Download A theory of trade in a global production network

A theory of trade in a global production network
Maarten Bosker∗ and Bastian Westbrock„
September 2016
Abstract
This paper argues that the determinants of the welfare gains from trade have
fundamentally changed with the emergence of a global production network. Towards
this end, we develop a novel counterfactual approach to decompose the welfare effects
of any small trade cost variation in any general equilibrium model of international
trade. Our findings stress a unique feature of supply chain trade: the gains from a
further integration of the global production network are not so much determined by
a country’s own local conditions or those of its direct trade partners. Instead, the
economic prospects of a country depend on its connections to important trade intermediaries, countries that provide indirect access to the demand and supply of many
other nations. We complement our theoretical findings by an easy-to-implement empirical strategy and identify each country’s key intermediaries.
Keywords: Global production network, gains from trade, trade intermediation (JEL
codes: F10, F11).
1
Introduction
Global supply chains are a defining feature of the modern world economy. This paper
theoretically substantiates the intuitive notion that their emergence has intertwined the
fortunes of all our nations to an unprecedented extent. In particular, we argue that the
established determinants of the gains from trade, notably a country’s own local conditions
(its endowments, technologies, and trade connections) as well as those of its direct trade
partners, have little to say about the expected welfare gains from a further increase in
the extent of international production fragmentation. Instead, a country’s connections to
∗
„
Erasmus University Rotterdam, The Netherlands, Tinbergen Institute and CEPR. [email protected]
Utrecht University School of Economics, The Netherlands. [email protected]
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important trade intermediaries, countries that provide indirect access to the demand and
supply of many other countries, have become the key determinants of these gains.
Towards this end, we study a number of counterfactual exercises in an off-the-shelf
model of the global production network (GPN).1 In these exercises, we consider previously
unexplored counterfactual changes in the extent of international production fragmentation
at the local, regional, and global level.2 The true novelty of our paper is, however, that
it disentangles the important drivers behind the resulting welfare effects. We develop
a counterfactual approach that establishes the importance of each country’s initial local
conditions versus its network exposure to the local conditions of its direct and indirect
trade partners in determining these gains. Our approach combines classic comparative
statics analysis (e.g., Mas-Collel et al., 1995) with new analytical tools from the social and
economic networks literature (Minabe, 1966; Ballester et al., 2006; Temurshoev, 2010). It
enables us to arrive at an exact mathematical expression for how a shock to any number
of nodes or links in the GPN propagates to all nations that are part of this network. We
can, in fact, impose any trade cost or technology shock onto our model and obtain explicit
solutions for the resulting changes in trade flows, prices, and welfare that only feature
exogenous parameters and initial, ex-ante, values of the endogenous variables.
Importantly, the effects on these variables can be linearly decomposed into several
meaningful channels, most notably into the effects that can be attributed to a country’s
own local conditions versus those related to its network exposure. Moreover, all the expressions we derive can be easily taken to the data by using no more than readily available
data on gross trade flows of final and intermediate goods and estimates of the model’s
elasticity parameters. This allows us to complement our theoretical results by quantitative
predictions based on real world data. To give an outlook on our main findings: in all
the counterfactual scenarios considered in this paper, a country’s network exposure to its
indirect trade partners, what we will refer to as its exposure through trade intermediation, is the central determinant of the resulting welfare effects. In some exercises, trade
1
The model used in this paper is representative for the class of studies that either uses an Armingtontype (e.g., Allen et al., 2014; Johnson and Noguera, 2016) or a Ricardian-type model (Eaton and Kortum,
2002; Caliendo and Parro, 2015) with international input-output linkages and perfect competition in each
sector. Nevertheless, our methodology and findings also speak to models of input-output trade with
imperfect competition such as, e.g., in Costinot and Rodrı́guez-Clare (2013) or Hsieh and Ossa (2016).
2
Throughout the paper, we interchangeably refer to a “global supply chain”, a “global production
network”, or an “internationally fragmented production process” to denote a multi-stage, multi-country
spanning process that leads to different varieties of the same final output. Similarly, we interchangeably
refer to the trend towards a more integrated international production process as “increased international
production fragmentation” or a more “integrated global production network”.
2
intermediation is even all that matters.
Section 2 of the paper starts by defining what we mean by network exposure and
trade intermediation. Section 3, then, presents the model and Section 4 our counterfactual
approach. In Section 5, we establish the importance of network exposure and trade intermediation for understanding the gains from trade in a GPN. We do this based on three
counterfactual scenarios:
First, we study a worldwide decrease in the costs of sourcing foreign intermediate inputs,
which in our interpretation is one of the main drivers behind the ongoing trend of global
production fragmentation. Our findings show that trade intermediation is the single-most
important determinant of the predicted per capita income gains in each and every nation
(Proposition 2). The intuition is that such a worldwide trade cost reduction gives no
country a competitive edge in its direct sales markets. The welfare effects experienced in a
country are, thus, dependent on its trade partners’ improved access to the demand in the
“downstream” stages of the supply chain and their improved access to the intermediate
inputs in the “upstream” stages. In other words, the welfare effects are proportional to
a country’s network exposure to the initial local conditions of its indirect trade partners,
and not to its own or to those of its direct trade partners. An interesting implication of
this result is that a country’s economic prospects are much more correlated with the initial
levels of welfare in other parts of the world than with its own initial welfare. This bears the
potential for a reduction of world income inequality as hypothesized earlier, for example,
by Feenstra (1998) or Baldwin (2011); something that is impossible in our model in the
absence of cross-border production linkages.
Second, we consider an export cost reduction on all final and intermediate goods shipped
along a single trade route. Due to the intensified competition in the markets of the two
involved trade partners, one of the consequences is a demand reduction for products from
a third country. This is known as the import competition effect. In a GPN, the same
cost reduction can have a countervailing positive effect on third country demand, because
of the trade partners’ increased demand for intermediate inputs. We call this a scale
effect (Hamermesh, 1993). What we show is that this externality might even exceed the
demand increase in the country lowering its export costs. This is particularly the case
when this exporter adds little value to its own exports, but mainly acts as an intermediary
of other nations’ value added (Proposition 3); in other words when this country is a “pure
intermediary”.
Finally, we identify the important trade intermediaries in the world economy. For this
3
purpose, we study the flip-side of the classic gains from trade analysis, where the focus is on
the isolated country itself (e.g., Arkolakis et al., 2012; Ossa, 2015; Blaum et al., 2015), and
ask how a country’s isolation affects all other nations. Based on our effect decomposition,
we identify a number of countries that derive their importance for other nations’ welfare
not so much by adding value themselves, but by intermediating other countries’ valued
added. Importantly, being well-connected to these key intermediaries is the crucial welfare
determinant in a gradually integrating world economy (Proposition 4).
Of course, we are not the first to study the economics of global supply chains. Already
the early theories of Ethier (1979, 1982), and later Krugman and Venables (1995), Eaton
and Kortum (2002), Yi (2003), and Costinot et al. (2013), have made clear that production
fragmentation has important implications for the sensitivity of national incomes to changes
in trade barriers or factor costs.3 Our contribution to that literature is that we offer, by
means of our novel counterfactual approach, a comprehensive investigation of the drivers
behind these effects. In this regard, our work is also related to another strand in the
international economics literature with similar ambitions. The focus there is, however, on
the different adjustment margins within a country, i.e., the welfare gains arising from the
improved access to goods, new services, or the exit of inefficient firms (Arkolakis et al., 2008;
Feenstra, 2010; Melitz and Redding, 2014). We, in contrast, are interested in disentangling
the importance of a country’s local conditions vis-à-vis its exposure to the local conditions
in other countries that lie beyond its direct control, as the determinants of a country’s
economic prospects.
On the technical side, two alternatives to our counterfactual approach are the “equilibrium in changes” of Dekle et al. (2008) and the local and global comparative statics of
Allen et al. (2014). These approaches are, unlike ours, also suited to look at the effects of
large (trade) shocks to the world economy. Yet, as they partially rely on either numerical
solutions, or on difficult to interpret expressions involving pseudo-inverses, they do not
lend themselves to the type of effect decomposition that we need in this study.
Our findings also speak to a, by now, sizable empirical literature documenting the extent
of international production fragmentation.4 Despite their descriptive value, a shortcoming
of the measures developed there is that they are not well grounded in general equilibrium
theory, rendering them unsuitable for policy evaluations or welfare analyses. Indeed, from
3
Another stream in the literature investigates the consequences of production fragmentation for distinct
groups of laborers within a nation (Feenstra and Hanson, 1996; Antràs et al., 2006; Grossman and RossiHansberg, 2008).
4
Important studies in this field are Hummels et al. (2001); Daudin et al. (2011); Antràs et al. (2012);
Johnson and Noguera (2012a); Koopman et al. (2014), and Los et al. (2015).
4
the viewpoint of our model, these existing measures provide at best a partial picture of the
welfare effects at play. In particular, while they are of some use for assessing the effects
of certain policies on a country’s factor demand, the consequences for an economy’s prices
are not at all captured by the existing measures. Our findings suggest straightforward
extensions to also predict these price effects.
Finally, this paper is related in spirit and methodology to a literature investigating the
consequences of embeddedness in social and economic networks. It is particularly close to
a group of studies on the relationship between the network structure of a national supply
chain and macroeconomic outcomes (see e.g., Acemoglu et al., 2012, 2015; Baqaee, 2016;
Oberfield, 2013). One of the main findings similar to ours is that a productivity shock to
an economic sector propagates more swiftly, the stronger the input-output connections between that sector and key intermediaries in the national economy. There are, nevertheless,
two important differences: first, we study a geographically dispersed production network
with imperfect labor mobility across space. The absence of international labor mobility
has profound implications for worldwide income inequality in our model, even when people
are perfectly mobile between sectors. Second, while the above-mentioned papers mainly
study the propagation of shocks to nodes in the network, our focus is on the links that
bind the network together.
2
Network exposure and trade intermediation
Our definitions of network exposure and trade intermediation are illustrated in the network
graph of Figure 1. The nodes in this graph depict three representative countries in the
GPN. They are characterized by a number of local conditions, in particular a country’s
resources, technologies, consumer expenditures, etc. These local conditions will partially
determine the level and change of a country-specific welfare measure in our model.
Additionally, welfare will depend on existing trade relationships. Countries trade final
and intermediate goods with each other. Their trade connections are depicted by the outgoing and incoming links in the graph: an outgoing link indicates value added exports, an
incoming link indicates value added imports. The values zij ≥ 0 measure the strength of
the international trade connections; zii ≥ 0 measures the strength of domestic connections.
These strengths are determined by the level of (internal) trade costs, consumer preferences
for, and production technologies requiring, specific goods produced in specific other countries. But, they also depend on the local conditions of the two involved trade partners.
5
k
exposure through
intermediation
neighborhood
exposure
j
local
conditions
i
Figure 1: Network exposure and trade intermediation
A country will, for example, rely less on foreign value added in its production processes,
the better its own production capabilities, rendering its incoming trade links weaker. Conversely, a country’s outgoing trade links will become stronger, the better its know-how and
the cheaper its domestic production factors.
Since each country uses part of its value added imports in subsequent production steps,
all countries are linked in a multi-stage production process. As a result, the level and
change of our country-specific welfare measure is not only shaped by a country’s own local
conditions, but also by a country’s network exposure to the local conditions of its direct
and indirect trade partners. Definition 1 formalizes this:
Definition 1. Let yi be a country-specific outcome variable of interest (e.g., the level or
change of per capita income). This variable is a function of a country’s exposure to its own
local conditions and its network exposure to the local conditions of its direct and indirect
trade partners. In particular, the column vector y = {y1 , ..., yn } is given by the function
f ∶ R3n → Rn :
network exposure
y=f(
Z1 ξ
°
(direct exposure to)
local conditions
³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹µ
,
Z2 ξ
,
Z3 × ξ
)
°
²
to direct
neighborhood
(1)
through trade
intermediation
where ξ = {ξ1 , ..., ξn } is the column vector of every country’s local conditions, Z1 is the
6
diagonal matrix of every country’s domestic trade connections, Z2 a full matrix with zeroes
on the diagonal capturing each country’s direct trade connections to all other nations, and
Z3 a full matrix of every country’s indirect trade connections to all nations.
As can be seen from the definition, we single out a country’s exposure to its own local
conditions, via an internal trade connection, and subsume it under the direct impact of its
local conditions on the outcome variable. Naturally, internal connections can be viewed as
part of a country’s local economic conditions.
We further subdivide network exposure into what we define as direct neighborhood exposure and indirect exposure through trade intermediation. The former relates the outcome
variable to the local conditions of a country’s direct trade partners, i.e., nations in its
direct neighborhood in a graph-theoretic sense. The latter, instead, associates it with the
local conditions in nations that lie beyond a country’s direct neighborhood, to which it is
exposed through any chain of at least two trade connections.5
The matrix Z3 in Definition 1 is our shorthand notation for all these chains of indirect
trade connections. In the three-country example of Figure 1, for instance, there are multiple
ways for the exports of country i to reach country k via j. The shortest connection consists
of the two links ij and jk, while there are multiple longer routes which all consist of one or
more loops on the connection ij and/or jk. We also subsume a country’s indirect exposure
to its own local conditions, via two or more (internal) trade links, under the construct
of trade intermediation. This means that, in Figure 1, country k might be identical to
country i and even i = j = k is in accordance with our definition of trade intermediation.
Our main motivation is to simplify the exposition. Nevertheless, do note that this type of
indirect exposure is also only present in a (globally) fragmented production process.
In the following sections, we will provide a more careful theoretical foundation for our
measures of network exposure and trade intermediation in a fully fledged general equilibrium trade model. In particular, we will show that they are the important determinants
of the welfare gains from a further increase in the extent of production fragmentation.
5
Note that our definition of trade intermediation differs from that in e.g. Ahn et al. (2011); Antràs and
Costinot (2011), and Rauch and Watson (2004). All these papers consider a separate intermediary sector
that facilitates matching between producers and foreign buyers. In our setting, all countries add value to
their exports. But, they are at the same time intermediaries of the foreign value added content in their
products, which was embodied in the intermediate inputs used in their production of these goods.
7
3
A model of the global production network
Consider a world economy consisting of i ∈ N = {1, 2, ..., n} countries. Each country is
home to two distinct sectors. One sector produces varieties of a final consumption good,
the other one varieties of an intermediate product. Let a final goods variety be denoted
by η f and an intermediate goods variety by η i . We assume that each variety η f ∈ Hf
and η i ∈ Hi is assembled by a distinct, perfectly competitive firm. Moreover, these firms
operate with a constant returns to scale technology and use labor as their sole original
production factor. Labor is inelastically supplied, immobile across countries, but perfectly
mobile between sectors. Labor endowment in country i is fixed at li > 0 and the economy
wide wage rate is wi > 0.
Preferences. Workers have constant elasticity of substitution (CES) preferences over
consumption goods with a substitution elasticity of α > 1
ui = ( ∑ q(η f )
α−1
α
η f ∈Hf
)
α
α−1
(2)
That is, they consume a utility maximizing quantity q(η f ) ≥ 0 of every variety η f , subject
to the constraint that their total expenditure must not exceed wi .
Technologies. Producers operate in a GPN. They employ nested CES production functions that combine labor and an aggregate intermediate input at an elasticity of β ≥ 1, and
substitute between the different intermediate varieties at an elasticity of γ > 1. Specifically, a producer of a final (intermediate) goods variety located in country i, who wants
to produce qi (η t ) > 0 units, t ∈ {f, i}, employs a cost-minimizing combination of labor and
intermediate inputs under the technological constraint that
β−1
β
qi (η t ) = mi (η t )[κi li
+ κii (( ∑ q(η i )
η i ∈Hi
γ−1
γ
)
γ
γ−1
)
β−1
β
]
β
β−1
(3)
Here, mi (η t ) > 0 defines a country’s total factor productivity, and κi > 0 and κii > 0 define
the relative factor productivities of labor and intermediate inputs, respectively. In the
special case of Cobb-Douglas technologies (β = 1), the factor productivities of labor and
intermediate inputs additionally satisfy 0 < κi < 1 and κii = 1 − κi to ensure constant returns
to scale under any parameter constellation.
8
This production function specifies an, in principal infinitely long, roundabout production process, where the output of one intermediate goods producer enters another firm’s
(and even its own) products. One complication with this production technology is its
possibility to “explode”: an intermediate goods producer combines labor with imported
inputs to produce another intermediate product, whereby the imported inputs augment
the productivity of local labor. The output is, in turn, used by other producers of intermediate goods, which do the same. Their outputs come back as imports and further increase
labor productivity. Thus, without any further constraint(s), this roundabout production
process could augment the productivity of labor towards infinity, similar to the Romer
(1990) model of increasing input varieties. This does, however, not occur when an input
price reduction primarily substitutes away other intermediate inputs and less so local labor,
i.e., when assuming γ ≥ β. As we show in the appendix (see Proposition 5 in Appendix
B.1), all producer prices converge to finite values in this case. We will therefore make this
assumption from now on.
Trade costs. Trade inside and between countries involves exporter-importer specific
trade costs. They take the standard “iceberg” form in our model, whereby we additionally
allow for potentially distinct costs for final and intermediate goods shipments.
Formally, for every unit of a good shipped from country i to j, only 1/τijt ≤ 1 units
t
arrive, t ∈ {f, i}, where τijt is such that (i) τijt ≤ τikt τkj
for any third country k, i.e., trade
6
costs are lowest on the direct routes, and (ii) the absence of a trade relationship is implied
by τijt → ∞.
Balanced trade. Denote by xtij ≡ ∑ηt ∈Hijt xt (η), for t ∈ {f, i}, the value of all final
t
(intermediate) goods exports from country i to country j, where Hij
is the set of final
(intermediate) varieties exported from i to j. We say that trade in final and intermediate
goods is balanced, if in all countries i
∑ xji = li wi
f
j∈N
and
∑ xiji + li wi = ∑ (xij + xiij )
f
j∈N
(4)
j∈N
All the above assumptions together describe our model. Despite the simplicity of this
setting, it is general enough to encompass many of the input-output trade models from
the literature. It nests the Armington models with supply chain linkages of, for example,
6
t
We could dispense with assumption (i), if we interpret τij
as the lowest shipping cost on the route
between countries i and j, indirect paths included.
9
Costinot and Rodrı́guez-Clare (2013) and Allen et al. (2014). They are obtained by setting
α = γ > β = 1. It also encompasses Ricardian models of input-output trade, such as
Eaton and Kortum (2002) or (a two sector version of) Caliendo and Parro (2015). In these
models, a comparative static variation of a trade cost parameter has additional implications
for the number of goods exported by a nation, on top of the intensive margin adjustments
considered in the Armington case. Yet, despite their richer micro-foundation, an economy’s
welfare can, just as in the Armington case, be entirely captured by the wage rate wi and
CES price indexes depicting the cost of a unit of output, pi , an aggregate intermediate
input, pii , and a consumption unit, pfi .7
Our setting can be considered to be even more general, because it allows for endogenous
labor cost shares as an additional adjustment margin to a trade cost shock (as long as β ≠ 1).
This has interesting, and empirically relevant,8 implications for the cross-country division
of labor: an import barrier reduction, for example, that is accompanied by the “offshoring”
of previously local value added is, of course, bad news for local laborers. But it also implies
that production elsewhere rises, which can feed back onto all other countries in terms of an
additional demand for their intermediate products. This scale effect (Hamermesh, 1993)
may even offset the losses in the country lowering its import barriers. It is, in fact, the
main mechanism behind our Propositions 2 and 3.
3.1
A simplified version of the model
To highlight the role of network exposure and trade intermediation in our model, without
getting lost in some of its intricacies, we focus on a particularly tractable version in the
main text. The analysis of the full model is presented in Appendix B.1. We will always be
explicit about which (parts) of our results carry over to the general case.
The simplified version of our model makes the following additional assumptions:
7
The equivalence between the different models is most clearly pointed out in Allen et al. (2015): the
identity of the economy wide price indexes in an Armington model and the Eaton & Kortum model
becomes obvious from their Example 1. In contrast, Example 2 shows that the full Caliendo & Parro
model is more general than both our setting as well as the Armington model. Nevertheless, our setting
can be interpreted as a two sector version of that model with flexible labor cost shares in every sector.
8
Recent empirical evidence, indeed, suggests that labor cost shares have not been constant over the past
decades (e.g., Feenstra, 1998; Karabarbounis and Neiman, 2014; Johnson and Noguera, 2016). Moreover,
they do appear to be endogenous to trade cost shocks (Hasan et al., 2007). Our model captures this
additional adjustment channel. Johnson and Noguera (2016) and Caliendo and Parro (2015), the latter
in their appendix, also present quantitative trade model with input-output linkages and “nested CES”
production functions. Both do not, however, allude to the implications of this departure from the standard
assumption of constant labor cost shares.
10
A1: Armington with uniform elasticities. First, we make the Armington assumption
that for every final and intermediate goods variety η f ∈ Hf and η i ∈ Hi there is one, and
only one, country able to produce it. We index varieties by the country index for this
purpose, and assume that for any t ∈ {f, i} it holds mi (ηjt ) = 1 for j = i, and mi (ηjt ) = 0
otherwise. Moreover, we assume that α = β = γ > 1 and κii = 1. That is, the relative
productivity of labor versus any one of the intermediate input varieties is only determined
by the labor productivity parameter κi , whereas the elasticity of substitution between labor
and intermediate inputs is identical to that between any two intermediate or final goods.9
A2: lower bound on intermediate goods trade costs. Second, we assume that the
GPN is not too integrated in its upstream stages, neither in the initial state nor after any
imposed trade cost variation. That is, let τiji and (τiji )′ denote the initial and counterfactual
trade cost on intermediate goods between any two countries i, j ∈ N respectively, then
1−γ
}<1
min { max ∑ zij1−γ , max ∑ zji
i∈N
j∈N
i∈N
for zij ∈ {τiji , (τiji )′ }
(5)
j∈N
As we will see below, this assumption ensures that our roundabout production process does
not explode in this version of the model with γ = β. One limitation is that it precludes the
counterfactual scenario of free trade between all nations ((τiji )′ = 1 for all ij) or free trade
inside a country ((τiii )′ = 1). Do note, however, that condition (5) becomes obsolete in the
general model of Appendix B.1.
The major advantage of making these two additional assumptions is that it allows
for explicit solutions for all goods prices and quantities in a balanced trade equilibrium.
In fact, wage rates remain the only implicitly defined variables in our model. This not
only simplifies the characterization of the equilibrium, but, more importantly, also the
counterfactual analysis, as we no longer have to deal with an implicitly defined set of price
equations.
Equilibrium in the goods markets. In a balanced-trade equilibrium, the producer
prices and the values of all intermediate goods shipments need to be internally consistent.
9
Even though the assumption of uniform elasticities seems quite restrictive, note that intermediate
inputs can be seen as the output of another country’s labor. Thus, β = γ could be justified from this point
of view. The assumption α = γ, on the other hand, is common in the literature (see e.g., Krugman and
Venables, 1995; Eaton and Kortum, 2002; Ossa, 2015).
11
That is, let p denote the column vector of producer price indexes (p1 , p2 , ..., pn ), corresponding to the cost-minimizing input bundle of production function (3), and let xi denote
the column vector of intermediate goods outputs, (∑j∈N xi1j , ∑j∈N xi2j , ..., ∑j∈N xinj ). Then,
balanced trade requires that:
pi =
(κγi wi1−γ
1/(1−γ)
i 1−γ
+ ∑ (pj τji ) )
j∈N
∀i ∈ N
(6)
∀i ∈ N
(7)
and
xii = ∑ (
j∈N
(pi τiji )1−γ
(pj )1−γ
f
∑ (xjk + xijk ))
k∈N
where (6) directly follows from cost-minimization under the technological constraint (3),
and (7) from applying Shephard’s lemma to p.
Combined with the uniform elasticity assumption β = γ, these two equations allow us
to express all prices and trade volumes as an explicit function of wages wi . Taking both
sides of (6) to the power of (1 − γ), one obtains the following expression for the augmented
producer price index p1−γ
= κγi wi1−γ + ∑j∈N pj1−γ (τjii )1−γ . In vector notation,
i
p1−γ = K γ w1−γ + AT p1−γ
(8)
where K γ w1−γ is the column vector of every country’s “value added” and AT is the transpose of the full matrix
A = ((τiji )1−γ ) ∈ R+n×n
(9)
We will refer to this matrix as the trade intensity matrix of intermediate goods from now
on. An entry aij ≥ 0 in this matrix is inversely related to the trade cost along the route ij.
Equation (8) shows the often encountered interdependence between the producer price
indexes of different countries. The specific functional form for the production technology
used in previous studies (e.g., Eaton and Kortum, 2002; Caliendo and Parro, 2015) did
not allow for an explicit solution, however. Here, we can solve for p1−γ by means of the
inversion of matrix I − AT , where I denotes the identity matrix.
The resulting inverse matrix, [I −AT ]−1 , can be interpreted as the Leontief inverse of our
geographically dispersed production network: every intermediate goods-producing nation
contributes to the productivity in other nations through its value added exports. The value
12
added of a producer from country i is used in other countries directly, by producers that
share a direct trade connection with i, as well as indirectly, embodied in the intermediate
products of yet one or more other countries. The AT -matrix determines the intensity of
the direct usages. The indirect usages, on the other hand, depend on multiples of this
matrix. This becomes more clear from a Neuman series expansion of the [I − AT ]−1 -matrix
which, by assumption A2, converges and is given by:10
[I − AT ]
−1
2
3
∞
= I + AT + [AT ] + [AT ] + ... = ∑ [AT ]
h
h=0
Hence, in our model, the vector of augmented producer price indexes can be solved for by
∞
h
p1−γ = ∑ [AT ] K γ w1−γ
(10)
h=0
From here, we immediately arrive at the vector of consumer price indexes corresponding
to utility function (2)
(pf )1−γ = B T p1−γ
(11)
where, in analogy to the A-matrix, B = ((τijf )1−γ ) ∈ Rn×n
defines the full matrix of final
+
goods trade intensities.
Turning to the trade volumes, if we spell out (7) in vector notation and apply the same
approach as for the prices, we arrive at the following column vectors that collect the final
and intermediate goods outputs (in “quantities”) assembled in every nation
∞
qi = (P γ−1 )xi = ∑ Ah qf
(12)
h=1
qf = (P γ−1 )xf = B(P f )γ−1 Lw
whereby P γ−1 , (P f )γ−1 , and L are the diagonal matrices of every country’s augmented price
indexes and population sizes, respectively. Obviously, outputs also depend on a variant of
10
This assumption is essentially a norm condition on the A-matrix. A less stringent requirement for a
converging Neuman series is based on the largest characteristic root of AT , which should be smaller than
one in absolute terms (Debreu and Herstein, 1953, Theorem V).
Note also that the convergence is automatically satisfied in Leontief’s original input-output model, which
is the special case of γ = 0 in our model, as well as in variants of this model with Cobb-Douglas technologies
(e.g., Acemoglu et al., 2012; Costinot and Rodrı́guez-Clare, 2013). In the former case, this is due to the
assumption of fixed technical coefficients which guarantee, in our notation, that ∑j∈N aji < 1. In the latter
case, the reason is the constant labor cost share in production.
13
the Leontief inverse. The difference is that for prices the intensity of a country’s incoming
links mattered (see (11)). Here it is instead the intensity of a country’s outgoing links that
matter for the access to other countries’ “final demand” —as given by the column vector
(P f )γ−1 Lw—. In other words, as long as the A- and the B-matrices are not symmetric, a
country is differently exposed to other nations on its supply and its demand side.
Labor market equilibrium. To close the equilibrium of our model, we can make use
of the balanced trade condition (4) in combination with the expression for the labor cost
share, λi = ∂ ln(pi )/∂ ln(wi ) = κγi wi1−γ /(pi )1−γ . Together, they imply that a wage vector
w ∈ Rn++ clears every country’s labor market, if it satisfies Lw = ld (w), whereby labor
demand is given by
∞
ld (w) = K γ W 1−γ (qf + qi ) = K γ W 1−γ ∑ Ah qf
(13)
h=0
Despite the fact that we have an explicit solution for qf in terms of no more than
exogenous parameters and wages (see (12)), it is still impossible to explicitly solve (13) for
an equilibrium wage vector. This can most easily be seen from the fact that, even if we
disregard (P f )γ−1 in qf the remaining equation system (13) is still non-linear in wages.
Nevertheless, we show in Appendix B.4 that the model admits for a “unique” marketclearing wage vector, whereby uniqueness is ensured only up to a scale factor x > 0, so
that every multiple xw of an equilibrium wage w is an equilibrium as well. This, in turn,
ensures that all goods are traded against a positive price (as long as the trade costs on a
route permit), and that all nations add value in the production of these goods.
Welfare. We define welfare in each country by its real income per capita, ui (w) =
wi /pfi (w). Since population sizes are fixed, markets are perfectly competitive, and full
employment is assumed, our findings for this measure carry, however, also over to other
welfare criteria such as total labor income or national income. Moreover, since ui (w) is
unaffected by a proportional wage change in all countries, welfare is uniquely determined
in equilibrium.
For a column vector z, let ln(z) be the corresponding vector with entries ln(zi ). Based
on equations (11) and (13), every nation’s real income per capita can be expressed in terms
14
of the following column vectors
∞
ln(u) = ln (L−1 K γ W 1−γ [B̂ + B̃ + ∑ Ah B](P f )γ−1 Lw)
(14)
h=1
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
exposure to final demand
+
∞
1
h
ln ([B̂ T + B̃ T + B T ∑ [AT ] ]K γ w1−γ )
γ−1
h=1
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
exposure to value added
where B̂ is the diagonal matrix with entries b̂ii = bii for all i ∈ N , while B̃ is the complementary matrix B̃ = B − B̂.
In terms of our definitions of Section 2, identity (14) states that the level of welfare in
every nation is an increasing function of a country’s exposure to its own local conditions,
i.e., final demand and value added, as well as its network exposure to these very same
conditions in other countries. This exposure is crucially determined by the intensity of
a country’s trade connections, as captured by the A- and B-matrices or their tranpose
(in case of supply-side exposure). In a world without production linkages (i.e. A = 0),
a country is solely exposed to its own local conditions and to those of its direct direct
trade partners. In an integrated GPN, in contrast, welfare is additionally dependent on
a country’s exposure to its indirect partners through trade intermediation. The matrix
∞
∑h=1 Ah B and its transpose capture the size of this indirect exposure.
On the (labor) demand side, a country’s exposure additionally depends on some of its
own local economic conditions, because only a share of the demand for domestic products
accrues to a local worker. The rest is claimed by other workers or by the producers of
intermediate inputs used in domestic production. As indicated by the diagonal matrix
L−1 K γ W 1−γ , a local worker’s share of this demand is larger, the smaller the domestic
workforce, the lower the prevailing wage rate, and the higher her labor productivity.
Our interpretation of identity (14) has prominent precursors in the international ecoh f
nomics literature: Redding and Venables (2004), for example, define qf + qi = ∑∞
h=0 A q
h
T
γ 1−γ as the supplier access of every
as the market access and (pf )1−γ = B T ∑∞
h=0 [A ] K w
country. They argue that these constructs of a country’s connectivity to other nations’ demand and supply, define a country’s welfare level —just as we do in (14)—. In fact, their
supplier access is identical to our measure of exposure to other nations’ value added. Market access is also almost identical to our measure of exposure to all nations’ final demand:
the sole difference is that market access does not include the part of this exposure defined
15
by L−1 K γ W 1−γ . Daudin et al. (2011) and Johnson and Noguera (2012a), on the other hand,
define a value added trade measure to trace back the total value content of a final product
to all the nations that contributed in its production. The equivalent of their measure in our
f
h f
f
f
n×n
model is K γ W 1−γ ∑∞
h=0 A Q , where Q denotes the full matrix Q = (qij ) ∈ R+ . Value
added trade can, thus, be viewed as a measure of workers’ exposure to the final demand
in particular other nations.
What we show in the following is that, despite their usefulness in explaining the observed
differences in welfare levels, these previous measures are of limited use when it comes to
predicting the welfare gains from a further increase in the extent of international production
fragmentation. The key predictor of these gains is a country’s network exposure through
trade intermediation only.
4
Counterfactual approach
The methodology behind our counterfactual exercises consists of a combination of tools
from classic comparative statics analysis and recent results from network economics on the
exact solution for the propagation of shocks through a Leontief inverse matrix.
The total differential. In line with classic comparative statics analysis, we linearly
approximate the effects of any trade cost shock on welfare in each country by d ln(u) =
d ln(w) − d ln(pf ). This approximation is viable, as long as the trade cost shock does
not have dramatic implications for the country’s welfare such as, e.g., under a full trade
embargo against it.
The income effects, in turn, depend on the adjustments to the entire system of price
and labor market equations. Let dZ denote a, potentially infra-marginal, change in any
matrix from the old state Z to the new state Z ′ . The total differentials of (10) and (13)
can be written as
d ln(w) = dln(w) ln(ld ) × d ln(w) + dln(pf ) ln(ld ) × d ln(pf ) + dA ln(ld )
(15)
T
d ln(pf ) = (dln(w) ln(pf )) × d ln(w) + dAT ln(pf )
Here, dA ln(ld ) and dAT ln(pf ) denote the column vectors of the direct effects of any variation to the A-matrix and its transpose AT on the labor market and price equation, (13)
and (11) respectively. The indirect general equilibrium effects, that arise from the interaction between the price and wages adjustments, are split into the three Jacobian matrices
16
dln(w) ln(ld ), dln(pf ) ln(ld ), and (dln(w) ln(pf ))T . These matrices capture, respectively, expenditure effects (changes in every country’s labor demands due to changes in foreign consumer expenditures), import competition effects (changes in labor demand due to pressure
on prices), and wage-induced price changes (in- or deflation).
Solving system (15) for the endogenous variables, we arrive at
[I − Φ] × d ln(w) = dln(pf ) ln(ld ) × dAT ln(pf ) + dA ln(ld )
(16)
T
d ln(pf ) = (dln(w) ln(pf )) × d ln(w) + dAT ln(pf )
where Φ = [dln(w) ln(ld ) + dln(pf ) ln(ld ) × (dln(w) ln(pf ))T ] and where d ln(pf ) is still a
function of the endogenous wage changes, an issue we solve below.
Direct effects. We first answer the question of how a variation to the A-matrix affects
labor demands and prices through the Leontief inverses in (10) and (13). In particular, we
wish to express —in the spirit of comparative statics analysis— the difference [I − A′ ]−1 −
[I −A]−1 = A′ −A+(A′ )2 −A2 +... in terms of the initial state of the matrix, [I −A]−1 , and the
imposed change dA = A′ −A. Towards this end, we can expand on established results on the
exact solution for matrix inverses (Henderson and Searle, 1981) and their applications in
the economics (Minabe, 1966; Temurshoev, 2010) and social networks literature (Ballester
et al., 2006).
Lemma 1, which is presented in Appendix A.1, summarizes our extensions of these
results for three particular variations: (1) a small proportional change to an arbitrary
number of cells in the A-matrix, (2.) a large change to a single cell aij , and (3.) a large
proportional variation of all cells in row i and column i. For these cases, the lemma can be
summarized as follows: For square matrices Z and dZ, such that [I −Z]−1 and [I −Z −dZ]−1
exist, we can write
dZ [I − Z]−1 = x[I − Z]−1 dZ [I − Z]−1
(17)
where the scalar x ∈ R and the matrix dZ depend on the specific type of variation.
The value of Lemma 1 is threefold: first, it lays the foundation for our decomposition
of any counterfactual income effect that distinguishes the role of a country’s exposure
to its own local conditions from that of its network exposure in predicting this effect.
Second, because the available data on the GPN only reflects its current and past state,
the lemma is essential for the empirical implementation of our approach. Third, even
17
though for the simple model analyzed here, matrix [I − Z ′ ]−1 can be computed exactly for
a specific counterfactual variation to the A-matrix by [I − A − dA]−1 , this is impossible for
the full model studied in Appendix B.1. The reason is that there Z ′ consists of additional
endogenous variables, next to the exogenous trade intensities.
The Jacobian matrices. The model-specific versions of the Jacobian matrices in (16)
can be easily derived from equations (10) and (13). Their solutions are presented in
Proposition 1 below. Yet, the invertibility of the matrix
I − Φ = I − dln(w) ln(ld ) − dln(pf ) ln(ld ) × (dln(w) ln(pf ))T
(18)
remains an issue. We use a well-known result in comparative statics analysis (Proposition
17.G.3 of Mas-Collel et al., 1995, p. 618) to establish this, which has as the sole requirement
that the labor market equation, Lw = ld (w), satisfies the sufficient conditions for a unique
Walrasian equilibrium (as established in Appendix B.4). To give a sketch of the argument:
because Lw − ld (w) is homogeneous of degree zero, we are free to hold the wage rate in
any one country i∗ ∈ N fixed, so that dwi∗ = 0. Moreover, we can leave out i∗ ’s labor
market equation from the first line in (16), which by Walras’ Law is redundant, focus on
the remaining equations, and still capture the effect of any exogenous shock to the entire
∗
system. We introduce the superindexes −i∗ and +i∗ for this purpose, whereby Z −i means
∗
that row i∗ and column i∗ are deleted from matrix Z, and Z +i indicates that a vector of
zeros is inserted before row i∗ and before column i∗ .
∗
Invertibility of {I − Φ}−i follows, now, from the fact that labor demand, ld (w), is
homogeneous of degree one, so that the row-sum norm of Φ is one. And, because labor
demand also satisfies the gross-substitutes property (as shown in Appendix B.4), the row∗
sum norm of {Φ}−i is smaller than one, which is sufficient for invertibility.
The foreign trade multiplier. Since {Φ}−i satisfies the norm condition, the matrix
∗
−i∗ ]h . This shows
[{I − Φ}−i ]−1 can alternatively be written as a Neumann series, ∑∞
h=0 [{Φ}
more clearly that this inverse matrix can be interpreted as a multi-country version of the
Harrod-Keynes foreign trade multiplier. In our neoclassical model, this multiplier contains
two additional feedback channels of a changing nominal wage rate, next to the classic
Keynesian multiplier governing the adjustment of consumer expenditures: notably, the
feedbacks on the intensity of import competition in every country, and on the use of labor
∗
18
in production.11
The following proposition summarizes our approach:
Proposition 1. Every country’s per capita income change associated with an arbitrary
variation of the A-matrix is given by the column vector
d ln(u) = Ψi dA ln(ld ) + Ωi dAT ln(pf )
∗
∗
where
dAT ln(pf ) =
−1
−1
(P f )γ−1 B T dAT [I − AT ] K γ w1−γ
γ−1
−1
dA ln(ld ) = [LW ]−1 K γ W 1−γ dA [I − A]
qf
and where Ψi and Ωi are the n × n-dimensional full matrices
∗
∗
i∗
Ψ
Ωi
∗
T
−i∗ −1
= (I − (dln(w) ln(pf )) ) {[{I − Φ}
+i∗
] }
(19)
= Ψi × dln(pf ) ln(ld ) − I
∗
with
Φ = dln(w) ln(ld ) + dln(pf ) ln(ld ) × (dln(w) ln(pf ))T
−1
dln(w) ln(ld ) = −(γ − 1)I + [LW ]−1 K γ W 1−γ [I − A] Qf
(20)
−1
dln(pf ) ln(ld ) = (γ − 1)[LW ]−1 K γ W 1−γ [I − A] Qf × (dln(w) ln(pf ))T
−1
(dln(w) ln(pf ))T = (P f )γ−1 B T [I − AT ] K γ W 1−γ
and Qf is the full matrix of final goods trade flows (in quantities), Qf = (qijf ) ∈ Rn×n
+ .
In the next section, we make use of Proposition 1 and perform three particular counterfactual exercises that together highlight the salient features of trade in a GPN. Besides
deriving several general propositions, we also put numbers on these predictions using real
world data and the following simple empirical approach.
11
In that sense, the multiplier already captures an interdependence between every country’s welfare
even in the absence of any input-output linkages. What we focus on in the following are the consequences
of production interdependencies beyond that point, i.e., that only arise in the presence of input-output
linkages.
19
Empirical implementation. Just as the equilibrium in changes approach of Dekle et al.
(2008), our empirical implementation does not need any estimation (when taking an estimate of γ from the literature, see footnote 14), nor information on prices, wages, trade
costs or productivities. All we require is readily available data on gross trade flows of final
and intermediate products and an estimate of the elasticity parameter γ. Appendix A.2
shows how one can simply rewrite all expressions in Proposition 1 in terms of these inputs.
In all empirical illustrations, we use UN COMTRADE data on final and intermediate
goods trade flows for 100 countries in the year 2005.12 Final and intermediate goods are
distinguished based on the UN’s BEC classification: the BEC class “consumption goods”
are considered as our final products and the combined BEC classes “capital goods” and
“intermediate goods” as our intermediate products.13 One practical complication is that
expenditures on domestically produced final and intermediate goods, i.e., internal trade
flows, are typically not reported. They can, however, be inferred from the macroeconomic
identities xfii = (Ind.V A)i − ∑j≠i [xfij + xiij − xiji ] and xiii = (Ind.P rod)i − (Ind.V A)i − ∑j≠i xiji ,
using the UN COMTRADE trade data in combination with data on total industrial value
added and production that we obtain from the WDI and UNIDO respectively.
Finally, we need to fix a value for γ. Our default specification is γ = 5. We do
verify the robustness of our result with regard to this choice (see Appendix B.3).14 The
precise choice of γ mainly influences the absolute size of the counterfactual welfare effects.
The relative welfare effects in different countries, as well as the relative importance of a
country’s exposure to local conditions vis-a-vis its network exposure in determining these
effects, remain largely unaffected.
12
Data coverage is most comprehensive in this year, especially for developing economies, allowing us
to include as many countries as possible in our empirical illustrations. To be included, a country should
report its total value added, the value of its total production as well as at least one import or export flow
from/to another country.
13
Results excluding capital goods from this definition are available upon request.
14
We could instead estimate γ using notably the method developed in Caliendo and Parro (2015).
Their method uses variation in tariff rates across sectors and bilateral trade partners to identify the trade
elasticity (1 − γ in our case). Tariffs are, however, set at a much finer-grained level than the two rather
stylized categories of final and intermediate products that feature in our simple model. Of course, we could
aggregate tariffs up to these broader categories. But this aggregation can be done in different ways, each
using different assumptions, and each resulting in a different estimate. We, therefore, keep things simple
and fix a value for γ that lies in the middle of the range of estimated trade elasticities in the literature
(Caliendo and Parro, 2015; Eaton and Kortum, 2002; Romalis, 2007).
20
5
5.1
Counterfactual results
The gains from production fragmentation
We saw before that, in equilibrium, a country’s level of welfare can be explained by its
exposure to its own local conditions, its direct neighborhood exposure, and its exposure
through trade intermediation (see equation (14)). Here, we ask whether the very same
statistics also predict the changes in welfare associated with a further integration of the
GPN.
Technically, we examine a small, proportional increase in all off-diagonal cells of the
A-matrix: dA = xÃ, where ãii = 0 for all i ∈ N and ãij = aij for all i ≠ j and x → 0+ . One
might think of, for example, a worldwide cost reduction on intermediate goods shipments
or a technical innovation that increases their productivity in subsequent production steps.
Our preferred interpretation is, however, what Baldwin (2011) dubs the “2nd unbundling”:
the still ongoing revolution in ICT that started in the mid 1980s and that has since then
gradually eroded the role of proximity in the execution of interdependent production tasks.
General predictions. The following result shows that the predicted per capita income
effects only depend on a country’s exposure to the final demand and value added of its
indirect trade partners, i.e. its exposure through trade intermediation:
Proposition 2. Consider a worldwide trade cost reduction on all intermediate goods exports and imports, dA = xà where ãii = 0 for all i ∈ N and ãij = aij for all i ≠ j and x → 0+ .
The resulting per capita income effect in every nation is a linear function of a country’s
exposure through trade intermediation, as defined in Definition 1:
d ln(u) = Ψi dA ln(ld ) + Ωi dAT ln(pf )
∗
∗
where
∞
1
f γ−1 T
(P ) B ∑ [AT ]h ÃT p1−γ
dAT ln(p ) = − x
γ−1
h=0
f
−1
∞
∞
h=0
h=0
(21)
dA ln(ld ) = x[LW ] K γ W 1−γ ∑ Ah à ∑ Ah qf
The proof proceeds in two steps: first, we derive the expressions in (21) from Property
(1.) of Lemma 1, for dZ = Ã, in combination with Proposition 1 and the expressions for
21
h
T
γ 1−γ , and outputs qf = B(P f )γ−1 Lw. It remains to
producer prices, p1−γ = ∑∞
h=0 [A ] K w
be seen that equation (21) expresses d ln(ui ) —in line with Definition 1 of Section 2— in
terms of a linear function of the final demand and value added of all nations j ∈ N intermediated through at least two trade links, so that we can speak of trade intermediation. This
follows immediately from inspection of the two direct effects in (21), in combination with
the expressions for p1−γ and qf (see the discussion below). The general equilibrium effects
∗
∗
contained in the Ψi − and Ωi −matrices only imply even higher order network exposure, as
becomes evident from (19) and (20). Note moreover that, because equation (21) expresses
income effects in terms of percentage changes, network exposure is inversely related to
the initial levels of prices, (P f )1−γ , and incomes, LW , in a country, whereby the exposure
on the demand side is —just as the welfare level (14)— additionally dependent on the
prevailing labor cost share, K γ W 1−γ .
To see that the two direct effects in (21), indeed, only depend on a country’s exposure through trade intermediation, note first that this is straightforward to establish for
the effect of the trade cost reduction, dA = xÃ, on the prices of consumption goods,
dAT ln(pf ). Obviously, consumers benefit only indirectly from it —in proportion to their
exposure through trade intermediation—, because the trade cost reduction improves in a
first instance their suppliers’ access to foreign intermediate inputs.
The intuition behind the labor demand effect, dA ln(ld ), is less straighforward. Since
the trade cost reduction lowers the prices of all foreign intermediate inputs, it scales up
their input share in subsequent production steps. The total demand effect in every country
is given by the column vector x(qf + qi ). Foreign intermediate goods producers access it
in proportion to their initial trade intensities with the downstream buyers, that is, in
proportion to their entries in the A-matrix. The demand effect is also experienced “further
up the supply chain”, by countries that themselves supply intermediate inputs to these
producers; yet, at a rate discounted by their trade intensity with the intermediate goods
h
producers, which is captured by the second and higher-order summands of the ∑∞
h=0 A matrix. The total effect on the labor demand in any nation i is, thus, given by15
To better understand the origin of the demand effect, it is worth splitting up the impact of dA = xÃ
onto the market share of a typical foreign intermediate goods producer and the importers’ optimal cost
i
i
shares in production. Country i’s initial sales in country j are proportional to πij
(1−λj ), where πij
denotes
country i’s share in j’s expenditures on intermediate goods and 1 − λj denotes j’s intermediate input cost
share. The first-order effect of the trade cost reduction —leaving the higher order summands in (22) aside
for a moment— can, thus, be decomposed into
15
i
i
i
dA (πij
(1 − λj )) = dA πij
× (1 − λj ) + πij
× dA (1 − λj )
22
dA lid = xκγi wi1−γ ( ∑ aij + ( ∑ aik + ∑ ail ∑ alk + ...) ∑ akj )(qjf + qji )
j≠i
k∈N
l∈N
k∈N
(22)
j≠k
which is nothing but the absolute rate of change in row i of vector dA ln(ld ) after noting
h f
that (qf + qi ) = ∑∞
h=0 A q .
There are three important things to note about equation (22): first, since qjf and qji
measure the trade intensity-weighted access to other countries’ final demand, the equation
suggests that the demand effect in any country i is entirely dependent on its network
exposure through trade intermediation, as defined in Section 2. Second, the demand effect
is partially related to a country’s domestic consumer demand. However, as equation (22)
makes clear, country i only depends indirectly on it, through a chain of trade links with
at least one trade partner k at an intermediate step. Third, the magnitude of the demand
effect is related to a country’s “upstreamness” in the GPN: take a country i that operates at
the second-to-last production step so that it sells all its intermediate outputs to producers of
final goods. That country would only benefit in proportion to ∑j≠i aij qjf from the worldwide
trade cost reduction dA = xÃ. In contrast, as suggested by the remaining summands in
(22), countries operating further up the supply chain benefit from the additional demand
created at every single intermediate production step. It is, therefore, not surprising that the
expressions in (21) or (22) are closely related to the upstreamness measure, as introduced
by Antràs et al. (2012) into the empirical supply chain literature.16
A global trade cost reduction on final goods. Overall, Proposition 2 suggests that
an integrated GPN shifts the focus from a country’s own local conditions and those of its
Note, first, that dA improves the market share of seller i at the expense of country j’s own intermediate
i
goods producers. In particular, seller i steals part of their market share, πjj
, in proportion to its own
i
i
i i
i 1−γ
inital sales share, πij , that is, dA πij = xπij πjj (pj ) . At the same time, since intermediate inputs
become more cost-efficient vis-à-vis domestic labor, country j raises their share in production at a rate
i
i
i
dA (1 − λj ) = x(1 − λj ) − xπjj
(pij )1−γ . The total effect is, thus, given by dA (πij
(1 − λj )) = xπij
(1 − λj ),
which is by identity (35) nothing but entry ij of matrix A.
Turning to the higher order effects, the demand increase in country i is also shared by its upstream suppliers,
who gain in proportion to their initial market shares as well as country i’s own share of intermediate inputs
i
in production, i.e., πki
(1 − λi ). This is, again, nothing but entry ki of the A-matrix.
16
The equivalence with the Antràs et al. (2012) measure becomes most clear, when we start from (21) and
spell out the effect of a 100% intermediate goods trade cost reduction as dA ld = K γ W 1−γ [I − A]−1 Ã[I −
A]−1 qf . The Antràs et al. (2012) measure of upstreamness, on the other hand, can be written in our
notation as [I − A]−1 [I − A]−1 qf . The sole difference stems from the fact that Antràs et al. (2012) think of
a symmetric productivity shock in all sectors, whereas we envision the propagation of a worldwide trade
cost reduction.
23
direct trade partners, to its exposure to the local conditions of its indirect trade partners as
the central determinant of its economic prospects. To further stress this distinct feature, it
is worthwhile comparing this result to the welfare effects of an analogous worldwide trade
cost reduction on the imports and exports of all final goods, dB = xB̃.
This has the following direct effects on supply and demand:
dB T pf = −
x
x
B̃ T p1−γ ≈ −
(pf )1−γ
γ−1
γ−1
∞
dB (ld ) = xK γ W 1−γ ∑ Ah q̃f ≈ xK γ W 1−γ (qf + qi )
h=0
f
f
f
). Thus, quite contrary to the analogous increase
, ..., ∑j≠n qnj
, ∑j≠2 q2j
where q̃f = (∑j≠1 q1j
in the A-matrix, prices drop and labor demands increase in proportion to a country’s initial
exposure to its own local conditions, its direct neighborhood exposure as well as its network
exposure through trade intermediation. In other words, all countries gain approximately in
proportion to their initial levels of welfare, regardless of the exact input-output structure
of the GPN (compare with formula (14)).17
Quantitative predictions. An interesting implication of Proposition 2 is that a more
integrated GPN bears the potential for a reduction of world income inequality. To see this,
recall from equation (14) that each country’s welfare level can be expressed as the product
h f
of its supplier access, (pf )1−γ = B T p1−γ , and market access, ∑∞
h=0 A q . When substituting
these expressions into (21) it follows directly that a country’s predicted welfare change,
due to a further integration of the GPN, is related to the market and supplier access (and
thus welfare levels) of all its direct and indirect trade partners. A country’s own initial
real income does not predict these changes.
To put perspective on this possibility, Figure 2 plots the predicted real income changes
associated with dA = 0.01Ã against the 2005 level of real GDP per capita for the 100
countries in our dataset. Overall, this trade cost reduction is predicted to result in a
widening of the worldwide income gap: rich countries gain on average more than poor
countries.18 However, this average picture still hides substantive heterogeneity. Many
17
The proportionality with the initial welfare level is exact for a trade cost reduction on all final goods
shipments, dB = xB. This is not surprising as such a cost reduction is equivalent to a Hicks-neutral
technical innovation, leaving the marginal product of all inputs, including labor, unaffected, i.e., dB ln(ld ) =
0. The real income effect is, thus, entirely dependent on a country’s improved access to consumption goods,
which is identical across countries and given by dB ln(u) = (x/(γ − 1))1.
18
As depicted by the regression line in Figure 2, a 1% higher initial GDP per capita is associated with
24
.3
Figure 2: The gains from a more integrated production network
SGP
MYS
EST
HUN
CZE
LUX
BEL
% increase real GDPpc
.1
.2
NLD
IRL
SVK
LVA
PHL
GHA
KAZ
THA
UKR
MNG
BDI
BGR
KGZ
TUN
PRY
BOL
LTU
POL
BLR
MAR
LKA
KHM
CMR
KEN
MWI
BGD
SEN
CHN
ARM
GEO
IND PAK
NOR
GBR
ITA
FJI
TZA
MDG
CAN
KOR
ROM
IDN
YEM
DNK
MLT
FRA
CRI
MDA
ETH
AUT
SWE
FIN
DEU
CHE
SVN
VNM
ZAFDZA MEX RUS
MKD
MUS
ECU
URY
IRN
PER
AZE
TUR
JOR
COL BRA
PRT
ISL
TTO
NZL
ESPSAU
AUS
ARE
USA
JPN
LBN
MAC
EGY
HKG
BWA
SDN
TON
PAN
SUR
CYP
0
GMB
6
8
10
12
ln real GDPpc 2005
Notes: The figure shows predicted per capita income changes due to a 1% trade cost reduction on
all intermediate goods exports and imports, dA = 0.01Ã. The predictions are plotted against real
GDP per capita in 2005.
low-middle income countries experience larger welfare gains than notably the U.S., GreatBritain or Japan. In light of Proposition 2, the only explanation for these heterogenous
welfare effects is that some countries are better exposed to the network through their
connections to countries that intermediate other nations’ value added and demand. In
Section 5.3, we shed further light on this by identifying the key intermediaries in the GPN.
Robustness. A final caveat to note is that Proposition 2 does not fully carry over to
the general model introduced in Section 3. As shown in Appendix B.3, the welfare effects
of a worldwide trade cost reduction on all intermediate goods exports and imports, dA =
xÃ, will additionally depend on a country’s direct neighborhood exposure. However, when
taking the predictions of the general model to the data, a country’s exposure through trade
a 0.02ppt [s.d. 0.005] higher counterfactual income gain. On the other hand, an analogous trade cost
reduction on all final goods exports and imports, dB = 0.01B̃, increases real incomes by about the same
0.05% in each and every nation.
25
intermediation always overshadows this additional effect. In fact, Figure 5 in the appendix
shows that, for a wide range of assumed elasticity parameters, the average country’s welfare
gains can for more than 80% be explained by trade intermediation.
5.2
The externalities of export cost reductions
Our second counterfactual explores the welfare effects of a more subtle trade cost variation.
Specifically, we investigate the effect of a one-sided export cost reduction along a single
trade route. Naturally, this improves the exporter’s market access in the importer country
and the importer’s access to the exporter’s supply of intermediate and final outputs. In an
integrated GPN, these supply and demand effects will, however, also partially spill over to
third countries not directly involved in the link. Our ambition is to compare the magnitude
of the exporter’s gains with the magnitude of these network externalities.19
General predictions. Suppose dA = xaij Iij and dB = ybij Iij for an exporter i and an
importer j ≠ i, where Iij is an n×n-matrix with a one in cell ij and zero everywhere else, and
where x, y > 0. Based on Property (2.) of Lemma 1, Proposition 1, and the expressions
for prices and outputs, (10) and (12) respectively, the per capita income change in any
country k ∈ N is given by
d ln(u) = Ψi dij ln(ld ) + Ωi dij ln(pf )
where
dij ln(pf ) =
1
(P f )γ−1 (
γ−1
ybij Iji
´¹¹ ¹¸¹ ¹ ¹¶
+
(i) direct supply
exposure
x T ∞ T h
B ∑ [A ] aij Iji )p1−γ
z
h=0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
(ii) exposure to
intermediated supply
∞
−1
dij ln(ld ) = [LW ] ( K γ W 1−γ Iij
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
+ K γ W 1−γ ∑ Ah Iij )(yqijf +
h=1
(iii) direct demand
exposure
19
(23)
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
x i
q )1
z ij
(iv) exposure to
intermediated demand
Of course, our counterfactual approach would also allow for the assessment of realistic trade agreements
like NAFTA or TTIP. The exact same mechanisms play a role as in the simpler counterfactual considered
here. In fact, Lemma 1 in the appendix states that the effect of marginal changes to multiple cells in the
A-matrix on the Leontief inverse, [I − A]−1 , is nothing but the sum of the effects of changing each cell
individually.
26
[h]
[h]
∞
where z = (1 − xaij ∑∞
h=0 aji ) > 0 is a scalar and where ∑h=0 aij denotes cell ij of matrix
∞
∑h=0 Ah .
Equation (23) makes clear that the network externalities of this cost reduction depend
crucially on a country’s network exposure to the demand and supply of the two directly
affected trade partners. To see this, let us, for comparison, first imagine a world without
production linkages, i.e., A = 0. Then, the sole effect of the export cost reduction ij
relevant for wages in all other countries k ∈ N /{i} is effect (i), which enters the import
competition matrix contained in Ωi (see (19)). All other effects are either zero or can be
ignored, since we treat exporter i’s labor demand as our reference and, thus, fix dwi = 0
(relevant for effect (iii)). The implication of this increased import competition is that wages
in countries k ∈ N /{i} will decline: as country i exports more to importer j, other nations
face tougher competition for their products. This leads to a wage decline in those nations
∗
followed —based on the logic of the trade multiplier matrix [{I − Φ}−i ]−1 , which has all
its entries positive when A = 0— by further rounds of wage reductions.20 The upside of
this is that consumers from all nations benefit from a price reduction on final products.
Introducing an integrated GPN into the picture gives rise to two additional positive
externalities: first, effect (ii) in (23) shows that consumers additionally benefit from the
improved access of country j’s producers to the intermediate goods assembled in country i.
This cost reduction is partially passed on to final goods producers in the rest of the world,
either directly or indirectly through a chain of producers at intermediate production steps.
Second, workers from all nations see the demand for their intermediate outputs increase
(effect iv). The reason is the extra demand for intermediate inputs in country i needed in
the assembly of the additional goods shipped along the link ij. The better a country is
connected to country i, the larger the externality on its domestic demand for labor.
Hence, in a GPN, there is room for wage increases in countries that are not directly
affected by an export cost reduction. In fact, the expressions in (23) allow for constellations,
where a workers from third countries might actually benefit more than the exporter’s own
workers. A particularly interesting case in this regard is the one, where the exporter
adds relatively little value to its own exports, but mainly acts as an intermediary of other
nations’ value added. Proposition 3 shows that this is, indeed, sufficient for at least one
other country to face at least the same demand effect as the country reducing its export
20
This is essentially an application of a well-known result in comparative statics analysis (e.g., Mas-Collel
et al., 1995, p. 618) stating that, if one good becomes more abundant (in the importer country j), the
relative demand for all other goods must decline.
27
cost:
Proposition 3. Consider a unilateral export cost reduction between exporter i and importer
j, and suppose exporter i’s labor productivity is marginally small. That is, suppose that
ceteris paribus, κi → 0. There is at least one country k ∈ N /{i} for which dij lim wk ≥
dij lim wi .
The result is a consequence of the fact that country i does not add value in the limit
economy: κγi wi1−γ → 0. As such, country i’s own workers do only marginally benefit from
the export cost reduction, because the demand effect is proportional to their cost share in
production. The entire effect, thus, accrues in countries that supply intermediate inputs
to i and that make up its entire value added share in production.
For the formal proof of Proposition 3, we take advantage of the fact that in our Walrasian economy we are always free to fix wages in one country at wi = 1; also in the
transition to our limit economy where κi → 0. From this, it follows immediately that
lim[κγi wi1−γ ] = 0. Hence, in the limit, country i’s entry in vector dij ln(ld ) and its entries in
∗
row i of matrix [{I − Φ}−i ]−1 are zero. As a result, we get dij lim wi = 0. The proposition
is, then, a consequence of the fact that we are still free to fix dij lim wk = 0 for any country
k ≠ i. Thus, there is at least one other country k that faces the same demand effect as
exporter i, and potentially a third country with a strictly positive demand effect.
Quantitative predictions. To explore the possibility of positive wage and income externalities in today’s GPN, we calculated the average welfare and wage externality, ∑k∉i duk /(n−
1) and ∑k∉i dwk /(n − 1), associated with a 1% increase in the export intensity along each
of the 9,900 exporter-importer pairs in our dataset.
In 56% of these cases, the export cost reduction generates an average positive welfare
externality, compared to only 29% when hypothetically switching off all supply chain linkages (A = 0). We even find a number of exporter-importer pairs, be it only a handful, with
an average wage externality that is larger than the wage effect experienced by the country lowering its export costs; something that would be impossible without supply chain
linkages. On the exporter’s side of these pairs, we typically find the same few countries.
Among them, Hong Kong and Panama stand out. In line with Proposition 3, a commonality of these “pure intermediaries” is that their trade volume by far exceeds their own
value added in production, so that they predominantly intermediate other nations’ value
and demand.
28
Robustness. Proposition 3 also carries over to a Cobb-Douglas specification (β = 1) for
our general model (see Appendix B.3). In all other cases, exporter i’s demand for local
labor still depends, even when κi → 0, on the effect of the export cost reduction on the
wages rates of other countries. This makes it impossible to unambiguously pin down the
size of the wage externalities relative to the wage effect in the country lowering its export
costs. However, our quantitative predictions in Figure 6 of the appendix show, that the
number of cases where the average wage externality exceeds exporter i’s own wage effect
only increases, when we relax the assumption α = γ = β.
5.3
Identifying key trade partners
The previous counterfactuals highlighted countries’ network exposure through trade intermediation as the crucial welfare determinant in a gradually integrating GPN. Our goal here
is to identify the important trade intermediaries, i.e., those countries providing indirect access to the supply and demand of many other countries. We do this by hypothetically
isolating one country after the other from the network and calculating the magnitude, but
more importantly also the sources, of the income losses inflicted on the remaining nations.21
General predictions. Formally, we set all of a country’s entries in the A- and B-matrices
to zero. Based on Property (iii) of Lemma 1 and Proposition 1, the isolation of country
i ∈ N has the following effect on the per capita income in any j ∈ N /{i}:
d ln(u) = Ψi d−i ln(ld ) + Ωi d−i ln(pf )
∗
∗
21
Our approach borrows from the concept of a key industry, as developed in the regional economics
literature (Rasmussen, 1956; Hirschman, 1958) and applied later in the social network literature (Ballester
et al., 2006). The idea is to quantify the value of a sector by the forward and backward linkages that are
severed, when this sector is hypothetically isolated from a national input-output network. In the same
spirit, we define the importance of a country in terms of the value added and final demand it intermediates
between the nations participating in the GPN.
It should be clear, however, that this exercise is not only of hypothetical value. It can also be used to
simulate the effects of trade embargoes or natural disasters with severe consequences for a single nation’s
productive capacities.
29
where
d−i ln(pf ) =
∞
1
1 ∞
1
(P f )γ−1 B T ∑ [AT ]h ( Iii K γ w1−γ + Iii ∑ Ah K γ w1−γ )
γ−1
z
z
h=0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹h=1
¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
(i) ctr. i’s
value added
∞
−1
d−i ln(ld ) = − [LW ] K γ W 1−γ ∑ Ah (
h=0
qfi
®
(iii) ctr. i’s
demand
+
(24)
(ii) intermediated
value added
1
Iii qf−i
z
´¹¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¶
(iv) intermediated
demand
+
1 ∞ h f
Iii ∑ A q−i )
z h=1
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
(v) intermediated
demand
f
f
f T
where Iii is a square matrix with a one in cell ii and zero everywhere else, qfi = (q1i
, q2i
, ..., qni
)
[h]
f
f
f
and qf−i = (∑k≠i q1k , ∑k≠i q2k , ..., ∑k≠i qnk )T are column vectors, and z = ∑∞
h=0 aii is a scalar,
[h]
∞
∞
h
where ∑h=0 aii denotes cell ii of matrix ∑h=0 A . Appendix A.3 provides more details on
the derivation.22
Equation (24) distinguishes five channels through which welfare in other countries is
affected: on the demand side, workers need to accept wage cuts due to the foregone access to country i’s own final demand (effect iii) and to the final demand that country i
intermediated from elsewhere (effects iv+v). The magnitude of the latter two effects is
h f
determined by Iii ∑∞
h=0 A q−i , which can be interpreted as our model-specific version of
vertical specialization trade, as introduced into the empirical supply chain literature by
Hummels et al. (2001).23 On the other hand, the import competition effect incorporated
∗
in the Ωi -matrix (see expression (19)) —in combination with effects (i) and (ii)— suggests
that a country’s isolation relaxes competition in the world markets, with positive spillovers
on other countries’ wage incomes. In other words, this effect captures the idea that every
nation is to some extent dispensable, because others can fill its gap.
Turning to the price effect, there are two additional direct channels by which the isolation of country i affects prices. Next to the wage-induced price effect incorporated in the
∗
Ψi -matrix (see expression (19)): consumers lose access to country i’s local value added
22
To calculate the welfare losses inflicted on the isolated nation i itself, our linear approximations are of
little use, as the losses are certainly more than marginal. However, we can follow Arkolakis et al. (2012)
aii
ii
ii
and calculate the loss by u′i /ui = ( 1−a
. This can be estimated by u′i /ui = ( T ot.prod
. Note
i )
i )
ii qii
i −xii
that these terms differ from the ones presented in their appendix (p.115), because of our specification of
a model with flexible labor cost shares (β > 1).
23
h f
To be precise, AIii ∑∞
h=0 A q−i measures every nation’s access to the vertical specialization trade of
country i. The equivalence with the empirical measure, (1 − λi )(∑j≠i xfij + ∑j≠i xiij ), follows from the
h f
f
i
identity ∑∞
h=0 A q = q + q and the alternative expression for the A-matrix in (35).
qf
xf
γ−1
30
γ−1
(effect i) and to the value added of other nations that is incorporated in the products
assembled in country i (effect ii).
The particular value of the effect decomposition in (24) is that it allows us to determine each country’s contribution to every other country’s network exposure through trade
intermediation, which as highlighted in Section 5.1, is the key welfare determinant in a
gradually integrating GPN. Specifically, the contribution of trade partner i to every other
country’s welfare gains is given by the extent to which country i intermediates value added
and demand to other nations (effects ii and v in (24) respectively). Country i’s own value
added (effect i) and the access it grants to the final demand in its direct neighborhood
(effects iii+iv), in contrast, are irrelevant. In other words, if we isolate one nation after
the other and sum up effects (ii) and (v), we approximately retain the expressions in (21).
To make the link with Section 5.1 even more salient, we retain exact equivalence, if we
consider, just as there, a marginal trade cost variation on all of a country’s trade ties:
Proposition 4. Suppose dA−i = −x( 12 Iii A + 12 AIii − aii Iii ) and dA = xÃ, where ãij = aij
for all i ≠ j and ãii = 0 for i ∈ N and x → 0+ . The per capita income changes associated
with a worldwide trade cost reduction on all intermediate goods exports and imports are
equivalently determined by
dA ln(u) = − ∑ dA−i ln(u)
i∈N
The result, which fully carries over to the general model introduced in Section 3, follows
immediately from Property (1.) of Lemma 1 and the additivity of the effects in (24) for
different isolated nations.
Quantitative predictions. Figure 3 shows the 32 most valuable trade partners in our
2005 dataset, when considering all effects in (24). The average real income loss is largest
when severing all trade ties with Germany, the overall key trade partner: 1.2%. Isolating
the average nation shown in Figure 3 causes an average welfare loss of only 0.3%.
More interestingly, an average 60% of this loss is due to the foregone access to the
isolated country’s own local value added and demand (effects i and iii in (24)). The other
40% stems from the isolated country no longer intermediating other nations’ demand and
value added (effects ii, iv, and v). These averages hide substantive variation between
countries, however. Some countries, e.g. USA, Japan, Spain, India or Turkey, are primarily
contributors of local value added and demand. Others, e.g. Singapore, Malaysia, Belgium,
31
Figure 3: Key trade partners and Key intermediaries
DEU
USA
FRA
CHN
GBR
ITA
NLD
JPN
BEL
RUS
SGP
ESP
CHE
KOR
SWE
intermediated
THA
BRA
local
POL
MYS
IND
AUT
AUS
TUR
FIN
CZE
ZAF
UKR
CAN
DNK
HUN
IDN
IRL
−1.2
−1
−.8
−.6
−.4
−.2
0
avg. % welfare change in other countries when isolating country i
Notes: The figure shows the 32 countries whose isolation causes the largest average per capita
income loss in the remaining 99 nations in our 2005 dataset. The loss is broken down into two
channels: foregone local value added and demand (effects i and iii in (24)) and foregone access to
intermediated value added and demand (effects ii, iv, and v).
Thailand, and Ireland, derive their importance primarily from being intermediaries.
To connect to our analysis in Section 5.1, Germany, China, the Netherlands, France,
Belgium, the USA, and Singapore contribute, in absolute terms, most to other nations’
welfare based on their role as intermediaries. But, these totals do not tell us for which
particular countries they are so valuable. The upper panel of Figure 4 provides this piece
of information. Here, we position all 100 countries in our data set in a network graph.
An arrow pointing from i to j indicates that country i is either the single-most important
trade intermediary for country j or that country i contributes to a significant extent (at
least 1%) to welfare in j by acting as that country’s intermediary.
The figure suggests that —quite different from the network of overall key trade partners
shown in the bottom panel of Figure 4— trade intermediation is a geographically confined
phenomenon. In the Americas, the U.S. is the major intermediary for almost all other
countries. Yet, the U.S. does not share any important intermediation links with countries
32
Figure 4: Key Intermediaries and Key Players by Nation
(a) Key Intermediaries
TZA
TON
MWI
BLR
UKR
NZL
LBN
ZAF
MKD
FJI
MDA
CYP
GHA
HUN
BGR
ROM
KEN
RUS
AUT
ARE
AUS
LTU
BDI
YEM
CHE
SAU
TUR
LUX
POL
TUN
MAC
GMB
JOR
DEU
LVA
IRN
KAZ
SDN
MDG
GEO
MNG
KOR
CHN
PAN
JPN
DZA
ETH
FRA
HKG
KGZ
SEN
SVN
NLD
CZE
ITA
THA
SGP
PRT
SUR
BEL
IDN
IND
AZE
MYS
EGY
ARM
PAK
PHL
LKA
ESP
SWE
SVK
MLT
BGD
CMR
MAR
ECU
VNM
DNK
IRL
GBR
NOR
USA
KHM
FIN
MEX
BRA
ISL
MUS
BWA
EST
TTO
URY
BOL
ARG
PRY
PER
COL
CRI
CAN
(b) Key Players
UKR
POL
TZA
MWI
MDA
GEO
GHA
HUN
VNM
LTU
NZL
KGZ
MAR
CHE
MKD
PHL
KOR
ARE
KAZ
LUX
CHN
CMR
FRA
ETH
SEN
DEU
TUN
ARM
ESP
BGD
IND
MLT
KEN
IDN
MAC
EGY
AZE
CZE
PRT
THA
SGP
PAK
NLD
JPN
HKG
LKA
TUR
SVN
IRN
YEM
DZA
MDG
ROM
SVK
PAN
MNG
RUS
ZAF
LVA
AUS
CYP
LBN
AUT
FJI
TON
BLR
BDI
BGR
Pajek
JOR
MYS
ITA
KHM
IRL
DNK
SWE
FIN
USA
BEL
SDN
ISL
SAU
PER
EST
NOR
GBR
BRA
SUR
ECU
URY
BWA
MUS
GMB
BOL
PRY
ARG
CRI
COL
CAN
MEX
TTO
Notes: The figures position the 100 countries in our 2005 dataset in two network graphs. For each country,
the upper (lower) panel shows its most important trade intermediary (trade partner), indicated by an arrow
pointing from the intermediary (trade partner) to the focal country. On top of this, we add the top 1
percentile of intermediation (overall welfare) links, which are associated with a contribution of at least 1%
(2%) to the focal nation’s per capita income. Arrow sizes are proportional to welfare contributions, ranging
from 1 − 4% (2 − 8%). Node sizes show a country’s overall importance as an intermediary (trade partner).
33
Pajek
on other continents. A similar picture emerges for the key intermediaries in Europe, Asia,
and to a lesser extent Sub-Saharan Africa: Germany, the Netherlands, Belgium, Singapore,
and South Africa feature significantly in their respective regions, but play virtually no role
elsewhere. Only China, France, Great Britain, and Italy are examples of countries that
intermediate value beyond their immediate geographical neighborhood. All this is in line
with the empirical pattern found elsewhere (Johnson and Noguera, 2012b; Los et al., 2015)
leading Baldwin and Lopez-Gonzalez (2014) to divide the world into Factory Asia, America,
and Europe.
More importantly, the network graph provides the missing piece of information to fully
understand our findings in Figure 2, where we saw that a further integration of the global
production network offers little prospects for the world’s poorest nations. As can be seen
from the upper panel of Figure 4, the reason for this is that many developing countries in
Africa, Central Asia, and the Middle East, simply lack the same good access to important
trade intermediaries that the more developed nations enjoy.
6
Conclusion
In this paper, we present a number of novel theoretical results concerning the welfare
consequences of a gradually integrating global production network. Our findings show
that established determinants of a country’s welfare, notably its exposure to its own local
conditions and to those of its direct trade partners, are poor predictors of these welfare consequences. Instead, countries well-connected to important trade intermediaries, countries
that provide indirect access to the value added and final demand of many other countries,
have gained most from this development.
Beyond the theoretical contributions of this paper, the counterfactual approach that
we develop to disentangle the different determinants of any counterfactual welfare change
might also prove useful in other models of input-output trade or economic geography.
The more so, as it readily lends itself for empirical implementation using no more than
observable gross trade flows and estimates of the model’s elasticity parameters. In this
regard, our approach might even inspire new empirical strategies to estimate these elasticities. The system of total differentials that lies at its heart establishes a linear relationship
between observable macroeconomic data, some in levels and some in changes, with the
model’s important elasticities as coefficients (see Section 4 or Appendix B.2). Using a
large scale trade cost shock as a quasi-experimental setting, one could estimate these co-
34
efficients using fairly straightforward methods. In light of our focus on network effects,
it seems particularly worthwhile to complement existing studies on the direct effects in
countries experiencing a trade cost shock (e.g., Autor et al., 2013; Caliendo and Parro,
2015) by looking at the effects in third countries whose trade costs remained unaffected.
Our findings also have some practical implications. For example, they can be used for
the assessment of trade policies or economic sanctions, or for the identification of countries
with systemic importance for the world economy. In particular, our effect decomposition
into exposure to local conditions vs. network exposure puts countries on the economic
landscape, not for the value they add to the production of goods or services, but for
their role as trade intermediaries. Our findings stress the need to not only consider how
a particular trade agreement exposes oneself to the economic conditions of the directly
involved trade partner(s), but also —and possibly more important— how it will affect one’s
indirect exposure to the local conditions of the latter’s own trade partners. One valuable
extension of our model in this direction is a richer input-output structure consisting of
more than just two sectors, as in, e.g., Caliendo and Parro (2015) or Ossa (2015). This
would allow for a more fine-grained analysis of the role of network exposure at the level
of individual sectors, or even firms. Our conjecture is that, similar to the studies of
Hirschman (1958) or Oberfield (2013) on national supply chains, such a model would
highlight those sectors that are most effective in spreading innovations throughout the
entire world economy.
A
A.1
Appendix for print
Lemma 1 and Proof
The following lemma shows how to relate —in the spirit of comparative statics analysis—
a new inverse matrix, [I − Z ′ ]−1 , to the initial state of this matrix, [I − Z]−1 , and the
imposed change dZ = Z ′ − Z:
Lemma 1. Consider square matrices Z and Z ′ and a scalar x ∈ R, such that [I − Z]−1 and
[I − Z ′ ]−1 exist:
1. For Z ′ = Z + xdZ with x → 0+ , it holds
[I − Z − xdZ]
−1
= [I − Z]
−1
35
−1
−1
+ x[I − Z] dZ [I − Z]
(25)
2. For Z ′ = Z + xzij Iij , where Iij is a square matrix with a one in cell ij and zero
[h]
−1
everywhere else and where ∑∞
h=0 zij denotes cell ij of matrix [I − Z] , it holds
[I − Z − xzij Iij ]
−1
= [I − Z]
−1
xzij
+
−1
[h]
1 − xzij ∑∞
h=0 zji
[I − Z] Iij [I − Z]
−1
(26)
3. For Z ′ = I−i ZI−i , where I−i = (I − Iii ) and Iii denotes a square matrix with a one in
[h]
−1
cell ii and zero everywhere else and where ∑∞
h=0 zii denotes cell ii of matrix [I −Z] ,
it holds
[I − I−i ZI−i ]
−1
= Iii + [I − Z]
−1
−
1
−1
[h]
∞
∑h=0 zii
[I − Z] Iii [I − Z]
−1
(27)
Before we proof the claim, we first review some established solutions for the inverse of
a sum of matrices:
Henderson and Searle (1981). Let X be a nonsingular square matrix, and U, Y and
V be (possibly rectangular) matrices such that U Y V is a square matrix. It holds
[X + U Y V ]
−1
−1
= X −1 − X −1 U [I + Y V X −1 U ] Y V X −1
(28)
The following identities are useful special cases:
Minabe (1966, p.58). By successive application of (28) for a nonsingular square matrix
X and a square matrix Y , such that all characteristic roots ρ of X −1 Y satisfy ∣ρ ∣ < 1, we
get
[X − Y ]
−1
∞
h
= X −1 + ∑ (X −1 Y ) X −1
(29)
h=1
Neumann’s series expansion. Expanding on Minabe (1966), we get for X = I
−1
[I − Y ]
−1
= I + Y [I − Y ]
−1
∞
= I + [I − Y ] Y = ∑ Y h
(30)
h=0
Sherman and Morrison (1949, 1950). For y ∈ R, a column vector u, and a row vector
vT of identical length
[X + yuvT ]
−1
= X −1 −
y
X −1 uvT X −1
1 + yvT X −1 u
36
(31)
Equipped with these results, we are ready to the proof the claim:
Proof of part 1. Applying (29) for X = I − Z and Y = xdZ, we get
−1
−1
[I − Z − xdZ]
= [I − Z]
∞
−1
h
+ ∑ ([I − Z] xdZ) [I − Z]
−1
h=1
−1
Suppose, now, that x → 0+ . Then, in the limit, all characteristic roots of [I − Z] xdZ
clearly satisfy limx→0+ ∣ρ ∣ < 1. We moreover obtain
h
1 ∞
−1
−1
−1
−1
lim+ ∑ ([I − Z] xdZ) [I − Z] = [I − Z] dZ [I − Z]
x→0 x
h=1
Thus, expression (25) is nothing but a first-order Taylor approximation around [I − Z]−1 .
Proof of part 2. The expression in (26) is an immediate corollary of the result by
Sherman and Morrison (1949, 1950) when setting X = I − Z, y = −xzij , u = (u1 = 0, u2 =
0, ..., ui = 1, ui+1 = 0, un = 0), and v = (v1 = 0, v2 = 0, ..., vj = 1, vj+1 = 0, vn = 0).
Proof of part 3. We prove here a more general version of the claim, where Z ′ = Ixi ZIyi
with Ixi = (I + xIii ), Iyi = (I + yIii ), x, y ∈ R, and Iii is a square matrix with a one in cell ii
and zero everywhere else.
Applying (28) for X = I, Y = −Z, U = Ixi , and V = Iyi , we get
[I − Ixi ZIyi ]
−1
−1
= I + Ixi [I − ZIxi Iyi ] ZIyi
(32)
−1
= I + Ixi [I − Z − Z(x + y + xy)Iii ] ZIyi
Applying (28) again, this time for X = I − Z, Y = Iii , U = −Z(x + y + xy)Iii , and V = I, we
find
−1
[I − Z − Z(x + y + xy)Iii ]
= [I − Z]
−1
−1
+ [I − Z] Z(x + y + xy)
(33)
−1
−1
−1
× Iii [I − Iii [I − Z] Z(x + y + xy)Iii ] Iii [I − Z]
where we have made use of the fact that Iii is idempotent, Iii = Iii Iii . Finally, we can write
−1
−1
Iii [I − Iii [I − Z] Z(x + y + xy)Iii ] Iii =
37
1
[h]
1 − ∑∞
h=1 zii (x + y
+ xy)
Iii
(34)
[h]
∞
h
where ∑∞
h=1 zii denotes cell ii of matrix (∑h=0 Z )Z.
For x = y = −1, the combination of (32)-(34) yields expression (27) of the lemma, since
[I − I−i ZI−i ]
−1
= I + I−i ([I − Z]
−1
−1
− [I − Z] Z
−1
= Iii + I−i [I − Z] I−i − [I − Z]
−1
−1
1
I [I
[h] ii
∞
∑h=0 zii
1
−1
∞
−1
− Z] )ZI−i
I [I − Z]
[h] ii
−1
∑h=0 zii
+ I−i [I − Z] Iii + Iii [I − Z] I−i
1
−1
−1
−1
= Iii + [I − Z] − ∞ [h] [I − Z] Iii [I − Z]
∑h=0 zii
where, for the second equality, we use I−i [I − Z]−1 ZIii = I−i [I − Z]−1 Iii = [I − Z]−1 Iii −
[h]
[h]
∞
−1
−1
−1 − I
Iii ∑∞
ii ∑h=0 zii . ∎
h=0 zii and similar Iii [I − Z] ZI−i = Iii [I − Z] I−i = Iii [I − Z]
Expanding on the identity of Sherman and Morrison (1949, 1950) one can also trace
back arbitrary large variations in the Leontief inverse to the original state of this matrix
in a sequence of k ≥ 1 functional mappings.
Let X be an invertible matrix. Moreover, let Iis js be a square matrix with a one in
cell is js and zero everywhere else, and let xs ∈ R. Based on Sherman and Morrison (1949,
1950), define the endomorphic function
fis js (X −1 ) = X −1 + yis js X −1 Iis js X −1
where the scalar yis js is the output of the function yis js = g(xs , X −1 , Iis js ). We can then
write
k
[I − Z − ∑ xs Iis js ]
s=1
−1
= fik jk (fik−1 jk−1 (fik−2 jk−2 (...fi1 j1 ([I − Z]−1 ))))
whereby at any step s ≥ 1, fis js can be written in the form
fis js ([I − Z]−1 ) = [I − Z]−1 + [I − Z]−1 Ys [I − Z]−1
Here, Ys is given by Ys = zis js ( ∑st=1 Iit jt + ∑i∈Ns Iii ), where Ns ⊆ N is the subset of all
countries involved in at least one of the trade cost changes up until step s and scalar zis js
is the output of the function zis js = h((xt )t=1,...,s , [I − Z]−1 , (Iit jt )t=1,...,s ).
A.2
Empirical implementation
In this appendix, we explain how we numerically implement our counterfactual exercises in
the simple Armington model we focus on in most of the paper. The full model introduced in
Section 2 and analyzed in Appendix B.1 can be implemented in the exact same way. In fact,
38
all equations in Appendix B.1 are already expressed in terms of observable macroeconomic
constructs and the model’s elasticity parameters, for which estimates are readily available
from the literature.
To put numbers on our counterfactuals, we use readily available data on gross bilateral
trade flows of final and intermediate goods, and total industrial GDP and production in
each of the 100 countries in our 2005 dataset. The model equivalents to these data are
X f , X i , Lw and (X f + X i )1, respectively. Moreover, we obtain each country’s labor cost
share, λi , by simply dividing its industrial GDP by total industrial production.
One can directly relate the unobserved A− and B−matrices, the unobserved augmented
price indexes for final and intermediate goods, (pf )1−γ and p1−γ , and the unobserved
augmented labor productivities and wages, κγ and w1−γ , to these observed data. To see
this, first note that we can equivalently write A and B as
A = P γ−1 Πi (I − Λ)P 1−γ
(35)
B = P γ−1 Πf (P f )1−γ
(36)
where P 1−γ and (P f )1−γ are the diagonal matrices of every country’s augmented producer
and consumer price indexes, and Πf and Πi are the full matrices of every country i’s share in
every other country j’s total expenditure on final and intermediate products, respectively.
i
These identities follow immediately from the following expressions for πji
, λi , and πijf
λi =
f
πji
=
i
πji
=
∂ ln(pi ) κγi wi1−γ
= 1−γ
∂ ln(wi )
pi
∂ ln(pfi )
∂ ln(pfji )
=
(pj τjif )1−γ
(pfi )1−γ
(37)
∂ ln(pii ) (pj τjii )1−γ
=
∂ ln(pji )
(pii )1−γ
f 1−γ
which are obtained from applying Shephard’s lemma to p1−γ
, and (pii )1−γ .
i , (pi )
An implication of (35) is the following: we can express [I − A]−1 and [I − AT ]−1 in terms
of a product of three matrices:
[I − A]−1 = P γ−1 [I − Πi (I − Λ)]−1 P 1−γ
[I − AT ]−1 = P 1−γ [I − (I − Λ)(Πi )T ]−1 P γ−1
(38)
Moreover, based on (37), we can write
K γ W 1−γ = P 1−γ Λ
Qf = P γ−1 X f
(39)
The right hand sides of the expressions in (35)-(39) feature only data, X f , Λ, Πf ,
and Πi , with the exception of the unobserved price indexes. Nevertheless, we show in the
39
following that, when substituting (35)-(39) into the expressions of Proposition 1, which
determine the counterfactual welfare effects in our model, all unobserved price indexes
cancel out, leaving us with expressions that only involve observable data and γ.
Direct effects of dA: Using (35)-(39) and relying on Lemma 1, the direct effects of an
arbitrary variation dA can be written as
−1
−1
−1
(P f )γ−1 B T [I − AT ] dAT [I − AT ] K γ w1−γ
γ−1
−1
−1
−1
(Πf )T [I − (I − Λ)(Πi )T ] (P γ−1 dAT P 1−γ )[I − (I − Λ)(Πi )T ] λ
=
γ−1
dAT ln(pf ) =
and
dA ln(ld ) = [LW ]−1 K γ W 1−γ [I − A]
−1
−1
dA [I − A] qf
−1
−1
= [LW ]−1 Λ[I − Πi (I − Λ)] (P 1−γ dAP γ−1 )[I − Πi (I − Λ)] xf
where all components of these vectors involve only observable data and γ except for
P γ−1 dAT P 1−γ = P γ−1 ((AT )′ − AT )P 1−γ
(40)
P 1−γ dA P γ−1 = P 1−γ (A′ − A)P γ−1
We can, however, make use of the identities P γ−1 AT P 1−γ = (I −Λ)(Πi )T and P 1−γ A P γ−1 =
Πi (I − Λ) to substitute the initial states of the matrices in (40) by their data equivalents.
Moreover, we can fill in arbitrary values for the new states of A′ and (AT )′ , as long as the
compound matrices P γ−1 (AT )′ P 1−γ and P 1−γ A′ P γ−1 retain their economic interpretation.
′
Formally, this requires that P γ−1 (AT )′ P 1−γ 1 ≈ ((I − Λ)(Πi )T ) 1 < 1 and 1T P 1−γ A′ P γ−1 ≈
′
1T (Πi (I − Λ)) < 1T .
Direct effects of dB: The empirical equivalents of the direct effects of an arbitrary
variation in one or more cells of the B-matrix can be written as
dB T ln(pf ) =
−1
((P f )γ−1 dB T p1−γ )
γ−1
−1
dB ln(ld ) = [LW ]−1 K γ W 1−γ [I − A] dB (P f )γ−1 Lw
−1
= [LW ]−1 Λ[I − (Πi )(I − Λ)] (P 1−γ dB (P f )γ−1 )Lw
40
Similar to above, we can use the fact that
(P f )γ−1 dB T p1−γ = (P f )γ−1 (B T )′ p1−γ − (Πf )T 1 = (P f )γ−1 (B T )′ p1−γ − 1
P 1−γ dB (P f )γ−1 = P 1−γ B ′ (P f )γ−1 − Πf
to obtain empirical equivalents for the remaining unobserved components. Again, we can
fill in arbitrary values for the new states of B ′ and (B T )′ , as long as (P f )γ−1 (B T )′ p1−γ ≈
′
′
((Πf )T ) 1 < 1 and 1T P 1−γ B ′ (P f )γ−1 ≈ 1T (Πf ) < 1T .
Indirect effects: The expressions for the general equilibrium effects in Proposition 1
have the following empirical equivalents
dln(w) ln(ld ) = −(γ − 1)I + [LW ]−1 V
dln(pf ) ln(ld ) = (γ − 1)[LW ]−1 V (dln(w) ln(pf ))T
−1
(dln(w) ln(pf ))T = (P f )γ−1 B T [I − AT ] K γ W 1−γ = (Πf )T [I − (I − Λ)(Πi )T ]−1 Λ
where V is defined as:
V
−1
−1
= K γ W 1−γ [I − A] Qf = Λ[I − Πi (I − Λ)] X f
In all the above expressions, only γ remains unobserved. In all the empirical results
presented in this paper, we take an estimate for γ from the literature (see Section 3.1
for details), but verify the robustness of our findings with respect to this choice.
A.3
Identifying key countries
To obtain the welfare expressions in (24), let us first determine the impact of isolating
country i on matrix [I − A]−1 B. Based on Lemma 1 Part (3.), this impact is given by
∞
[I − I−i AI−i ]−1 I−i BI−i − [I − A]−1 B = ( ∑ Ah −
h=0
1
∞
[h]
∑h=0 aii
∞
∞
h=0
h=0
∑ Ah Iii ∑ Ah + Iii )
∞
× (B − Iii B − BIii + Iii BIii ) − ∑ Ah B
= −
1
∞
[h]
∑h=0 aii
∞
∞
h=0
h=0
h=0
(41)
∞
∑ Ah Iii ∑ Ah B(I − Iii ) − ∑ Ah BIii
h=0
where, for the final line, we have made use of the fact that Iii is an idempotent matrix,
[h]
∞
h
i.e., Iii = Iii Iii , and the fact that Iii ∑∞
h=0 A Iii = Iii ∑h=0 aii .
41
Applying (41) on the price vector (11), we arrive at the following supply effect
d−i ln(pf ) =
=
∞
∞
1
1 ∞
(P f )γ−1 ((I − Iii )B T ∑ [AT ]h Iii ∑ [AT ]h − Iii B T ∑ [AT ]h )K γ w1−γ
γ−1
z h=0
h=0
h=0
∞
1
(P f )γ−1 (I − Iii )B T ∑ [AT ]h (
γ−1
h=0
1 ∞ T h
Iii ∑ [A ]
z h=1
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
intermediated value added
+
+
1
Iii
z
±
)K γ w1−γ
(i) ctr. i’s
value added
1
Iii 1
γ−1
[h]
In the first line, we defined the scalar z = ∑∞
h=0 aii . In the second line, we decomposed
the first summand from line one into the channels (i)-(ii) highlighted in (24). The final
summand in line three is omitted in (24), because (a) we ignore the welfare effect on the
isolated country i itself and (b) we also ignore the relaxed competition in country i, as no
country j ≠ i is going to sell in country i after the isolation anyhow.
Applying (41), in turn, onto the labor demand equation (13), we arrive at the demand
effect
∞
1 ∞
d−i ln(ld ) = − [LW ]−1 K γ W 1−γ ∑ Ah ( Iii ∑ Ah B(I − Iii )(P f )γ−1 Lw + BIii (P f )γ−1 Lw)
z h=0
h=0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
ctr i’s demand
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
intermediated demand
The summands are nothing but channels (iii)-(v) of (24). Similar to the empirical approach
presented in Appendix A.2, one easily obtains empirical equivalents for the above supply
and demand effects by means of (35) and (36) in combination with Lemma 1.
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44
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45
B
B.1
Appendix: potentially for online publication
The general model
Here, we verify the robustness of our main findings from the text in the context of the full
model introduced in Section 3. This means, we let go of the assumptions of an Armington
model with α = β = γ > 1 (A1) and a lower bound on intermediate goods trade costs (A2)
and replace them by one of the following two alternatives:
Alternative A3: Armington with flexible elasticities. Suppose that for every final
and intermediate goods variety η f ∈ Hf and η i ∈ Hi there is one, and only one, country able
to produce it. The elasticities of substitution between two consumption varieties, between
labor and an aggregate intermediate input, and between two intermediate input varieties
are denoted by α, β, and γ, respectively, whereby α, γ ≥ β ≥ 1 and α, γ > 1.
Alternative A4: Ricardo with flexible elasticities. Suppose an infinite number of
varieties, Hf = {1, ..., ∞} and Hi = {1, ..., ∞}. Each variety η ∈ Hf ∪ Hi can potentially
be produced in every nation i ∈ N . Buyers purchase, however, only from the lowest
cost producer, trade costs included, so that not every variety is produced everywhere.
Instead, for each t ∈ {f, i}, buyers from country j choose the producer i for which i =
arg mini∈N {pi τijt /mi (η)}, whereby pi denotes country i’s producer price index. Thus, variety
η is supplied by the nation that has a comparative advantage in its production and delivery.
Every nation’s comparative advantages are exogenously determined. In particular, the
productivity of country i in the production of variety η, mi (η), is the outcome of a random
variable m ≥ 0, which is independently drawn for each country from a Fréchet distribution
Fit (m) = exp[−mti m1−θ ]
t
where mti > 0 and θt > α, γ ≥ β ≥ 1 and θt > 2 for t ∈ {f, i}.
Demands, prices, and expenditure shares. In the Ricardian specification for our
model, the prices of all goods consumed and produced can be summarized by economywide price indexes for an output,
pi = (κβi wi1−β + (κii )β (pii )1−β )1/(1−β)
an aggregate intermediate input,
pii = ( ∑ (pj )1−θ aji )1/(1−θ )
i
i
j∈N
and a consumption unit,
pfi = ( ∑ (pj )1−θ bji )1/(1−θ
f
j∈N
46
f)
Here, aij ≡ mii (ρi τiji )1−θ and bij ≡ mfi (ρf τijf )1−θ denote the generalized trade intensities,
which also contain the country-specific, mti , and homogeneous productivity parameters,
ρf = h(θf , α) > 0 and ρi = h(θi , γ) > 0, inherited from the Fréchet distribution.
Equilibrium wage rates wi are fixed by the labor market equation
i
f
∞
h
Lw = l (w) = Λ(x + x ) = Λ ∑ [Πi (I − Λ)] Πf Lw
d
f
i
(42)
h=0
i
h
where Λ is the diagonal matrix of labor cost shares, (λi )i∈N , and ∑∞
h=0 [Π (I − Λ] is the
Leontief inverse corresponding to the full matrix of intermediate input expenditure shares,
(πiji )ij∈N ×N , and the diagonal matrix of intermediate input cost shares, (1 − λi )i∈N .
By Shephard’s lemma, the entries in these matrices are given by
∂ ln(pij ) pi1−θi aij
∂ ln(pi ) κβi wi1−β
i
and πij =
= 1−β
=
λi =
∂ ln(wi )
∂ ln(pi ) (pij )1−θi
pi
(43)
∂ ln(pfj ) pi1−θf bij
∂ ln(pi ) (κii )β (pii )1−β
f
= f
1 − λi =
=
and πij =
∂ ln(pii )
∂
ln(p
)
p1−β
(pj )1−θf
i
i
Even though, in the Ricardian model, the elasticity parameters α and γ are superseded
by the shape parameters θf and θi of the Fréchet distribution, the prices and expenditure
shares presented above are identical to the ones of an Armington model with α = θf ,
t
γ = θi , and mti (ρt )1−θ = 1. This implies, in other words, that the welfare predictions of any
counterfactual trade cost variation to the two models are identical. We, therefore, continue
with the notation of the simpler Armington model in the following.
Equilibrium. From the preceding it follows that in equilibrium, if it exists, per capita
incomes are given by
ln(u) = ln (L−1 K β W 1−β (xf + xi )) +
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
exposure to final demand
1
β−1
ln (p) +
ln ((pf )1−γ )
β
γ−1
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
(44)
exposure to value added
This means that, in line with our Definition 1 from the main text, welfare levels depend on
a country’s exposure to its local consumer demand and value added as well as its network
exposure to every other country’s final demand and value added.
The following results, inspired by Alvarez and Lucas (2007), show that a trade balancing
wage and price vector indeed exists. Extending on their Theorem 1, we first show that the
vector of producer price index functions g(p, w), with entries
gi (p, w) =
(κβi wi1−β
+ (κii )β ( ∑
j∈N
47
1−β
1−γ
p1−γ
)
j aji )
1
1−β
(45)
defines a contraction mapping on a compact and complete subspace of Rn . By the Contraction Mapping Theorem, it then follows that there exists a unique p(w) > 0 consistent
with p = g(p, w):
Proposition 5. For any w ∈ Rn++ and either (i) γ > β ≥ 1 or (ii) γ = β > 1 and ∣A(K i )β ∣ < 1,
where ∣A(K i )β ∣ denotes the matrix norm of A(K i )β , there is a unique p(w) > 0 that
satisfies (45) for all i ∈ N .
The proof of this statement is delegated to Appendix B.4. It immediately implies that,
for given wages, the prices of all final and intermediate varieties are uniquely determined.
Furthermore, the existence of a wage equilibrium reduces to finding a wage vector that
satisfies Lw = ld (w). The latter follows from the following result, which is an extension of
Theorem 2 in Alvarez and Lucas (2007):
Proposition 6. There is at least one w ∈ Rn++ that satisfies Lw = ld (w).
Also, the proof of that statement can be found in Appendix B.4. If we focus on the
model specification of Section 3.1, where we considered the case α = β = γ > 1, and
furthermore normalize the wage rates in all countries by the sum of “world incomes”, we
can even show that the equilibrium wage vector is:
1
Proposition 7. Suppose that α = β = γ > 1. For all w ∈ Rn++ , let w̄ = ∣w∣
w, where
d
∣w∣ = ∑i∈N wi is the vector norm of w. There is a unique w̄ that satisfies Lw̄ = l (w̄).
Again, see Appendix B.4 for the proof. There are a few noteworthy points about these
results:
First, different from the model of Section 3.1, the existence of a unique price equilibrium,
as established in Proposition 5, is entirely satisfied by γ > β. We, thus, do not require any
further maximum bound on the intermediate goods trade intensities, aij , even if labor cost
shares are endogenous, β > 1. The intuition is that, when γ > β intermediate inputs are
primarily substituted against each other and less against local labor, so that labor and
total factor productivity are not augmented towards infinity in a roundabout production
process.
Second, the existence of at least one wage equilibrium in Proposition 6 follows from
the fact that the labor market equation, Lw = ld (w), satisfies the formal requirements
of a Walrasian exchange economy. Yet, we cannot generally rule out the possibility of
multiple wage equilibria. This is only guaranteed in the case of α = β = γ > 1, as stated in
Proposition 7.24 Here, uniqueness follows from the fact that labor demand, ld (w), satisfies
the gross substitutes property, i.e., all off-diagonal cells j ≠ i of the Jacobian matrix dw ld
are strictly positive. We can, then, apply established results for Walrasian economies (e.g.,
Proposition 17.F.3 of Mas-Collel et al., 1995, p.613), whereby uniqueness is ensured only
up to a scale factor x > 0, so that every multiple xw̄ of w̄ is an equilibrium as well.
Alvarez and Lucas (2007, Theorem 3) also prove uniqueness for the case of α = γ > 1 and β = 1 (CobbDouglas technologies), which however requires additional parameter restrictions on the size of trade costs
and the (exogenous) labor cost share.
24
48
In contrast, we cannot establish the gross substitutes property for the general case.
Intuitively, because one country’s output enters another country’s production process, a
wage increase in the former raises the latter’s unit costs. A wage increase in one country
has, thus, a negative impact on another country’s total output and, in turn, its demand
for labor. In other words, in a fragmented production process, labor demands turn “complementary”.25
B.2
Counterfactual approach
The total differential. Expanding on the ideas of local comparative statics, we can
investigate small counterfactual variations around any one of the equilibrium points of
Proposition 6, regardless of the number of equilibria.
The following system of total differentials generalizes our counterfactual approach of
Section 4. It summarizes the effects of any small change to any number of cells in the Aor B-matrix on an equilibrium wage and price vector.
d ln(w) − Φ′ 1 = dA ln(ld ) + dB ln(ld )
dA ln(ld ) + dB ln(ld ) =
(46)
∞
d ln(λ) + [LW ]−1 Λ ∑ [Πi (I − Λ)]
h
(47)
h=0
× (dΠi (I − Λ) + Πi d(I − Λ) + dΠf Λ)(xf + xi )
d ln(pf ) = dB T ln(pf ) + (Πf )T (dAT ln(p) + dln(w) ln(p) d ln(w)) (48)
dln(pi ) = dAT ln(pi ) + (Πi )T (dAT ln(p) + dln(w) ln(p) d ln(w)) (49)
∞
h
dAT ln(p) = ∑ [(I − Λ)(Πi )T ] (I − Λ)dAT ln(pi )
h=0
∞
h
dln(w) ln(p) × d ln(w) = ∑ [(I − Λ)(Πi )T ] Λ d ln(w)
(50)
(51)
h=0
For the price total differentials in (48)-(51), we have made use of the properties (43) for
CES price indexes. For the labor market total differential in (46), on the other hand, we
have made use of Lemma 1 Property (1.) to determine the direct effects, as shown in
This complementarity is switched off in the special case of α = β = γ. The reason is that any country
1’s access to the ‘final demand’ of country m can be written as
25
f
i
λ1 × π12
(1 − λ2 ) × ... × π(m−1)m
lm wm =
(p(m−1) )1−γ b(m−1)m
κγ1 w11−γ (p1 )1−γ a12 (pi2 )1−γ
×
×
...
×
lm wm
i
(p1 )1−γ
(p2 )1−γ (p2 )1−γ
(pfm )1−γ
so that the scale effects (Hamermesh, 1993) on (pi )1−γ , which feature in the numerators of the expeni
ditures share terms πi(i+1)
, for i = 1, ..., (m − 1), cancel against the substitution effects on (pi )1−γ in the
denominators of the cost share terms λi and 1 − λi .
49
i
(47). In particular, that lemma allows us to determine the total differential of ∑∞
h=0 [Π (I −
h
Λ)] . Based on the properties (43) for CES price indexes, the direct effects can be further
decomposed into
d ln(λ) = (β − 1)dAT ln(P )1
dΠi = dA Πi − (γ − 1)(dAT ln(P ) × Πi − Πi × dAT ln(P i ))
(52)
d(I − Λ) = − (β − 1)(dAT ln(P i ) − dAT ln(P ))(I − Λ)
dΠf = dB Πf − (α − 1)(dAT ln(P ) × Πf − Πf × (dAT ln(P f ) + dB T ln(P f )))
Here, dAT ln(P ), dAT ln(P i ), dAT ln(P f ), and dB T ln(P f ), denote the diagonal matrices
corresponding to the vectors of price differentials in (48)-(50). More specifically, these
matrices only contain the direct effects of a variation to the A- and B-matrix. The indirect
wage effects of (51) are delegated to matrix Φ′ in (46), which generalizes the matrix of
general equilibrium effects (18) of Section 4 and is characterized in the following.
The Jacobian matrix Φ′ . The full matrix Φ′ captures all the indirect, wage-induced
effects on labor demand ld . Making repeated use of the alternative labor market clearing
condition Lw = ΛΠf Lw + ΛΠi (I − Λ)[Λ]−1 Lw, the matrix Φ′ can be further decomposed
into Φ′ = Φ′′ + Φ′′′ , where
∞
h
Φ′′ = [LW ]−1 Λ ∑ [Πi (I − Λ)] × ((γ − 1)Πi dln(w) ln(P i ) (I − Λ)[Λ]−1 LW
h=0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
import competition effect on
intermediate input sales
(β − 1)Πi ( dln(w) ln(P ) −
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
scale effect on
input cost share
)(I − Λ)[Λ]−1 LW
dln(w) ln(P i )
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
(53)
substitution effect on
input cost share
∞
h
+ (α − 1)Πf dln(w) ln(P f ) LW ) + [LW ]−1 Λ ∑ [Πi (I − Λ)] Πf LW d ln(W )
h=0
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
import competition effect on
spendable income effect
final sales
Φ′′′ = −(β − 1)(
d ln(W )
´¹¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¶
scale effect on
labor cost share
−
dln(w) ln(P )
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶
∞
) − [LW ]−1 Λ ∑ [Πi (I − Λ)]
h=0
substitution effect on
labor cost share
× ((γ − 1) dln(w) ln(P ) Πi (I − Λ)[Λ]−1 LW + (α − 1) dln(w) ln(P ) Πf LW )
´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶
scale effect on intermediate input sales
50
scale effect on final sales
h
and where d ln(W ) denotes the diagonal matrix (d ln(wi ))i∈N and
∞
h
dln(w) ln(P ) = diag{ ∑ [(I − Λ)(Πi )T ] Λ d ln(w)}
h=0
dln(w) ln(P ) = (Π ) dln(w) ln(P )
dln(w) ln(P f ) = (Πf )T dln(w) ln(P )
i
i T
(54)
The foreign trade multiplier [{I − Φ}−i ]−1 . As shown in Appendix B.4, labor market
equation (42) satisfies Walras’ Law and homogeneity of degree zero. Thus, we are free to
set all entries in row i∗ and column i∗ of the Jacobian Φ′ to zero for any i∗ ∈ N , and still
able capture the effects on the labor demand system properly.
However, unlike in the simpler model of Section 3.1, the problem arises that multiple
d ln(w) might be consistent with the labor market total differential (46). This is, because
∗
(i) d ln(W ) enters Φ′′′ pre- and post-multiplied by some full matrices, and
(ii) it is not clear that a matrix inverse to {I − Φ′ [d ln(W )]−1 }−i exists, as the labor
demand in our general model does not satisfy the gross substitutes property, which ensured
invertibility in Section 3.1.
∗
Both these issues are solved, i.e., d ln(w) − Φ′ 1 = dA ln(ld ) + dB ln(ld ) is consistent with
a unique d ln(w) (up to a scale factor x > 0), under the following additional assumptions:
(i) α = γ, and
−i∗
(ii) for any i∗ ∈ N , all characteristic roots ρ of {Φ′ [d ln(W )]−1 }
satisfy ∣ρ∣ < 1.
Assumption (i) is common in models of input-output trade (e.g., Krugman and Venables, 1995; Eaton and Kortum, 2002; Ossa, 2015). Its implication is that Φ′′′ 1 simplifies
to
Φ′′′ 1 = − (β − 1)(d ln(W ) − dln(w) ln(P ))1
∞
−1
h
− (γ − 1)[LW ] Λ ∑ [Πi (I − Λ)] [Λ]−1 LW dln(w) ln(P )1
h=0
Hence, we can write Φ′ 1 = Φd ln(w), where matrix Φ is independent of d ln(W ). This also
∗
means that if an inverse to {I − Φ}−i exists,
d ln(w) = {[{I − Φ}−i ]−1 }
∗
+i∗
(dA ln(ld ) + dB ln(ld ))
produces a unique prediction for d ln(w) (up to a scale factor x > 0). Invertibility is,
however, satisfied by Assumption (ii). More specifically, it follows from established results
51
in spectral theory (e.g., Debreu and Herstein, 1953, Theorem V) that [{I − Φ}−i ]−1 can
be written as a converging Neuman series and, hence, the matrix receives the meaning of
a foreign trade multiplier.
∗
Empirical implementation. The following counterfactual exercises can be brought to
the data in the exact same way as our exercises for the model of Section 3.1. We need no
further estimation nor additional information on prices, wages, trade costs or productivities. In fact, all the equations shown above are already expressed in terms of observable
macroeconomic constructs (trade shares, labor cost shares, total industrial GDP and production) and the model’s elasticity parameters, for which estimates are readily available in
the literature.
B.3
Robustness of counterfactual results
Worldwide increase in production fragmentation. Consider a worldwide trade cost
reduction on all intermediate goods imports and exports, so that dB Πf = 0, dB T ln(pf ) = 0,
and
x
x
T
(Π̃i ) 1 = −
(P i )γ−1 ÃT p1−γ
γ−1
γ−1
= xΠ̃i = xP 1−γ Ã(P i )γ−1
dAT ln(pi ) = −
dA Πi
(55)
(56)
where for Z ∈ {A, AT }, Z̃ denotes the matrix with z̃ij = zij for i ≠ j and z̃ii = 0.
Contrary to Proposition 2, the per capita income changes are no longer solely dependent
on a country’s exposure through trade intermediation. On the price side, consumers still
gain only indirectly via their suppliers’ improved access to intermediate inputs (apply (55)
to (48)). Yet, on the demand side, the prospects of workers partially depend on the initial
access of their employers to their direct upstream suppliers. Specifically, the “neighborhood”
component in the labor market total differential (46) is given by26
d ln(λ) − [LW ]−1 Λ((γ − 1)dAT ln(P ) Πi (I − Λ) + (α − 1)dAT ln(P ) Πf Λ)(xf + xi )
= (β − 1)(I − Λ)dAT ln(pi ) − (γ − 1)(I − Λ)dAT ln(pi )
26
(57)
To establish the identity in (57), note that
d ln(λ) − [LW ]−1 Λ((γ − 1)dAT ln(P ) Πi (I − Λ) + (α − 1)dAT ln(P ) Πf Λ)(xf + xi )
= d ln(λ) − (γ − 1)dAT ln(P )[LW ]−1 Λ(xf + xi )
= d ln(λ) − (γ − 1)dAT ln(p)
where we made use of the assumption α = γ and the equilibrium condition Lw = Λ(xf + xi ) = ΛΠf Λ(xf +
xi ) + ΛΠi (I − Λ)(xf + xi ). The “neighborhood” component inside d ln(λ) follows, finally, from (50) and
is given by (I − Λ)dAT ln(pi ).
52
.6
median share
trade intermediation in d(ln u)/dA
.7
.8
.9
1
Figure 5: Trade intermediation vs. direct neighborhood exposure
1
2
3
4
5
6
β
α=γ=2
α=γ=7
7
α=γ=3
α = γ = 11
8
9
10
11
α=γ=5
Notes: For each combination of α = γ > 1 and γ ≥ β ≥ 1, the figure shows the median share
(among the 100 countries in our 2005 dataset) of trade intermediation in total network exposure in
determining the welfare effects of a worldwide 1% trade cost reduction on all international shipments
of intermediate goods.
Obviously, Proposition 2 is only valid, when β = γ. In any other case, the wage effect
additionally depends upon the initial exposure of a country’s producers to their input
suppliers’ value added (apply (55) to (57) and recall the definition of p1−γ ).
Nevertheless, Figure 5 shows that for a wide range of parameter specifications, the average country’s welfare gains associated with a 1% global trade cost reduction on intermediate
goods are for more than 80% accounted for by a country’s exposure to intermediated supply
and demand.
In contrast, the welfare effects due to a similar trade cost reduction on all final goods
shipments are, just as in the model of Section 3.1, proportional to the initial level of welfare
in a nation, i.e., the absolute size of the effect is related to a country’s initial exposure to
itself, to the countries in its neighborhood, and its indirect trade partners. For dB = xB,
for example, we get d ln(u) = (x/(α − 1))1 > 0.
A unilateral export cost reduction. Consider a trade cost reduction on the shipments
of exporter i to importer j, so that
x i
π Iji 1
γ − 1 ij
y
dB T ln(pf ) = −
πijf Iji 1
α−1
dAT ln(pi ) = −
53
and dA Πi = xπiji Iij
and dB Πf = yπijf Iij
Suppose, in a first instance, a world without production linkages, i.e. Λ = I and Πi = 0 in
the system of total differentials (46)-(53). The labor market differential simplifies to
∞
+i
d ln(w) = { ∑ [{Φ}−i ]h } [LW ]−1 (dB Πf Lw + (α − 1)Πf LW dB T ln(pf ))
h=0
Fixing dwi = 0 for exporter i and noting that Φ has all its entries positive in this case (see
(53)), we immediately arrive at d ln(wk ) < 0 for any k ≠ i.
Consider, now, the same cost reduction but without the constraints on Λ and Πi . Then,
contrary to above, the wage rate in any country k ≠ i might increase. Indeed, if we consider
an exporter i that adds little value of its own to its output, i.e., limκi →0 [λi ] = 0, the direct
effect on i’s labor market total differential becomes dij lim[ln(lid )] = d lim[ln(λi )] ≤ 0 in
the limit. On the other hand, workers from another country k might still benefit from
the export cost reduction, because of country i’s increased demand for intermediate inputs
in the assembly of the goods traded along the link ij. In fact, in the case of CobbDouglas technologies (β = 1), we find that dij lim[ln(lid )] = 0 and that all entries in row i
∗
of lim[{I − Φ}−i ]−1 are zero. It, thus, follows dij lim[ln(wi )] = 0. Moreover, since we are
still free to keep the wage rate in any one country l constant at dij lim[wl ] = 0, there is at
least one other country l ∈ N /{i} with the same wage effect. And due to the additional
demand for inputs in country i, there is potentially another country k with a larger labor
demand effect, i.e., dij lim[wk ] ≥ dij lim[wi ]. In other words, for the case of β = 1, we can
re-establish Proposition 3 on the existence of a “pure intermediary”.
We cannot establish the same result for the intermediate cases 1 < β < α = γ. The
reason is that, even in the limit of κi → 0, exporter i’s demand for local labor still depends
on the effects of the export cost reduction on the wage rates of other countries (see, in
particular, the substitution effect on the labor cost share in the Jacobian matrix (53)).
However, as our quantitative predictions shown in the left panel of Figure 6 make clear,
there is a considerable number of exporters in the 2005 trade network that qualifies as
a pure intermediary. We find most of them, when assuming a Cobb-Douglas technology
(α = γ > β = 1), and the least for the specification of Section 3.1 (α = β = γ > 1). In the
light of our theoretical results for these cases, this begs for some additional explanation.
The question is how a demand impulse in a country k that supplies intermediate inputs to
exporter i affects that country’s demand for local labor. With Cobb-Douglas technologies,
producers from k will maintain a constant labor cost share. Thus, a positive demand
impulse will be fully absorbed by a wage increase. With flexible labor cost shares (β > 1),
in contrast, this wage increase will be partially mitigated by firms offshoring parts of their
production. Hence, wages rise most in the Cobb-Douglas case.
The right panel of Figure 6, in contrast, shows that —while generating most instances
of a positive wage externality— the average wage externality on all nations k ≠ i is most
negative in the Cobb-Douglas case. The intuition is the following: with Cobb-Douglas
technologies, only the sales shares Πi and Πf remain as an adjustment margin in the labor
market total differential (46). Yet, since the sum of all exporters’ market shares in the
importer country j must, by definition, always stay constant, exporter i’s gains will be
54
0
avg. wage externality (%) of ij exp.cost ↓
−.005 −.004 −.003 −.002 −.001
0
% ij exp.cost ↓: >=1 wage externality +
5
10
15
Figure 6: Wage externalities of an export cost reduction
1
2
3
4
α=γ=2
α=γ=7
5
6
β
7
α=γ=3
α = γ = 11
8
9
10
11
α=γ=5
1
2
3
4
5
α=γ=2
α=γ=7
6
β
7
α=γ=3
α = γ = 11
8
9
10
11
α=γ=5
Notes: For each combination of α = γ > 1 and γ ≥ β ≥ 1, the figure plots: [right panel] the average
wage externality (∑k≠i dwk )/(n − 1), resulting from a 1% export intensity increase along each of
the 9,900 exporter-importer trade links ij in our dataset, and [left panel] the share of these 9,900
unilateral export cost reductions that generate at least 1 positive wage externality in country k ≠ i.
offset by at least one other country’s losses. A flexible labor cost share introduces an
interesting additional adjustment margin: the import cost reduction in country j pushes
the local labor cost share down which is, of course, bad news for local laborers. But their
substitution by foreign intermediate inputs also means that production elsewhere rises. In
an integrated GPN, this output increase feeds back onto all other countries in terms of
an additional demand for their intermediate products. Indeed, as suggested by the right
panel of Figure 6, the additional labor demand created in this way in third countries k ≠ i
increases in the elasticity of the labor cost share.
Key intermediaries. The analysis so far suggested that, just as in the simpler model
from the main text, a country’s exposure to trade intermediation is also a crucial welfare
determinant in the general setting considered here. This raises the question about the
identity of the most valuable trade partners in this regard, i.e., the key intermediaries.
Independent of the precise parameter constellation, they can be found by substituting
1
1
dA Πi = −x( Iii Πi + Πi Iii − πiii Iii )
2
2
1
T
(dA Πi ) 1
dAT ln(pi ) = −
γ−1
(58)
into the total differentials of (46)-(53). As these total differentials are all additively decomposable, it immediate follows that Proposition 4 from the main text carries over. We can,
therefore, reconstruct a country’s welfare gains from a worldwide increase in production
fragmentation, i.e., the 1st counterfactual, by adding up the incremental welfare losses from
isolating one nation after the other from the network.
55
B.4
Equilibrium in the general model
Proof of Proposition 5. We verify here that the the log-linearized vector function
ln g(ln(p)), with row entries
ln gi (ln(p)) =
1−β
1
ln (κβi wi1−β + (κii )β ( ∑ exp [(1 − γ) ln(pj )]aji ) 1−γ )
1−β
j∈N
(59)
is endomorphic on the compact and complete space P ⊂ Rn , i.e., ln g ∶ P → P, and a
contraction mapping.
Existence of a unique ln(p) ∈ P such that ln(p) = ln g(ln(p)) and, thus, a unique p > 0
such that p = g(p), then, follows from the Contraction Mapping Theorem.
First, note that if β = γ and if the norm condition ∣(K i )γ AT ∣ < 1 is satisfied, (59) is
solved for by
ln(p) =
1
ln ([I − (K i )γ AT ]−1 K γ w1−γ )
1−γ
Hence, ln(p) is uniquely defined.
To confirm the endomorphism in all the other cases γ > β ≥ 1, note that (59) is
monotonically declining in aji . Hence, a conservative upper bound for gi (p) is given by
β
1−β
ḡi = κi
wi
such that gi (p) ≤ ḡi ≡ p̄i . On the other hand, when we define ā = max{aij ∣ij ∈ N × N } > 0,
a conservative lower bound for gi (p) is given by
g 1−γ
i
=
(κβi wi1−β
1−β
+ (κii )β ā 1−γ (
∑
1−β
1−γ
p1−γ
)
j )
1−γ
1−β
(60)
j∈N
1−γ
such that gi1−γ (p) ≤ g 1−γ
and summing up, we get
i . Inserting ā into all other g i
∑
i∈N
g 1−γ
i
= h( ∑
i∈N
p1−γ
i
, ā) ≡ ∑
(κβi wi1−β
i∈N
1−β
+ (κii )β ā 1−γ (
∑
1−β
1−γ
)
p1−γ
i )
1−γ
1−β
(61)
i∈N
It can be verified that the function h ∶ R++ × R++ → R++ is increasing and concave in its
27 By
first argument. Hence, h has a unique positive fixed point x̄ ≡ ∑i∈N g 1−γ
= ∑i∈N p1−γ
i
i .
Function h might also have a fixed point in the positive domain, when γ < β, so that h′′ > 0. Just as
in the case γ = β, this requires an additional upper bound on the trade intensities aij and the productivity
parameters κii . If such a bound is given, however, the following arguments also carry over to a model with
γ < β.
27
56
(60), we then find a pi ≡ g i (x̄), which by (61) satisfies ∑i∈N pi 1−γ = x̄. Moreover, since (60)
and (61) are monotonically increasing in aij , it follows for ā > 0 and all i ∈ N that p̄i > pi .
It remains to be seen that, if we define the compact and complete space
P = [ln(p1 ) , ln(p̄1 )] × [ln(p2 ) , ln(p̄2 )] × ... × [ln(pn ) , ln(p̄n )]
then, because ln(gi ) is monotonically increasing in pi and because ∑i∈N pi 1−γ = x̄, it follows
that ln(g) maps P into itself.
To establish the contraction property, note that for ln(p̄) = max{ln(p̄i )∣i ∈ N } and
ln(p) = min{ln(pi )∣i ∈ N }, it holds
ln(p) + ( ln(p̄) − ln(p))1 ≥ ln(p′ )
(62)
for all ln(p), ln(p′ ) ∈ P. Thus, let us write x ≡ ( ln(p̄) − ln(p)). It then follows
ln g( ln(p′ )) − ln g( ln(p)) ≤ ln g( ln(p) + x1) − ln g( ln(p))
= dln(p) ln g( ln(p) + xz) x1
= (I − Λ)(Πi )T x1 = x(I − Λ)1
The inequality in the first line follows from (61). In the second line, we have applied the
Mean Value Theorem for some z ∈ (0, 1)n and, in the third line, we have made use of
the functional properties (43) for CES price indexes. We finally receive the contraction
property from the fact that I − Λ ≤ (1 − λ)I < I, where the modulus (1 − λ) < 1 is defined
by the minimum labor cost share compatible with the compact price set P. ∎
Proof of Proposition 6. To prove existence of at least one wage equilibrium, we verify
that there is at least one w ∈ Rn++ such that
∞
h
0 = W z(w) ≡ Λ(xf + xi ) − Lw = Λ ∑ [Πi (I − Λ)] Πf Lw
(63)
h=0
satisfies the following properties: for all rows i of z(w) and all vectors w
i) zi (w) is continuous,
ii) zi (w) is homogeneous of degree zero,
iii) ∑i∈N wi zi (w) = 0 for all w ∈ Rn++ (Walras’ Law),
iv) for k = maxj lj > 0, zi (w) > −k for all w ∈ Rn++ and
0
v) if w → w0 , where w−i
≠ 0 and wi0 = 0 for some i, then maxj zj (w) → ∞.
Existence then follows from Proposition 17.C.1 of Mas-Collel et al. (1995, p.585), which is
an application of Kakutani’s Fixed-Point Theorem.
57
i) The continuity of zi (w) follows immediately from the fact that all price indexes p(w),
pf (w), and pi (w), are continuous in w.
ii) Note, first, that p(w), pf (w), and pi (w) are all homogeneous of degree one with
regard to a proportional wage change. To see this, start from the identities
−1
λ = [I − (I − Λ)(Πi )T ] × [I − (I − Λ)(Πi )T ] λ = [I − (I − Λ)(Πi )T ]1
−1
so that [I −(I −Λ)(Πi )T ] λ = 1. Applying the identity to the price total differential in (50),
i T h
it immediately follows that dx1 ln(p) = x ∑∞
h=0 [(I − Λ)(Π ) ] λ = x1. Hence, by Euler’s
Theorem, p(w) is homogeneous of degree one. The homogeneity of the price indexes pf (w)
and pi (w), then, follows immediately from their definition.
Substituting dx1 ln(p) = dx1 ln(pf ) = dx1 ln(pi ) = x1, in turn, into the Jacobian matrix
′
Φ in (53), we can see that for dln(w) = x1 all effects but the spendable income effect cancel
h
i
f
out. Yet, because labor market clearing implies that [LW ]−1 Λ ∑∞
h=0 [Π (I −Λ)] Π LW x1 =
′
x1, we can immediately conclude that for dln(w) = x1 it is Φ 1 = x1. By Euler’s theorem,
ld (w) − Lw is, thus, homogeneous of degree zero.
iii) To verify Walras’ Law, note that (63) is nothing but the requirement (4) of balanced
trade in each country. On the world market, this implies
∑ wi zi (w) = ∑ ( ∑ xij (w) + ∑ xiij (w) − ∑ xiji (w) − li wi )
f
i∈N
i∈N
j∈N
j∈N
j∈N
so that ∑i∈N wi zi (w) = 0 follows from the fact that consumer expenditures satisfy ∑j∈N xfji (w) =
li wi in each country.
iv) A lower bound on zi (w) is implied for by zi (w) > −li for all w ∈ Rn++ . Thus, it holds
zi (w) > − maxj∈N lj for all i ∈ N .
0
(v) Suppose that w → w0 , where w−i
≠ 0 and wi0 = 0. For any country i and w ∈ Rn++ it
holds
zi (w) ≥
max {wi−1
j∈N
λi (w)
xfij (w)}
− max lk = max {
k∈N
j∈N
κβi p1−α
bij lj wj
i
wiβ p1−β
(pfj )1−α
i
} − max lk
k∈N
(64)
0
Because pj and pfj are strictly positive for w−i
≠ 0, it follows that the denominator of the
right-hand term in (64) approaches zero in the limit as wi goes to zero. This implies that
limw→w0 zi (w) → ∞ and, therefore, —by Proposition 17.C.1 of Mas-Collel et al. (1995, p.
585)— we have established existence of at least one equilibrium. ∎
Proof of Proposition 7. For the model of Section 3.1 with α = β = γ > 1, we can
additionally prove uniqueness of the wage equilibrium of Proposition 6 (up to a scale
58
factor x > 0). This follows from the fact that zi (w) defined in (63) satisfies the gross
substitutes property, that is for all j ≠ i and w ∈ Rn++ it holds ∂zi /∂wj > 0. To see this, note
that the Jacobian matrix Φ′ in (53) simplifies to
∞
h
Φ′ = −(γ − 1) d ln(W ) + [LW ]−1 Λ ∑ [Πi (I − Λ)] Πf LW (d ln(W ) + (γ − 1) dln(w) ln(P f ))
h=0
Because dln(w) ln(P f ) > 0, it follows that Φ′ has all its off-diagonal elements positive.
Uniqueness, then, follows from Proposition 17.F.3 in Mas-Collel et al. (1995, p.613), where
additionally the homogeneity of degree zero of zi (w) is exploited. ∎
59