DISCUSSION PAPER SERIESDP14250RECONSIDERING THE MARKET SIZE
EFFECT IN INNOVATION AND GROWTH
Helene Latzer, Kiminori Matsuyama and Mathieu
Parenti
MACROECONOMICS AND GROWTH

ISSN 0265-8003

RECONSIDERING THE MARKET SIZE EFFECT IN
INNOVATION AND GROWTH
Helene Latzer, Kiminori Matsuyama and Mathieu Parenti
Discussion Paper DP14250
Published 25 December 2019
Submitted 22 December 2019
Centre for Economic Policy Research
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Copyright: Helene Latzer, Kiminori Matsuyama and Mathieu Parenti

RECONSIDERING THE MARKET SIZE EFFECT IN
INNOVATION AND GROWTH
Abstract
In the standard horizontal innovation model of endogenous growth, larger economies innovate
more and grow faster. Due to the homotheticity of preferences, however, it does not matter
whether the large market size comes from a large population or a high per capita expenditure. In
this paper, we extend the standard model to allow for nonhomothetic preferences. Among others,
we show that, holding the size fixed, economies with higher per capita expenditure and smaller
populations innovate more and grow faster for the empirically relevant case of incomplete passthrough, strategic complementarity in pricing, and procompetitive entry.
JEL Classification: O11, O31, O33
Keywords: Endogenous Growth, balanced growth, Horizontal innovation, Nonhomothetic
preferences, Directly explicitly additive (DEA) preferences, Demand composition, Incomplete passthrough, Strategic complementarity in pricing, Procompetitive entry, Competition and growth
Helene Latzer - helene.latzer@uclouvain.be
Paris-I, CNRS, France and University Saint-Louis, Belgium
Kiminori Matsuyama - k-matsuyama@northwestern.edu
Northwestern University and CEPR
Mathieu Parenti - mathieu.parenti@ulb.ac.be
Université Libre de Bruxelles and CEPR

Acknowledgements
We thank Philippe Aghion, Gino Gancia, Chad Jones, Xavier Vives, the workshop participants at University of Tokyo, Keio
University, Chicago Fed, as well as those at joint seminars in Saint Peterburg and Vienna, and at the conferences at Hitotsubashi
University and Peking University for their feedback. The usual disclaimer applies.

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Reconsidering the Market Size Effect in Innovation and Growth

Reconsidering the Market Size Effect in Innovation and Growth

Hélène Latzera

Kiminori Matsuyamab

Mathieu Parentic

December 21, 2019; 09:20

Abstract
In the standard horizontal innovation model of endogenous growth, larger
economies innovate more and grow faster. Due to the homotheticity ofpreferences, however, it does not matter whether the large market size comesfrom a large population or a high per capita expenditure. In this paper, we extend
the standard model to allow for nonhomothetic preferences. Among others, weshow that, holding the size fixed, economies with higher per capita expenditureand smaller populations innovate more and grow faster for the empirically
relevant case of incomplete pass-through, strategic complementarity in pricing,and procompetitive entry.
Keywords: Endogenous growth, Balanced growth, Horizontal innovation,
Nonhomothetic preferences, Directly explicitly additive (DEA) preferences,
Demand composition, Incomplete pass-through, Strategic complementarity inpricing, Procompetitive entry, Competition and growth
JEL classification: O11, O31, O33___________________aParis-I, CNRS, France & Université Saint-Louis, Belgium.
Northwestern University, USA. Corresponding author, email: k-matsuyama@northwestern.edu.
cUniversité Libre de Bruxelles, Belgium.
b*We thank Philippe Aghion, Gino Gancia, Chad Jones, Xavier Vives, the workshop participants at University ofTokyo, Keio University, Chicago Fed, as well as those at joint seminars in Saint Peterburg and Vienna, and at theconferences at Hitotsubashi University and Peking University for their feedback. The usual disclaimer applies.

1

Reconsidering the Market Size Effect in Innovation and Growth1.

Introduction
The standard horizontal innovation models of endogenous growth—say, Romer (1990),

Grossman and Helpman (1993, ch.3), Gancia and Zilibotti (2005), and Acemoglu (2008; ch.13.4)
just to name a few--, predict that countries with larger market sizes innovate more and hence
grow faster. However, due to the assumption of consumer demand homotheticity, thecomposition of the aggregate demand has no effect. In particular, it does not matter whether thelarge market size comes from a larger population size or a higher per capita expenditure. Oncethe aggregate market size is controlled for, poorer countries with larger population sizes innovate
as much as richer countries with smaller population sizes. Furthermore, without any effects ofthe demand composition, the (demand-side) market size effect becomes indistinguishable fromthe (supply-side) scale effect, for which there exists little supporting evidence.
Another well-known feature of the standard horizontal innovation models of endogenous
growth is that the markup rate at which innovated products are sold is determined solely by anexogenous preference parameter. Thus, the markup rate can change only by changing thepreferences, which makes it impossible to conduct any welfare analysis of markup rate changes.
Furthermore, any (exogenous) increase in the markup rate leads to an increase in the innovation
rate, contrary to some empirical evidence suggesting that competition fosters innovation andhence growth.1 Another implication of this feature is that a change in the production cost ofinnovated products has no effect on the profit share earned by the innovators, and hence on theinnovation and growth rates of the economy.
In this paper, we extend textbook horizontal innovation models of endogenous growth toallow for nonhomothetic preferences to examine the effect of the market size composition on
innovation and growth. We consider an economy, populated by 𝑁 identical agents, eachendowed with ℎ units of labor.2 For the preferences, we follow the footsteps of Dixit and Stiglitz(1977). Virtually all the existing horizontal innovation models of endogenous growth build on
their well-known model of monopolistic competition with homothetic CES preferences in DixitStiglitz (1977; Section I). Instead, we build on their lesser-known model of monopolistic
1

See, e.g., Blundell, Griffith, van Reenen (1995, 1999). Some economists view this as prima facie evidence against
horizontal innovation models of endogenous growth, in favor of vertical innovation and creative destruction models
of endogenous growth: see, Aghion et. al. (2005), Aghion, Akcigit, and Howitt (2015), and Acemoglu et. al. (2018).
2
Thus, ℎ measures the worker efficiency, and can be interpreted as the level of human capital or the quality of thelabor force. The total labor endowment of the economy is hence equal to 𝐿 = ℎ𝑁. With labor being the only factor
of production, 𝐿 is also the size of the economy.

2

Reconsidering the Market Size Effect in Innovation and Growthcompetition with directly explicitly additive (DEA) nonhomothetic preferences in Dixit-Stiglitz(1977; Section II), which contains homothetic CES as a knife-edge case. 3 A distinctive featureof monopolistic competition model with DEA is that the price elasticity of demand each firm
faces is a function of per capita consumption of its product only. We use this class of
nonhomothetic preferences for two reasons. First, it allows us to develop a horizontal innovation
model of endogenous growth that preserves the balanced growth property in spite ofnonhomotheticity. Second, it allows us to account for the following empirical observations: 4i)Incomplete (less than 100%) Pass-Through; An increase in the production cost reduces
the markup rate of monopolistically competitive firms.

ii)

Strategic Complementarity in Pricing; Each firm responds by increasing its price whencompeting firms increase their prices.

iii)

Procompetitive Entry; Other things being equal, the presence of more firms reduces themarkup rates,by departing from homothetic CES within this class to allow the elasticity of substitution
between products to be smaller at higher indifference curves. 5
Here are our main results. First, after controlling for the size 𝐿 = ℎ𝑁, a richer country
with a smaller population (a higher ℎ with a smaller 𝑁) innovates more and hence grows faster.
Hence, the composition of the aggregate demand matters, and the demand-side market size effectcan be distinguished from the supply-side scale effect. Second, even though both the markup rate3

See also Zhelobodko et al. (2012). Although Dixit and Stiglitz (1977) called Section II “Variable Elasticity Case”,
the well-known Bergson’s Law states that, within this class of preferences, they are homothetic if and only if they
are CES. In other words, any departure from CES within this class introduces nonhomotheticity. Hence, one could
equally call Section II “Nonhomothetic Case.” Of course, nonhomotheticity and non-CES are generally distinct
properties of preferences. Indeed, it is possible to have nonhomothetic CES as well as homothetic non-CES in a
broader class of preferences. However, such preferences would be incompatible with the balanced growth property,
which we would like to preserve in order to keep our departure from the standard model to the minimum.
4
See Berman et al (2012) as well as the review provided by De Loecker and Goldberg (2014) for incomplete passthrough; Fontagné et al (2018) and Amiti et al (2019) for strategic complementarity in pricing; Campbell andHopenhayn (2005) and Feenstra and Weinstein (2017) for procompetitive entry. The Dixit-Stiglitz (1977, Section I)
model with homothetic CES has been widely criticized because they are at odds with these empirical observations;the markup rate never changes in response to a change in the production cost, or in the market environment (i.e.,
pricing and entry of other firms).
5
Zhelobodko et.al. (2012) called it the case of “increasing relative love for variety (RLV).” It turns out that this isequivalent to assuming that demand for each product is more price elastic at a higher price (Marshall’s second lawof demand) within DEA. Of course, homothetic CES is a knife-edge case within DEA, so that, if we departed in the
opposite direction of decreasing RLV (i.e., the elasticity of substitution being larger at higher indifference curves),
we would have obtained the opposite results of more than 100% pass-through, strategic substitutes in pricing, andanti-competitive effects of entry. We view such a departure as empirically irrelevant.

3

Reconsidering the Market Size Effect in Innovation and Growth

and the innovation (and growth) rate stay constant over time along the balanced growth path,they are both endogenous and hence they could either be positively or negatively correlated
across the balanced growth paths, depending on the sources of underlying variations. We showthat the markup and innovation rates move in the same direction if changes are caused byexogenous variations in production cost or per capita expenditure, while they move in the
opposite directions if changes are caused by variations in the discount rate, the innovation cost or
the population size. This implies, in particular, that the measure of competitiveness and thegrowth rate are positively correlated in cross-sections of countries, if countries differ mostly in
the innovation (or firm entry) cost.
It should be noted that the (demand-side) market size effect on innovation and growth westudy is conceptually distinct from the (supply-side) scale effect, whose empirical validity has
been questioned by Jones (1995) and many others. Nonhomotheticity provides one natural wayof distinguishing these two effects. 6 Indeed, our results suggest that the difference in per capita
income across countries could potentially be one reason why there is little supporting evidence
for the scale effect. We also show that correlations between competition and growth acrosscountries depend on the sources of variations across countries. As such, our results suggest thathorizontal innovation models of growth can also contribute to the debate regarding competition
and growth.
Boucekkine et. al. (2017) also investigated the implications of departing from CES
preferences on the markup and growth rates in a balanced growth model of horizontal
innovation. However, their framework is too restrictive to study the effect of the demand
composition. Moreover, under the class of preferences they consider, the markup rule of thefirms is independent of the market environment (i.e., pricing and entry of other firms). Hence, it
can account neither for procompetitive effects of entry nor for strategic complementarity inpricing. In contrast, our framework is consistent with these two empirical findings, in addition to
incomplete pass-through.
2.

Back to the Basics: Innovation and Growth under CES

6

An alternative approach to distinguish the two has been pursued in the directed technological change literature,using multi-sector, multi-factor extensions of endogenous growth models, see, e.g., Acemoglu (2008, ch.15),
Aghion and Howitt (2008, ch.8), and Gancia and Zilibotti (2009).

4

Reconsidering the Market Size Effect in Innovation and Growth

We start with a benchmark balanced growth model with homothetic CES, which
encompasses two versions of the textbook models (Grossman and Helpman, 1993; Gancia andZilibotti, 2005) in order to facilitate comparisons with our model with DEA preferences.
2.1.

Intratemporal problem
Labor is the only factor of production. We consider an economy populated by

𝑁 identical agents, each supplying inelastically ℎ units of labor measured in efficiency units.
Hence, the total labor supply measured in efficiency units is given by 𝐿 = ℎ𝑁, which is also thesize of the economy.
Time is continuous and extends from 𝑡 = 0 to infinity. Intertemporal preferences of eachagent take the following form:
𝒰 = ∫ 𝑙𝑜𝑔 𝑈(𝐱 ) 𝑒𝑑𝑡,

where 𝑈(𝐱 ) is the intratemporal utility, 𝐱 = {𝑥 (𝜔); 𝜔 ∈ [0, 𝑉 ]} is the consumption profile,with 𝑉 being the range of the products that have been innovated by time t, and 𝑥 (𝜔) denoting
the consumption of product 𝜔 ∈ [0, 𝑉 ].
At time 𝑡, each agent earns a wage income equal to 𝑤 ℎ, and spends 𝐸 , and chooses 𝐱 to
maximize 𝑈(𝐱 ) subject to the intratemporal budget constraint,𝑝 (𝜔)𝑥 (𝜔)𝑑𝜔 = 𝐸 ,(1)

where 𝑝 (𝜔) denotes the price of product 𝜔 ∈ [0, 𝑉 ]. When 𝑈(𝐱 ) is a CES with the elasticity
of substitution 𝜎 > 1:
𝑈(𝐱 ) =

𝑥 (𝜔)

𝑑𝜔,

(2)

this intratemporal maximization problem yields the per capita demand curve for each product:[𝑝 (𝜔)]
𝐸,(𝑃 )with the price elasticity being constant and equal to 𝜎 > 1, where𝑥 (𝜔) =(𝑃 )≡[𝑝 (𝜔 )]𝑑𝜔 .

The total demand for product 𝜔 is simply given by 𝑞 (𝜔) = 𝑁𝑥 (𝜔).
2.2.

Firms’ intratemporal problem

5(3)

Reconsidering the Market Size Effect in Innovation and Growth

Each product 𝜔 ∈ [0, 𝑉 ] is produced and sold exclusively by a single firm, which is also
indexed by 𝜔 ∈ [0, 𝑉 ]. Producing one unit of each product requires 𝜓 efficiency units of labor.
Each firm chooses its price, 𝑝 (𝜔) or the quantity, 𝑞 (𝜔) = 𝑁𝑥 (𝜔), to maximize the profit,
𝜋 (𝜔) ≡ (𝑝 (𝜔) − 𝑤 𝜓 )𝑞 (𝜔) = (𝑝 (𝜔) − 𝑤 𝜓 )𝑁𝑥 (𝜔)subject to eq.(3) taking 𝑤 , 𝜓 , 𝑃 , and 𝐸 as given. This profit maximization problem has aunique solution, and hence all firms adopt the same pricing rule:𝑝 (𝜔) 1 −1
= 𝑤 𝜓 ⟺ 𝑝 (𝜔) = 𝑀𝑤 𝜓 ≡ 𝑝 ,𝜎where 𝑀 ≡ 𝜎⁄(𝜎 − 1) is the markup rate, which is exogenously constant under CES.
Because all firms set the same price, firm symmetry entails that all products are produced
and consumed by the same amount, and all firms earn the same level of profits:𝑞 (𝜔) = 𝑁𝑥 (𝜔) = 𝑁𝑥 = 𝑞 ; 𝜋 (𝜔) = 𝜋
and the intratemporal budget constraint, eq.(1), becomes simplified to:𝑝𝑥𝑉 =𝐸.
The above mark-up rule also implies that the share of the aggregate expenditure that goesto the firms’ profits is also exogenously constant and given by:(𝑝 − 𝑤 𝜓 )𝑞 𝑉
𝜋𝑉
𝑝 −𝑤 𝜓
1
(4)
=
=
=
𝑁𝐸
𝑝𝑞𝑉
𝑝
𝜎
Likewise, the share of the aggregate expenditure that goes to the wage payment in the production
sector is also exogenously constant, and given by:where 𝐿

𝑤 𝐿
1 1
(5)
= 1− =𝑁𝐸𝜎 𝑀= 𝜓 𝑞 𝑉 = 𝜓 𝑁𝑥 𝑉 denotes the total number of efficiency units of labor employedin the production of the existing products.
2.3.

R&D and resource constraints
Because firms are symmetric, the market value of each firm is the same and equal to𝐵 ≡∫ 𝜋 𝑒 ()

𝑑𝑠, where 𝑅 ≡ ∫ 𝑟 𝑑𝜏 is the cumulative interest rate and 𝑟 is theinstantaneous one. Log-differentiating this expression of 𝐵 with respect to t, we obtain:
𝐵̇ + 𝜋=𝑟.
𝐵
Innovating per unit of new products requires 𝐹 efficiency units of labor:
𝐹 𝑉̇ = 𝐿 ,6(6)

Reconsidering the Market Size Effect in Innovation and Growth

where 𝐿is the number of units of labor being employed in the R&D sector at time 𝑡. There is

free entry in the R&D sector. Hence, whenever the R&D sector is active, net returns from R&D,
𝐵 𝑉̇ − 𝑤 𝐿= (𝐵 − 𝑤 𝐹 )𝑉̇ , are equal to zero, which means that the cost of creating a

product (the R&D cost) and the value of creating a product (the value of a firm) are equalized:(7)
𝐵 =𝑤 𝐹.
Finally, the labor resource constraint, or the labor market equilibrium condition, is given by:ℎ𝑁 = 𝐿 = 𝐿
2.4.

(8)

= 𝐹 𝑉̇ + 𝜓 𝑁𝑥 𝑉 .

+𝐿Intertemporal problem
To describe the intertemporal maximization problem of the agent, we first derive theintertemporal budget constraint. Each agent holds 1/𝑁 fraction of the ownership shares of the
profit-making firms, hence their asset holding is 𝑎 = 𝐵 𝑉 /𝑁. At time 𝑡, an agent earns thewage income 𝑤 ℎ and the profit income 𝜋 𝑉 /𝑁, spends 𝐸 = 𝑝 𝑥 𝑉 and purchases assets (theownership shares of the new profit-making firms) by 𝐵 𝑉̇ /𝑁. The flow budget constraint ishence:
𝐵 𝑉̇ ⁄𝑁 + 𝐸 = 𝑤 ℎ + 𝜋 𝑉 ⁄𝑁
By adding the capital gains 𝐵̇ 𝑉 ⁄𝑁 on both sides, and using eq.(6) and the fact that 𝑎 =𝐵 𝑉 /𝑁, the above expression can be written as:𝑎̇ + 𝐸 = 𝑤 ℎ + 𝑟 𝑎
By integrating this expression from 𝑡 = 0 to infinity, we obtain the intertemporal budget
constraint:𝐸𝑒

𝑑𝑡 ≤ 𝑎 +with the no-Ponzi scheme condition, lim 𝑎 𝑒

𝑤 ℎ𝑒≥ 0. Subject to this intertemporal budget⟶

constraint, eq.(9), agents choose an expenditure path, {𝐸 }𝒰 =𝑙𝑜𝑔 𝑈(𝐱 ) 𝑒𝑑𝑡 =𝑙𝑜𝑔 𝑉 (𝑥 )

𝑒

=𝜆 𝑒

7

, so as to maximize:
𝑑𝑡 =The first-order condition is given by
1
𝑒
𝐸

(9)

𝑑𝑡,

,

𝑙𝑜𝑔 𝑉

𝐸
𝑝𝑉

𝑒

𝑑𝑡.

Reconsidering the Market Size Effect in Innovation and Growth

where 𝜆 is the Lagrange multiplier associated with eq.(9). Log-differentiating this first-ordercondition with respect to 𝑡 leads to the familiar Euler equation:(10)𝐸̇= 𝑟 − 𝜌.
𝐸
2.5.

The Balanced Growth Path
The balanced growth path (BGP) is defined as an equilibrium path satisfying thefollowing three conditions:i)

The growth rate of the range of products 𝑔 ≡ 𝑉̇ ⁄𝑉 is constant and positive.

ii)

The allocation of labor between the production and R&D sectors is constant: 𝐿
and 𝐿

iii)

= 𝐿∗

= 𝐿∗ .

The markup rate, 𝑀 ≡ 𝜎⁄(𝜎 − 1) , is constant, satisfied automatically under CES.
To guarantee the existence of such a BGP, we follow Grossman and Helpman (1993) and

Gancia and Zilibotti (2005) and many others by assuming that knowledge spillovers from past
R&D experiences reduce the cost of R&D as follows:𝐹 =which implies that 𝐿

𝐹
,
𝑉

(11)

= 𝐹 𝑉̇ = 𝐹𝑔 .

Regarding the production cost, 𝜓 , Grossman and Helpman (1993) assume 𝜓 = 𝜓 sothat knowledge spillovers are limited to R&D. In contrast, Gancia and Zilibotti (2005) assume
𝜓 = 𝜓/𝑉 so that they benefit both R&D and production equally. As will become clear below,
however, neither of these assumptions play any role in ensuring the existence of a BGP under
CES, so we intentionally leave 𝜓 unspecified in this section.
We now derive the law of motion for this economy under the assumption that the R&D
sector is active: 𝐿= 𝐹 𝑉̇ = 𝐹𝑔 > 0. By inserting eq.(6) into the Euler equation, eq.(10), we

obtain𝐸̇𝐵̇ + 𝜋=− 𝜌.
𝐸
𝐵Using eqs. (4), (7) and (11), this can be written as
𝐸̇
𝑤̇
𝑁 𝐸
=
−𝑔 +
− 𝜌.
𝐸
𝑤
𝜎𝐹 𝑤
By defining ℰ ≡ 𝑁𝐸 ⁄𝑤 , this can be simplified to:8

Reconsidering the Market Size Effect in Innovation and Growthℰ̇ℰ=− 𝑔 − 𝜌,ℰ
𝜎𝐹
while the labor market equilibrium condition eq.(8) becomes, using eq.(5):

(12)

1
ℰ.
𝜎
By combining eq.(12) and eq.(13), we obtain the law of motion for ℰ :

(13)

𝐿=𝐿

+𝐿

= 𝐹𝑔 + 1 −

(14)

ℰ̇
ℰ − ℰ∗=, where ℰ ∗ ≡ 𝐿 + 𝜌𝐹.
ℰ
𝐹

Since ℰ̇ > 0 for ℰ > ℰ ∗ and ℰ̇ < 0 for ℰ < ℰ ∗ , eq.(14) would imply divergence, leading to aviolation of the equilibrium conditions, unless the economy jumps immediately to ℰ = ℰ ∗ andstays at ℰ = ℰ ∗ . This in turns implies 𝐿= (1 − 1⁄𝜎 )ℰ ∗ ≡ 𝐿∗ and 𝐿= 𝐹𝑔 = 𝐿 − 𝐿∗ areall constant along the only equilibrium path and the economy stays on the balanced growth path,
as long as the parameters are such that the R&D sector is active: 𝐿∗ = 𝐿 − 𝐿∗ = 𝐿 −(1 − 1⁄𝜎 )ℰ ∗ > 0 ⟺ 𝑀 ≡ 𝜎⁄(𝜎 − 1) > 1 + 𝜌𝐹 ⁄𝐿 . Hence, we have:
Proposition 1A: Balanced Growth Path under CES
Suppose 𝑀 ≡ 𝜎⁄(𝜎 − 1) > 1 + 𝜌𝐹 ⁄𝐿 . Then, the economy jumps immediately to thebalanced growth path along which1
𝐿 + 𝜌𝐹(𝐿 + 𝜌𝐹) =
< 𝐿;
𝜎
𝑀
𝐿
1
1
𝜌𝐹
𝐿 = 𝐿∗ = − 1 − 𝜌𝐹 = 1 −
𝐿−
> 0;
𝜎
𝜎
𝑀
𝑀
𝐿
1
1 𝐿 𝜌
𝑔 = 𝑔∗ =− 1− 𝜌 = 1−
− > 0.
𝜎𝐹
𝜎
𝑀 𝐹 𝑀
𝐿

= 𝐿∗ = 1 −From Proposition 1A, one could immediately show
Proposition 1B: Comparative statics under CES
In the benchmark CES case,i)

Both an increase in the discount rate 𝜌 and in the R&D cost 𝐹 leave the markup rate 𝑀
unchanged, increase 𝐿∗ , decrease 𝐿∗ , and decrease the growth rate 𝑔∗ ;ii)

An increase in the total labor supply 𝐿 = ℎ𝑁 leaves the markup rate M unchanged,
and increases 𝐿∗ , 𝐿∗ , and 𝑔∗ ;iii)

An increase in the elasticity of substitution 𝜎 decreases the markup rate 𝑀, increases
𝐿∗ , and decreases 𝐿∗ , and 𝑔∗ .
9

Reconsidering the Market Size Effect in Innovation and Growth

Table 1 summarizes Proposition 1B.
Table 1: Comparative Statics under CES (Proposition 1B)𝜌↑𝐹↑
𝐿 = ℎ𝑁 ↑
𝜎↑

𝑀
0
0
0
−

𝐿∗
+
+
+
+

𝐿∗
−
−
+
−

𝑔∗ = 𝐿∗ ⁄𝐹−−+−

Three features of these results under CES deserve special emphasis. First, the per capita
labor endowment ℎ and the population size 𝑁 enter in the law of motion for ℰ , eq.(14), as wellas the expressions for 𝐿∗ = 𝐿 − 𝐿∗ and 𝑔∗ only through their product 𝐿 = ℎ𝑁. In short, whatmatters is the aggregate market size, not its composition. Once the country size, 𝐿 = ℎ𝑁, iscontrolled for, a richer country with a higher ℎ and a lower 𝑁 innovates as much as a poorer
country with a lower ℎ and a higher 𝑁. This also implies that the (demand-side) market size
effect is indistinguishable from the (supply-side) scale effect. This property is due to thehomotheticity of preferences and does not hold under nonhomothetic preferences. Second, theproduction cost, 𝜓 , has no effect on the aggregate dynamics. This neutrality of 𝜓 is due to theexogeneity of the markup rate, which depends solely on the preference parameter 𝜎. Thus, theshare of the aggregate expenditure accruing to the firms’ profits never change when the
production cost changes. Hence, the production cost never affects incentive to innovate, andhence it has no impact on the labor allocation between the production and R&D sectors. This is
the reason why we left it unspecified in this section. 7 Again, this feature will disappear as we
depart from CES. Third, the markup rate changes only with a variation in 𝜎, which leads to a
positive correlation between the markup and growth rates. This result is at odds with some
empirical evidence suggesting that competition fosters innovation and growth, if we use themarkup rate as an inverse measure of competition, as commonly done.
3.

Innovation and Growth under Directly Explicitly Additive (DEA) Preferences

7

Note however that the time path of 𝜓 affects that of 𝑥 (and hence the welfare), because 𝐿∗ = 𝜓 𝑁𝑥 𝑉 . For
example, if we assume 𝜓 = 𝜓, as in Grossman and Helpman (1993), 𝑥 must shrink at the rate equal to 𝑔∗ , so that𝑥 𝑉 stays constant. Instead, if we assume 𝜓 = 𝜓/𝑉 , as in Gancia and Zilibotti (2005), 𝑥 stays constant.

10

Reconsidering the Market Size Effect in Innovation and Growth

We now depart from CES preferences and consider a broader class of DEA preferences,which admits CES as a limit case.
3.1.

Intratemporal problem
The intratemporal preferences are called directly explicitly additive (DEA) if the direct

utility function is explicitly additive:𝑈(𝐱 ) ≡𝑢(𝑥 (𝜔))𝑑𝜔,

where the sub-utility function, 𝑢(∙), satisfies 𝑢(0) = 0, 𝑢 (𝑥) > 0, and 𝑢"(𝑥) < 0. For atechnical reason, it is assumed to be thrice-differentiable. The agents maximize thisintratemporal utility subject to their intratemporal budget constraint, eq.(1), which yields theinverse demand curve for each product;𝑝 (𝜔) =(15)𝑢 𝑥 (𝜔) 𝐸
,
Δ

where
Δ ≡

𝑢 𝑥 (𝜔 ) 𝑥 (𝜔 )𝑑𝜔captures the effects of the competing firms in the market.
The firms choose 𝑝 (𝜔) or 𝑞 (𝜔) = 𝑁𝑥 (𝜔) to maximize the profit 𝜋 (𝜔) =(𝑝 (𝜔) − 𝑤 𝜓 )𝑁𝑥 (𝜔) subject to eq.(15), taking 𝑤 , 𝜓 , Δ , and 𝐸 as given, which yields thepricing rule:𝑢 𝑥 (𝜔) 𝐸Δ1−

1
𝜎 𝑥 (𝜔)= 𝑝 (𝜔) 1 −1𝜎 𝑥 (𝜔)=𝑤𝜓 ,or𝑢 𝑥 (𝜔) 𝐸
𝑀 𝑥 (𝜔) Δ

=𝑤𝜓,

(16)

where
𝜎(𝑥) ≡ −𝑢 (𝑥)>1𝑥𝑢”(𝑥)

is the price elasticity, and 𝑀(𝑥) ≡ 𝜎(𝑥)⁄[𝜎(𝑥) − 1] > 1 is the markup rate.8

8

Without the condition, 𝜎(𝑥) > 1, the firm’s profit-maximization problem, its pricing rule, eq.(16), and its markup
rate, 𝑀(𝑥) > 1 would not be well-defined. It is equivalent to assuming that the firm’s revenue, 𝑝 (𝜔)𝑞 (𝜔) =𝑢 𝑥 (𝜔) 𝑥 (𝜔)(𝑁𝐸 ⁄Δ ) is increasing in 𝑥 (𝜔), and hence its marginal revenue, the LHS of eq.(16), is positive.

11

Reconsidering the Market Size Effect in Innovation and Growth

A distinctive feature of monopolistic competition under DEA is that the price elasticity ofdemand each firm faces, 𝜎(𝑥), is a function of per capita consumption of its product and nothingelse. Notice that, for any differentiable price elasticity function 𝜎(∙) > 1, one could define thesub-utility function as 𝑢(𝑥) = ∫ 𝑒𝑥𝑝 − ∫( )𝑑𝑦, which satisfies 𝑢(0) = 0; 𝑢 (𝑥) > 0,𝑢"(𝑥) < 0, and is thrice-differentiable. Hence, one could also use the price elasticity function𝜎(∙) as the primitive of the DEA preferences. Clearly, CES is a special case, where 𝜎(𝑥) = 𝜎 >1 ⟺ 𝑢(𝑥) = 𝐴(𝑥), with 𝐴 being a positive constant.

In what follows, we consider the subclass of DEA preferences that satisfy the following
two assumptions:
Assumption (D1):1𝑥𝑀 (𝑥)
+
>0
𝜎(𝑥)
𝑀(𝑥)

This inequality is equivalent to assuming that 𝑢 (𝑥)⁄𝑀(𝑥) is decreasing in 𝑥. In words, thefirm’s marginal revenue, the LHS of eq.(16), is decreasing in 𝑥 (𝜔). Hence, eq.(16) has aunique solution, 𝑥 (𝜔) = 𝑥 , which is decreasing in 𝑤 𝜓 Δ /𝐸 . This implies the symmetry ofequilibrium across firms and products, 𝑝 (𝜔) = 𝑝 , 𝑞 (𝜔) = 𝑞 , and 𝜋 (𝜔) = 𝜋 . Furthermore,(D1) ensures that the balanced growth path is the only equilibrium path of the economy: see
Proposition 2A.
Assumption (D2):𝜎 (𝑥) < 0 ⟺ 𝑀 (𝑥) > 0

In words, the demand for each product becomes more price elastic as one moves up along thedemand curve, eq.(15), (i.e., for a higher price/a lower quantity), which is sometimes called“Marshall’s Second Law of Demand”. (D2) is also equivalent to assuming that the elasticity of
substitution between any two products, 𝜔 and 𝜔 evaluated at 𝑥(𝜔 ) = 𝑥(𝜔 ) = 𝑥 is decreasing
in 𝑥.9 Obviously, CES is a borderline case of the subclass of DEA satisfying (D2). Note alsothat (D2) implies (D1).

9

Zhelobodkho et.al. (2012) studied what they called “relative love for variety (RLV),” which is nothing but theinverse of 𝜎(∙). Thus, (D2) is equivalent to assuming that RLV is increasing.

12

Reconsidering the Market Size Effect in Innovation and Growth(D2) plays a crucial role in the comparative static results in Proposition 2B. It is thus
important to understand its empirical implications. Since (D2) implies (D1), the unique solutionof eq.(16), 𝑥 (𝜔), is strictly decreasing in 𝜓 Δ . Hence, we can show:i)Imperfect Pass-Through: A higher production cost, 𝜓 , reduces 𝑥 (𝜔). This leads to alower markup rate, 𝑀 𝑥 (𝜔) under (D2);ii)

Strategic Complementarity in Pricing; If the competitors reduced their prices andincreased their sales, Δ would go up, which would reduce 𝑥 (𝜔). This would lead to alower markup rate, 𝑀 𝑥 (𝜔) , and a lower price, 𝑝 (𝜔) under (D2);iii)

Procompetitive Entry; The presence of more firms, an exogenous increase in 𝑉 would
lead to a higher Δ , which would reduce 𝑥 (𝜔). This would lead to a lower markup,
𝑀 𝑥 (𝜔) under (D2),for which there exists ample empirical evidence, as already discussed in the introduction. To
further underscore the importance of (D2), consider the implications of maintaining (D1) but
assuming the opposite of (D2), so that

>−
( )

( )
( )

> 0. Then, a decline in 𝑥 (𝜔), caused byan increase in 𝜓 or Δ , would lead to an increase in 𝑀 𝑥 (𝜔) with 𝑀 (𝑥) < 0. Thus, it would
imply more than 100% pass-through, strategic substitutes in pricing, and anti-competitive entry,
contrary to the empirical evidence. This is the reason why we depart from CES in the directionof (D2). It is worth noting that, in other classes of non-CES preferences, assuming Marshall’s
Second Law of Demand, i.e., demand for each product is more price elastic at a higher price,
may not imply strategic complementarity in pricing, and procompetitive entry. 10 This is anotherreason why we use DEA.
Because (D1) ensures the symmetry of equilibrium, the pricing rule now becomes1(17)= 𝑤 𝜓 ⟺ 𝑝 = 𝑀(𝑥 )𝑤 𝜓 .
𝜎(𝑥 )Using this expression, and following the same steps as in the CES case, the profit share in the𝑝1−

aggregate expenditure, eq.(4) now becomes
𝜋𝑉
1
1
=
≡1−
,
𝑁𝐸
𝜎(𝑥 )
𝑀(𝑥 )

10

For example, in a class of preferences used in Boucekkine et.al. (2017), the pricing rule of each firm isindependent of the pricing behaviors of other firms or the number of firms competing.

13

(18)

Reconsidering the Market Size Effect in Innovation and Growth

while the share of the wage payment to the production sector in the aggregate expenditure, eq.(5)becomes
𝑤 𝐿
1
1
(19)
= 1−
≡
.
)
)
𝑁𝐸
𝜎(𝑥
𝑀(𝑥
Note that departing from CES to DEA does not change the relations between these shares and themarkup rate. However, these shares are endogenous under DEA, because the markup rate is afunction of per capita consumption of each product, which is increasing under (D2).
3.2.

Intertemporal problem
Under DEA, the intertemporal utility is now given by
𝒰 =𝑙𝑜𝑔 𝑉 𝑢

𝐸
𝑝𝑉

𝑒

𝑑𝑡,

which agents maximize subject to the intertemporal budget constraint eq.(9). This leads to thefirst-order condition given by
𝜁(𝑥 )
𝑒
𝐸

=𝜆 𝑒

,

where 𝜁(𝑥) ≡ 𝑢 (𝑥)𝑥⁄𝑢(𝑥) > 0, while 𝜆 is again the Lagrange multiplier associated witheq.(9). Log-differentiating this first-order condition with respect to 𝑡 yields an augmented Euler
equation,(20)𝐸̇𝜁 ̇ (𝑥 )1𝑥̇=𝑟 −𝜌+=𝑟 −𝜌+ 1−− 𝜁(𝑥 ) ,
𝐸
𝜁(𝑥 )
𝜎(𝑥 )
𝑥
which features an additional term, which is absent in the original Euler equation, eq.(10). 11
3.3.

Balanced Growth Path
Following the same definition for the BGP as before, a constant markup rate requires that𝑥 must be constant. Furthermore, from 𝐿= 𝜓 𝑁𝑥 𝑉 , a constant 𝑥 and a constant 𝐿

requiresthat 𝜓 𝑉 , must be constant. Thus, in order to ensure the existence of a BGP, it is now necessary
to assume along Gancia and Zilibotti (2005) that knowledge spillovers improve productivity not
only in R&D but also in production, as 𝜓 = 𝜓/𝑉 .12

11

For CES, 𝜎(𝑥 ) = 𝜎 and 𝜁(𝑥 ) = 1 − 1⁄𝜎, so that the last term disappears.
This assumption has been made in other horizontal innovation models of endogenous growth, such as Foellmi and
Zweimuller (2006). Note also that it is isomorphic to assuming that 𝐹 = 𝐹, 𝜓 = 𝜓, and ℎ = ℎ𝑉 .
12

14

Reconsidering the Market Size Effect in Innovation and Growth

Again, we derive the law of motion for the economy. Following the same steps as in the
CES case, but noticing that the augmented Euler equation, eq.(20) now has the additional term,eq.(12) is now modified to:
ℰ̇
ℰ
1
𝑥̇
=
−𝑔 −𝜌+ 1−− 𝜁(𝑥 ) ,ℰ
𝜎(𝑥 )𝐹𝜎(𝑥 )𝑥while eq.(13) is modified to
𝐿=𝐿+𝐿= 𝐹𝑔 + 𝜓𝑁𝑥 .

Combining these two equations, and using eq.(18), ℰ ≡ 𝑁𝐸 ⁄𝑤 = 𝑁𝜓𝑥 𝑀(𝑥 ), we obtain thefollowing law of motion for 𝑥 :(21)1𝑥 𝑀 (𝑥 ) 𝑥 ̇𝑁𝜓𝑥 𝑀(𝑥 ) − (𝐿 + 𝜌𝐹)
+
=
.
𝜎(𝑥 )
𝑀(𝑥 ) 𝑥𝐹⁄
(D1) implies that the bracket term in front of 𝑥 ̇ 𝑥 on the LHS of eq.(21) is positive. (D1) also
𝜁(𝑥 ) +

implies that 𝑥 𝑀(𝑥 ) is increasing in 𝑥 . Thus, 𝑥̇ > 0 for 𝑥 > 𝑥 ∗ and 𝑥̇ < 0 for 𝑥 < 𝑥 ∗ ,where 𝑥 ∗ is defined implicitly by
𝐿 + 𝜌𝐹 ℎ + 𝜌𝐹 ⁄𝑁
(22)
=
.
𝑁𝜓
𝜓
Thus, eq.(21) would imply divergence, leading to a violation of the equilibrium conditions,𝑥 ∗ 𝑀(𝑥 ∗ ) =unless the economy jumps immediately to 𝑥 = 𝑥 ∗ and stays at 𝑥 = 𝑥 ∗ . This in turns implies𝐿= 𝜓𝑁𝑥 ∗ and 𝐿= 𝐹𝑔 = 𝐿 − 𝐿= 𝐿 − 𝜓𝑁𝑥 ∗ are all constant along the only equilibrium

path and the economy stays on the balanced growth path, as long as the parameters are such that
the R&D sector is active: 𝐿= 𝐹𝑔 = 𝐿 − 𝜓𝑁𝑥 ∗ = 𝑁(ℎ − 𝜓𝑥 ∗ ) > 0. Since (D1) implies that

the LHS of eq.(22) is increasing in 𝑥 ∗ , this condition can be written as 𝑀(ℎ/𝜓) > 1 + 𝜌𝐹/𝐿.
Hence, we have:
Proposition 2A: Balanced Growth Path under DEA with (D1)
Suppose 𝑀(ℎ/𝜓) > 1 + 𝜌𝐹/𝐿. Then, the economy jumps immediately to the balanced growth
path along which
𝐿= 𝐿∗ = 1 −1
𝐿 + 𝜌𝐹
(𝐿
+
𝜌𝐹)
=
< 𝐿;
𝜎(𝑥 ∗ )𝑀(𝑥 ∗ )

= 𝐿∗ =

𝐿
1
1
𝜌𝐹
−
1
−
𝜌𝐹
=
1
−
𝐿
−
> 0;
𝜎(𝑥 ∗ )𝜎(𝑥 ∗ )𝑀(𝑥 ∗ )𝑀(𝑥 ∗ )

𝑔 = 𝑔∗ =

𝐿
1
1
𝐿
𝜌
−
1
−
𝜌
=
1
−
−
> 0,
𝜎(𝑥 ∗ )𝐹𝜎(𝑥 ∗ )𝑀(𝑥 ∗ ) 𝐹 𝑀(𝑥 ∗ )𝐿15

Reconsidering the Market Size Effect in Innovation and Growth

where 𝑥 ∗ is defined implicitly by eq.(22).
Note that this proposition requires (D1), but not (D2). Note also that, by comparing Proposition
1A and Proposition 2A, departing from CES to DEA does not alter the functional relations
between 𝐿∗ , 𝐿∗ , 𝑔∗ on one hand and 𝜎(𝑥 ∗ ) and 𝑀(𝑥 ∗ ) on the other hand. The only but
significant difference is that, under DEA, the parameters, 𝜌, 𝐹, ℎ, 𝑁, and 𝜓 affect 𝜎(𝑥 ∗ ) and𝑀(𝑥 ∗ ) through eq.(22).
From Proposition 2A, it is straightforward to show:
Proposition 2B: Comparative Statics under DEA with (D2)Under DEA preferences with (D2),i)

Both an increase in the discount rate 𝜌 and in the R&D cost 𝐹 increase per capita per
product consumption 𝑥 ∗ and the markup rate 𝑀(𝑥 ∗ ), increase 𝐿∗ , and decrease 𝐿∗ andthe growth rate 𝑔∗ ;ii)

An increase in the population size 𝑁 decreases 𝑥 ∗ and 𝑀(𝑥 ∗ ), increases 𝐿∗ , 𝐿∗ , and 𝑔∗ ;iii)

An increase in per capita labor endowment ℎ increases 𝑥 ∗ and 𝑀(𝑥 ∗ ), increases 𝐿∗ , 𝐿∗ ,and 𝑔∗ ;iv)

Both an increase in 𝑁 (a decrease in ℎ) for a fixed ℎ𝑁 = 𝐿, and an increase in 𝜓
decrease 𝑥 ∗ and 𝑀(𝑥 ∗ ), increase 𝐿∗ , and decrease 𝐿∗ and 𝑔∗ .

Table 2 summarizes Proposition 2B. The signs in the shaded boxes are the consequences of(D2): 𝜎 (𝑥) < 0 (𝑀 (𝑥) > 0). If we had instead assumed 𝜎 (𝑥) > 0 (𝑀 (𝑥) < 0), there would
be the opposite signs in the shaded boxes. Under CES, they would be “0”.
Table 2: Comparative Statics under DEA with (D2): Proposition 2B𝑥 ∗ 𝑀(𝑥 ∗ )
+
+
𝜌↑
+
+
𝐹↑
𝑁↑
−
−
+
+
ℎ↑
𝑁 = 𝐿/ℎ ↑ , fixed 𝐿
−
−
𝜓↑
−
−
An immediate corollary of Proposition 2B is:𝐿∗ = 𝜓𝑁𝑥 ∗
+
+
+
+
+
+

𝐿∗
−
−
+
+
−
−

𝑔∗ = 𝐿∗ /𝐹
−
−
+
+
−
−

Corollary: Correlations between the Markup and Growth Rates under DEA with (D2)Under DEA preferences with (D2),16

Reconsidering the Market Size Effect in Innovation and Growthv)

A change in the discount rate 𝜌, the R&D cost 𝐹, or the population size 𝑁 causes themarkup rate 𝑀(𝑥 ∗ ) and the growth rate 𝑔∗ to move in the opposite direction.

vi)

A change in per capita labor endowment ℎ or the production cost 𝜓 causes the markup
rate 𝑀(𝑥 ∗ ) and the growth rate 𝑔∗ to move in the same direction.
We now discuss the implications of departing from CES within DEA in the direction of(D2), by comparing Proposition 1B with Proposition 2B and its corollary.
Just as in CES, both an increase in 𝜌 and in 𝐹 discourage R&D, which causes thereallocation of labor from the R&D sector to the production sector. This causes an increase in𝑥 ∗ , per capita consumption of each product. Unlike in CES, this increases the markup rate𝑀(𝑥 ∗ ) under (D2). This secondary effect mitigates the impact on labor reallocation, but notenough to overturn it. However, this causes the negative correlations between the markup rate𝑀(𝑥 ∗ ) and the innovation and growth rate, 𝑔∗ . In particular, this implies that the measure ofcompetitiveness and the growth rate are positively correlated in cross-section of countries, if
countries differ mostly in the innovation (or firm entry) cost.
Just as in CES, both an increase in 𝑁 and in ℎ, by increasing the total labor supply, leadto an increase in the labor supply to both the production and the R&D sectors. The latter leads toan increase in the growth rate, due to the familiar scale effect. However, as seen in eq.(22), they
have opposite impacts on 𝑥 ∗ and hence on 𝑀(𝑥 ∗ ) under (D2). A higher 𝑁 leads to a lower 𝑥 ∗ ,which in turn leads to a lower 𝑀(𝑥 ∗ ) under (D2), mitigating the effect on the growth rate as wellas generating the negative correlations between 𝑀(𝑥 ∗ ) and 𝑔∗ . In contrast, a higher ℎ leads to ahigher 𝑥 ∗ , which in turn leads to a higher 𝑀(𝑥 ∗ ) under (D2), amplifying the effect on the growthrate as well as generating the positive correlations.
To see such differential effects of changes in ℎ and in 𝑁 more clearly, consider the effectof increasing 𝑁 and decreasing ℎ simultaneously to keep the total labor supply 𝐿 = ℎ𝑁
unchanged. This removes the scale effect. Without a change in 𝐿, an increase in 𝑁 would
necessitate a decline in per capita consumption of each product. This would have no impact on
the markup rate and the allocation of labor between the production and R&D sectors under CES.
However, under (D2), this causes the markup rate to decline, reducing the incentive to innovate,
which causes the reallocation of labor from the R&D sector to the production sector, and a
decline in the innovation and growth rates. This result suggests that, once the aggregate market
17

Reconsidering the Market Size Effect in Innovation and Growth

size is controlled for, richer countries with smaller population sizes innovate more and hence
grow faster than poorer countries with larger population sizes.
Indeed, this effect of an increase in 𝑁 without an increase in 𝐿 is completely isomorphic
to the effect of an increase in 𝜓, as clearly seen from eq.(22). Without a change in 𝐿, a higher 𝜓,just like a higher 𝑁, necessitates per capita consumption of each product to decline. This change
would be neutral under CES. Under (D2), however, it leads to a decline in the markup rate
through imperfect pass-through, which discourages R&D, causing labor to reallocate to the
production sector and the growth rate to decline.
It is worth pointing out that, though Proposition 2A does not require (D2), thecomparative statics results reported in Proposition 2B and its corollary depend on (D2). If we
had departed from CES in the opposite direction, many of these results would be overturned, as
indicated by the shaded area in Table 2. That is why we have disciplined our departure fromhomotheticity by the empirical evidence of imperfect pass-through, strategic complementarity inpricing and procompetitive effects of entry.
4.

Concluding Remarks
In the standard horizontal innovation model of endogenous growth, larger economies

innovate more and hence grow faster. Due to the homotheticity of consumer demands, however,it does not matter whether the large market size comes from a large population or a high per
capita expenditure. In this paper, we extended the standard textbook model by building on the
Dixit-Stiglitz (1977; Section II) model of monopolistic competition with directly explicitly
additive (DEA) nonhomothetic preferences. In spite of nonhomotheticity, the model preserves
the balanced growth property of the standard model. Furthermore, we discipline our departurefrom homotheticity by the empirical evidence of incomplete pass-through, strategic
complementarity in pricing, and procompetitive entry. Among others, it has been shown that,once the aggregate market size is controlled for, richer countries with smaller populations
innovate more and grow faster, which suggests that the difference in per capita income acrosscountries could potentially be one reason why there is little supporting evidence for the scaleeffect. It has also been shown that the correlations between the markup and growth rates acrossthe balanced growth paths could be either positive or negative, depending on the source ofvariations. In particular, the measure of competitiveness and the growth rate are positively
18

Reconsidering the Market Size Effect in Innovation and Growthcorrelated in cross-sections of countries, if countries differ mostly in the innovation (or firm
entry) cost.

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