base64="YE9Ya1ECqEVngB3uGIx7/eKKv8k=">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</latexit>⇣H<latexit sha1_base64="R/hARMtEXLAIw3KtVg7nL8GI/zk=">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</latexit>

Figure 4: Three cases.
used for expositional purposes.8 In the case of ζ̂2 in the figure, ζ < ζ̂ < ζ ensures the existenceof an interior equilibrium. A sufficiently large L or/and a sufficiently small ρ leads to ζ̂1 withmultiple values of ζ in steady state, as in the main model. However, ζ̂ < ζ does not necessarily
guarantee the existence of equilibrium values of ζ, which we denote ζL ≤ ζH . The following
lemma summarises the condition required for it:
Lemma 7. Define
ϕ2I gI
δE
> 0,
B≡
>0
2δE
3


5
ϕ2 gI
C ≡ρ+
L̃ − (1 − γ)
δE > 0
6
2δE


λ−1
5
D≡
+
δE > 01 − sE
6
A≡

where C > 0 due to (70). Rearranging Ω (ζ) = Θ (ζ), one obtains
A1 ζ 2 + A2 ζ + A3 = 0
where


A1 = 8AD + AB + DL̃ − C > 0,
A2 = −6AB − 6AD − 7 DL̃ − C < 0,

A3 = 6 DL̃ − C > 0.

(71)
(72)
(73)

Assume L is large enough such that DL̃ − C > 0, implying that the inequalities in (71), (72)8

If ζ̂ changes its position, then so do ζ and ζ. But we ignore this to illustrate the three cases in the same
diagram.

27

and (73) hold. Then, ζL and ζH are given by ζL , ζH =

−A2 ±

p

only if A22 ≥ 4A1 A3 .

A22 − 4A1 A3, and exist if and
2A1

We can solve for gE , using the above equations:

gE =
where

1
λ−
6

−1
ΥE (ζ)

ΥE (ζ) ≡ δE (λ − 1) L̃ − ρ + 1 − γ +(74)ζ
3 (ζ − 1)



ϕ2 gI
> 0.
2

(75)

ΥE (ζ) is decreasing in ζ. Intuitively, a smaller ζ thickens the Pareto tail, i.e. the likelihoodthat entrants grow big gets larger in the sense that the average number of products per firm
increases. One can easily show that A2 < 0 means ΥE (ζ) > 0.

A.3

Stability Analysis

Linearise (64) and (65) around the steady state (Z ∗ , ζ ∗ ):
Ż
ζ̇

!

Z − Z∗

ΘZ (Z ∗ , ζ ∗ ) Θζ (Z ∗ , ζ ∗ )|{z}ζ − ζ∗!
=ΩZ (Z ∗ , ζ ∗ ) Ωζ (Z ∗ , ζ ∗ )

!
(76)

J

where


δE
1
λ−
> 0,
λ
6
ϕ2 gI
1
< 0,
Ωζ (Z ∗ , ζ ∗ ) = −Z ∗ I
6 (ζ ∗ − 1)2


δE
ζ∗
∗ ∗
ΘZ (Z , ζ ) =1−(ζ ∗ − 1) (ζ ∗ )2 > 0,λ6(2ζ − 3) (2ζ − 1) 2
(ζ − 1) ζ 2
∗ ∗
∗
ϕ
g
Θζ (Z , ζ ) = Γ (Z ) +> 0.
I6(ζ − 1)2ΩZ (Z ∗ , ζ ∗ ) = Z ∗We focus on the case of a large L such that Θζ > 0. In this case, the determinant and traceof the Jacobian, J in (76), are given by
Slope of Θ (ζ)
Slope of Ω (ζ)


z
z
}|
{
}|
{




 





(+)
(−)



z}|{
z}|{ 
(+) (+) z}|{z}|{  
Θ
Ω


ζ 
ζ 
Det (J) = ΩZ ΘZ − −
 + − , ΘZ   ΩZ  |{z}|{z} 








(+)
(+)






(+)

(+)

z}|{ z}|{Tr (J) = ΩZ + Θζ .

28

Let us first consider the case of ζ̂2 in Figure 4. Because of the assumption Θζ > 0, the slopeof Θ (ζ) is negative, as in the figure. This means Det (J) > 0 and Tr (J) > 0, hence (ζ ∗ , Z ∗ )is an unstable source. Next, let us turn to the case of ζ̂1 in Figure 4. Regarding (ζH , Z), its
stability property is the same as in the case of ζ̂2 in the figure. Consider ζL . The slopes of
0 = Ω (ζ) and 0 = Θ (ζ) are both positive, and the former is less steeper than the latter. It
means Det (J) < 0, hence (ζH , Z ∗ ) is a saddle point.

A.4

Zipf ’s Law

Consider ζL in the case of ζ̂1 in Figure 4. A smaller ϕ reduces ζ. This should be clear from thefact that (67) and (32) essentially takes the same form, and the LHS of (67) indeed increases∂ϕ2 gI
with a smaller ϕ because> 0 from (47). For the same reason, Ω (ζ) increases as ϕ falls∂ϕ(see (68)) and ζ̂ decreases (see (69)), implying that the 0 = Ż = Ω (ζ) curve in Figure 4 shifts
leftward. Therefore, ζL can be made arbitrarily close to 1, as ϕ decreases.

29