HIAS-E-95

A New Impetus for Endogenous Growth:
R&D Oﬀshoring via Virtual Labor Mobility
Noritsugu Nakanishi(a) and Ngo Van Long(b),(c)(a) Kobe University, Rokkodai-cho 2-1, Nada-ku, Kobe 657-8501, Japan(b) McGill University, Montreal, Canada(c) Hitotsubashi Institute for Advanced Study, Hitotsubashi University

February, 2020

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A New Impetus for Endogenous Growth:
R&D Oﬀshoring via Virtual Labor Mobility
NoritsuguNAKANISH*BOENgoVanLONG

Abstract
We develop a simple North-South model of quality ladders to showthat the virtual mobility of labor across time zones, facilitated by
the advance in communication technology, can raise the endogenous
growth rate of the world economy. The unique balanced growth rateis increasing in the endowments of skilled labor in both countries anddecreasing in the rate of impatience. Moreover, we find that partial
R&D oﬀshoring to the South has initially a negative eﬀect on thelevel of skilled wages in the North, but this is compensated for by its
positive eﬀect on the growth rate in both North and South.

JEL-Classification: F43, O41
Keywords: R&D oﬀshoring, Virtual labor mobility, Time zone diﬀerence,
Endogenous growth, North-South trade/BLBOJTIJ Kobe University, Rokkodaicho 2-1, Nada-ku, Kobe 657-8501, Japan. Email: nakanishi@econ.kobe-u.ac.jp. Phone:+81-78-803-6837 (Oﬃce direct).
†-POH McGill University,Montreal Canada;and Hitotsubashi Institute for Advanced Study, Hitotsubashi University.
Email:
ngo.long@mcgill.ca.
∗

1Data Availability Statement
Datasharingisnotapplicable

Acknowledgement
Part of this paper was written when Nakanishi was visiting McGill University and CIRANO. Nakanishi is grateful to the hospitality of both institutions. Nakanishi also acknowledges the financial support of the Japan Society for the Promotion of Science: Grant-in-Aid for Scientific Research (A),
No.16H02016.

2

1

Introduction

The oﬀshoring of tasks to low-wage economies has generated much debate.
On one hand, many economists have argued that, from a long run perspective, free trade in goods and services (tasks) is potentially welfare-enhancing.
They point to the well-established proposition that under ideal conditions thegainers from free trade can compensate the losers and remain better oﬀ. On
the other hand, other economists have reminded us two inescapable facts oflife: (i) there are significant short-run adjustment costs,1 and (ii) in reality,
losers are never fully compensated (Autor, Dorn, and Hanson, 2013).2 Several influential authors have suggested that oﬀshoring is probably one of thesignificant factors that have contributed to job polarisation, falling wages
and declining employment opportunities in developed economies (Krugman,2008; Autor, Dorn, and Hanson, 2015; Acemoglu, Autor, Dorn, and Hanson,2015).
While the bulk of oﬀshoring occurs in the manufacturing sector, there isa growing trend of oﬀshoring of tasks, which has been made possible by drastic improvements in the information and communication technology (ICT).
For example, thanks to ICT, several hundred US hospitals have used overseas tele-radiology services (Wachter, 2006). When an emergency CT scan
is taken at midnight in a hospital located in New York, it is 10:30 a.m. in
1

Adam Smith was well aware of these costs. In the Wealth of Nations, referring toadjustment costs that accompany a sudden influx of imports, he wrote that “Humanity
may in this case require that freedom of trade should be restored only by slow gradations
and with a good deal of reserve and circumspection. Were those duties (i.e., tariﬀs) andprohibitions (i.e., quotas) taken away all at once, cheaper foreign goods of the same kindmight be poured so fast into the home market, as to deprive all at once many thousandsof our people of their ordinary employment and means of subsistence.”2
For the US, the medium-run adjustment costs have recently been found to be more substantial than many economists had thought. Using US local labor market data covering theperiod 1990–2007, Autor, Dorn, and Hanson (2013, p.2159) found adverse distributional
consequences and significant medium-run eﬃciency losses associated with adjustment toincreased imports from China. As reported in the New York Times (17 May 2015), Professor Gordon Hanson admitted that “I think what we have learned is that the US labormarkets aren’t as flexible and self-correcting as I think we had presumed. The uneasinessI have about the way we’ve handled globalization is not so much about globalization itself.
It’s that if we don’t have the right safety net, you’re going to impose an enormous amountof hardship” (quoted by Binyamin Appelbaum in his NYT article, “Perils of Globalization
When Factories Close and Towns Struggle,” May 17, 2015).

3

Bangalore, India, where radiologists employed by companies such as Teleradiology Solutions and NH Teleradiology 24/7 do the reading and reporttheir findings back to the US hospital, with an average turn around of one
hour. This stark example illustrates the fact that, where the virtual labormobility is available, time zone diﬀerences among countries come into play
as a new driving force for international trade in services. Other examples include software industry located in the US and India (Cairncross, 1997), anda semi-conductor chip manufacturer that keeps 24-hour chip design systems
by locating design teams separately in the US, India and Europe (Brown and
Linden, 2009). In this context, many people were struck by the sound of analarm bell: in a well cited paper, Blinder (2006) argued that the oﬀshoringof service sector jobs is likely to become a major concern in the decades tocome, thanks to the advance in ICT. These services range from computer
programming and chip designs to legal services and medical tasks such asradiology. According to an estimate by Blinder (2006), the number of US
jobs that are potentially oﬀshorable is in the range of 42 to 56 million, of
which around 28–40 million are in the service sector. In a follow-up paper,
Blinder (2009) stresses that since ICT keeps getting both better and cheaper,
the scope for oﬀshoring will increase inexorably, and wages of skilled workers
in the advanced industrialized economies (the North) will suﬀer a setback.
On the contrary, there has been some empirical evidence that R&D oﬀshoring have brought gains to oﬀshoring countries. Indeed, some empiricalstudies have pointed out some interesting features in the globalization of the
R&D activities, which provide us with a fresh perspective for R&D oﬀshoringthat is quite diﬀerent from Blinder (2006, 2009)’s anxieties about negative impacts of oﬀshoring. Branstetter, Glennon, and Jensen (2018) have reported
the transformation of landscape of global R&D expenditure by US-based
multinational enterprises: (i) the total amount of the global R&D expenditure from 1999 to 2014 has increased at an average annual rate of 5.6%, much
faster than the growth rate in US multinational employment of 2.1%; (ii) the
share of traditional destinations of the US foreign R&D, such as the UK,
Germany, Japan, France, and Canada, has declined from 74% in 1989 down
to 43% in 2014, whereas the share of emerging countries such as China, India,
4and Israel has grown rapidly by 2014—the emergence of new R&D hubs; (iii)
R&D oﬀshoring is particularly important in ICT- and/or software-intensive
countries and it is carried out by ICT- and/or software-intensive firms. Paying special attention both to the diﬀerent modes of governance and to two
types of innovation outcomes, Nieto and Rodriguez (2011) have shown that
there is a positive and significant correlation between R&D oﬀshoring andinnovation performance. Similarly, Bertrand and Mol (2013) have found apositive and significant innovative eﬀect of R&D oﬀshore outsourcing, whichis greater than that of R&D domestic outsourcing, and they have argued
that the key to innovation is heterogeneity of knowledge inputs that is madeavailable to firms by the R&D oﬀshore outsourcing.
In a recent paper, Rodriguez and Nieto (2016) investigated the rolesof R&D oﬀshoring in the sales growth of small- and medium-sized enterprises (SMEs) and found a positive significant correlation between R&D
oﬀshoring and firm growth. In a related study D’Agostino, Laursen, andSantangelo (2013) examined the impact of R&D oﬀshoring from advanced
OECD countries/regions to emerging economies such as Brazil, Russia, India, China, Singapore, and Taiwan (BRICST) on the home knowledge production. They found that “regions that both oﬀshore R&D in medium/low
technology-intensive sectors to BRICST and carry out above median level
R&D at home, produce more knowledge than regions that either only oﬀshore R&D in medium/low technology-intensive sectors to BRICST, or only
conduct above median R&D at home” (p.166). Similarly, conducting an empirical analysis on R&D oﬀshoring and the productivity growth of European
regions, Castellani and Pieri (2013) found that “oﬀshoring regions exhibit
higher growth both in value added and employment than the non oﬀshoring
ones, but the growth in output is larger than the one of employment, thus
determining positive productivity eﬀects” (p.1588).
Motivated by both Blinder’s concerns and the apparently more positive
empirical evidence, in this paper we formalize a theoretical model of endogenous growth driven by partial oﬀshoring of R&D from North to South. A
key ingredient of the model is the role of time zone diﬀerence, which makes aNorthern firm’s R&D workers in North and in South complementary rather
5than substitute factors of production. Our main finding is that even though
the partial oﬀshoring of R&D tasks to South will have a (short-run) negativeeﬀect on the skilled wage in North, it will raise the rate of innovation andconsequently boost the growth rate of Northern skilled workers’ wage. The
mechanism behind our model is that with the improved access to virtual
skilled researchers from diﬀerent time zones, especially from countries withlower wages, Northern firms will have incentives to give a greater push totheir R&D activities, even though R&D successes are random.
There are two distinctive features of R&D that our model takes into
account. One is that R&D is a cumulative process: the exchange and refinement of ideas come in consecutive steps. A project is completed only after
a sequence of eﬀorts. This feature of R&D renders it very sensitive to theease with which researchers can interact with each other via communications
networks. The other is that the fruit of R&D is uncertain by nature: even if
we complete all the scheduled steps of an R&D project, we may fail to obtain
the desired results. A firm’s research department, which hires researchers,
must take account of the uncertainty pertaining to R&D when it decides if
a research project should be carried out.
To examine the eﬀects of R&D outsourcing over diﬀerent time zones, weconstruct a variant of the Schumpeterian endogenous growth model developed by Aghion and Howitt (1992, 2009); see also Gersbach et al. (2013). We
show that the partial oﬀshoring of R&D to the South will increase the growthrate of the North, but decreases the Northern skilled wages temporarily. The
skilled workers (i.e., researchers or scientists) in the North will suﬀer froma sudden decrease in their income in the short run, but ultimately they will
gain from a higher growth rate of their wages, in the long run.
The plan of our paper is as follows. A brief review of the related literature is presented in the next section. In Section 3, we develop a North-Southmodel where Northern firms engage in R&D activities to improve the quality of their intermediate inputs. In Section 4, we elaborate on how R&D
decisions are made in a stochastic environment. In Section 5, we consider
a situation where no virtual skilled labor though communications networkfrom South is available to the Northern R&D firms, which we call “com6munications autarky.” We show how the market for skilled labor is cleared
and how the skilled wage rate and the rate of growth are determined along abalanced growth path in communications autarky. Section 6 considers a situation where the advancement of communication technology enables Northern
R&D firms to oﬀshore their R&D activities to South located in a diﬀerent
time zone, which we call “communications liberalization.” We show how
the communications liberalization leads to a short-term fall in the wages ofNorthern skilled workers, but an increase in the long-run growth rate of theirwages. In Section 7, by taking account of full general equilibrium conditions,
we show how the relative price of the final agriculture good in terms of the
final manufacturing good (i.e., the terms-of-trade of South) and the national
incomes of the countries are determined. Section 8 discusses the welfare
implications of our findings and points to possible extensions of the model.
Section 9 concludes.

2

Related Literature

Our paper is built on recent works in two streams of literature: (a) the identification of various channels through which time zone diﬀerence constitutes
an important source of gain from trade, and (b) the analytics of trade intasks and task oﬀshoring.
Parallel to the advance of communication technology is the striking emergence of a series of insightful papers on the economics of time zones. Marjit(2007) was the first to prove that the time zone diﬀerence is a new independent source of comparative advantage. He considered a Ricardian modelin which the production process of the time-zone-related good is verticallydivided into two consecutive stages, each of which takes one normal working
day. If a firm performs the production in its home country by employing
only local workers, it takes two working days to complete producing one unitof the good because people normally work only in the daytime.3 In contrast, if a firm performs the first stage in the daytime of the home country
and outsources the second stage via the communications networks to foreign3

For simplicity, we assume away the night-shift working.

7

workers residing in a country located in the opposite time zone, the firm can
get the finalized good one-day earlier than when the virtual labor mobility
is not available. Without the virtual labor mobility, the consecutive production process is interrupted by the resting time every night and, therefore, thecompletion is delayed; on the other hand, with the virtual labor mobility, theproduction process can be made continuous and the completion time will berendered shorter. Because of the existence of time discounting, the shorterproduction time enables the firm to obtain a higher profit. This continuity
eﬀect can be a cause of comparative advantage.
Focusing on the roles of night-shift working, Matsuoka and Fukushima(2010) pointed out another cause of comparative advantage related to thetime zone diﬀerence. They assumed that the production process is verticallydivided into two consecutive stages and that, unlike Marjit (2007), thesestages must be performed continuously without any interruption. If a firm
wants to continue the production during the night by hiring local workers
in the home country, the firm has to pay a nighttime premium to the home
workers in addition to the normal daytime wage rate; in contrast, if the firm
can hire (indirectly through the communications networks) foreign workers
in a country located in the opposite time zone during the nighttime in thehome country (the foreign workers are in their daytime), the firm can save thenighttime premium payment because the foreign workers do the jobs in theirown daytime. This cost-reduction eﬀect can be another cause of comparativeadvantage, too.
As clearly shown by Marjit (2007), Matsuoka and Fukushima (2010) andothers,4 both the continuity and cost-reduction eﬀects of the virtual labormobility improve the static eﬃciency of the production of time-zone-related
goods. In view of this, one might think it natural that these eﬀects also contribute to enhance the dynamic eﬃciency of the economy and to increase theeconomic growth rate. So far, however, only a few studies have dealt withthe growth eﬀects of time-zone-related trade in services and/or virtual labormobility. Kikuchi and Marjit (2011) considered a simple two-country AK
4

For other contributions concerning the time zone diﬀerence and comparative advantage, see Kikuchi (2006, 2009) and Kikuchi and Iwasa (2010).

8

growth model that incorporates intermediate business services, which canbe traded across diﬀerent time zones. They showed that a reduction of thetransaction cost of trade in services due to the advancement of the communications networks increases the endogenous growth rates of both countries.
Marjit and Mandal (2017) extended the Kikuchi-Marjit model to an AK-type
optimal endogenous growth model. They showed that the liberalization of
trade in services can raise not only the level of the GDP of the country but
also the growth rate. They also examined the labor market implications of
the liberalization of trade in services across diﬀerent time zones.5 The key
factor that enhances growth in the Kikuchi-Marjit and Marjit-Mandal models is the expanded availability of the intermediate business services, whichraises the marginal productivity of capital and, thereby, increases the growthrate. Their models, however, do not take account of R&D activities. Needless to say, R&D is one of the most important and vital driving forces of theeconomic growth. In this paper, we will focus on the roles of R&D, in particular, on how the oﬀshoring of R&D through the communications networks
taking advantage of time zone diﬀerences aﬀects the growth rate.6
The recent literature on task oﬀshoring receives a great push from the authoritative work of Grossman and Rossi-Hansberg (2008) (G&RH for short).
These authors were the first to identify three eﬀects of oﬀshoring: the productivity eﬀect, the relative-price (terms-of-trade: TOT) eﬀect, and the laborsupply eﬀect. In G&RH’s main model, fragmentation is modelled as a decline
in the cost of trading a continuum of unskilled tasks. This leads to a positive
productivity eﬀect that increases the wage of North’s unskilled workers, anda positive TOT eﬀect for North. In a variant specification, G&RH supposethat skilled tasks can also be oﬀshored. In this case, oﬀshoring can lead toa deterioration in North’s TOT. Although we also find these eﬀects, the fea5

In static settings, both Kikuchi and Long (2011) and Nakanishi and Long (2015) haveexamined the implications of trade in services across diﬀerent time zone on the income
distribution. The former is based on the standard Heckscher-Ohlin-Samuelson model; thelatter on the Specific-Factors model.
6
Within the framework of endogenous growth model, Davis and Hashimoto (2018) havedeveloped a model in which firms are free to choose the locations of production activity and
R&D activity separately. They focused on the roles of occupational choice by (potential)
workers and diﬀerence in the relative abundance of assets of the countries.

9

tures of our model are considerably diﬀerent from those of G&RH’s model.
First, G&RH assume perfectly competitive markets (p.1979), while we havea monopolistic competition sector. Second, they assume that “foreign andlocal tasks of a given type are perfect substitutes in production” (p.1982,
footnote 12), while in our model the two R&D teams (one in North and theother in South) are complements when both teams are employed (i.e., whenpart of an R&D project is oﬀshored). This is a distinctive feature of ourmodel due to the time zone diﬀerence, which is absent in G&RH. Third,in the first half of their paper, they used the “small country assumption,”
which they acknowledge (in their footnote 17) as a “somewhat artificial assumption,” while we consider the case of two large economies in which theterms of trade is determined endogenously. Finally, G&RH did not consider
endogenous growth, while our emphasis is on the eﬀect of partial oﬀshoringof R&D on the rate of growth of the world economy.
Rodriguez-Clare (2010) (RC for short) compares short-run and long-run
eﬀect of oﬀshoring. Similar to the current paper, RC considers a dynamic
model in which technology levels are endogenous. Workers can choose towork in the production sector or to do research (p.230). In RC’s model,
both countries have a research sector, which is a diﬀerent point from ourmodel where research only occurs in North. Each country has a “stock ofideas,” denoted by T , and it is assumed that the richer country has a higher
stock of ideas per workers, T /L. In the short run, oﬀshoring production
reduces North’s wage, but in the long run, as a larger fraction of North’s
workers become researchers, North can sustain a higher T /L, which translates
into higher wage in North in the long run. RC concludes that “increased
fragmentation could have negative eﬀects for rich countries, but these eﬀects
dissipate in time, so that the long-run eﬀects are always positive for thecountries doing the oﬀshoring” (p.231). In RC’s model, by assumption, North
does not oﬀshore research to South. In our model, North is able to employ
South’s researchers via virtual labor mobility. While this is a form of R&D
oﬀshoring, Southern researchers are complements rather than substitutes forNorthern ones, because in our model, South’s scientists working for North inSouth’s daytime (i.e., in North’s nightime), building on each other’s previous
10

steps.
In our model, there is no oﬀshoring of production activities. If we extend
the model to allow for oﬀshoring of production, there would be additional
considerations to take into account. For example, as argued by Naghavi
and Ottaviano (2009), oﬀshoring production activities may weaken the information feedback from production to R&D. Some authors, such as Ernst(2006) and Macher and Mowery (2008), have voiced concerns about “actualoﬀshoring” of R&D activities. It should be noted that “actual oﬀshoring”(in the sense of replacing Northern R&D scientists with Southern ones) does
not occur in our model, as they are complements, not substitutes.
In line with the empirical findings reported in Section 1, our model showsthat partial oﬀshoring of R&D activities can enhance growth. Moreover,
our approach highlights the importance role of time zone diﬀerences which
turn foreign and domestic researchers into complementary factors rather thansubstitutes. We show how both North and South benefit from the endogenous
growth generated by the increase in worldwide R&D activities.

3

Model

There are two countries, North and South. North is endowed with L units
of unskilled labor and H units of skilled labor. We assume that North’s
unskilled labor can only be used as an input of a final manufacturing good.
North is capable of producing a final manufacturing good (i.e., the Northgood) under a constant-returns-to-scale technology, using as inputs (i) unskilled labor and (ii) a continuum of intermediate inputs indexed by z, wherez belongs to the real interval [0, 1]. Each unit of intermediate good z is produced by a firm, also denoted by z, using one unit of skilled labor. Skilledlabor can also be employed in the R&D sector. The quantity of skilled workers employed in the production of intermediate goods (respectively, in the
R&D sector) in period t is denoted by HI,t (respectively, HR,t ).7 The full7

We assume a discrete-time dynamic model. Subscript “t” attached to some variables
indicates “period t.”

11

employment condition for skilled labor is
HI,t + HR,t = H.
As in Romer (1990), the final manufacturing good is produced under
perfect competition, while the firms that produce intermediate goods operate
under monopolistic competition and earn positive profits. Intermediate goods
are not durable nor tradable goods. As for the formulation of R&D decisions,
we assume that the incumbent firm z is the only one that conducts R&D
activities for that input.8 We will explain later how the demands for skilledlabor in the intermediate good production and in the R&D activities are
determined.
South is endowed with L∗ units of unskilled labor and H ∗ units of skilledlabor. South specializes in a traditional agriculture good (i.e., the Southgood), which is produced under a Ricardian technology by using only unskilled labor as input; the labor input-coeﬃcient is assumed to be one. Since,in the absence of R&D oﬀshoring from North, there is no job for the Southernskilled workers, South’s skilled labor can only work as unskilled labor andearn South’s unskilled wage. Thus, in the communications autarchy scenario,
the whole labor force in South, L∗ + H ∗ , is employed as unskilled labor in the
agricultural sector and, accordingly, the wage rates received by two diﬀerent
types of labor in South are equalized. In this scenario, South’s output of the
agriculture good, denoted by yS , becomesyS = L∗ + H ∗ .
South exports some of its agricultural output to North and imports
North’s final manufacturing good. By assumption, North is not capable ofproducing the agricultural good, and South is not capable of producing the8

There is another possible formulation of R&D decisions: in each intermediate input
sector z, there is a pool of potential innovators who are attempting to improve the qualityof z; if an innovator succeeds in the quality improvement, she captures the current market
for the very intermediate input z and becomes a monopoly. Because this alternative
formulation of R&D decision has some theoretical diﬃculties, we adopt the formulation
described in the text.

12

final manufacturing good nor the intermediate good. All firms (i.e., manufacturing firms, agriculture firms, intermediate-good/R&D firms) do not moveacross countries.

3.1

Household Sector

Preferences of households/consumers are the same in North and South. We
assume that consumers are infinitely-lived individuals. Without loss of generality, we assume that in each country, the population size is equal to unity.
Because the same argument equally applies to both countries, we omit thesuperscript (N or S) that indicates the identity of a country in this subsection.
In period t, any individual who consumes cS,t units of the agriculture
good and cN,t units of the final manufacturing good derives the (temporal)
utility from the following Cobb-Douglas function:
U (cS,t , cN,t ) = δ ln [cS,t ] + (1 − δ) ln [cN,t ](1)

where 0 < δ < 1.
In principle, in any period, a consumer can consume more or less thanher current income. Let Et be the consumption expenditure in period t by
the representative consumer. In any period t, given her chosen consumption
expenditure level Et , the consumer chooses the quantities of two consumption
goods, cS,t and cN,t , to maximize her utility function (Eq. (1)) subject to the(temporal) budget constraint:
pt cS,t + cN,t = Et ,(2)

where pt is the price of agriculture good in terms of the manufacturing goodin period t. This yields the demand functions for the final consumption
goods:
δEt,
pt
cN,t = (1 − δ)Et .
cS,t =

13

(3)
(4)

Substituting the above results back into Eq. (1), we obtain the (temporal)
indirect utility function:
V (Et , pt ) ≡ ln [Et ] − (1 − δ) ln [pt ] + κ,

(5)where κ ≡ δ ln δ + (1 − δ) ln(1 − δ) is a constant.

To determine the consumer’s optimal time path of expenditure Et for

t = 0, 1, 2, . . . , ∞, we posit that the consumer solves her intertemporal optimization problem:

max E0
{Et }

∞
!

1
V (Et , pt ),
t
(1
+
ρ)
t=0

where ρ is the consumer’s degree of impatience (i.e., the utility discountrate) and E0 is the expectation operator. The intertemporal optimization is

subject to the intertemporal budget constraint that the present value of theexpenditure stream (i.e., Et for t = 0, 1, 2, . . . ) be equal to the present valueof the income stream (i.e., It for t = 0, 1, 2, . . . ):
E0
where

#t

τ =0 (1 + rτ )

−1

∞ "
t
!

t=0 τ =0

(1 + rτ )−1 [Et − It ] = 0,

is the discount factor and rτ is the equilibrium interest

rate in period τ .
In later sections, we will focus on balanced growth paths. Along a balanced growth path, the rate of interest becomes a constant, r, which is endogenously determined via the well-known Keynes-Ramsey equation:9
r =ρ+

gE
,
σ

(6)

where gE is the rate of growth of consumption expenditure and σ is theelasticity of intertemporal substitution. In balanced growth, gE is equal to
the growth rate of income. In fact we will show that, in our model, thegrowth rate of national income is the same for both countries. Further, it
should be noted that, we have σ = 1 because of the logarithmic indirect
utility function derived in Eq. (5).
9For the derivation of the Keynes-Ramsey equation, see Appendix A.

14

Concerning the financing of R&D expenditure by intermediate good firms,
we note that in our model, the firm’s expenditure on R&D may be higher orlower than its flow of revenue from sales net of production cost. To finance
R&D, the firm can issue bonds (I.O.U.’s) to consumers in North. This does
not create any diﬃculty to our analysis, because in a model with identicalconsumers, all consumers in North know that the firm’s debt is not part oftheir net wealth; even bonds issued by North’s government, if any, are notconsidered by the identical consumers in North as part of their net wealth,as Barro (1974) pointed out in his famous paper on Ricardian equivalence.

3.2

The Manufacturing Good Firms

The manufacturing good (the North good) is produced by perfectly competitive firms, under the production function$ 11−αyN,t = Lu,t
[At (z)]1−α [xt (z)]α dz.

(7)0where Lu,t is unskilled labor input, xt (z) is the quantity of intermediate goodz, and At (z) is the quality of intermediate input z in period t. We assume thatthe manufacturing good is the numéraire and its price is normalized to unity.
Let wu,t denote the unskilled wage rate and qt (z) the price of intermediate
good z in period t. The representative manufacturing firm takes wu,t andqt (z) as given, for all z ∈ [0, 1]. Its profit maximization problem is
$ 1
$ 1
1−α
1−α
α
max Lu,t[At (z)] [xt (z)] dz −qt (z)xt (z)dz − wu,t Lu,t0

0

The first-order conditions are such that each factor is chosen at a level that
equates its marginal product to its factor price:
$ 1
yN,t
−α
wu,t = (1 − α)Lu,t[At (z)]1−α [xt (z)]α dz = (1 − α)
Lu,t
0
1−α
α−1 1−αqt (z) = α[At (z)] [xt (z)] Lu .

(8)
(9)

The above system of equations implicitly defines the demand for unskilledlabor, denoted by LDu,t , and that for intermediate inputs.
15

We assume that the economy’s available unskilled labor is supplied inelastically and equal to L > 0. Then, the market equilibrium condition forunskilled labor in North becomesLDu,t = L.
Accordingly, Eqs. (8) and (9) can be rewritten as follows:yN,t,L
qt (z) = α[At (z)L]1−α [xt (z)]α−1 .
wu,t = (1 − α)The second condition gives the demand function for intermediate input z:
1
%
& 1−α
α
xt (z) =At (z)L.
(10)
qt (z)
Notice that xi (t) is increasing in L, as expected. Indeed, the size of theunskilled labor force, L, is a good indicator of the size of the manufacturingsector. The larger is manufacturing sector, the greater is the demand forintermediate inputs.

3.3

The Intermediate Good Firms

Each intermediate good firm is a monopoly in each sector; one unit of intermediate input production requires one unit of skilled labor. Let wt denotethe skilled wage rate in period t. Each firm sets its price qt (z) to maximize
its profit πt (z) ≡ [qt (z) − wt ] xt (z) subject to the intermediate good demand(Eq. (10)). The solution yields the constant mark-up pricing rule:
qt (z) =wt.
αThis rule implies that in equilibrium, the demand for each intermediate input
is decreasing in the skilled wage rate, and increasing in its quality level:1% 2 & 1−ααxt (z) =At (z)L.
(11)
wt
Then, the maximized profit of the intermediate good firm is−απt (z) ≡ θ (wt ) 1−α At (z)L,
16

(12)

1+α

where θ ≡ (1 − α)α 1−α is a positive constant. Note that both the profit

maximizing output of the intermediate good and the maximized profit arelinear in the quality parameter At (z).

4Research and Development

We model R&D as follows. We assume that the incumbent intermediategood firm z is the only one that conducts R&D activities for that input. Its
research department employs researchers to carry out a research project. A
project needs to be implemented in a sequence of steps. The second stepcannot begin unless the first step has been completed, and so on. Each step
takes 12 consecutive hours of a researcher. For simplicity, we assume thata research team consists of only one researcher. (If the research department
employs two researchers on the same day to work on the same project, their
total research output is not greater than that of a single researcher.)
Researchers work only during their local daytime. If a research firm inNorth relies only on Northern researchers, then at the end of the year, its
research project will have completed 365 steps. On the other hand, if the firmuses both Northern and Southern researchers (located in the opposite timezone) then at the end of the year, its research project will have completed
730 steps. We assume that one period of time consists of 365 days (i.e., one
period is equal to one year) and that a single project requires 365 steps andthe result of one project can be obtained only after finishing the 365th step.
Therefore, a firm can complete at most one project in one period if it uses
only Northern researchers, while it can complete up to two projects in one
period if it can use both Northern and Southern researchers.

4.1

R&D Indicator

Let R be the domestic R&D indicator; similarly, R∗ be the foreign-outsourcing
R&D indicator. We assume that R and R∗ are binary variables, not continuous variables:
R, R∗ ∈ {0, 1}.
17

One unit of research project is divided into 365 steps; each step takes one
regular working day (12 hours). The (s+1)-th step cannot begin unless the sth step has been completed; and the result can be obtained only after finishingthe 365th step. Then, without partial oﬀshoring of R&D, each research
department can accomplish at most one research project per year. Whenpartial oﬀshoring of R&D is possible, in contrast, a research department canundertake two research projects in the following manner: the 1st step of one
project is carried out in the home country in the first day of a year; the 2nd
step is outsourced to a foreign country located in the opposite time zone in
the first day (but, the nighttime in home); the 3rd step is carried out in thehome country in the second day, the 4th step is outsourced to the foreigncountry in the second day (the nighttime in home) and so on. In this way,
the length of time for completion of one project is compressed to half a year.
Then, by using the second half of a year, the research firm can undertakeone more project.

4.2

R&D Productivity

By conducting R&D, each research department can increase the growth rate
gt (z) of At (z). The relation between gt (z) and the research indicators isrepresented by
gt (z) = λ[R + R∗ ],R, R∗ ∈ {0, 1},(13)

where λ ≥ 0 stands for the productivity of R&D in terms of the size of

quality innovation. Further, we assume that λ is a stochastic variable thathas a compact support [λL , λH ] where 0 < λL < λH . (More precisely, we
should write λt,z because the realized values of λ diﬀer across firms andacross periods; however we omit these subscripts to lighten notation.) The
distribution function F of λ must satisfy F (λL ) = 0 and F (λH ) = 1 andF ′ (λ) > 0 for λ ∈ [λL , λH ]. For this to hold, we propose the following

18

distribution10

F (λ) ≡⎧
ββ⎪⎪
⎨1 + λ − λ
H

1
⎪
⎪
⎩
0

if λ ∈ [λL , λH ],if λ > λH ,if λ < λL ,where β ≡ λH λL /(λH − λL ). Then, the density function f becomes⎧⎨ β
if λ ∈ [λL , λH ],f (λ) =λ2⎩0if λ ∈/ [λ , λ ].
L

(14)

(15)

H

The expected value of λ isE(λ) =

$ λH
λL

% &
β
λH
λ · 2 dλ = β ln.
λλL

Taking account of the definition of growth rate, we can rewrite Eq. (13)
as follows:
At+1 (z) − At (z) = λ[R + R∗ ]At (z),which means that an increment of the quality of intermediate good z is proportional to the product of the skilled labor input [R + R∗ ] and the current
quality At (z). This is reminiscent of what Rivera-Batiz and Romer (1991a)called the knowledge-driven specification of R&D. In their model, the stateof technology is represented by a closed interval of continuous range of intermediate goods with its upper bound being A, which also reflects the current
knowledge level, and technological progress due to R&D is represented by thediﬀerential equation Ȧ = δHA, where H is the skilled labor input to R&Dand δ is a positive constant. The resemblance between the R&D function inRivera-Batiz and Romer (1991a) and that in our model is obvious.11
There is, however, a subtle but important diﬀerence between the R&D
function in Rivera-Batiz and Romer (1991a) and that in our model. In the10This distribution is a modified version of the standard Pareto distribution, whereF (λ) = 1 − (B/λ)a for λ ≥ B and F (λ) = 0 for λ < B, where B is a positive parameter.
11Rivera-Batiz and Romer (1991a) have also proposed an alternative specification of
R&D called the lab equipment specification, which depends on human capital, unskilledlabor, and capital goods, but not on the current level of knowledge per se (i.e., A).

19

former case, each skilled worker in H can be considered a perfect substitutefor other skilled workers in H. In our case, although the skilled workers inNorth and South enter the R&D production function in the additive form,
this does not mean that the Northern skilled workers R and the Southernskilled workers R∗ are substitutes for each other. Because the Southern skilledworkers are available to the Northern research firms only in the nighttime ofNorth, they do not directly compete with the Northern skilled workers who
are active only in the daytime of North. The Southern skilled workers are
irreplaceable with the Northern skilled workers and vice versa. The Northern and Southern skilled workers are, in a sense, complements rather thansubstitutes.

4.3

Expected profits from R&D investment

At the beginning of period t, each firm’s research department draws its
R&D productivity λ from F . After knowing λ, the firm makes a decision
on R&D. If firm z decides to do R&D, it can improve the quality of intermediate good z according to the R&D function, Eq. (13), and the realized λ. The next period’s quality after this period’s R&D is given by
At+1 (z) = (1 + gt (z)) At (z) ≡ (1 + λ [R + R∗ ]) At (z). For simplicity, we as-sume that the skilled-labor input coeﬃcient for R&D is equal to unity. Concerning the cost of oﬀshoring, Grossman and Rossi-Hansberg (2008) argued
that “[oﬀshoring of tasks] is costly, because remote performance of a task
limits the opportunities for monitoring and coordinating workers” (p.1978).
Along the lines with their argument, we introduce a parameter that represents the communication cost between the head oﬃce in North and the
R&D workers in South. Then, the expected net returns from period t’s R&D
investment becomes−αθ (wt+1 ) 1−α (1 + λ[R + R∗ ]) At (z)L − (1 + rt ) (wt R + τ wt∗ R∗ ) ,(16)

where rt , wt∗ , and τ ≥ 1 represent the interest rate, the foreign skilled wage
rate, and the iceberg-type communication cost, respectively.12
12

Formally, Grossman and Rossi-Hansberg (2008) assumed that aLj βtj (i) units of foreign labor is required to perform task i in producing a unit of good j and that βtj (i) ≥ 1

20

On the other hand, if the firm does not conduct R&D, the quality of z
does not improve, i.e., At+1 (z) = At (z), and the profit in period t + 1 issimply−αθ (wt+1 ) 1−α At (z)L.

(17)

If the profit in Eq. (16) is greater than the profit in Eq. (17), the firm will
carry out the R&D project.

4.4

R&D under Communications Autarky in North

Under communications autarky, firms cannot oﬀshore R&D. Therefore, wehave R∗ = 0 a priori. The condition for domestic R&D to be carried out(i.e., the condition for R = 1) is that the profit in Eq. (16) is greater thanthe profit in Eq. (17). With appropriate arrangements, we have R = 1 if andonly ifαφwt (wt+1 ) 1−αλ>,At (z)L

(18)

where φ ≡ (1 + rt )θ−1 . Although φ depends upon “period t” via the rate ofinterest rt , we omit an explicit indication of t from φ for notational simplicity.
Furthermore, for any sector z, let
α1−α+z,t ≡ φwt (wt+1 )λ.
At (z)L

Then, the decision on R&D by firm z can be summarized by the following
decision function h:+z,t ) ≡h(λ| λ,+z,t ,
1 if λ > λ+z,t .
0 if λ ≤ λ(19)As λ is a stochastic variable drawn from the distribution function F , theprobability for R = 1 can be written as follows:.
+z,tProb[R = 1] = Prob λ > λfor all i and j. Their βtj (i), in particular, their parameter β and our τ play a similar role.
To quote Grossman and Rossi-Hansberg (2008): “Reductions in this parameter represent
improvements in communication and transportation technology that reduce proportionallythe cost of oﬀshoring all tasks performed by low skilled labor. With this parameterization,
we can address an important and topical question, namely: how do improvements in theopportunities for oﬀshoring aﬀect the wages and well-being of diﬀerent types of labor?”21

More specifically,.
+z,t= 1 − Prob λ ≤ λ/ 0+z,t .
=1−F λ

⎧
⎪
⎪ 1
⎨ β
/ 0 ⎪
β
+
−
Prob[R = 1] = 1 − F λz,t =+z,t λH
⎪
λ
⎪
⎪
⎩ 0

+z,t < λL ,if λ+z,t ∈ [λL , λH ],if λ+z,t > λH .
if λNote that this probability also gives the demand for domestic skilled labor
in R&D from sector z.

4.5

Skilled Labor Market in North

Let H be the domestic supply of the skilled workers. These skilled workers
are either employed in the production of intermediate inputs, or hired as
researchers. The skilled-labor demand for the production of intermediate
goods, HI,t , is represented by a function DI of wt :1
21
1
2
$ 1 ,% 2 & 1−α
α 1−α 0 At (z)Ldz
αHI,t = DI (wt ) ≡At (z)L dz ≡.
1wt0(wt ) 1−αConcerning the demand for researchers, let us partition the set of firms
into three subsets, denoted by ΩH,t , ΩM,t , and ΩL,t , where
3
,
1
α
3
φwt (wt+1 ) 1−α3ΩH,t ≡ z ∈ [0, 1] 3 At (z)L >
,
3
λL
3
,
1
α
α
3 φw (w ) 1−αφwt (wt+1 ) 1−α3tt+1ΩM,t ≡ z ∈ [0, 1] 3
≥ At (z)L ≥
,
3
λL
λH
3
,
1
α
3 φw (w ) 1−α3tt+1ΩL,t ≡ z ∈ [0, 1] 3
> At (z)L .
3
λH

Firms in the set ΩH,t have very high At (z) and therefore will do R&D, regardless of the value of their random draw λ (which we know is always inthe set [λL , λH ]). These firms will set R = 1. Firms in the set ΩL,t have22very low At (z) and will not do R&D, regardless of the value of their randomdraw λ. Lastly, firms in the set ΩM,t will do R&D with a positive probability+z,t − β/λH .
β/λWithout loss of generality, re-label firms so that each of ΩH,t , ΩM,t , andΩL,t is a connected subset of the unit interval [0, 1]. Thus, the expecteddemand for researchers, HR,t , is represented by a function DR of wt andwt+1 :
HR,t = DR (wt , wt+1 ) ≡ |ΩH,t | +

$

ΩM,t

4

βAt (z)Lβ−αλH
φwt (wt+1 ) 1−α5dz,

where |ΩH,t | denotes the length of the interval ΩH,t .

In what follows, for simplicity, we assume that the sets ΩH,t and ΩL,t are

empty, implying that ΩM,t = [0, 1].13 This assumption would hold if no firm
has a long sequence of highly favorable (or highly unfavorable) realizationsof λ. Then7αφwt (wt+1 ) 1−αDR (wt , wt+1 ) ≡ 1 −Fdz
At (z)L
0
&
$ 1%
$ 1
β
βAt (z)L
=1−
1+
dz +
α dz
λH
0
0 φwt (wt+1 ) 1−α21
β 0 At (z)Ldz
β
=
−
.
α
λH
φwt (wt+1 ) 1−α$ 1

6Clearly, both DI and DR are decreasing in the current skilled wage rate
wt and the next-period skilled wage rate wt+1 . The domestic skilled-labormarket equilibrium obtains if
HI,t + HR,t = H,
13

(20)

Even if ΩL,0 = ∅ initially, some sector z may be trapped into ΩL,t in the course of
sectoral growth at a certain period t as a result of several unfavorable draws of λ, implyingΩL,t ̸= ∅. However, if ΩL,t becomes large enough, the demand for skilled labor shrinks and,αconsequently, the skilled wage rate falls. In turn, the threshold value φwt (wt+1 ) 1−α /λH
between ΩL,t and ΩM,t decreases as the skilled wage rate falls. Then, some sectors in ΩL,t
may be back into ΩM,t . It may be the case that ΩL,t has a certain positive measure in thelong run. In this case, we can exclude those (range of) sectors in ΩL,t from the definitionof intermediate good sectors from the very outset.

23where H > 0 is a constant inelastic supply of skilled labor in North.14 By
taking account of the definitions of DI and DR , we can rewrite Eq. (20) as
follows:2

α 1−α Āt L
(wt )

1
1−α

+

β Āt L
β
=H+
,
α
λH
φwt (wt+1 ) 1−α

where
Āt ≡
denotes the average quality level.

$ 1

(21)

At (z)dz.

0

The skilled labor market equilibrium condition, Eq. (20), suggests that
there is a potential trade-oﬀ between the production of intermediate goods
and the R&D activities. If the economy employs more researchers in any
period t, then, in that period, the output of the intermediate good has to be
reduced. This is also true in the long run, but the reduction in the quantityof intermediate goods will be compensated for by the higher quality of theintermediate goods, which has been realized through R&D conducted in theprevious periods.

5

Balanced Growth Path

We define the balanced growth path as a path along which the average quality
level Āt grows at a certain constant rate g. We will soon show that, along abalanced growth path, the skilled and unskilled wage rates and the national
income grow at a common rate, denoted by gw , which is also equal to thegrowth rate of expenditure, gE , as appeared in the Keynes-Ramsey equation(i.e., Eq. (6)). Along a balanced growth path, the (constant) rate of interest
induced from Eq. (6) with σ = 1 is r = ρ + gw . Then, φ now becomes
φ = (ρ + 1 + gw )θ−1 .

(22)

Clearly, φ is increasing in gw .
14

For the skilled labor in North to be supplied to the Northern skilled labor market, itis necessary that the skilled wage rate wt is not lower than the unskilled wage rate wu,tin North; otherwise the skilled workers in North opt to work as unskilled labor and theskilled labor market collapses. We can ensure that wu,t < wt if the endowment of unskilledlabor L is very large relative to the endowment of skilled labor H.

24

5.1

Growth Rate and the Skilled-Labor Market

From the skilled labor market equilibrium condition Eq. (21), we recognizethat if the average quality level were to grow at a certain constant rate gon the balanced growth path, so must the skilled wage rate. Let gw be thegrowth rate of the skilled wage rate. Then, if the balanced growth path were
to exist, we must have the solutions Āt = A(1+g)t and wt = w(1+gw )t , where
A and w are the initial values of Āt and wt . Substituting these solutions into
Eq. (21), we obtain
AL
w

1
1−α

4

α

2
1−α

β
+
α
φ(1 + gw ) 1−α58

1+g(1 + gw )

1
1−α

9t

=H+

β
.
λH

For this condition to hold for all t along the balanced growth path, the term
inside the square brackets must be equal to unity, that is,
1

1 + g = (1 + gw ) 1−α .
It is easy to see that, because 0 < α < 1, the growth rate of the average
quality level g is higher than that of the skilled wage rate gw .
Now, the skilled labor market equilibrium condition reduces to
%
&
2
AL
β
β
α 1−α +=H+,1
α(1 + g) φ
λH
w 1−α

(23)

or, equivalently,
w=

%4

α

2
1−α

β
+
(1 + g)α φ

5ALλH
HλH + β

&1−α

,

(24)

where φ = (1 + r)θ−1 = (1 + ρ + gw )θ−1 = (ρ + [1 + g]1−α )θ−1 ; φ can beseen as an increasing function of g. Eq. (23) describes the relation between
the growth rate g and the (initial) skilled wage rate w. This relationshipis monotone decreasing, which is depicted by curve S in Figure 1 with theslope:
3dw 33−(1 − α)βw{(1 + g)α φg + α(1 + g)α−1 φ}=< 0,
2
dg 3skilled market(1 + g)α φ{α 1−α (1 + g)α φ + β}where φg denotes the derivative of φ with respect to g.
25

(25)

Figure 1: The balanced growth rate and the skilled wage rate in North undercommunications autarky
The intuition behind the negative relationship between the growth rate gand the (initial) skilled wage rate w can be explained as follows. It should be
noted that in our model, the growth of an economy is brought about by both
quality improvements through R&D and increases in the skilled labor inputs
in the final good production. Suppose that skilled wage rate w decreases.
Then, for a given initial quality A, a lower skilled wage rate w would reducethe cost of R&D. More intermediate good firms carry out R&D, stimulating
growth in quality. At the same time, a low skilled wage provides intermediate
good firms with more incentives to expand their production, which in turnwould increase the output of the final good. Because of these positive eﬀects,
the growth rate g increases.

5.2

Consistency Condition

The growth rate of the quality in a single sector (say, sector z) is given by+z,t ).
gt (z) ≡ λh(λ| λ26

Then, the expected growth rate of quality in sector z in period t becomes. $ λH++z,t )f (λ)dλE [gt (z)] = E λh(λ| λz,t ) =λh(λ| λ
λL
8
9
(26)
$ λH
$ λH
β
λH
=
λf (λ)dλ =dλ = β ln+z,t!z,t!z,t λλλλNow we impose the following condition: for all z ∈ [0, 1] and for all t,
g = E[gt (z)].

(27)

That is, along the balanced growth path, the expected growth rates of thesectors coincides with the actual aggregate growth rate of the overall economy(i.e., the growth rate of the average quality level Āt ). This can be seen as aconsistency condition. The actual (or realized) growth rates of some sectors
may be greater and those of others may be smaller than g. Even if so, ourconsistency condition requires that the expected value of the growth rate beconsistent with the actual aggregate growth rate.15
Combining Eqs. (26) and (27), we have
% &
g
λH
exp
=
.
+z,t
β
λ

1

+z,t along the balanced growth path where (wt ) 1−α grows at rate
Expanding λg, we have

φw

1
1−α

% &
g
(1 + g) (1 + g) exp= λH At (z)L.
βαtNote that the left-hand side of the above equation is independent of z. By
integrating the above expression over all z, we get
% &
1
g
α
t
φw 1−α (1 + g) (1 + g) exp= λH Āt L = λH AL(1 + g)t .
β
Finally, we obtain1

φw 1−α (1 + g)α exp[g/β] = λH AL,
15

One may think of Eq. (27) as a result of the law of large number. Unfortunately,
Eq. (27) cannot be derived from the law of large number formally; that is why we simply
impose it on our model as an independent condition.

27

or, equivalently,

%

ALλH
w=
(1 + g)α φ exp[g/β]

&1−α

.

(28)

Eq. (28) describes the relation between the growth rate g and the (initial)
skilled wage rate w based on the consistency requirement. This is also amonotone decreasing relationship:3
dw 33−(1 − α)w{(1 + g)βφg + αβφ + (1 + g)φ}=
< 0.
3
dg consistency(1 + g)βφ(29)

We depict this relationship by curve C in Figure 1. By comparing Eq. (25)and Eq. (29), we can show that curve C is steeper than curve S:
3
3
dw 33
dw 33
<
< 0.
dg 3consistency
dg 3skilled market

Note that the derivation of Eq. (28) is based on the calculation of the ex-pected growth rate of each single intermediate good sector. Unlike Eq. (27),the growth rate g that appears in Eq. (28) only reflects the growth eﬀects
of quality improvements through R&D, but is separated from the indirect
growth eﬀects that come from increased production of the intermediate inputs. Therefore, responding to any given decrease in w, the growth rate in
Eq. (27) increases more than that in Eq. (28). This explains the diﬀerencesin the slopes of curve C and curve S in Figure 1.

5.3

Equilibrium in Communications Autarky

Combining both Eq. (24) and Eq. (28), we can determine the growth rate
and the (initial) skilled wage rate simultaneously and endogenously. Figure 1
illustrates the situation. Curve S represents the skilled labor market equilibrium, Eq. (24), and curve C represents the consistency requirement, Eq. (28).
From Eq. (25) and Eq. (29), we know that both curves are downward-sloping
and that curve C is “steeper” than curve S; accordingly, the intersection ofboth curves is unique.16 At point e, where both curves intersect, the growthrate g and the (initial) skilled wage rate w are determined.
16

The proof of the existence of a unique equilibrium is given in Appendix B.

28

To see the properties of the model, we conduct a comparative statics
analysis. Suppose that the supply of the skilled labor H increases. As
the right-hand-side (RHS) of Eq. (23) increases, the left-hand-side (LHS)
must increase, too. Accordingly, g and/or w have to decrease, implying adownward-shift of curve S to curve S ′ in Figure 1. Point e′ becomes the new
equilibrium point. We state this result more formally in Lemma 1.
Lemma 1. Under communications autarky, there exists a unique growth rateg > 0 if HλH > (1 + ρ)α/(1 − α). The growth rate g is increasing in North’s

endowment of skilled labor, H, in the maximal productivity parameter, λH ,and is decreasing in the rate of impatience, ρ; further, it is independent ofNorth’s endowment of unskilled labor, L.
For the proof, see Appendix B.1.
Notice that whereas the growth rate g increases in response to an increasein H, the skilled wage rate must temporarily decrease, as shown in Figure 1.
It should be noted that an increase in the skilled labor supply brings about anegative level eﬀect, but it brings about a positive growth eﬀect on the skilled
wage rate, because the growth rate of the skilled wage rate gw must increase
following the increased growth rate g (as far as we are concerned with thebalanced growth path).
With the above results, we can calculate the unskilled wage rate in North
in balanced growth. By substituting Eq. (7) and Eq. (11) into Eq. (8) andby taking account of the balanced growth conditions, we obtain
8: ; 19α$ 1
α2 1−α1−α−αwu,t = (1 − α)L
[At (z)]At (z)L dz
wt0= (1 − α)α

2α
1−α

(wt )

−α
1−α

(30)

Āt = wu (1 + gw )t ,

where
2α

−α

wu ≡ (1 − α)α 1−α Aw 1−α(31)

is the initial value of the unskilled wage rate. On one hand, Eq. (30) showsthat the unskilled wage rate in North grows at the same rate as the skilled
wage rate. On the other hand, Eq. (31) indicates that the initial value of29

unskilled wage rate, wu , is negatively related to the initial value of skilled
wage rate w.
We can also calculate the total (average) profit of the intermediate goodfirms in balanced growth, which is denoted by π̄t . From Eq. (12), we obtain
$ 1
−α
π̄t ≡
πt (z)dz = θ(wt ) 1−α Āt L = π̄(1 + gw )t ,
(32)
0

where
−α

π̄ ≡ θALw 1−α(33)

is the initial value of the total profit of the intermediate firms. Similar to
the unskilled wage rate π̄t grows at the rate gw , while the initial value π̄ isnegatively related to w.

6

Communications Liberalization

Now let us consider a situation where skilled labor in South becomes available to the research firms in North due to development of the communications
networks such as the Internet—we call this situation “communications liberalization.” In South, there is a pool of skilled workers, but no intermediate
good sectors nor R&D firms.17 Without communications liberalization, theskilled workers in South cannot bring their ability into full play, instead theywork as unskilled labor in a certain traditional sector.

6.1

R&D under Communications Liberalization

Under communications liberalization, each firm z faces several options on
how to conduct R&D. There are four options: (i) do no R&D project; (ii) do
only one R&D project by hiring only skilled workers in North; (iii) do only
17

In our model, the only motivation of Northern firms for R&D oﬀshoring is to hireSouthern skilled workers via virtual labor mobility by taking advantage of time zone
diﬀerences. If we consider a diﬀerent situation where there are some Southern R&D firms,
the possibility of cooperation and competition in R&D between Northern and Southern
R&D firms may provide those firms with an alternative incentive for R&D oﬀshoring. For
cooperative and non-cooperative incentives for R&D, see, for example, Marjit (1991) and
Kabiraj and Chattopadhyay (2014).

30

one R&D project by oﬀshoring its research activities during the nighttimes
in North to the skilled workers in South; or (iv) do two consecutive R&D
projects by hiring skilled workers in both North and South and by letting
them work day and night continuously. With a slight abuse of notation, let
us denote the research firm’s strategies that correspond to cases (i), (ii), (iii),and (iv) by 0, R, R∗ , and RR∗ , respectively.18
Let π0 , πR , πR∗ and πRR∗ be the profits of corresponding strategies, respectively. By definition, we have−απ0 = θ(wt+1 ) 1−α At (z)L,−απR = θ(wt+1 ) 1−α (1 + λ)At (z)L − (1 + rt )wt ,−απR∗ = θ(wt+1 ) 1−α (1 + λ)At (z)L − (1 + rt )τ wt∗ ,−απRR∗ = θ(wt+1 ) 1−α (1 + 2λ)At (z)L − (1 + rt ) (wt + τ wt∗ ) .
Let us assume πR > πR∗ , which is equivalent to wt < τ wt∗ .19 This assumptionmeans that if a research firm were to conduct only one R&D project, it
hires only Northern skilled workers. When a research firm conducts two
R&D projects, it oﬀshores part of each project to South even if the Southernskilled wage rate is higher than the Northern skilled wage rate. Because
the Southern skilled workers process the assigned part of a project during
the nighttime in North (when the Northern skilled workers are sleeping) andvice versa, the Southern and Northern skilled workers can be considered tobe “complements” rather than “substitutes.” By having two teams of skilledworkers (one in North and the other in South) for one project, a researchfirm can reduce the number of days to complete a project.
If π0 < πR , πRR∗ , then a research firm would carry out at least one R&D
project. In addition, if πR > πRR∗ , then the research firm conducts only one(domestic) R&D project. By the definitions of π functions, we can show that
π0 < πR if and only if
+z,t
λ>λ

With the R&D indicators, strategy 0 corresponds to “R = R∗ = 0” ; strategy R to
“R = 1 and R∗ = 0” ; R∗ to “R = 0 and R∗ = 1”; and RR∗ to “R = R∗ = 1” .
19This assumption of πR > πR∗ (equivalently, wt < τ wt∗ ) is not essential to our later
analysis; it is introduced only for the ease of interpretation. If πR < πR∗ , we have anothercase to be examined, but we only obtain essentially the same results.
18

31

and πR > πRR∗ if and only if
where

+∗ ,
λ<λ
z,t

α

∗
1−α
+∗ ≡ φ(τ wt )(wt+1 )λ.
z,tAt (z)LIn what follows, we assume that wt < τ wt∗ . Then we have+z,t < λ
+∗ .
λ
z,t

With these two threshold values of λ, the R&D decision of a research firm on
the number of projects can be summarized by the following decision function:⎧+⎪⎨ 0 if λ < λz,t ,∗+z,t , λ+ )=+z,t ≤ λ ≤ λ+∗h(λ| λ1 if λ
z,t
z,t
⎪
⎩
∗
+
2 if λ > λz,t .

6.2

Demand for Research Workers

The demand function for the domestic research workers in North under communications liberalization is the same as in the case of communications autarky:
DR (wt , wt+1 ) =β Āt Lβ−.
αλHφwt (wt+1 ) 1−α

To derive the demand function for (foreign) research workers in South,
let us consider the probability that a firm z chooses strategy R∗ (in addition
to R). The probability of R = R∗ = 1 becomes20
.
.
∗
∗
∗
+
+
Prob[R = R = 1] = Prob λ > λz,t = 1 − Prob λ ≤ λz,t
/ 0+∗=1−F λz,t=20

βAt (z)Lβ−.
αλHφ(τ wt∗ )(wt+1 ) 1−α+∗ implies λ > λ+z,t .
It should be noted that under our maintained assumption, λ > λz,t32

By integrating both sides of the above equation over all z, we obtain the
demand function for Southern research workers:&$ 1%
βAt (z)Lβ∗DR∗ (wt , wt+1 ) ≡
−
dz
α
λH
φ(τ wt∗ )(wt+1 ) 1−α0
β Āt L
β
=
−
,
α
∗
λH
φ(τ wt )(wt+1 ) 1−αThis shows that the demand depends negatively on the (current) skilled wage
rate in South and the future skilled wage rate in North.21

6.3

Skilled Labor Market in SouthThe equilibrium condition for the skilled labor market in North is the same
as before; that is, it is described by Eq. (21). Then, let us turn to examine
the skilled labor market in South.
In South, there is no other industries nor sectors that can eﬀectively employ the skilled workers. The demand for skilled workers in South only comesfrom the research firms in North. Let H ∗ > 0 be the supply of skilled workers in South. Then, the equilibrium condition for the skilled labor market inSouth becomes22
DR∗ (wt∗ , wt+1 ) = H ∗or, equivalently,
β Āt L
β
∗
=
H
+
.
α
λH
φ(τ wt∗ )(wt+1 ) 1−α(34)

From the skilled labor market equilibrium condition in North, we havealready shown that if a balanced growth path were to exist, both Āt and1(wt ) 1−α have to grow at a common rate g. Similarly, for Eq. (34) to holdfor all t on a balanced growth path under communications liberalization, we
21

Similar to the case of communications autarky, we are assuming here the sets ΩH,tand ΩL,t are empty.
22
As in the case of skilled labor market in North in communications autarky (see footnote 14), it is necessary that under communications liberalization the Southern skilled∗
wage rate wt∗ is not less than the Southern unskilled wage rate wu,t
. Because, as we willshow later, the Southern skilled wage rate is higher than the Southern unskilled wage
rate along the balanced growth path, all of H ∗ are employed by Northern firms as skilledworkers.

33

recognize that both wt∗ and wt+1 have to grow at another common rate gw .
Taking account of the balanced growth condition, we can rewrite Eq. (34) as
follows:βAL
β= H∗ +,αλH
φτ w∗ w 1−α (1 + g)αwhere w∗ denotes the initial value of the skilled wage rate in South.

6.4

(35)

Consistency Condition under Communications Liberalization

The expected growth rate of sector z is given by$+z,t , λ+∗ )f (λ)dλE[gt (z)] = λh(λ| λz,t=$ λ!∗z,t!z,tλ= β ln= β ln88

λf (λ)dλ +
+∗
λ
z,t
+z,t
λ

9

$ λH
!∗
λ
z,t

+ 2β ln(λH )2+z,t · λ
+∗z,t
λ

9

8

.

2λf (λ)dλ
λH
+∗z,t
λ

9

(36)

Similar to the communications autarky case, we impose the following consistency condition: for all z ∈ [0, 1] and for all t,
g = E[gt (z)].

(37)

The implication of Eq. (37) is the same as Eq. (27). Combining Eqs. (36)and (37), we have% &g(λH )2
exp=.
+z,t · λ+∗z,tβλBy expanding the above expression and by integrating the result over all z,
we have<√αφ( τ w∗ w)w 1−α (1 + g)α (1 + g)t exp [g/β] = λH Āt L.

Finally, we obtain
/√
0 α
<
φ
τ w∗ w w 1−α (1 + g)α exp [g/β] = ALλH .
34

(38)

6.5

Equilibrium under Communications Liberalization

The skilled labor market equilibrium in North (Eq. (21)), the skilled market
equilibrium in South (Eq. (34)), and the consistency condition (Eq. (38))
together describe the equilibrium under communications liberalization. In
equilibrium, the skilled wage rate in North w, the skilled wage rate in Southw∗ , and the growth rate g are determined endogenously.
Combining Eq. (34) and Eq. (38), we can eliminate w∗ and obtain%

ALλH {(λH /β)H ∗ + 1}w=
φ(1 + g)α exp[g/β]

&1−α

.

(39)

This represents the consistency condition augmented by the skilled labormarket equilibrium condition in South. By carefully examining, we recognize that the functional form of Eq. (39) is almost the same as that of theconsistency condition Eq. (28) under communications autarky: only diﬀerence between them is the term (λH /β)H ∗ + 1 in the numerator of Eq. (39).
If H ∗ = 0 (i.e., if the Southern skilled workers are not available to the Northern research firms), both consistency conditions coincide. Technically, atleast from the viewpoint of North, the communications liberalization can be
identified with an increase in H ∗ from 0.23
Figure 2 illustrates the eﬀects of communications liberalization on thegrowth rate and the Northern skilled wage rate. Curve S and curve C represent the skilled labor market equilibrium in North and the consistency
condition under communications autarky, respectively. The initial equilibrium under communications autarky is point e, where curve S and curve Cintersect. Now, let us consider communications liberalization. Communications liberalization does not aﬀect the skilled labor market equilibrium
condition in North; therefore, curve S remains unchanged. On the other
hand, by communications liberalization, the skilled labor market in South is23

As illustrated in Figures 1 and 2, an increase in H induces a downward-shift of curve S,
while an increase in H ∗ induces an upward-shift of curve C. Although the eﬀects ofincreases in H and H ∗ on the growth rate and the skilled wage rate are qualitatively thesame, an increase in H ∗ in South cannot be identified with an exogenous increase in H in
North. In particular, as we will discuss later, Southern skilled wage rate is diﬀerent from
Northern skilled wage rate.

35

Figure 2: The balanced growth rate and the skilled wage rate in North undercommunications liberalization
introduced and the consistency condition changes from Eq. (38) to Eq. (39).
Accordingly, curve C shifts up as curve C ′ in the figure. The new equilibrium
obtains at point e′ . Obviously, the growth rate increases, while the Northernskilled wage rate decreases. We state this result as Proposition 1, a formal
proof of which can be found in Appendix B.2.
Proposition 1. Communications liberalization lowers the (initial) skilled
wage rate in North, while raises the growth rate.
Similar to Lemma 1, we can show the following corollary.
Corollary 1. Under communications liberalization, the growth rate g is increasing in both North and South endowments of skilled labor, but is independent of both North and South endowments of unskilled labor.
It is well recognized in the context of endogenous growth theory that
economic integrations have significant eﬀects on the growth rate of a country or that of the world. The mechanism driving the above result is similar
to the scale eﬀect shown by Romer (1990), which says that an increase in
the supply of human capital (i.e., skilled workers) brings about an increasein the growth rate. There are, however, some diﬀerences between Romer(1990)’s scale eﬀect and our result. In our case of communications liberalization, the additional supply of human capital comes from South located36

in the opposite time zone to North via virtual labor mobility. Further, as
explained in Subsection 4.2, the Northern and Southern skilled workers are
considered complements for each other. This complementarity between theSouthern and Northern skilled workers enables the R&D production function(Eq. (13)) to work properly. That is, after the communications liberalization, each Northern research firm can employ additional units of Southernskilled workers R∗ for the production of R&D without aﬀecting the quantity and productivity of the Northern skilled workers R. Consequently, theNorthern research firms can eﬀectively double the number of R&D projects
and, thereby, increase the growth rate. The virtual labor mobility opens a
new channel through which the scale eﬀect spreads across distant countries
located in diﬀerent time zones.24
Because of a decrease in the skilled wage rate due to communications
liberalization, the intermediate goods become less expensive and will be used
more intensively in the production of the Northern final good. Accordingly,
the marginal productivity of unskilled labor is increased. Therefore, theNorthern unskilled workers now experience not only a hike of the initial
unskilled wage rate in the short run, as indicated by Eq. (31), but also
an increase in the growth rate in the long run. That is, communications
liberalization is inevitably beneficial to the Northern unskilled workers.
On average, a decrease in the skilled wage rate induced by the communications liberalization increases the total gross profit of the intermediate firms
and also increases the growth rate of the total profit as shown in Eqs. (32)–(33). The situation, however, is diﬀerent from firm to firm. If an intermediate
good firm does not conduct R&D, then communications liberalization is beneficial to this firm, because the firm can hire skilled workers by paying a lower
24

Rivera-Batiz and Romer (1991b) have considered an endogenous growth model with
two symmetric developed countries and shown that changes in the growth rate (due totrade restrictions) can be decomposed into three diﬀerent eﬀects: the integration eﬀect, theredundancy eﬀect, and the allocation eﬀect. In our asymmetric North-South trade setting,because the R&D activities are carried out only in North and the trade pattern of the final
goods is assumed a priori, the redundancy and allocation eﬀects have been assumed away
at the very outset. Only the integration eﬀect of communications liberalization works in
our model and it enables the Northern research firms to enjoy the scale eﬀect of increasingreturns-to-scale in R&D production.

37

skilled wage rate, but does not incur any R&D cost. On the other hand, if anintermediate good firm conducts R&D, it has to incur the R&D cost, while
the benefit from the quality improvement by R&D will be materialized one
period later. Therefore, even though the firm can hire the skilled workers
with lower payments and receive a higher gross profit, the net benefit thatthe firm can receive immediately after the communications liberalization canbe negative.

7

General Equilibrium Consideration

What will happen to the skilled workers in South under communications
liberalization? A superficial observation goes as follows. By eliminating gfrom Eq. (23) and Eq. (35), we obtain2−αHλH + βALα 1−α λH
1−α ,
w =
·
w
−
·
w
τ {H ∗ λH + β}
τ {H ∗ λH + β}

∗

(40)

which indicates a positive relation between w and w∗ .25 With this result,one may conclude that communications liberalization lowers the skilled wage
rate w∗ in South, because communications liberalization induces a decrease
in w as shown in Proposition 1. However, this is a false conclusion.
In our setting, because there is no skilled labor market in South before
communications liberalization, potential skilled workers in South had to work
as unskilled workers in the traditional sector and received the unskilled wage
rate in communications autarky. Communications liberalization enables thepotential skilled workers in South to work for the research firms in North as
the actual skilled workers through the virtual labor mobility. By communications liberalization, the skilled workers in South do not necessarily experiencea decrease in the skilled wage rate, but face a switch in the wage rate fromunskilled to skilled. In order to answer the aforementioned question appropriately, we have to take explicit account of the endogenous determinationof the unskilled wage rate in South and that of the terms of trade between
countries in a fully general equilibrium framework.
25

Note that the RHS of Eq. (40) is concave in w and the second term asymptoticallyapproaches zero as w goes infinity.

38

7.1

The National Income of North

Let ItN and YtN denote the National Income and the Gross Domestic Product(GDP) of North in period t, respectively. The GDP of North is defined by
the sum of the unskilled wage income, the skilled wage income, and the total
profits of the intermediate firms:
YtN ≡ wu,t Lu + wt H +$ 1

πt (z)dz.

(41)

0

By substituting the equilibrium conditions and the balanced growth conditions, we obtain the following results:
2α

−α

−α

YtN = (1 − α)α 1−α w 1−α AL(1 + gw )t + w(1 + gw )t H + θw 1−α AL(1 + gw )t
.
−α
1−α
= BLw
+ wH · (1 + gw )t .

(42)

2α

1+α

where B ≡ A(1 − α){α 1−α + α 1−α }. Since all the components of the GDP
grow at the same rate as the skilled wage rate gw , the GDP itself grows at
gw along a balanced growth path.
In communications autarky (i.e., in a situation without R&D oﬀshoring),
the National Income and the GDP coincide with each other. In contrast,
under communications liberalization, there is a gap between ItN and YtN ,which corresponds to the skilled wage payment by the research firms to theSouthern skilled workers.
,
YtN
N
It =
YtN − τ wt∗ H ∗7.2(communications autarky),(communications liberalization).

(43)

The National Income of SouthSouth specializes in the agricultural good (the South good), which is produced by using unskilled labor with a Ricardian technology. We assume thelabor input coeﬃcient is equal to unity. Let pt be the relative price of theSouth good in terms of the North good in period t, which corresponds to theterms of trade (TOT) of South. It should be noted that pt can be identified
with the unskilled wage rate in South. Further, let yS and L∗ be the supply of
39

the South good and the endowment of (intrinsic) unskilled workers in South,
respectively.
In communications autarky, because the skilled workers have to work as
unskilled labor, the total supply of unskilled labor is equal to the sum of
the unskilled and skilled workers: L∗ + H ∗ . As we assumed the Ricardiantechnology with unit labor coeﬃcient, the supply of the South good can be
identified with the supply of unskilled labor:yS = L∗ + H ∗ .

(44)

The supply of the South good does not grow over time. The national income
ItS of South in communications autarky in period t becomes
ItS = pt (L∗ + H ∗ ).

(45)

As is obvious from Eq. (45), the national income of South, along a balanced
growth path, must grow at the same rate as the TOT pt .
Under communications liberalization, on the other hand, the skilled workers work for the research firms in North as skilled labor and receive the skilled
wage rate wt∗ , and only the (intrinsic) unskilled workers participate in the production of the South good. Accordingly, the supply of the South good andthe national income of South under communications liberalization become
as follows:yS = L∗ ,(46)

ItS = pt L∗ + wt∗ H ∗ .

(47)

One simple fact derived from Eqs. (44) and (46) is that communications
liberalization decreases the supply of the South good. The growth of theSouthern national income under communications liberalization depends bothon the evolution of the TOT and on that of the skilled wage rate in South.

7.3

Trade in Goods and the Terms of Trade

As shown in Subsection 3.1, a representative consumer in each country maximizes her temporal utility (Eq. (1)) subject to the temporal budget constraint40

(Eq. (2)). This yields the demand functions for both final goods (Eqs. (3)and (4)). The demand function for the South good becomes as follows: fork = N, S,
ckS,t =

δEtk
,
pt

(48)

where superscript k (= N or S) indicates the identity of each country, Etk is
the consumption expenditure, which is determined through the intertemporalutility maximization, and pt is the relative price of the agriculture good (i.e.,
the TOT of South) in terms of the manufacturing good in period t. Undercertain conditions, we can show that Etk is equal to the National Income Itkfor all t along a balanced growth path.
S
The demand-supply equation for the South good, cN
S,t + cS,t = yS,t , deter-

mines the equilibrium TOT pt .26 From Eqs. (42) through (48), we can write
the trade equilibrium condition in communications autarky as follows:>δ = N
Yt + pt (L∗ + H ∗ ) = L∗ + H ∗ .
pt

By solving it for pt , we obtain

pt =

δYtN
.
(1 − δ)(L∗ + H ∗ )With this result, we can conclude that the equilibrium TOT in communications autarky has to grow at the same rate as YtN (i.e., gw ) along the balanced
growth path. Therefore, for some initial p, we must have pt = p(1 + gw )t for
all t. Substituting this and Eq. (42) into the above result, we obtain
p=

−α
δH
δBL
·w+
· w 1−α .
∗∗∗∗(1 − δ)(L + H )(1 − δ)(L + H )(49)

Eq. (49) indicates the relation between the initial TOT p and the equilibriumskilled wage rate w in North in communications autarky. It should be notedthat the RHS of Eq. (49) is convex in w and that the second term asymptotically approaches zero as w goes infinity. This implies that for suﬃciently
large w, the RHS of Eq. (49) can be approximated by the first term, which
is linear in w.
26

By the Walras law, we can omit the demand-supply equation for the North good.

41

Let us turn to the case of communications liberalization. From Eqs. (42),(43), (46), (47), and (48), we can write the trade equilibrium condition undercommunications liberalization as follows:>δ = N
Yt − τ wt∗ H ∗ + pt L∗ + wt∗ H ∗ = L∗ .
pt
By solving it for pt , we also obtainpt =YtN − (τ − 1)wt∗ H ∗.
(1 − δ)L∗

Because YtN and wt∗ grow at the rate gw along the balanced growth pathunder communications liberalization, pt has to grow at the same rate gw .
Similar to the communications autarky case, we obtain%&δH
δ(τ − 1)H ∗ {HλH + β}p=−·w
(1 − δ)L∗ τ (1 − δ)L∗ {H ∗ λH + β}
8
9
2
−α
δBL
δ(τ − 1)Aα 1−α H ∗ λH++
· w 1−α∗∗∗(1 − δ)Lτ (1 − δ)L {H λH + β}(50)

The RHS of Eq. (50) resembles that of Eq. (49). It is convex in w and thesecond term also asymptotically approaches zero as w goes infinity.

7.4

The Skilled Wage Rate in SouthBy making use of Eqs. (40), (49), and (50), let us examine the eﬀects ofcommunications liberalization on the TOT (which can be identified with theunskilled wage rate in South) and the skilled wage rate in South.
In Figure 3, the skilled wage rate in North is measured horizontally, andboth the TOT and the skilled wage rate in South are measured vertically.
Bold curves W , P , and Q are the graphs of Eq. (40), Eq. (49), and Eq. (50),respectively. Note that these curves have their respective linear asymptotes,
which are represented by three dotted lines originating from point O. Here,
we are assuming that the slope of the asymptote for curve W is the steepest;that for curve Q is in the middle; and that for curve P is the gentlest: that
is,
δH
δH
δ(τ − 1)H ∗ {HλH + β}
HλH + β
<
−
<
.
∗
∗
∗
∗
∗
(1 − δ)(L + H )(1 − δ)Lτ (1 − δ)L {H λH + β}
τ {H ∗ λH + β}(51)42Figure 3: The determination of the terms of trade and the skilled wage rate
in South: A case where both types of Southern workers gain.

Figure 4: The determination of the terms of trade and the skilled wage rate
in South: A case where both types of Southern workers lose.
By inspection, we recognize that the inequalities in Eq. (51) tend to besatisfied if (i) the communication cost is suﬃciently small (i.e., τ is close
to unity), (ii) the endowment of the unskilled workers in South (i.e., L∗ ) islarge, (iii) the expenditure share on the South good (i.e., δ) is small.27
Suppose that the equilibrium skilled wage rate in North in communications autarky is w as in Figure 3. The height of the corresponding point e on
curve P stands for the equilibrium TOT in communications autarky. We denote it by pA . Now suppose that communications liberalization takes place.
27

In particular, if τ = 1, the first inequality is automatically satisfied.

43

From Proposition 1, we know that the equilibrium skilled wage rate in Northdecreases down to, say, w′ as illustrated in the figure. Under communications
liberalization, the TOT follows from Eq. (50) and the skilled wage rate inSouth follows from Eq. (40). Therefore, the equilibrium TOT, denoted by
pL , and the skilled wage rate in South, denoted by w∗ , under communications
liberalization can be found by reading points corresponding to w′ on curves Q
and W , respectively. Clearly, we have pL < w∗ . That is, by communications
liberalization, the skilled workers in South come to receive a higher wage rate
than the unskilled workers in South. Otherwise the skilled workers in South
would not work for the Northern research firms; rather they would prefer toworking as unskilled workers in the traditional sector in South.
Figure 3 illustrates a case where we have pA < pL < w∗ . The intuition
behind this result can be explained as follows. The R&D oﬀshoring undercommunications liberalization creates a new market for the skilled labor inSouth. The Southern skilled workers, who have worked in the traditional
sector as unskilled labor, now can move to the R&D sector in order to earn ahigher wage rate and work for the Northern research firms via virtual labormobility. The labor supply in the traditional sector decreases and, accordingly, the supply of the traditional good (the South good) also decreases.
The short supply of the South good induces an increase in the relative price,which corresponds to an increase in the unskilled wage rate in South. In thiscase, both the skilled and unskilled workers in South can enjoy a positive
level eﬀect on their wage rates. Furthermore, because the growth rate also
increases, they can enjoy a positive growth eﬀect thereafter. This is in sharp
contrast with the case for the skilled workers in North, who suﬀer from anegative level eﬀect of communications liberalization on the wage rate.
Although it may seem natural that the R&D oﬀshoring induces an increase in the TOT (i.e., pL > pA ), this is not always the case. Dependingon the parameter values, we may have pL < pA . On one hand, the R&D
oﬀshoring induces a reduction in the supply of the South good as explainedin the previous paragraph. On the other hand, the R&D oﬀshoring lowers
the total income of the Northern skilled workers and, thereby, reduces thedemand for the South good. If the demand reduction eﬀect outweighs the44

supply reduction eﬀect, the relative price would fall after communications liberalization. When pL < pA , we have two subcases: one where pL < pA < w∗
and the other pL < w∗ < pA . In the former, the Southern skilled workers
gain from communications liberalization, while the Southern unskilled workers do not. In the latter, which is illustrated in Figure 4, both the skilled
and unskilled workers in South as well as the skilled workers in North suﬀer
from communications liberalization.

8

Discussion

In this section, we discuss some limitations of our model and some possible
avenues for extension.
What are the welfare implications of our model? As we assumed in ourmodel that all agents in both countries have the same homothetic temporalutility function (Eq. (1)), we can meaningfully think of the aggregate welfareof each country or that of the world as whole.28 Then, we can ask thefollowing legitimate question: How does communications liberalization aﬀect
the aggregate welfare of a country or the world, in particular, in the shortrun? Now, suppose that initially the world is in communications autarky,
and there is only trade in the two final goods. Both countries are growing
at the same rate. Suppose that suddenly, there is an unexpected advance incommunication technology, which allows North to make use of South’s skilledworkers. There is then a sudden drop in skilled wage in North. In South,
skilled workers gain; their real income in terms of the manufacturing goodrises as we have shown that w∗ > pL after communications liberalization.
Short-run welfare eﬀects seem ambiguous, because while Northern skilledincome falls in the short run, these workers are also shareholders of North’s
intermediate good firms, and these firms experience an increase in profit due
to the lower wage (see Eq. (12)).
Can we say that world’s short-run welfare jumps up at the time of communications liberalization? It seems that the result on world short-run welfare
28

A concise explanation of the conditions for aggregation of preferences and their relation
to the notion of social utility function can be found in Nakanishi (2019, Chapter 4).

45

could be ambiguous. Just after communications liberalization, the world
supply of agriculture good decreases by the amount of H ∗ , which by itself
induces a decrease in the aggregate world welfare. In contrast, the world
supply of manufacturing good increases.29 This increase in the manufacturing good, however, may or may not be suﬃcient to compensate the world
welfare loss caused by the decrease in the agriculture good. Of course, in thelong run, both countries benefit from communications liberalization, because
both countries will get on a new balanced growth path with a higher growthrate. Obviously, the growth eﬀects dominate the costs of short-run loss inwage income.
A limitation of our model is that we have considered only two types ofworkers, skilled workers and unskilled workers. This is in keeping with mosttheoretical papers on skilled-unskilled wage diﬀerential. However, in reality,
there are many skill levels, and the impact of trade liberalization on wages
may depend on the level of aggregation of worker types. As documented inAutor and Dorn (2013) the wage profile has experienced substantial changes.
Wages of highly skilled workers have risen, and so have wages of very lowskilled workers, whereas the wages of workers in the middle range of skill
have declined. While in their paper, Autor and Dorn (2013) attributed thesechanges to technological changes, in their co-authored works with Hanson(Autor, Dorn, and Hanson, 2013, 2015) they also referred to the impactof trade with China. The oﬀshoring of middle-skilled tasks from North toSouth might have contributed to the increase in the Northern wages of highly
skilled workers, via a possible complementarity between oﬀshored tasks andhigh-skilled tasks.30 On the other hand, middle-skilled labor in South may
be near perfect substitutes for middle-skilled labor in North, so that when(cheaper) middle-skilled labor in South can be employed by Northern firms
29

By simple calculation, we can show that the supply of manufacturing good along a−2α−αbalanced growth path in period t is yN,t = α 1−α w 1−α AL(1 + g)(1−α)t . Suppose that
communications liberalization suddenly occurs in period t. Then, the world economy
jumps to a new balanced growth path with a higher g and a lower w. Consequently,
the supply of manufacturing good, yN,t , increases along a new balanced growth path inperiod t.
30
We thank a referee for this insightful suggestion.

46

(either indirectly, via production of in Southern factories, or directly, via
communication network, e.g., labor in computer programming, testing ofcodes and so on), the wages of middle-skilled labor in North must fall.
We outline below how our model could be extended to account for theAutor-Dorn findings. We can suppose that (i) highly skilled labor can workin research and development or high-end manufacturing, (ii) middle-skilledlabor are “grinders” that work in routine manufacturing jobs, and (iii) very
low-skilled labor can only work in the low-end of the service industry. If weassume that tasks of middle-skilled labor can be oﬀshored to South where
the wage is lower, while North’s middle-skilled labor have limited mobility
between occupations, then oﬀshoring of these tasks will definitely result infalling relative wages of middle-skilled labor. As for the rising employment
and wages in the low-end service sector, that can be explained by a combination of (a) a higher growth rate when economies switch from a low-growth
path with communications autarky to a high-growth path thanks to communications liberalization with complementary North and South research teams,and (b) non-homothetic preferences, such that as average income rises, theshare of expenditures on service rises (e.g., people go to restaurants or nailsaloons more often), a feature that has been incorporated in recent growthmodels such as Ngai and Pissarides (2007), Kongsamut et al. (2001), andAlvarez-Cuadrado, Long, and Poschke (2018).
Our model abstracts from trade in intermediate goods. Only the final
manufacturing good, produced solely in North, and the final agriculture
good, produced in solely in South, enter the trade in goods. If we relax
this assumption and allow intermediate goods to be traded, some countries
that previously do R&D for quality improvement in their own intermediate
goods may drop R&D altogether if high-quality foreign intermediate goods
become importable at reasonable prices. The role of imported intermediate
inputs as a replacement of domestic intermediate inputs has been empiricallyinvestigated by Li (2019); see also the references in Li (2019).

47

9Concluding Remarks

We have shown that communications liberalization can increase the growthrate of both North and South, because it enables the R&D departments ofNorthern firms to tap into the pool of South’s skilled labor. While the immediate impact eﬀect of the access to South’s skilled labor pool on the skilled
wage in North is negative, we show that Northern skilled workers benefit in
the long run, because the long-run growth rate of their wage is higher undercommunications liberalization than under communications autarky. Extending the model to account for trade in final goods, we find that South’s terms oftrade in balanced growth under communications liberalization may improveor worsen, as compared with its terms of trade in balanced growth undercommunications autarky. But even in the case of a worsening of the termsof trade, the loss that arises from it is likely to be more than compensatedfor by the the pro-growth eﬀect of communications liberalization.
We also find that, in the short run, a transition from balanced growthunder communications autarky to communications liberalization may depress
the wage rates of the skilled workers in North and both types of workers inSouth, while making the owners of monopolistic competition firms better oﬀ.
This is consistent with the empirical finding that in the past few decades thelabor income share has declined in most OECD countries.31

31

See, e.g., Alvarez-Cuadrado, Long, and Poschke (2018) and the references citedtherein.

48

Appendix
A

Derivation of the Keynes-Ramsey Equation

Without loss of generality, we assume that in each country, the population
size is unity. In what follows, because the same argument equally applies toboth countries, we omit the superscript (N or S) that indicates the identityof country. Following Obstfeld and Rogoﬀ (1996), we suppose that a representative consumer holds a stock of net foreign assets, denoted by Bt , whichearns the rate of interest rt (to be determined in equilibrium). The diﬀerence
between her income, It , and her expenditure, Et , is used to buy additional
foreign assets. Thus the dynamic equation for Bt is
Bt+1 = (1 + rt )Bt + It − Et .

(52)

Using the dynamic programming approach, let J(Bt ) denote the value function of the consumer’s intertemporal optimization problem. The Bellman
equation is%&1
Et−1 J(Bt ) = max V (Et , pt ) +Et (J(Bt+1 )) ,
Et
1+ρ

(53)

where Et is the expectation operator, Bt+1 is as given by Eq. (52) and ρ > 0

is the consumer’s degree of impatience.32 At the beginning of period t, theconsumer, facing the given market rate of interest rt , makes a decision on
her consumption expenditure Et . The optimal decision must maximize the
RHS of Eq. (53). We obtain the FOC:
(1 + ρ)∂V (Et , pt )− Et [J ′ (Bt+1 )] = 0.
∂Et

(54)

Applying the Envelope theorem, Eq. (53) yields
Et−1 J ′ (Bt ) =(1 + rt )Et [J ′ (Bt+1 )] .
1+ρ

32

(55)

For the dynamic programming approach and the Bellman equation, see Stokey andLucas (1989).

49

Using Eqs. (54) and (55), we get(1 + ρ)∂V (Et−1 , pt−1 )∂V (Et , pt )= (1 + rt )
∂Et−1
∂Et

Applying the logarithmic operator, we get the Euler’s equation:%&%&∂V (Et−1 , pt−1 )∂V (Et , pt )ln
− ln
= ln [1 + rt ] − ln [1 + ρ] .
∂Et−1
∂Et
Applying the linear approximation to the above Euler’s equation, we get thefollowing results consecutively:33%&
VEt − VEt−1
rt − ρ = −VE t%&
VEE × ∆Et + VEp × ∆pt=−
V%& %E & %&%&E × VEE∆Et
p × VEp ∆pt
=−
−
.
VE
Et
VE
pt
This is the Keynes-Ramsey equation, -which. relates the rate of interest rtto (i) the growth rate of consumption∆Et
Et, multiplied by 1/σ, where σ is.
VE
the elasticity of intertemporal substitution, σ ≡ − E×V> 0 and (ii) theEE- .
t
growth rate of the terms of trade, ∆p, multiplied by the price elasticity ofptmarginal indirect utility of expenditure. The indirect utility function derived
in Eq. (5) implies that the price elasticity of marginal indirect utility of
expenditure is identically zero. Thus the Keynes-Ramsey rule reduces to theusual textbook formula:

%
&
1 ∆Et
rt = ρ +σ Et

Under balanced growth, the growth rate of expenditure is equal to the growthrate of income, which is the same for both North and South as we have shownin the main text:

33

%

& %
&
∆Et
∆It
=
≡ gw .
Et
It

The expressions VE and VEE represent the first-order and second-order derivatives of
V with respect to E. Similarly, VEp represents the second-order cross derivative of V withrespect to E and p.

50

Thus, along the balanced growth path, we have
r =ρ+

gw
,
σ

where σ = 1 because of the the indirect utility function that we derived in
Eq. (5).

B

Existence of a Unique Equilibrium

B.1

Communications Autarky

We show the proof of the existence of a unique equilibrium of the balanced
growth path and the skilled wage rate under communications autarky.
From Eq. (24) and Eq. (28), we have both
%
&
2
AL
β
β
α 1−α +=H+1αφ(1 + g)
λH
w 1−α
and

% &
g
AL
φ(1 + g) exp= λH 1 .
β
w 1−α
α

1

By eliminating AL/w 1−α from the above expressions, we obtain
% &
.
2
g
α 1−α
φ(1 + g) α
+ β exp= HλH + β,β(56)where φ = (ρ + [1 + g]1−α )θ−1 . The RHS is a constant (with respect to g) andthe LHS of the above expression is a function ψ(g), which is monotonicallyincreasing in g, with
ψ(0) = (1 + ρ)α/(1 − α) + β

andlim ψ(g) = +∞.

g→+∞

Therefore, if
HλH > (1 + ρ)α/(1 − α),(57)

we have a unique positive balanced growth rate that solves Eq. (56). From
the above argument, it is easy to see that the unique growth rate becomes
higher if the LHS of Eq. (57) is larger or if the RHS is smaller. Further,because Eq. (56) is independent of the unskilled labor endowment L, so isthe equilibrium growth rate g. This proves Lemma 1.
51

B.2

Communications Liberalization

Let us turn to the existence of a unique equilibrium of the balanced growth
path and the skilled wage rate under communications liberalization.
By squaring both sides of Eq. (38), we get? α@2
φ2 τ w∗ w w 1−α (1 + g)α exp[g/β] = (ALλH )2?
@α
1
φτ w∗ w 1−α (1 + g)α φw 1−α (1 + g)α exp[g/β] = (ALλH )2 .

Thus

1

φw 1−α (1 + g)α exp[g/β]
ALλH
=
.
α
∗
ALλH
φτ w w 1−α (1 + g)αOn the other hand, from Eq. (34), we obtainAλH
H ∗ λH + β=.
αβφτ w∗ w 1−α (1 + g)αCombining these two results, we obtain1

φw 1−α (1 + g)α exp[g/β]H ∗ λH + β
=
ALλH
β
1

βφw 1−α (1 + g)α exp[g/β]= H ∗ λH + β.
ALλHTherefore,βφ(1 + g)α exp[g/β].
1H ∗ λH + βw 1−αFurthermore, from Eq. (21), we have
%
&
2
β
ALλH
α 1−α += HλH + β.
1αφ(1 + g)
w 1−α
ALλH

=

(58)

(59)

1

By eliminating the term ALλH /w 1−α from Eq. (58) and Eq. (59), we finallyobtain?

αβ φ(1 + g) α

2
1−α

@

+ β exp[g/β] = (HλH + β)(H ∗ λH + β),(60)

where φ = (ρ + [1 + g]1−α )θ−1 . The RHS is a constant (with respect to g) andthe LHS is a monotonically increasing function βψ(g) of g, which is equal to
the LHS of Eq. (56) multiplied by β, with the properties thatβψ(0) = β [(1 + ρ)α/(1 − α) + β] > 0 and52

lim βψ(g) = +∞.

g→+∞Therefore, if
β {(1 + ρ)α/(1 − α) + β} < (HλH + β)(H ∗ λH + β),(61)

there exists a unique positive balanced growth rate under communications
liberalization. Note that if H ∗ = 0 (which corresponds to communications
autarky), the condition Eq. (61) reduces to Eq. (57).
Now we prove Proposition 1. Let g a be the solution to Eq. (56) and g ℓbe the solution to Eq. (60). We have to show g a < g ℓ . Dividing both sidesof Eq. (60) by β, we obtain%&H ∗ λHψ(g) = (HλH + β)
+1 .
β

(62)

The LHS of the above equation is the same as Eq. (56) and the RHS isgreater than HλH + β (i.e., greater than the RHS of Eq. (56)). Since ψ ismonotonically increasing in g, then we obtain g a < g ℓ . Corollary 1 follows
from the facts that Eq. (60) is independent of both L and L∗ and, that the
RHS of Eq. (62) is monotonically increasing in both H and H ∗ .

53

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