DP2013-24

The Effec t of Globalization in an
End ogenous Growth Model with
Heterogeneous Firms a nd Endogenous
International Spillovers: Note*
Katsufumi FUKUDA
August 22, 2013

* The Discussion Papers are a series of research papers in their draft form, circulated to encourage
discussion and comment. Citation and use of such a paper should take account of its provisional character.
In some cases, a written consent of the author may be required.

The Effect of Globalization in an Endogenous Growth Model with HeterogeneousFirms and Endogenous International Spillovers: Note

August 22, 2013

Katsufumi Fukuda 1

Abstract
This paper shows that globalization increases (decreases) the growth rate if and only if
the beachhead cost for the domestic market is strictly higher (lower) than that for theforeign market in a endogenous growth model with firm heterogeneity, international
trade, and endogenous international spillover under specified necessary and sufficientconditions for exporting firms being more productive than non-exporting firms.

JEL Classification: F12, F15, O30, O33
Keywords: heterogeneous firms, endogenous international spillovers,
endogenous growth theory

1

Katsufumi Fukuda, Graduate School of Economics, Kobe University, 2-1Rokkodai-cho, Nada-ku,

Kobe, Hyogo 657-8501, Japan. E-mail: 2katsufumi.fukuda@gmail.com. Tel: +81-78-803-7250. We
would like to thank David Denslow and Yan Ma.

1

1. Introduction
How does globalization affect economic growth and welfare? Many economists
studying international trade and economic growth have tackled this theme empiricallyand theoretically. Empirical papers show that the relationship between globalization andeconomic growth is ambiguous. Frankel and Romer (1999) show that globalization
increases the growth rate, but Vamvakidis (2002), Clemens and Williamson (2004), andMinier and Unel (2013) finds that globalization increases or decreases the growth rate.
International trade increases welfare through many ways. First, new imported varieties
increase welfare. For instance Broda and Weinstein (2006) argue that new varieties
increased US GDP by 2.6 % between 1972 and 2001 and Coe and Helpman (1995)
argue that international spillover depends on the amount of imported goods from trade
partner. Second, as trade reallocates resources from the less productive non-exporting tomore productive exporting firms, industrial productivity rises (Bernard, Jensen, andSchott, 2006) 2
Recently, many economists have analyzed the role of firm heterogeneity ininternational trade and economic growth. Baldwin and Robert-Nicoud (2008) extend
Grossman and Helpman (1991) into an R&D-based growth model with firm
heterogeneity as modeled by Melitz (2003). They show that globalization reduces

2

Melitz (2003) construct a monopolistic competition model with firm heterogeneity which isconsistent with this empirical evidence.
2

economic growth in an economy both with exogenous and with the Coe and Helpman
type endogenous international knowledge spillovers 3 .

In the former model,

globalization leads to higher R&D costs and a lower growth rate due to greater
competition through importing under a sufficient condition that productivity forexporters is higher than that for non-exporters, that is, the sunk cost for the foreign
market is lower than that for the domestic market and to higher or lower welfare due tohigher productivity of production firms and higher R&D cost. In the latter model, undera sufficient condition that the productivity of exporting firms be higher than that for
non-exporters, the negative effect of rises in R&D costs unambiguously dominates the
positive effects of the decrease in R&D costs that results from increases in international
knowledge spillover. Thus, economic growth rate decreases. Moreover, there is an
ambiguous effect on welfare because there is a negative channel of decreases in thegrowth rate and higher R&D costs and there is a positive effect through increases in theweighted average of productivity among production firms, but they do not derive aparameter condition for gains from trade.

Unel (2010) extends Romer (1990) into an

R&D-based growth model with firm heterogeneity and endogenous international
knowledge spillover 4. They show that globalization increases or decreases economic
growth and welfare, but his paper also does not derive a condition for gains from trade.
Moreover, Gustafsson and Segerstrom (2010) show that due to diminishing returns toknowledge in the R&D sector globalization does not affect economic growth.
Dinopoulos and Unel (2011) show that globalization affects economic growth3

The international knowledge spillover depends on the fraction of exporting firms amongproduction firms, that is, the cutoff point for the foreign market relative to the domestic market inthis model.
4 Unel (2010) assumes that international knowledge spillover depends on the value of total traderelative to the value of intermediate goods produced domestically.
3

ambiguously because there is a positive effect that the number of researchers devoted
increases due to an increase in the mark-up rate and a negative effect of higher R&D
costs. They find that per capita expenditures and growth rates vary inversely, but they
do not derive a condition for gains from trade.
This paper derives the necessary and sufficient conditions under which globalization
increases (decreases) economic growth when the sunk beachhead cost for the domestic
market is higher (lower) than that for the foreign market in a model with firm
heterogeneity and endogenous international spillover 5. This result is more relevant to
the empirical evidence cited above. This paper shows that endogenous international
spillover plays an important role in economic growth because Baldwin andRobert-Nicoud (2008) and Unel (2010) show that further exposure to international tradenecessarily decreases economic growth in the exogenous international spillover modeland may increase economic growth in all other endogenous international spillover
models.
The rest of the paper is organized as follows: Section 2 describes the elements of themodel and derives the basic results. Section 3 offers concluding remarks.

2. The Model
2.1 Basic Structure

We follow Baldwin and Robert-Nicoud (2008) except that the sunk beachhead cost for
the foreign market may be higher than, equal to, or lower than that for the domestic
market.

There are two symmetric countries. Each consumer supplies inelastically one

We assume the sunk cost for the foreign market relative to that for the domestic
market is higher than a measure of free trade, 𝜏𝜏 1−𝜎𝜎 , where 𝜏𝜏 is the iceberg cost andσ > 1 is the elasticity of substitution between varieties.
54

unit of labor that is the only production factor and serves as the numeraire in eachperiod. There are monopolistically competitive goods and perfectly competitive R&D
sectors. Firms are heterogeneous with regard to the marginal cost of manufacturing,which is drawn from a Pareto distribution after incurring the sunk cost of variety
creation. To enter a market, each firm has to incur a sunk cost of variety creation, andthen chooses whether to exit the market immediately, serve the domestic market only or
to serve both the domestic and export markets. Each market requires expendituresassociated with fixed costs. Exporters face iceberg-type unit transportation costs as well.
In the R&D sector, there are constant returns to knowledge, and the growth rate is
endogenously determined. We follow Baldwin and Robert-Nicoud (2008) and Unel(2010) who assume endogenous international spillovers consistent with Coe andHelpman (1995).

2.2. Consumers

All consumers are identical. Each consumer supplies inelastically one unit oflabor in each time period. The total amount of labor supplied equals the size of the
population. Thus, labor supply can be denoted by 𝐿𝑡 . Each consumer earns income

from his or her assets and labor, and chooses the path of consumption expenditure andassets so as to maximize the sum of his or her discounted value of utility. The∞intertemporal utility function is given by 𝑢(𝑐𝑡 ) = ∫0 log 𝑐𝑡 𝑒 −𝜌𝑡 d𝑡, where 𝑐𝑡 is aconsumption index which depends on the consumption of a continuum of varieties,
given by 𝑐𝑡 = �∫Ω (𝑥𝐿𝐿𝑡 (𝑖))
𝑡

𝜎−1
𝜎

d𝑖 + ∫Ω (𝑥𝐸𝑡 (𝑖))
𝑡

5

𝜎−1
𝜎

𝜎
𝜎−1

d𝑖 �

, 𝜎𝜎 > 1, where 𝛺𝑡 is the set of

varieties that can be consumed, 𝑥𝐿𝐿𝑡 (𝑖 ) (𝑥𝐸𝑡 (𝑖 )) the individual demand for the i-th

variety produced domestically (produced abroad and exported), and 𝜌 the individual’s
subjective discount rate.

The inter-temporal budget constraint is 𝑆̇𝑡 = 𝑟𝑡 𝑆𝑡 + 𝑤𝑡 𝐿𝑡 − 𝐸𝑡 , where 𝑆𝑡 isaggregate assets, 𝑟𝑡 the rate of return on assets, 𝑤𝑡 the wage, and 𝐸𝑡 theeconomy-wide expenditure. Solving the dynamic optimization problem implies that𝐸̇𝑡𝐸𝑡

= 𝑟𝑡 − 𝜌. Static optimization yields the demand for each variety, given by
𝑝𝑗 (𝑖)−𝜎1−𝜎𝜎𝑥𝑡 (𝑖 ) = (𝑃 )1−𝜎 𝐸𝑡 , 𝑗 = 𝐿, 𝐸, where 𝑃𝑡 = �∫𝑖∈𝛺 [𝑝𝐿𝐿 (𝑖)]𝑡1−𝜎𝜎d𝑖 + ∫𝑖∈𝛺[𝑝𝐸 (𝑖)]

d𝑖�

1
1−𝜎

is

the price index, 𝑝𝐿𝐿 (𝑖 ) the price of varieties produced domestically, and 𝑝𝐸 (𝑖 ) the
price of the varieties produced abroad and exported.

2.3 Innovation
We next analyze firm behavior. For firms to enter the market, they have to pay the sunkcost of variety creation 𝑏𝐼𝑡 𝐹𝐼 , where 𝑏𝐼𝑡 ≡
𝐵

1

𝐵 𝑘
𝐵𝐿𝐿

𝑚𝑡 [1+� 𝐸 � ]represents the unit labor

requirement for knowledge creation, 𝑚𝑡 (𝐵𝐸 )𝑘𝑘 the international knowledge spillover,𝐿𝐿

and 𝑚𝑡 the number of varieties produced. This specification is consistent with theempirical findings of Coe and Helpman (1995). R&D researchers learn to generateknowledge more efficiently over time. Increased efficiency is reflected in lower unit
labor requirements. Then they find the unit labor requirement for manufacturing from a
𝐵

𝐵

Pareto distribution, given by 𝐺 (𝐵) = ∫0 𝑔(𝐵)𝑑𝐵 = ( � )𝑘𝑘 , where 1 + 𝑘𝑘 − 𝜎𝜎 > 0 and𝐵𝐵 ∈ [0, 𝐵�].

6

To enter the market, each firm has to pay the domestic and exporting sunk costs,
given by 𝑏𝐼𝑡 𝐹𝐿𝐿 and 𝑏𝐼𝑡 𝐹𝐸 , respectively. For one unit of good entering the domestic

and exporting markets, 1 and 𝜏𝜏units of goods, respectively, must be transported.

Firms with unit labor requirement 𝐵 > 𝐵𝐿𝐿 exit the market immediately. Firms withunit labor requirement 𝐵𝐸 < 𝐵 < 𝐵𝐿𝐿 enter the domestic market. Firms with unit labor
requirement 𝐵 < 𝐵𝐸 enter both markets.
2.4

Product Markets

If a firm with unit labor requirement 𝐵 enters the market, the firm earns profits
𝜋𝐿𝐿𝑡 (𝐵) = 𝑝𝐿𝐿𝑡 (𝐵)𝑥𝐿𝐿𝑡 (𝐵) − 𝐵𝑥𝐿𝐿𝑡 (𝐵). The profit-maximizing price is given by
𝑝𝐿𝐿𝑡 (𝐵) =

𝜎𝜎𝐵

𝜎𝜎−1

. Thus, the profit function is given by 𝜋𝐿𝐿𝑡 (𝐵) =𝜎𝜎 −𝜎 (𝜎𝜎−1)𝜎−1 𝐵 1−𝜎 𝐸𝑡(𝑃𝑡 )1−𝜎given consumer expenditure and price index, the profit function monotonically. For

decreases with the level of the unit labor requirement in manufacturing. For a firm toexport one unit of a good, the firm must produce 𝜏𝜏 > 1 units, and the
profit-maximizing price would be 𝑝𝐸𝑡 (𝐵) =exporting is given by 𝜋𝐸𝑡 (𝐵) =𝜏𝜎𝜎𝐵. Thus, the profit function for

𝜎𝜎−1

𝜏 1−𝜎 𝜎𝜎 −𝜎 (𝜎𝜎−1)𝜎−1 𝐵 1−𝜎 𝐸𝑡(𝑃𝑡 )1−𝜎. A consumer has two methods toaccumulate assets: firm shares and riskless bonds. The rate of return on the former typeof assets depends on dividends and capital loss (gain). The latter type comes frominterest. In equilibrium, the two rates of return are equalized. Thus, the following
𝜋𝑗𝑡 (𝐵)𝑉̇𝑗𝑡 (𝐵)no-arbitrage condition holds: 𝑉 (𝐵) + 𝑉 (𝐵) = 𝑟𝑡 , 𝑗 = 𝐿, 𝐸. This equation determines𝑗𝑡

𝑗𝑡

the value of a firm serving the market as a function of the level of its unit labor7requirement. The cost associated with serving the domestic market is 𝑏𝐼𝑡 𝐹𝐿𝐿 . Thus, thelocal cutoff 𝐵𝐿𝐿 is determined as follows:
𝑉𝐿𝐿𝑡 (𝐵𝐿𝐿 ) =𝑚̇𝜎𝜎 −𝜎𝜎 (𝜎𝜎 − 1)𝜎𝜎−1 (𝐵𝐿𝐿 )1−𝜎𝜎 𝐸𝑡= 𝑏𝐼𝑡 𝐹𝐿𝐿 ,(𝑃𝑡 )1−𝜎𝜎 (𝑟𝑡 + 𝑔)(1)

where 𝑔 ≡ 𝑚𝑡. Similarly, there exists a foreign cutoff 𝐵𝐸 satisfying𝑡

𝑉𝐸𝑡 (𝐵𝐸 ) =𝜏𝜏 1−𝜎𝜎 𝜎𝜎 −𝜎𝜎 (𝜎𝜎 − 1)𝜎𝜎−1 (𝐵𝐸 )1−𝜎𝜎 𝐸𝑡= 𝑏𝐼𝑡 𝐹𝐸 .
(𝑃𝑡 )1−𝜎𝜎 (𝑟𝑡 + 𝑔)(2)

Using the cutoff conditions (1) and (2), we obtain the cutoff ratio as a function of𝐵𝐸iceberg costs and the ratio of the second stage sunk costs 𝐵 = �𝐿𝐿1𝜏 1−𝜎𝜎𝜎𝐿𝐿 𝜎−1
𝜎𝜎𝐸

�

< 1.

Baldwin and Robert -Nicoud (2008) assume the following sufficient condition for𝜎𝜎exporters being more productive than non-exporters: 1 > 𝜎𝜎𝐸 > 𝜏𝜏 1−𝜎𝜎 . On the other𝐿𝐿

hand, we assume the necessary and sufficient condition for exporters being more
𝜎𝜎productive than non-exporters: 𝜎𝜎𝐸 > 𝜏𝜏 1−𝜎𝜎 .
2.5 Innovation Incentives

𝐿𝐿

A firm does not know its own productivity before entering the market, and decides
whether to enter or not by weighing its ex-ante value (benefits) against expected R&D
sunk cost. Due to free entry and exit, and the constant returns to scale technology in
𝐵

B

R&D, the free-entry condition is given by ∫0 𝐿𝐿 [𝑉𝐿𝐿 (𝐵) − 𝑏𝐼 𝐹𝐿𝐿 ]𝑔(𝐵)dB + ∫0 E [𝑉𝐸 (𝐵) −𝑏𝐼 𝐹𝐿𝐿 ]𝑔(𝐵)dB = 𝑏𝐼 𝐹𝐼 . It can be rewritten as𝜎𝜎 −𝜎𝜎 (𝜎𝜎 − 1)𝜎𝜎−1 ∆𝐸𝑡= 𝑏𝐼𝑡 𝐹� .
(𝑃𝑡 )1−𝜎𝜎 (𝜌 + 𝑔)

(3)

The left-hand side of (3) is the expected excess benefit from creating a new variety andthe right-hand side of (3) is the expected cost for creating a new variety;8𝐵∆≡ ∫0 𝐿𝐿 𝐵1−𝜎𝜎𝐵𝑔(𝐵)dB
𝐺(𝐵𝐿𝐿+ 𝜏𝜏 1−𝜎𝜎 ∫0 𝐸 𝐵1−𝜎𝜎)𝜎𝜎𝐵

productivity, 𝐹� ≡ 𝐺(𝐵𝐼 ) + ∫0 𝐿𝐿 𝐹𝐿𝐿
𝐿𝐿

𝑔(𝐵)dB𝑘𝑘(𝐵𝐿𝐿 )1−𝜎 Φ𝐵𝑔(𝐵)dB

𝐺(𝐵𝐿𝐿

𝑔(𝐵)dB
𝐺(𝐵𝐿𝐿=)

1+𝑘𝑘−𝜎𝜎+ ∫0 𝐸 𝐹𝐸)

𝐺(𝐵𝐿𝐿

is the weighted average of𝐵�𝑘𝑘= 𝐹𝐼 � � + 𝐹𝐿𝐿 Φ is the)
𝐵𝐿𝐿

expected cost of a producing firm’s creating a new variety, and Φ ≡ 1 + 𝜏𝜏

𝜎𝜎𝐸

−𝑘𝑘

�𝜎𝜎 �
𝐿𝐿

𝜎−1−𝑘
𝜎−1

measures the increase in R&D costs owing to globalization or the rise in competition
due to importing. The first term of the R&D cost is the cost of choosing the unit labor
requirement for manufacturing equal to or less than the domestic market cutoff level.
The second term of the last equation is the beachhead cost for the domestic market andthe final term of the last equation is the sunk beachhead cost for the foreign market.
𝐿𝐿

The production function for a new variety is given by 𝑚̇𝑡 = 𝑏 𝑅𝜎𝜎� , where
𝐼𝑡

𝐿𝐿𝐿𝐿

𝐿𝑅 represents the R&D researchers. R&D technology can be written as 𝑔 = 𝜎𝜎�𝑅 ,where 𝜒 ≡ 1 + 𝜏𝜏−𝑘𝑘𝑘

𝜎𝜎 𝜎−1
�𝜎𝜎𝐿𝐿 �𝐸

measures the degree of endogenous international spillover.

2.6 Full Employment Condition

The labor market is perfectly competitive, and labor is used for R&D or manufacturing.
𝐵

The full employment condition is 𝐿𝑡 = 𝐿𝑅 + 𝐿𝑥 where 𝐿𝑥 = ∫0 𝐿𝐿 𝐵𝑥𝐿𝐿 (𝐵) 𝑚𝑡
𝐵𝜏𝜏 ∫0 𝐸 𝐵𝑥𝐸 (𝐵)𝑚𝑡𝑔(𝐵)dB
𝐺(𝐵𝐿𝐿 )

𝐺(𝐵𝐿𝐿 )+

is the labor used for manufacturing. The price index is given by

𝐵

𝑔(𝐵)dB𝜎𝜎

1−𝜎𝜎(𝑃𝑡 )1−𝜎𝜎 = 𝑚𝑡 �∑𝑗=𝐿𝐿,𝐸 ∫0 𝑗 𝑃𝑗𝑡 (𝐵)1−𝜎𝜎
�=� �
𝐺(𝐵 )
𝜎𝜎−1
2.7

𝑔(𝐵)dB

Steady-State Equilibrium

𝐿𝐿

9𝑚𝑡 ∆. Thus, 𝐿𝑥 =

𝜎𝜎−1
𝜎𝜎

𝐸𝑡 .

Substituting the free-entry condition, costs for creating a new variety, the weighted
average of the unit labor requirement of production firms, and the Pareto distribution
into the local cut-off condition yields the unique solution of the local market cutoff level,
determined by
6

1+𝑘𝑘−𝜎𝜎level is given by

𝑘𝑘Φ

=

𝜎𝜎𝐿𝐿
� 𝑘
𝐵
𝜎𝜎𝐼 � � +𝜎𝜎𝐿𝐿 Φ
𝐵𝐿𝐿
1. The closed-form solution of the local cut-off(1+𝑘𝑘−𝜎𝜎)𝜎𝜎 𝑘 𝐵�
𝐵𝐿𝐿 = � (𝜎𝜎−1)𝜎𝜎 𝐼 � 1 .
𝐿𝐿Φ𝑘

The intuition behind this equation follows

Furusawa and Konishi (2012). A decrease in the iceberg cost leads to more competition
due to imports from abroad, and leads to lower demand for each variety and higher
demand for labor. Thus, a variety with lower productivity than the cutoff productivity
level cannot serve the domestic market.
Next we derive the closed-form solutions for per capita expenditure and R&D
difficulty. We obtain the first relationship between per capita expenditure and R&D
difficulty from full employment, which becomes 𝐿 =

𝜎𝜎−1
𝜎𝜎

𝐸+

𝑘𝑘𝜎𝜎𝐿𝐿 Φ𝑔(1+𝑘𝑘−𝜎𝜎)𝐿𝐿.

The full

employment condition is represented by a line with negative slope and a positive
E-intercept in the (𝐸, 𝑔) plane, because an increase in 𝐸 implies a larger number ofmanufacturing firms and a lower value of g implies a smaller number of researchers areneeded to maintain the equality. We obtain the second relationship between aggregate
expenditure and the growth rate from the free-entry condition𝐸

𝜎𝜎(𝜌+𝑔)=𝑘𝑘𝜎𝜎𝐿𝐿 Φ(1+𝑘𝑘−𝜎𝜎)𝐿𝐿. The

free-entry condition is an upward sloping curve with a positive E-intercept in the (𝐸, 𝑔)
plane, because an increase in 𝐸 implies that a larger benefit from creating a newvariety and associated costs must be offset by an increase in g to maintain the equality.

Using these conditions, the closed-form solutions for aggregate expenditure and the

6We assume

1 + 𝑘𝑘 − 𝜎𝜎 > 0.

10

growth rate7 are given by 𝐸 =𝐿𝐿(1+𝑘𝑘−𝜎𝜎)𝐿𝐿+𝜌𝑘𝑘𝜎𝜎𝐿𝐿 Φrespectively.

(1+𝑘𝑘−𝜎𝜎)𝐿𝐿

and 𝑔 =

(1+𝑘𝑘−𝜎𝜎)𝐿𝐿𝐿𝐿
𝑘𝑘𝜎𝜎𝜎𝜎𝐿𝐿 Φ−ρ(σ−1)𝜎𝜎,

We next analyze the effects of trade liberalization on the growth rate. We obtainhigher R&D knowledge spillover from a foreign country and an increase in R&D costby increasing the value of firms, which in turn increases the cost of creating a newvariety through decreasing the domestic cutoff level and increasing the exporting cutoff
level to satisfy the cutoff conditions. The condition for an increase in the growth rate
resulting from further exposure to trade is𝑘𝑘

1+𝑘𝑘−𝜎𝜎𝜕𝑔𝑘𝑘𝐿𝜏𝜏 −𝑘𝑘−1 (1 + 𝑘𝑘 − 𝜎𝜎) 𝐹𝐿𝐿 𝜎𝜎−1
𝐹𝐿𝐿 𝜎𝜎−1
−
>0=
�� �
−� �
� > 0.
2𝜕𝜏𝜏
𝑘𝑘𝜎𝜎𝐹𝐿𝐿 Φ
𝐹𝐸
𝐹𝐸
The growth rate increases (decreases) from further exposure to international trade when
the sunk beachhead cost for the domestic market is strictly larger (smaller) than the sunkcost for entering the foreign market. This is because there is a positive effect throughendogenous international spillover, which depends on the sunk cost for entering thedomestic market relative to that for the foreign market and a negative effect of anincrease in R&D cost through changes in cutoff points, which depends on the sunk costfor entering the domestic market relative to that for the foreign market. This result
seems to be relevant to empirical evidence between globalization and economic growth
because its empirical evidences shows conflicting results.
3. Conclusions
This paper shows the necessary and sufficient conditions under which further exposure
to international trade raises the growth rate in an endogenous growth model with firm

7

We assume(1+𝑘𝑘−𝜎𝜎)𝐿𝐿𝐿𝐿
𝑘𝑘𝜎𝜎𝜎𝜎𝐿𝐿 Φ−ρ(σ−1)
𝜎𝜎

> 0.

11

heterogeneity, endogenous international spillovers, and international trade. When the
sunk cost for the domestic market is strictly higher (lower) than that for the foreign
market, the growth rate increases (decreases). We can extend this model into a scale
invariant endogenous growth model by introducing population growth and diminishingreturns to knowledge in the R&D sector as in Jones (1995).

References
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