Global sourcing and business networks:
quality heterogeneity and firms’ efficiency
Giuseppe Vittucci Marzetti
∗
Maria Luigia Segnana
Chiara Tomasi
†
‡
November, 2008
Abstract
This paper develops an economic model on the relations between
final good producers and their suppliers, taking into account the role
of business and social networks as a channel influencing the easiness of
collecting information. Global sourcing can increase firms’ productivity
through the quality upgrading of intermediate inputs, but, because
of the quality heterogeneity of suppliers, it can also boost screening
costs given the need to search for the right supplier. We show that,
becuse of this, large firms can better exploit the potential gains from
quality upgrading. Moreover, business networks, via the reduction in
firms’ unit screening costs, can make both the overall production and
firms’ profitability increase. There are cumulative beneficial effects of
these networks and hence a developed network constitutes in itself an
incentive for firms to invest in its further development. Finally, we
sketch a possible extension of the model to analyze the choice of local
vs. global sourcing strategies and how their differences, in terms of
costs, suppliers’ heterogeneity and degree of embeddedness in networks,
actually affect firms’ choices and their efficiency.
Keywords: Delocalisation; Outsourcing; Heterogeneity; Business and
social networks; Search costs.
JEL Classification: F130; F230; L140; L240; D850.
∗
Department of Economics, University of Trento, via Inama 5, 38100 Trento, Italy, Phone:
+39 0461 882162 Fax: +39 0461 882222, E-mail: [email protected]
†
Department of Economics, University of Trento, via Inama 5, 38100 Trento, Italy,
Phone: +39 0461 882225, Fax: +39 0461 882222, E-mail: [email protected]
‡
Scuola Superiore Sant’Anna and University of Urbino, Email: [email protected]
1
1
Introduction
One of the distinct features of globalisation is the pivotal role played by the
internationalisation of production processes. During the last decade, firms
have started to locate different stages of production processes in different
countries in order to exploit location advantages, such as easier access to
local markets, to technological and organisational know-how and to lower
factor prices. Recent studies provide evidence of the impressive development
of the phenomenon, known as international sourcing, whereby previously
integrated production activities are spread over an international network
(e.g. Feenstra and Hanson, 1996; Hummels et al., 2001).
Several theoretical and empirical works have considered its impacts. Much
research has endeavored to investigate the consequences of global sourcing
on domestic labour markets, using sectoral and country data. More recently,
in line with the developments in trade theory, suggesting the importance of
plant-level heterogeneity (Melitz, 2003; Helpman et al., 2004), the attention
has been moved from aggregate to micro-level analyses. Empirical studies
have provided evidence on the effects of global sourcing on firms’ efficiency.
The literature has considered different channels through which it may in fact
raise firms’ productivity: specialisation, learning, variety and quality.
Following this literature, the present paper develops an economic model
to analyse the relationships between final good producers and their suppliers,
focusing on the quality heterogeneity of the latter as a channel influencing
firms’ efficiency. In fact, one of the mechanisms through which global sourcing
may affect firms’ productivity is the quality upgrading of intermediate inputs.
However, given the quality heterogeneity of suppliers, outsourcing firms
incur some screening costs, i.e. search costs for the best supplier. These
are fixed costs resulting from the need to search for the supplier, collect all
the relevant information on it and test its product. The large the quality
range of suppliers is, the higher these screening costs are. By expanding
suppliers’ heterogeneity, international sourcing may actually boost firms’
screening costs. Thus, potential positive effects on firms’ efficiency via the
quality upgrading mechanism can be partially temper by the increase of
screening efforts. We show that the increasing size of the market for these
firms is efficiency-enhancing for two reasons: the economies of scale due
to the decreasing fixed costs of searching for a partner and the increasing
average quality of the chosen supplier.
Moreover, as recently emphasized by economic literature, business and
social networks can impact on the relative easiness of collecting information
(e.g. Rauch, 2001; Rauch and Casella, 2001), thus being a source of efficiency
gains. We indeed show that, assuming quality heterogeneity of suppliers
and screening costs entailed in global sourcing strategies, these networks,
via the reduction in the unit screening cost, can make firms’ production
and profitability both increase. In addition, we show there are cumulative
2
beneficial effects at work for business and social networks, and hence a
developed network constitutes in itself an incentive for firms to invest in its
further development.
Starting from this general framework, we then analyse the mutual exclusive strategies of external sourcing: local vs. foreign, by considering the
differences between local and foreign suppliers and investigating how these
differences, in terms of costs, heterogeneity and degree of embeddedness
in networks, can affect firms’ strategies and their efficiency. In fact, local
systems, cluster analysis and inter-firm relational proximity have traditionally identified reasons for firms in industrial districts to perform better than
isolated ones, benefiting from the presence of external economies at the local
level. But the disintegration of international production processes and the
increased use of international suppliers do question this view, thus putting
under new lens the traditional linkages. In an extension of our model we
therefore try to focus upon the linkages between the characteristics of local
systems and firms’ sourcing strategies. In particular, by assuming that local
business networks are more developed than transnational ones and that
the heterogeneity of local suppliers is less pronounced, we investigate firms’
choice between local and global sourcing and how they interact with the
previous factors determining firms’ overall efficiency.
The paper is structured as follows. In Section 2 the debate on the growth
and the effects of global sourcing is summarized by showing how this notion is
in search for a theoretical and empirical assessment. Section 3 shows a model
where both the fragmentation of production and screening costs entailed
in quality upgrading are taken into account. In Section 3.1, we analyze
the global soucing strategies of a maximizing firm facing a qualitatively
heterogeneous supply of providers and incurring screening costs. In order to
investigate the factors behind the choice of the location, Section 3.2 instead
sketches a possible extension of the model where, along with the supply of
foreign providers, there is a supply of local providers which the firm can
alternatively resort to for the provision of the intermediate. Finally, Section 4
concludes by summarizing the main results and envisaging issues and future
lines of research.
2
The growth and the effects of international
sourcing
Despite common perceptions about the widespread diffusion of global sourcing, there is no a standardized definition of the phenomenon (Feenstra, 1998).
The term global (or international) sourcing has been used by economist with
slightly different meanings. Someone uses it mainly referring to international
partnerships, thus assuming a minimum level of relation durability (e.g.
Van Long, 2005). Some others utilize it in all the cases in which firms resort
3
to foreign markets to acquire intermediate inputs (Feenstra and Hanson,
1999). In this case the term is therefore just a synonym of international trade
in intermediate inputs. Some other economist uses the term outsourcing only
referring to cases in which what is actually outsourced is service provision
(Bhagwati et al., 2004). Sometimes the expression is used as a synonym
of delocalisation or offshoring, especially in the literature on international
trade (e.g. Hummels et al., 1998; Glass, 2004), referring to the international
fragmentation of production stages, being it due to international sourcing,
in a let’s say proper sense, or vertical FDI. In the latter case, the production
stage goes outside the country but still remains within the boundaries of
the firm, whereas in the former it crosses both the national and the firm
boundaries.
The meaning of global sourcing we stick here goes beyond these distinctions. We use the expression in a broad sense, including all subcontracting
relationships between firms and the hiring of external workers which occur
at an international level. The main concern of the present study is indeed
not the nature of relations across actors or the explanation of firm vertical
disintegration decision (the well-known “make or buy” question), but rather
whether the sourcing process occurs within the country where the firm is
located (domestic or local sourcing) or outside the country (international or
foreign sourcing).
The growing interest on global sourcing has spurred a number of empirical
works that have tried to weight up its magnitude and to identify the main
determinants and effects. However, empirical studies have been hampered so
far by a lack of systematic statistics, which in turn comes from the absence
of a common definition. Hence, different approaches have been used based
on macroeconomic data, both at a country and industry level, as well as on
microdata.1 With all the imperfect measures, the various studies have shown
that there have been a rapid expansion in global sourcing in several industries, including textiles, apparel, footwear, industrial machinery electrical
equipment, transportation equipment, chemical and similar products. They
all conclude that the foreign sourcing of intermediate goods and business
services is one of the most rapidly growing components of international trade,
being at the crossroad of the main features of economic integration.
What are the consequences of this increasing fragmentation? Previous
theoretical and empirical work consider several possible effects of global
sourcing on countries’ and firms’ performances. One of the more relevant
1
At an aggregate level several works have used information from trade statistics in
intermediate inputs and in parts and components (Feenstra and Hanson, 1996; Hummels
et al., 2001; Yeats, 1998). Some analysts suggest using intra-industry trade as a measure
of global sourcing (Jones and Kierzkowski, 2001). Others use data collected to register
specific kinds of trade in intermediates, such as the US offshore assembly program (OAP)
(Feenstra et al., 1999; Yeats, 1998) or the EU outward processing traffic (OPT) (Baldone
et al., 2001).
4
aspects concerns the effects on relative wages and labour demand. Since
foreign sourcing mainly emerge to exploit differences in factor costs across
countries, it usually implies a reallocations of unskilled labour-intensive parts
of the production process from relatively skilled labour abundant countries
to unskilled labour abundant ones. This shift in the labour composition is
expected to increase the relative demand for skilled workers and, consequently,
enlarge the skill wage premium while increasing wage inequalities.
Empirical evidence for the US (Feenstra and Hanson, 1996, 1999) and
some European countries (Egger and Egger, 2006; Hijzen et al., 2005; Amiti
and Wei, 2004) follows these theoretical intuitions. The results obtained from
aggregate statistics have also been confirmed by studies at the micro level,
which indeed show a positive correlation between the increase of offshore
activities and the shift in labour demand toward skilled workers (Head and
Ries, 2002; Görg and Hanley, 2005).
Other studies have instead focused on the effects of global sourcing on
firms’ productivity and efficiency. They resort to different theoretical arguments to explain the productivity advantages that may arise as a consequence
of contracting-out phases of the production process. global sourcing can in
fact raise productivity via several channels: specialisation, learning, variety
and quality. By eliminating inefficient parts of the production process, firms
may indeed specialise in the phases where they have comparative advantages.
The change in the workforce composition thus determines an increase in the
average productivity of the remaining workers (Amiti and Wei, 2006). Gains
could arise also from learning effects coming from the foreign technology
embodied in the imported intermediate inputs (Acharya and Keller, 2007;
Eaton and Kortum, 2001). Outsourcing firms may obtain benefits by accelerating the formation of innovative products and service. According to
Glass and Saggi’s (2001) model, global sourcing lowers the marginal cost of
production due to save costs, increases profits and creates greater incentives
for innovation. Theoretical models have also considered positive productivity
effects due to the access to more varieties of intermediate inputs and better
match between input mix and the desired technology or product charateristics (Kasahara and Rodrigue, 2005). Finally, import in intermediate goods
can increase productivity through higher quality inputs. Global sourcing
may in fact allow firms to purchase abroad higher quality inputs compared
to those domestically available (Grossman and Helpman, 1991; Markusen,
1989).
Nevertheless, quite a few empirical studies have thus far tested the
relationship between international sourcing and productivity. Among the
few, some works have used industry level data to investigate the productivity
effects on manufacturing and service sectors (Amiti and Wei, 2006; ten Raa
and Wolff, 2001; Fixler and Siegel, 1999). In particular, by using data on all
US manufacturing industries over the ’90s, Amiti and Wei (2006) find that
service and material foreign sourcing turns out to be positively correlated
5
with labour productivity; and similar results have been observed by Görg
et al. (2004) at a micro level.
3
An international sourcing model with quality
heterogeneity
As mentioned above, both the theoretical and empirical literature have
considered several channels through which global sourcing may actually
affect firms’ productivity. In our model, we emphasize the quality upgrading
mechanism. According to our setting, global sourcing, and more generally
the openness to international markets, may affect firms’ productivity through
a quality upgrading of intermediate inputs. The rapid expansion of global
sourcing and the enlargement of the market for intermediate inputs has
determined a raise in the number of suppliers and a high heterogeneity in
the quality of this supply.
The previous effects come as the source of firms’ efficiency gains. However,
according to our model, given the quality heterogeneity, a firm that decides
to outsource a phase of the production process or a service incurs some
screening costs. These costs derive from the testing procedures needed
to be implemented in order to incorporate the intermediate inputs in the
production process.2 Although firms can observe the quality of the supplied
inputs, they meet the screening costs related to the matching of in-house
production and the outsourced phase. Indeed, the larger the quality range
of intermediates is, the higher these screening costs will be. By expanding
quality heterogeneity, global sourcing may thus boost firms’ screening costs.
3.1
Foreign sourcing: quality heterogeneity and screening
costs
Let us first consider a monopolistic firm h producing a final good and foreign
sourcing the production of the needed intermediate input. For the sake of
simplicity, let us assume further that there is no other cost involved in the
production of the final good but the cost of the intermediate, and that both
the internal cost of its production and the cost of local sourcing for h are
always greater than the cost of foreign sourcing.3
Let us assume a unit measure of foreign suppliers m producing the
intermediate, uniformly distributed according to the quality of the latter and
indexed in descending order according to such quality. Let us suppose that
there is a one-to-one relation between this quality and the probability that
2
Likewise, Bartel et al. (2005, 2008) consider adjustment costs of outsourcing related
to the less than perfect match between the internal production process and the external
inputs resulting in misunderstandings, frictions, delivery lags or quality differences.
3
In the second part of our model, we will relax the latter assumption considering the
possibility of the firm to locally source.
6
the intermediate will eventually breakdown when employed in the production
of the final good. Let us denote this breakdown probability with µ. Apart
from this, all the supplied intermediates are homogeneous and their cif cost
is the same (c).
The quality and the related breakdown probability are not freely observable. In particular, at the beginning of the production period, the firm h
bears screening costs, s. Such costs are related to the need for searching for
a potential partner and testing its supplied intermediate. We assume that
these costs are directly proportional to the number of considered potential
partners, n (n ≥ 1):
(1)
s = αn
where α is the unit cost of screening, which includes the costs entailed in the
search for the supplier and the test of its product, encompassing also the
costs for collecting all the relevant information on it.
After n screenings, the firm outsources the production of the intermediate
to the supplier with the best quality among those actually contacted (m̄).
Given that the firm cannot freely obtain information on the suppliers and
given the assumptions on the distribution of suppliers, each screened supplier
can be considered by h a random drawing from a uniform distribution of
µ ranging from 0 to 1. Thus, after n screenings, the probability that the
intermediate provided by m̄ is characterized by breakdown probability µ̄ is
equal to:
(2)
g(µ̄) = nF 0 (µ̄)(1 − F (µ̄))n−1 = n(1 − µ̄)n−1
where F (µ) is the cumulative distribution function of µ.
Expected profits of firm h are given by:
(3)
E(p(q)q − C) = p(q)q − E(C)
where p is the price of the final good produced by h, q is the quantity sold
of the final good and C denotes firm’s total costs, whose expected value is:
(4)
E(C) = s + E(c)q = s + E(E(c|µ̄))q
In Equation (4) s denotes h’s screening costs and c its marginal cost. The
latter is simply equal to the cost of the intermediate given the simplifying
assumption that h incurs no production cost.4
4
Removing this assumption does not alter any of the results of the model.
7
The expected value of the variable unit costs conditional on µ̄ is:
(5)
E(c|µ̄) = (1 − µ̄)c + (1 − µ̄)µ̄2c + (1 − µ̄)µ̄2 3c + . . . =
∞
∞
∞
X
X
X
= c(1 − µ̄)
µ̄i (1 + i) = c(1 − µ̄)(
µ̄i +
µ̄i i) =
i=0
= c(1 − µ̄)
=
i=0
µ̄
1
+
1 − µ̄ (1 − µ̄)2
=c+
i=0
µ̄
c=
1 − µ̄
c
1 − µ̄
Hence, their unconditional expected value is:
Z 1
c
c
(6)
)=
g(µ̄)dµ̄ =
E(E(c|µ̄)) = E(
1 − µ̄
0 1 − µ̄
Z 1
n
(1 − µ̄)n−2 dµ̄ =
= cn
c
n−1
0
whereas the average quality of the chosen supplier is:
Z 1
Z 1
(7)
E(µ̄) =
µ̄ g(µ̄)dµ̄ =
µ̄ n(1 − µ̄)n−1 dµ̄ =
0
0
1
n+1
Assuming that h maximizes its expected profits by freely setting q and
n,5 its profit function is:
(8)
π = max p(q)q −
q,n
n
cq − αn
n−1
If we assume further that h faces an isoelastic demand function:6
(9)
q(p) = Ap−
5
For simplicity, we neglect the integer “problem” and treat the discrete number of
screened suppliers (n) as a continuous variable.
6
This demand function can be seen as the result of a maximizing rapresentative consumer
with a CES utility function as:
Z I
−1
U=
β(i)q(i) di
0
and whose walrasian demand function for good j is therefore:
q(j) = R I
0
W
β(i) p(i)1− di
p(j)−
where W denotes the wealth of such consumer. With N consumers in the market the
aggregate demand function faced by h is thus Equation (9), where A is given by:
A = RI
0
WN
β(i) p(i)1− di
8
with > 1, from the first order conditions of the maximization problem (8)
it follows:
−1 n−1 1
(10)
q=A
n
c
r
(11)
n=1+
cq
α
and therefore:
(12)
2−
n (n − 1)
A
=
α
−1
1
c−1
As proved in Appendix A, although the objective function is not concave,
the set of maximizers (q̄, n̄) is a singleton.
We can therefore analyze the effects of parameters change on the screening
efforts of the firm, its profits and the average quality of the chosen supplier.
In particular, two parameters will be considered: the market size of h (A) and
the unit cost of screening (α), which can be deemed as negatively affected by
the relative density and efficiency of business and social networks operating
across national borders (Rauch, 2001; Rauch and Trindade, 2003).
Let us note first that, given the presence of fixed costs directly proportional to the number of screened suppliers and the decreasing effect of an
increase in the screening efforts of the firm on the reduction of the expected
marginal cost, the supplier actually chosen will produce on average a product
whose quality is not maximal. By Equation (7), this average quality is
positively related to the optimal number of screened suppliers (n̄), which is
in turn related to A and α.
As for A, from Equation (12) it follows that the higher the demand faced
by h is, the greater the optimal number of screened suppliers – and hence
the quality of the intermediate – will be. Moreover, the marginal effect of an
increase of the market size on firm’s profits by the envelope theorem is equal
to:
−1
1 q̄ ∂π
1 − 1 −1 n̄ − 1 −1 1
(13)
=
=
∂A
A
n̄
c−1
which turns out to be positively related with n̄. Therefore, given that a
bigger A determines a greater n̄, the positive effect of marginal increases of
A on firm’s profits becomes larger the larger the initial level of A is.
Hence, according to our model, in presence of quality heterogeneity of
suppliers and screening costs, the differential efficiency gains from market
expansions tend to increase with the initial level. Thus, the biggest firms
9
can better exploit the potential gains from quality upgrading and has got an
advantage in the international arena.7
Let us consider now the effects of changes in the unit costs of screening
(α). From Equation (12) it follows that a decrease of α makes n̄, q̄ and the
quality of the chosen suppliers all increase. Thus, our model predicts that
“thicker” transnational business and social networks, via the increase in the
effectiveness of the screening efforts, can entail an increase of firm’s efficiency,
which comes from the increase in the average quality of the intermediate,
and an expansion of production.
Moreover, given that the effect of a marginal increase of α on firm’s
profits is simply equal to:
(14)
∂π
= −n̄
∂α
this effect interacts with the previous one creating cumulative incentives
for firms to invest in the development of transnational business and social
networks. In particular, given that the more the initial thickness of the
network is, the greater the optimal number of screened suppliers is, as a
result the higher will be the cost saving the firm would obtain by reducing
further α and thus its incentive to invest in the network.
Hence, our model predicts that transnational business and social networks
increase the overall production and firms’ profitability. Moreover, it shows
that a developed transnational network constitutes in itself an incentive for
firms to invest in its further development.8
3.2
Foreign vs. local sourcing
Let us suppose now that, along with the foreign providers (m), there is a
supply of local providers (ml ) that firm h can alternatively resort to for the
provision of the intermediate.
In case of local suppliers, their unit costs of screening for firm h (αl ) are
smaller than the correspondent costs for screening foreign suppliers (αl < α).
7
It is instructive to consider what happens instead in the more standard case of a
monopolistic firm incurring costant marginal costs (c) and facing an isoelastic demand
curve like (9). In this case, although the effect of a marginal increase of the market size on
firm’s profits is clearly still positive:
−1
∂π
1 “ q̄ ” 1
=
= p̄1− > 0
∂A
A
it remains costant across different initial market sizes:
„
„
«−1
«1−
∂π
1
1 −1
1
=
c
=
∂A
−1
c−1
8
The previous effects could be dimmed by pronounced decreasing returns to scale in
the investments made for network developments. They are instead emphasized assuming
seemingly increasing returns of such investments.
10
This can be seen as the result of, on the one hand, the geographical and
cultural proximity between h and its local suppliers; on the other hand, the
existence of local business and social networks usually more developed than
transnational ones.9
Moreover, because local providers are more spatially and technologically
concentrated, we assume that they are less heterogeneous in terms of quality
than the suppliers on international markets. In particular, we suppose that,
like foreign suppliers, there is continuum of local suppliers (ml ) producing
the intermediate suitable for h, which are uniformly distributed according to
its breakdown probability µ, but, unlike foreign suppliers, in case of local
providers the latter ranges from tm to tM , where 0 ≤ tm ≤ tM ≤ 1.
Finally, we assume that local providers sell the intermediate at a price (cl ),
not smaller than the cif cost of acquiring the intermediate on international
markets: cl ≥ c.
Firm h can make either local or foreign sourcing. In particular, h will
decide to make foreign sourcing if and only if:
(15)
π(A, α, c) > πl (A, αl , cl , tm , tM )
where π and πl are the expected profits in case of, respectively, foreign and
local sourcing.
The interplay between the differences in the previous features of local and
foreign providers, also in terms of the different business and social networks
they are embedded in, determines the location of the supplier chosen by firm
h.
Such differences can impact in opposite directions on the relative profitability of local vs foreign sourcing strategies. For istance, the higher unit
costs for screening foreign suppliers rather than local ones makes ceteris
paribus local sourcing more profitable. And the same happens because of the
ceiling threshold in the distribution of local suppliers in terms of breakdown
probability of the intermediate (tM ). On the contrary, the higher cost of the
intermediate supplied locally (cl ) and the floor threshold (tm ) both make
foreign sourcing a more efficient strategy. Thus, we can assume that there
exists a firm h to which the two strategies give on average the same profit
and we can thus analyze the impact of changes in the parameters on these
expected profits.
4
Final remarks
This paper has analysed the relationship between final good producers and
their intermediate suppliers, focusing on different (local and global) quality
9
This greater development of local networks compared to transnational ones can be
itself the result of the above proximity. Notwithstanding, it should be properly retained as
an independent factor that contributes towards reducing the unit costs of screening in case
of local suppliers.
11
heterogeneity as a channel affecting firms’ efficiency. When the new sourcing
opportunities are taken into account, this channel is shown to work through
a quality upgrading mechanism, along two directions: on the one side, by
increasing the optimal number of screened suppliers (and hence the quality
of the intermediate input) and, on the other, by increasing the effectiveness
of the screening efforts in search for the right supplier. The result is an
efficiency-enhancing effect.
Furthermore, two differential and cumulative effects are at work at the
firm level: (i) differential efficiency gains from market expansion. These gains
increase with the initial level of the market size such that bigger firms would
benefit of better exploitation of potential gains from quality upgrading; (ii)
differential incentives for creating or investing in transnational business and
social networks. Thicker networks imply higher cost saving and thus further
incentives to invest in network linkages, making them thicker and thicker.
The previous results thus show how the problem of sourcing at a local and
global level can be framed as a problem of matching between producers and
suppliers, a problem that can be addressed by comparing local and global
sourcing and investigating how their heterogeneity affect firm’s efficiency.
In a broader perspective, they suggest two final remarks. The first is
connected with the multi-dimensions of firms’ competitiveness, whereas the
second is more related with the Italian system of industrial districts and
their connections with the analysis of business and social networks.
First, in presence of global sourcing the sources of comparative advantage
refer to footloose property of countries and firms. Even the basis of firms’
and local systems’ productivity have now an international dimension. This
dimension has shifted the basis of industrial competitiveness towards new
ways of organizing production processes, accessing new markets in new
and unconventional ways. Both, firms and clusters, are potentially open
to international productivity transfers/spillovers interacting with the local
determinants of firms’ competitiveness.
Second, the Italian case can be a starting point for a more general research
on the role of networks in the international extension of production/demand
linkages. As a matter of fact, it is well known how, in the international arena,
some Italian SMEs are able to fill in the opening slots in the international
fragmentation of production and how, in some cases, they show a significant
persistence, a predominance and a remarkable role in the international
extension of the supply chains. This role suggests, first, that the effects of
international sourcing identified in this paper cannot be considered within
the multi-nationals framework only, because some SMEs are playing the role
of complementing the strategy and the technological supply chain of large,
international players. Second, this play in an international arena requires the
identification of transnational business networks and their influence on the
size of the market for a SME and on the costs of identifying and screening
international partners. In other words how business and social network
12
can overcome trade costs.10 Third, as international sourcing opportunities
are now open to local systems and industrial districts, they might imply
a changing role of the lead firm with respect to the local system. How
important are firms’ heterogeneity, their embedness in the local system and
their internationality compared to their local linkages in explaining this
changing role?
In conclusion, the need for a better understanding of local and global
linkages and their interactions is largely underestimated, both theoretically
and empirically and, surely, when talking about local it is now imperative to
analyze the global, because gaining economic sustainability depends crucially
on the capability of firms and local systems to cope with the new international
division of labour.
A
Appendix
The Hessian of the objective function of (8) is:
!
!
c
c
2p0 (q) + p00 (q)q (n−1)
− −1
A1/ q −(+1)/ (n−1)
2
2
2
(16) H =
=
2cq
2cq
c
c
− (n−1)
− (n−1)
3
3
(n−1)2
(n−1)2
Given that the upper left element of H is negative, if its determinant is
positive the matrix is negative definite. The condition is therefore:
1
2c( − 1) A c2
|H(q, n)| = 2
−
>0
(n − 1)3 q
(n − 1)4
This condition valued at (q̄, n̄) becomes:
1
− 1 n̄ − 1 A >
c
q̄
2
− 1 n̄ − 1
p̄ >
c
2
Given that:
n̄
c
− 1 n̄ − 1
it follows that the condition for a critical point (q̄, n̄) to be a local maximum
is:
p̄ =
(17)
2n̄ > Let us denote the left hand side of Equation (12) with r(n). Taking the
logarithm of this function and differentiating it with respect to n we obtain:
(18)
10
d log r(n)
d
2n − =
( log n + (2 − ) log(n − 1)) =
dn
dn
n(n − 1)
See the network of enabling transactions in Herrmann-Pillath (2006).
13
Figure 1: Critical points of n when > 2
which is stricly positive provided that condition (17) is satisfied. The limits
of r(n) are:
n
lim r(n) =
lim
lim (n − 1)2 = lim (n − 1)2 = +∞
n→+∞
n→+∞ n − 1
n→+∞
n→+∞
0
if ≤ 2
lim r(n) =
+
+∞
if > 2
n→1
Hence, we can have two cases. If 1 < ≤ 2, then r(n) is a strictly increasing
monotonic function approaching to 0 when n tends to its inferior limit. Thus,
there is one and only one possible value of n (with the associated value of q)
for which Equation (12) is satisfied and such value satisfies also condition
(17). On the contrary, if > 2, then r(n) is a convex function reaching
its minimum at n = /2. Within reasonable values of the parameters, the
situation will be the one depicted in Figure 1. Hence, in this case there are
two critical points, but the only one that satisfies condition (17) is the largest
one.
Quasiconcavity in the relevant subset has to be proved.
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