Technology, Networks and Trade (TNT) - The Ricardian Network ⇤ Glenn Magerman KU Leuven December 1, 2015 Latest version available here Abstract This paper proposes a simple model of strategic network formation between countries in inter- national trade. I present a dynamic version of a Ricardian model, in which perfectly competitive firms engage in frictionless international trade. In this setting, countries open up sequentially to trade, by adding or deleting one link at a time. I quantify the following results, using a simple example of three countries and two goods: I show that efficient networks arise from myopic network formation, and that stable equilibria exist. Goods are then optimally allocated across countries in the network. However, multiple equilibria exist. Importantly, there are negative network externalities: third countries not involved in a new trade link are strictly worse off than before, and these externalities remain in efficient networks. I then propose a generalization of the model towards many countries and many goods. Keywords: Externalities, Network Formation, Neoclassical Models of Trade. JEL Classification: D62, D85, F11. ⇤ I would like to thank Matthew O. Jackson and Kyle Bagwell at the Stanford Dept. of Economics for many helpful comments and discussions while setting up this project. I also thank Leonie Baumann, Andreas Bjerre-Nielssen and Theodoros Rapanos and participants at the Networks Discussion Group at Stanford for fruitful discussions. I gratefully acknowledge financial support from the Fulbright Foundation Belgium- Luxembourg, the Otlet-La Fontaine scholarship granted by Yves Moureau at the Department of Engineering at the University of Leuven, and the Junior Mobility grant by the University of Leuven. Part of this research is conducted while visiting the Stanford Dept. of Economics. E-mail: [email protected]. 1
Transcript
Technology, Networks and Trade (TNT) - The Ricardian Network
⇤
Glenn Magerman
KU Leuven
December 1, 2015
Latest version available here
Abstract
This paper proposes a simple model of strategic network formation between countries in inter-national trade. I present a dynamic version of a Ricardian model, in which perfectly competitivefirms engage in frictionless international trade. In this setting, countries open up sequentially totrade, by adding or deleting one link at a time. I quantify the following results, using a simpleexample of three countries and two goods: I show that efficient networks arise from myopicnetwork formation, and that stable equilibria exist. Goods are then optimally allocated acrosscountries in the network. However, multiple equilibria exist. Importantly, there are negativenetwork externalities: third countries not involved in a new trade link are strictly worse off thanbefore, and these externalities remain in efficient networks. I then propose a generalization ofthe model towards many countries and many goods.
⇤I would like to thank Matthew O. Jackson and Kyle Bagwell at the Stanford Dept. of Economics formany helpful comments and discussions while setting up this project. I also thank Leonie Baumann, AndreasBjerre-Nielssen and Theodoros Rapanos and participants at the Networks Discussion Group at Stanford forfruitful discussions. I gratefully acknowledge financial support from the Fulbright Foundation Belgium-Luxembourg, the Otlet-La Fontaine scholarship granted by Yves Moureau at the Department of Engineeringat the University of Leuven, and the Junior Mobility grant by the University of Leuven. Part of this researchis conducted while visiting the Stanford Dept. of Economics. E-mail: [email protected].
The entry of countries into the world market can have significant effects on the trade patternsbetween those countries. New trade relationships are formed, while existing trade patterns change.Classical models of international trade compare only two extreme states of the world: autarky andfull trade equilibrium. In reality however, countries sequentially open up to trade. Some examplesinclude the fall of the Berlin wall, the EU opening up to new Member States, the emergence of Chinaas an economic power etc. These examples constitute fairly exogenous shocks that reshuffle tradingpatterns through the driver of comparative advantage. If global trade indeed moves sequentiallyfrom equilibrium to equilibrium, this has an impact on goods prices, factor prices, factor allocation,production levels and welfare.
The main question I ask in this paper is how the entry of countries into the world marketimpacts trade flows between those countries. To attack this question, I develop a simple modelof Ricardian trade with multiple countries and multiple goods, embedded in a strategic networkformation game. In this setting, countries differ in technologies they posses to produce goods,and perfectly competitive firms engage in frictionless international trade. I derive a global marketclearing condition that allows me to map this trade model to the network formation game. In eachperiod, one trade link between two countries is either added or deleted. This process continuesuntil a stable network is attained, in which no country has an incentive to deviate. I show thatstable networks arise, starting either from autarky or complete networks. Moreover, these networksare also efficient, ensuring that goods are optimally allocated over the network of countries thatparticipate in international trade. Importantly however, this game reveals the existence of negativenetwork externalities: in any efficient network, countries would like to see the link between othercountries severed in order to be better off. In a simple example with three countries and two goods,I show that these externalities can largely offset the welfare gains from new trade links.
The main contributions can be summarized as follows. First, autarky is not stable nor efficient,and is never attained from network formation in this paper. Second, I show that there exists animproving path from autarky to global free trade, which maximizes global efficiency. Third, efficientnetworks are pairwise stable, providing optimal allocation of goods across the network, but multipleequilibria exist. Finally, negative externalities exist when going to free trade: a third country thatis not involved in a new trade connection is strictly worse off than before.
In Section 2, I define the network of international trade, in which countries represent nodesand trade flows represent links. Nodes and links in any given trade network are characterizedby the general equilibrium quantities of a simple Ricardian model of international trade. Generalequilibrium outcomes then result in stable networks if no new trade links are formed or severed.To determine stable equilibria, I use pairwise stability as a network equilibrium concept (Jacksonand Wolinsky (1996)). This concept is very natural in the application of international trade: twocountries trade if both are weakly better off, one country can unilaterally break the trade link if itis worse off than in the previous equilibrium. A pairwise stable network is then a network whereno countries have an incentive to deviate from the current state. The dynamics of the game are
2
captured by improving paths, which identify a sequence of networks that differ one link each timeuntil the network becomes stable. The question then becomes whether stable networks are alsoefficient, or whether sub-optimal stable networks exist.
Next, I define a simple Ricardian model of international trade in Section 3. There are n coun-tries and m goods in the world. Countries differ in technologies and labor supply, and firms insidecountries compete in perfect competition. Labor is the only factor of production, generating hori-zontally differentiated goods. I derive a global market clearing condition that allows me to map themodel of international trade to the network formation game.
As the model with n countries and m goods is generally impossible to solve, I show the mostimportant mechanics with a numerical example in Section 4. I characterize all possible outcomesin a simple setting with three countries and two goods. I show that myopic strategic networkformation in this setting leads to stable networks that are efficient and Pareto optimal. However,network formation leads to negative externalities for countries not involved in a new trade link.In this simple numerical example, the gains from trade for two new trade partners can be largelyoffset by a worse outcome for a third country. Moreover, multiple equilibria exist from this networkformation process. Some equilibria depend on costless re-exporting of goods in order to achieveoptimal allocation. These multiple equilibria exhibit the same efficient production and consumptionbundles, but exact trade flows are possibly undetermined.
In Section 5, I propose some general results on the characterization of the network formationgame. These proposals are only conjectures, further analysis is needed to prove general propositions.However, some results are intuitive. First of all, autarky is not pairwise stable nor efficient, andwill never be the outcome of the network formation game. Second, any connected network ispairwise stable and efficient. This generalizes the Arrow-Debreu framework of anonymous marketsto clearly defined network structures. This reduces the number of possible stable equilibria in thenetwork. Third, myopic strategic network formation leads to stable and efficient outcomes. This isan important result, as it implies that individual optimization leads to socially desirable outcomes.There are multiple equilibria, and the system exhibits hysteresis: starting from autarky towardsfree trade leads to other stable networks than starting from a complete network. The equilibriumoutcomes are identical however, and depend on costless re-exports. Finally, and most importantly,network formation leads to negative externalities for countries not involved in the new trade link.This has important policy implications, as the bilateral gains from a new trade agreement can belargely offset by losses incurred by countries not involved in this new agreement. One thus needs tomeasure the global impact of a new trade deal, not only the bilateral gains.
Section 6 proposes a more general framework to evaluate the impact of changes in the networkstructure of trade on global and country welfare. I extend the model of Eaton and Kortum (2002)to include network formation and dynamics, and show that all changes in network structure channelthrough relative wages. A change in the network structure of trade leads to changes in factor prices,which in turn lead to shifts in trade flows. This framework can then serve as a basis for welfareanalysis through the use of comparative statics of the network formation game. Section 7 concludes.
3
This paper mainly draws from the literature on international trade theory and network theory.First, there is a very long tradition of models of international trade based on Ricardo (1818). Inthese models, differences in technology across countries serve as a source of comparative advantagefor countries to specialize and trade, even in the absence of increasing returns to scale (Dornbusch,Fischer and Samuelson (1977), McKenzie (1954), Jones (1961), Eaton and Kortum (2002)). In allthese models, firms compete in perfect competition; Bernard, Eaton, Jensen and Kortum (2003)extend this setting to Bertrand competition with variable markups.
Second, there are few papers on network formation in international trade. There is a strand ofpapers that models the formation of free trade agreements and regional trade agreements. Freund(2000) proposes a model with three identical countries, in which firms compete in Cournot with sunkcosts of investment. She shows that goods flow disproportionally more between original membersof a regional trade agreement after opening up this agreement to a third country. Furthermore,welfare gains are larger for the original members of the agreement. Goyal and Joshi (2006) presenta model with many countries and homogeneous goods in Cournot competition. The main conclusionis that with symmetric countries, a free trade equilibrium is pairwise stable and unique. Furthercharacterization of equilibrium is impossible. Furusawa and Konishi (2007) propose a similar model,in which firms compete in monopolistic competition. The findings are identical to those in Goyaland Joshi (2006). Deltas, Desmet and Facchini (2012) model trade agreements between threecountries in a Heckscher-Ohlin setting. Here, goods can flow proportionally more or less betweenoriginal members of an agreement, depending on the exact network structure that is imposed. Whilemost of these papers exogenously enforce a trade agreement between countries, I complement thisliterature by developing a full-fledged trade model that considers endogenous network formation.Furthermore, this framework provides a more detailed characterization of equilibrium.
Third, there is a broad class of models on trade in networks, independent of internationaltrade models (see Manea (2015) for an excellent survey). These models consist of exogenouslydefined buyers, sellers and intermediaries that organize transactions between these buyers and sellers.Buyers only have valuation for the good that is sold and sellers only have valuation for cash. Thefocus in these models is how agents can exploit their position in the network in terms of extractingsurplus, and what kinds of structures of networks are efficient. In this literature, two papers arerelated to my setup. Kakade, Kearns and Ortiz (2004) look at different prices in networks in a“graphical” Arrow-Debreu setting. An endowment economy is set in a fixed network, and agents thatare connected in the network can trade without costs whenever a link is present. Multiple equilibriaexist, and algorithmic simulations can then provide approximate solutions to the system. Even-Dar, Kearns and Suri (2007) model a network matching game for bipartite exchange economies,in which buyers and sellers are again exogenously identified. This last paper fully characterizesNash equilibria for a bipartite network game. I expand on this literature by proposing a productioneconomy, in which whether a node buys or sells a good depends on the underlying Ricardian modelof trade. Furthermore I allow for endogenous network formation.
Tangent to this paper are the models of network formation in Jackson and Rogers (2007) and
4
Chaney (2014). In these papers, firms can only trade over the network they posses, and agents choosetheir connections through a mix of random encounters and existing connections. While these modelsare more flexible in terms of accessible network structures, I provide a model of strategic networkformation that additionally allows for welfare analysis.
2 Definitions
2.1 The network of international trade
There are n countries in the world; each country is labeled as i 2 N = {1, ..., n}. Let ij 2 g definea link, or equivalently an active trade relationship between i and j. In this paper, links are alwaysreciprocated as countries exchange goods. As ij = ji, links are then always undirected. Let g 2 G
define a network, in which nodes represent countries and links represent trade flows between thosecountries. G is then the set of all possible networks on N . A component g00 ✓ g is the distinctmaximal connected subgraph of a network g.
Let t = {0, ..., T} identify discrete time, and let k 2 K = {1, ...,m} define the set of goodsjointly produced by all i. The tuple ✓
it
2 ⇥
t
= {y,x,p,X,Uit
} then defines the set of quantities thatcharacterizes each node i at time t, where bold characters denote vectors of length k: y denotesquantities produced for each good k at time t, x denotes the consumption bundle, p denotes relativeprices, X denotes net export quantities, and U
it
denotes node i’s utility from the consumption of kgoods. Quantities in ✓
it
are derived from the general equilibrium outcomes of a Ricardian model ofinternational trade, to be established below. ⇥
t
= {✓1t, ..., ✓nt} characterizes all general equilibriumquantities in the network at time t, and g
t
2 G = {N,⇥t
} is then a fully characterized network g
at time t on the set of all possible networks G.I analyze fixed networks, defined by any g
t
2 G. I also analyze a simple strategic networkformation game with the following rules: in any g, two nodes i and j are allowed to form a linkij /2 g if both i and j agree. Alternatively, any i or j can unilaterally sever an existing link ij 2 g.Let g + ij be the network resulting from adding link ij to the existing network g and let g � ij bethe network resulting from deleting link ij from the existing network g. If g0 = g+ ij or g0 = g� ij,I say that g and g0 are adjacent. In this paper, I use g0 6= g to denote a network adjacent to g, andg00 6= g for any network in G.
2.2 Stability, efficiency and externalities
I use the concept of pairwise stability (Jackson and Wolinsky (1996)) to determine if general equi-librium outcomes ⇥
t
lead to stable networks gt
, such that no i has an incentive to deviate from thisoutcome. A network g is then pairwise stable with respect to U
i
if none of the following conditionsare fulfilled:
(i) g0 = g � ij for some ij 2 g such that Ui
(g0) > Ui
(g), or(ii) g0 = g+ ij for some ij /2 g such that U
i
(g0) � Ui
(g) and Uj
(g0) � Uj
(g) with at least one ofboth inequalities being strict.
5
If any of these conditions does hold, I say that g is defeated by g0. Then, an improving pathfrom g to g0 exists. An improving path from a network g to g0 is a finite sequence of networksg0, ..., gt, ...gT , such that each network g
t<T
is adjacent and defeated by the subsequent networkgt+1. A network g is then equivalently pairwise stable if and only if there is no improving path
emanating from g.
As a first measure of efficiency, I use a utilitarian notion of global welfare W (g) =P
n
i=1 Ui
(g) torank different g 2 G. I say that W (g00) is efficiency improving on W (g) if and only if
Pn
i=1 Ui
(g00) >P
n
i=1 Ui
(g). If no network g00 improves on g, g is efficient. I denote the change in welfare from goingfrom g to the adjacent g0, as dW = W (g0)�W (g).
As a second notion of efficiency, I use Pareto optimality. With a Pareto optimal network g, thereexists no g00 such that U
i
(g00) � Ui
(g) for all i and with strict inequality for at least one i. If utilitytransfers between nodes are possible, efficiency and Pareto optimality coincide. Without transfers,global welfare is a stronger notion of efficiency than Pareto optimality, as it rules out more efficientnetworks (Jackson and Wolinsky (1996)).
Next, I quantify the efficiency gains from going from g to g0 on an improving path by mutualgains. Mutual gains dV
ij
are the joint increase in the utility of nodes i and j from going from networkg to g0 by adding or deleting one link: dV
ij
= dUi
+ dUj
= [Ui
(g0)� Ui
(g)] + [Uj
(g0)� Uj
(g)].Finally, I define network externalities. A network externality in this paper is the change in U
h
for nodes h, generated from moving from g to g0 that includes two nodes i 6= h and j 6= h. I canthen decompose a change in welfare on an improving path into mutual gains (dV
ij
) and networkexternalities (dW
N\{i,j}), such that: dW = dVij
+ dWN\{i,j} = dV
ij
+
Pn
h 6=i,j
dUh
, where N \ {i, j}implies the set of nodes N excluding nodes i and j.
Some remarks. First, pairwise stability does not imply existence of equilibrium (Chakrabarti andGilles (2007)). Similarly, pairwise stability does not imply efficiency (Jackson and Wolinsky (1996)).It will turn out that myopic decisions of agents in a general equilibrium setting of international tradelead to stable and efficient networks.
Second, if there are no network externalities, dW = dVij
, and the welfare change on an improvingpath is completely due to mutual gains, not affecting other nodes in the network. If dW > dV
ij
,there are positive network externalities from link formation, and if dW < dV
ij
, there are negativenetwork externalities. It will turn out that the network formation in this paper can lead to negativeexternalities for nodes h, that are not involved the creation or deletion of a link ij.
Third, there can be several improving paths emanating from g, which include either link creationor link deletion. This makes it hard to generally evaluate the network formation game in a lineartime sense. Additionally, the cardinality of G increases as 2
n(n�1)2 . This exponential increase makes
it infeasible to completely characterize G for more than a few nodes.1 In most of the examples below,
1 With only n = 30, there are 2435 possible undirected networks, which exceeds most estimates of the number ofatoms in the universe (Chandrasekhar and Jackson (2012)).
6
I restrict analysis to a simple network with three countries, and calculate all possible outcomes giventhe model primitives. This allows me to trace all improving paths over G, and to derive some generalconjectures for settings with more countries and/or goods.
Finally, there are several possible extensions to this framework. For simplicity, I consider nocoalition formation, no strategic trade policy, no side payments or transfers, and I assume there isperfect information over the structure of the network and all quantities attached to it. Furthermore,from the definition of improving paths, changes from g to g0 are myopic decisions of all players, nota directly attained state from any preferred outcome.2
3 The Ricardian network
I start with the presentation of a simple Ricardian model of international trade without trade costsand derive a global market clearing condition that allows for the analysis of the network structure oftrade. I then derive autarky and free trade equilibria, and discuss the relationship between generalequilibrium and network formation.
3.1 Setup
Each country i is endowed with a labor force Li
and a country-specific technology for each good k,aik
. Technology aik
expresses the unit labor requirement for each good, which is the same for allfirms in i producing the same good k. Labor and technology are exogenous to the network and timeindependent. Each country i can produce any combination of goods k 2 K, according to a linearproduction possibility frontier (PPF) given by a full employment condition:
Li
=
mX
k=1
aik
yikt
(gt
) (1)
This is basic Ricardo, with the novel element that the amount of output for each good yikt
(gt
) nowexplicitly depends on the trade network g
t
at time t. There is only one factor of production, homo-geneous labor, paid in wages w
it
(gt
), and perfectly mobile across industries and perfectly immobileacross countries. There are many perfectly competitive firms inside each country, competing inprices. Free entry and exit ensures zero profits in equilibrium: ⇡
ikt
= 0 for all k and i. Using (1), Ican then write output of good k, y
ikt
, as:
yikt
(gt
) = �ikt
(gt
)
Li
aik
(2)
where �ikt
(gt
) 2 (0, 1) denotes the endogenous fraction of labor in i devoted to the production ofgood k, such that
Pm
k=1 �ikt(gt) = 1. If a country does not produce good k, �ikt
= 0. Intuitively,more output is generated for any good k the larger the country L
i
, the larger is the labor share
2 Several other equilibrium notions exist for the analysis of these alternative settings, such as Strong Nash Equilib-rium, Coalition Proof Nash Equilibrium and Pairwise Nash Equilibrium.
7
allocated to the production of good k, �ikt
(gt
), and the better technology country i possesses in theproduction of good k, a
ik
. Total production in i can then also be written as:
mX
k=1
yikt
(gt
) =
mX
k=1
�ikt
(gt
)
Li
aik
(3)
In autarky, prices are determined by the marginal cost of own production, such that pikt
(gt
) =
aik
wit
(gt
). With trade, prices settle internationally, such that
pikt
(gt
) = pkt
(gt
) = min{pikt
(gt
), for all i in the same component of gt
} (4)
Hence, country i will only produce good k if it is the cheapest supplier in the component at time t.Note that this implies that there is no distinction between the origins of the same good k. In thissimple setting, trade is frictionless, and there are no variable nor fixed costs of exporting.
Demand is homothetic and identical across countries. A representative consumer in i maximizesutility U
it
at time t over k goods, given by Cobb-Douglas preferences:
Uit
(xi1t, ..., ximt
, gt
) =
mY
k=1
(xikt
(gt
))
↵k (5)
where xikt
is the quantity consumed of good k at time t, ↵k
denotes the fraction of income spent ongood k, common across countries, and
Pm
k=1 ↵k
= 1 denotes constant returns to scale in consump-tion. Consumers maximize utility subject to a budget constraint given by
Pm
k=1 pikt(gt)xikt(gt) =
Yit
(gt
), where Yit
(gt
) = wit
(gt
)Li
is the expenditure of country i at time t. Marshallian demandsare then given by
xikt
(gt
) = ↵k
wit
(gt
)Li
pikt
(gt
)
(6)
All quantities in a given network gt
at time t are determined in general equilibrium from thetrade model. ✓
it
2 ⇥
t
= {y,x,p,X,Uit
} then denotes the set of equilibrium quantities for everynode i at time t. Given a network g
t
, all firms inside all countries decide simultaneously on thequantities to produce and consume. General equilibrium ⇥
t
is then defined by (i) profit maximiza-tion, (ii) utility maximization, (iii) global market clearing for all goods, and (iv) balanced trade foreach country. Trade balance ensures that each country’s budget constraint is satisfied. Equilibriumconditions can then be written as
(i) Profit maximization: max
pikt ⇡ik(gt)
(ii) Utility maximization: max
xikt Uit
(gt
)
(iii) Global market clearing:P
n
i=1 xikt(gt) =P
n
i=1 �ikt(gt)Liaik
, for all k(iv) Trade balance:
Pm
k=1 pikt(gt)xikt(gt) =P
m
k=1 pikt(gt)yikt(gt) = wit
(gt
)Li
, for all i
These four conditions pin down all quantities in ⇥
t
. The trade balance condition is the same as
8
the allocation rule in Jackson and Wolinsky (1996) when no side payments are possible.
3.2 Autarky equilibrium
In autarky equilibrium, domestic production equals domestic consumption, such that relative out-puts equal relative demands for all k, l 2 K:
yikt
yilt
=
xikt
xilt
=
↵k
pilt
↵l
pikt
(7)
where the second equation follows from Marshallian demands. Relative prices are pinned down bywages, such that
pikt
pilt
=
aik
ail
for all k, l 2 K (8)
Using(6), I can express demands completely in terms of exogenous variables:
xikt
= ↵k
Li
aik
for all k 2 K (9)
In autarky, this fully characterizes the autarky equilibrium ⇥0 in terms of exogenous variables, suchthat for all k, l 2 K and i 2 N :
✓i0 =
8>>>>>>>>><
>>>>>>>>>:
yiktyilt
=
↵kailt↵laikt
xikt
= ↵k
Liaik
piktpilt
=
aikail
Xkt
= 0
Uit
=
Qm
k=1
⇣↵k
Liaikt
⌘↵k
(10)
3.3 Trade equilibria
In general, explicitly solving the model with n countries and m goods is impossible without extraassumptions on the distribution of productivities and a continuum of goods k (Eaton and Kortum(2002, 2012)). It is however possible to evaluate the impact of the network structure on equilibriumquantities, which I exploit in deriving ✓
it
. Plugging absolute demands (6) and trade balance intoglobal market clearing, I can write the global market clearing condition for good k as (see AppendixA):
↵k
pkt
(gt
)
nX
i=1
mX
k=1
pkt
(gt
)
�ikt
(gt
)Li
aik
| {z }Global demand for k
=
nX
i=1
�ikt
(gt
)Li
aik
| {z }Global supply of k
(11)
Global demand for good k equals global supply of good k in equilibrium, and I have written globaldemand in terms of equilibrium outputs for all goods and all countries. The advantage of writingthe model in this way is that (i) it is completely defined in terms of primitives and only two
9
Figure 1: Three countries on a line.
A
B
C
endogenous parameters, �ikt
(gt
) and pkt
(gt
), and (ii) one can map network structures onto thismodel by exogenously shutting down trade channels: if j is not in the same component as i, demandfor good k in country i is independent of production of good k in country j. This is for example thecase when countries are in autarky, or when two separate components exist. The solution to (11)pins down all other quantities in ⇥
t
.At this point, the Ricardian model is a somewhat cumbersome to solve. The solution then
proceeds as follows: (i) assume a given specialization, calculate ⇥
t
, and confirm this is indeed anequilibrium; (ii) if not, incomplete specialization of one country pins down relative prices for bothgoods in a given component and one can calculate equilibrium outputs, from which ⇥
t
follows(Eaton and Kortum (2012)).
Consider for example a simple given network with three countries i 2 N = {A,B,C} and twogoods cloth and wine k 2 K = {c, w} as depicted in Figure 1. The three countries are arranged ona line, so that A can trade with B and B can trade with C, but A cannot directly trade with C (e.g.from political actions, prohibitively high fixed costs, trade embargoes, institutions). All countriesare part of the same component. Using (11) then facilitates deriving equilibrium quantities ⇥
t
ofthis network g
t
. Assume that a setting with complete specialization given by A and C producingcloth and B producing wine, is an equilibrium outcome. Complete specialization is just setting �
ikt
to 1 or 0 for each good and each country, after which I can solve for prices. Global market clearingfor wine can then be written as (normalizing p
wt
= p and pct
= 1):
↵w
p
✓LA
aAc
+ pLB
aBw
+
LC
aCc
◆
| {z }Global demand for wine
=
LB
aBw|{z}
Global supply of wine
(12)
Country B is the only supplier of wine in the network, which has to be balanced against globaldemand for wine. Demand for wine is in turn determined by the production of both cloth andwine in each country. (12) is one equation in one unknown (p). Given the exogenous parameters,one can directly solve for p and check whether this is an equilibrium outcome, by plugging p intooptimal demands and market clearing. If U
it
(gt
) � Ui0(g0) for all i, this outcome is an equilibrium.
If this is not an equilibrium, assume another specialization pattern and redo the analysis. If onecountry diversifies, (11) generates one equation in two unknowns: relative prices p and specializationpatterns �
ikt
. Relative prices over the component are then given by the country that diversifies, sothat pct
pwt=
aicaiw
, and demand then determines output quantities of each good. Note that the countryi that differentiates, then dictates world prices over the whole component.
10
3.4 General equilibrium and network formation
Let me further clarify the interaction between general equilibrium and network formation witha simple example. Consider a Ricardian model with 2 countries and k goods. At t = 0, bothcountries i 2 {1, 2} are in autarky, and ✓
it
2 ⇥0 represents quantities for all i in this state of theworld. Domestic production then equals domestic consumption, and g
t
2 G = {N,⇥0} can be seenas an empty network, in which all links ij are absent. All equilibrium outcomes then depend onlyon own (exogenous) characteristics L
i
and aik
for all k and i.At t = 1, both countries are allowed to trade. I derive ✓
it
2 ⇥1. First, observe that ⇥0 is notpairwise stable: from classical trade theory, we know that each of both countries is at least weaklybetter off under trade than under autarky, and one strictly better off. Hence, these countries decideto form a trade link. The change from g0 = {N,⇥0} to g1 = {N,⇥1} lies on an improving path.Since there are no alternative trajectories, ⇥1 is pairwise stable, as no country has an incentive todeviate to any adjacent network, which would imply going back to autarky as only possible option.
Both states of the world are general equilibrium outcomes in a classical sense, but they are notboth pairwise stable, only ⇥1 is. Furthermore, ⇥1 is efficient, as it maximizes W (g) on the set ofpossible networks G. ⇥1 is also Pareto optimal, as it is impossible to make one country strictlybetter off without making one country worse off. As there are only two countries, mutual gainsdV
ij
coincide with dW (g). We will see below that these results do not have to coincide when thereare more than two countries. There exist Pareto optimal networks that are not efficient, while thediscrepancy between mutual gains and the change in global welfare exactly identify the existence ofnetwork externalities.
4 Numerical example
I now turn to some simple numerical examples to illustrate the mechanisms in Section 3. I showthat efficiency and Pareto optimality do not have to coincide, and that network externalities exist.
4.1 Setup
There are three countries i 2 N = {A,B,C} and two goods k 2 K = {c, w}, representing cloth andwine. For simplicity ↵
c
= ↵w
= 1/2. Utility then collapses to Uit
(xict
, xiwt
, gt
) =
pxict
(gt
)xiwt
(gt
).The model primitives are given by the exogenous variables for technology a
ik
and country size Li
for each country in Table 1. Autarky quantities are then given by (10), and the fourth row of Table1 shows the autarky consumption bundles for each i. In autarky, A produces and consumes bothcloth and wine, similarly so for B and C. Relative prices depend on relative productivities in eachcountry, hence there is no common international price. Autarky relative prices range from 1/5 to 4.Country utility and global welfare then follow from plugging in optimal demands. Global welfarein autarky is W0 =
Pi
Ui0 ' 19.83.
One helpful tool in choosing an educated guess for potential specialization patterns in trade set-
11
Table 1: Primitives and autarky bundles.
A B Caic
16 1 2aiw
4 4 10Li
64 32 70Autarky bundles (x
ict
,xiwt
) (2, 8) (16, 4) (
35/2, 7/2)Relative prices (pct/pwt) 4
1/4 1/5Country utility (U
i0) 4 8 7.83Global welfare (W0) 19.83
tings with two goods is the chain of comparative advantage.3 I can then rank relative productivitiesas
aCc
aCw
<aBc
aBw
<aAc
aAw
=
2
10
<1
4
<16
4
(13)
The chain of comparative advantage then identifies which country specializes first and to whichgood, given relative demands. (13) shows that country C is relatively better at producing cloththan both other countries, country A is relatively better at producing wine, and country B will bethe first to diversify. With these primitives, I construct an example on two fixed networks, and Isubsequently explore a network formation game.
4.2 Fixed networks
In fixed networks, all links ij 2 gt
are exogenously determined and countries then settle ⇥
t
on thisgiven network. First, I consider Network 1 in Table 2. In this network, countries A and B aretrading, while C is in autarky. Relative prices settle between the two countries engaged in trade,while prices for C are determined by own productivities, as shown by the entries in the first row ofthe Table. Plugging relative prices into (11) pins down all equilibrium quantities ✓
it
2 ⇥
t
. In thesecond row, we see that country A specializes in wine, country B specializes in cloth, and country Cdiversifies. Both A and B consume 16 cloth and 8 wine, while for country C domestic demand equalsdomestic supply. A exports 8 wine to B, and imports 16 cloth from B (imports are representedas negative net exports). Both countries trade directly, and there are no re-exports (i.e. importedgoods that are exported again). Global welfare is W1 ' 30.45.
Next, I consider Network 2 in Table 3. In this network, A trades with B and B trades with C.General equilibrium is so that country A produces only wine, country B produces both cloth andwine, and country C produces only cloth. Then, the country that diversifies dictates relative pricesover the network. In this case, country B sets world prices as pct
pwt=
aBcaBw
=
14 . Production bundles
(yict
, yiwt
) and consumption bundles (xict
,xiwt
) in this network are given in the second and thirdrows of Table 2. This directly pins down trade flows over this network. Country A imports 32 cloth
3 See Costinot (2009) for an extension of this ranking technique to more goods and more countries, where in particularcases productivities can be ranked in terms of supermodularity. This result only holds however, for equilibria thatexhibit full specialization of all countries.
and exports 8 wine, and similar results for B and C are presented in the fourth row of the Table.Global welfare is now W2 = 32.75.
Importantly however, this equilibrium outcome depends crucially on re-exports: country Aimports 32 cloth from country B, but B can only send 29/2 of its cloth to A (=61/2 � 16), as itconsumes 16 cloth domestically. In order to induce trade with A, B needs to additionally re-export35/2 cloth from country C to country A. Vice versa, country C imports 8 wine, but B only has 29/8wine for exports (=4� 3/8) , so it needs to add another 35/8 of wine from A to serve C.
At this point, it is not yet clear whether either Network 1 and/or Network 2 are also Paretooptimal, stable or efficient. This requires the comparison of all possible networks g 2 G, which Ianalyze in the endogenous network formation game.
13
Figure 2: Endogenous network formation.
A
B C
c,w
c,w c,w
A A A
B B B
A A A
A
B B B
B
C C C
C C C
C
c,w
c,w c,w
w
c w c
w
c
w
c,w c
w
c,w c
w
c,w c
4.3 Endogenous network formation
I now turn to the full analysis of the network formation of trade with three countries and two goods.4
Starting from autarky, I let the game evolve over all possible improving paths, until a stable networkis attained, if any. All general equilibrium outcomes for each network and its improving paths arepresented in Appendix B; here I report the results and intuition from one improving path.
Figure 2 shows a representation of all eight possible networks on these three countries. The topcenter network represents an empty network g0, in which all countries are in autarky. Domesticproduction then equals domestic consumption, all countries produce both goods, and equilibriumconsumption quantities are given by the first row in Table 4. Global welfare in autarky is W0 ' 19.83.No links have been formed or severed yet, so changes in global welfare dW and mutual gains dV
ij
are zero. Hence, no externalities exist. This network is however not Pareto optimal, stable norefficient. There is an improving path from autarky to any network in which two countries starttrading.
In this example, there are three improving paths from g0. I pick one for exposition, in whichA and C start to trade, and B remains in autarky, defined by g1. A then specializes in wine, C in
4 Remember that the cardinality of G increases as 2n(n�1)
2 . Hence, for n = 3, there are 8 possible undirectednetworks. For n = 4, there are 64, and for n = 5, there are already 1024 possible networks. Given that generalequilibrium quantities have to be derived in every g, which in itself contains a dichotomous solution scheme, acomplete characterization of all possible networks for n > 3 is beyond the scope of this paper.
14
Table 4: Sequential network formation results.
Consumption bundles ExternalityA B C W
t
Wt
�Wt�1 dV
ij
(dWt
� dVij
)g0 (2, 8) (16, 4) (
35/2, 7/2) 19.83 0 0 0g1 (16, 8) (16, 4) (
35/2, 8) 31.66 11.83 11.83 0g2 (32, 8) (16, 4) (
35/2, 35/8) 32.75 1.09 4.17 �3.08
cloth. Global welfare increases to W1 ' 31.66, hence this move is efficiency improving. The changein welfare from going from g0 to g1 is 11.83. Both A and B gain strictly from trade, and the mutualgains are dV
ij
' 11.83. As the increase in global welfare and mutual gains coincide, there are noexternalities. Since g1 and g0 are adjacent networks and g0 is defeated by g1, this evolution lies onan improving path (represented by the red arrow in Figure 2).
Next, another link between these three countries is added or deleted, such that two networksare adjacent and that the previous network g1 is defeated by the next network g2. This results inan additional trade link between A and B. A, B and C now trade, and are in the same component.A specializes in wine, C in cloth, and B diversifies as before. Global welfare is W2 = 32.75, and thechange dW = 1.09. Importantly however, mutual gains are now dV
ij
= 4.17, much larger than thegain in global welfare. The difference is due to the negative externality on country C, which is notinvolved in the new trade link between A and B, and loses 3.08 units of utility. Furthermore, inthis example, the mutual gains are completely on the account of A, while B still consumes the samebundles as in autarky. I can furthermore show that there is no improving path from g2 to a fullyconnected network g3, as no link addition or deletion leads to U
it
(g3) > Uit
(g2) for any i. Therefore,g2 is stable. At the same time, the network is efficient, as it maximizes global welfare. This alsoimplies that C cannot react to this negative externality, and remains worse off in g2 than in g1.
An alternative improving path from g1 would have been a trade link between B and C insteadof A and C (indicated by the blue arrow in Figure 2). In this case, the resulting network is againstable and efficient, and now country A would be worse off under this new configuration. Again,this presents a network externality for the country not involved in the new trade link. Alternativeimproving paths generate the same results: From g0, any network with one link is on an improvingpath, and from there, any of the three networks in the third row of Figure 2 are on an improvingpath, generating stable and efficient network, that are also Pareto optimal.
Some remarks. First, it turns out that the networks in the same rows of Figure 2 share thesame network characteristics. While the exact quantities can be different over these networks, theirnetwork properties are identical, and summarized in Table 5. Autarky is not stable, efficient norPareto optimal, and there are no improving paths that end up in autarky. The networks in thesecond row of Figure 2 are all adjacent networks to the autarky network g0, on improving paths, andthese networks are all unambiguously welfare increasing compared to autarky. They are not stablenor efficient, as there exists improving paths from these networks, but they are Pareto optimal: it
15
Table 5: Network characterization.
Stable Efficient Pareto optimal Improving path Welfare increasing ExternalitiesRow 1 – autarky No No No No No NoRow 2 – one link No No Yes Yes Yes NoRow 3 – line Yes Yes Yes Yes Yes YesRow 4 – complete Yes Yes Yes No No No
is impossible to move to any other network without making at least one country worse off. Thenetworks in row three of the Figure are efficient: there are no alternative outcomes that generatehigher global welfare, and these outcomes are stable. Again, these outcomes are Pareto optimal.However, given that networks in row two are Pareto optimal, going from any network in row two toany network in row three implies that at least one country is strictly worse off than before, identifiedby the network externalities I quantified above.
The complete network in row four is stable, efficient and Pareto optimal. The exact trade flowsare undetermined however, as exports and imports for all countries can travel over two links for eachcountry. Additionally, there are no improving paths towards the complete network, as all networksin the third and fourth rows are stable. This implies that this network formation game exhibitshysteresis: starting in autarky, any network in row three can be achieved; conversely, starting fromthe complete network generates a stable network that remains complete.
Second, all three “line” networks are efficient, and network formation always ends up in one ofthese three. Only the network with country A in the middle exhibits no re-exports: country A canserve both B and C directly with own production and still induce trade. The other two networksinduce re-exports, while equilibrium quantities are the same.
Finally, in this setup, network formation is linearly growing: there are no situations where linksare broken. It is not clear how this changes with more countries or more goods, as improving pathsmay include link deletion.
5 Theoretical implications
From the model and the numerical examples above, I extrapolate some conjectures on the generalmechanisms of fixed network structures and network formation. Future rigorous mathematicalderivations are necessary to present general propositions at a later stage.
Conjecture 1. Autarky is not pairwise stable. Assuming Conjecture 1 holds, this implies thatstrategic network formation never results in a stable autarky network. From classical trade theory,we know that autarky is not Pareto optimal, nor efficient. The network dynamics in this papershow that autarky will never be a stable network. Proof of this conjecture would involve showingthat no improving path to autarky exists under any circumstance.
Conjecture 2. Any connected network is pairwise stable. If an improving path leads to a component
16
that involves all nodes in the network, this outcome is pairwise stable. Assuming Conjecture 2 holds,this implies that potentially different dynamics exist from autarky to a pairwise stable network, andthat potentially multiple stable networks exist. A next step in the analysis to characterize equilibria,is to quantify the number of networks in G that satisfy connectivity.
Conjecture 3. The Arrow-Debreu framework for anonymous markets extends to connected net-works. The Arrow-Debreu model states that under convex preferences, perfect competition andindependent demands, a price vector exists, such that markets clear for all goods in the economy.Additionally, any equilibrium is Pareto optimal and efficient. Assuming Conjecture 1 holds, thisestablishes existence of general equilibrium in any connected network g.
Corollary 1. Assuming Conjectures 2 and 3 hold, any connected network is Pareto optimal, efficientand pairwise stable.
Corollary 2. Existence and uniqueness of general equilibrium versus multiplicity of network equi-libria. Assuming Conjectures 2 and 3 hold, this implies that unique general equilibrium outcomesgenerate multiple stable equilibria, as efficient allocation of goods is frictionless over the componentor network.
Corollary 3. Irrelevance of network structure. Corollary 2 implies that all markets clear in acomponent of the network, independent of the number of links (network density) and the positionof countries (network centrality) in a component. This corollary implies that markets clear for nodesthat are only indirectly connected. Hence, any connected network (star, tree, line, circle,...) mapsto the same general equilibrium quantities as allocations are frictionless. This is clearly a strongstatement. With the existence of variable or fixed costs of link formation, networks will becomemuch more sparse, and starlike structures and cycles will possibly emerge instead, depending on thesize of these fixed costs. Proof of this corollary could involve using the fact that multiple improvingpaths exist from every state.
Corollary 4. Law of one price. Assuming Conjecture 1 holds, for any connected component, thelaw of one price holds.
Corollary 5. Relative prices and specialization. If incomplete specialization occurs, relative pricesover the whole network are dictated by the one country that diversifies, and demand then determinesthe relative outputs. Everyone gains from trade, except the one country that diversifies. At thesame time, efficiency increases.
Conjecture 4. Myopic strategic network formation leads to stable and efficient networks. Thisimplies that individual actions lead to globally efficient outcomes. There is always at least oneimproving path from autarky to networks with trade, and every improving path ends up in apairwise stable network. This stable network is efficient.
Conjecture 5. Strategic network formation generates negative externalities. Even while a stablenetwork is efficient, a country not benefiting from the mutual gains from a change along an improvingpath is always strictly worse off than in the previous equilibrium.
17
Conjecture 6. Variable trade costs and equilibria. The introduction of variable trade costs leadsto subsets of non-traded goods as in Dornbusch, Fischer and Samuelson (1977). Strategic networkformation with variable costs of trade still leads to stable and efficient outcomes as before. Equilibriawith re-exports disappear, as these involve paying trade costs twice. This follows from the triangularinequality theorem. Prices are then given by p
kt
(gt
) = min{aik
wit
(gt
)⌧ij
, for all i in the same component},where ⌧
ij
denotes iceberg-type variable trade costs. Unlike the basic setup above, this implies thatthe cheapest supplier of a good i can differ across destinations from the existence of ⌧
ij
.
Conjecture 7. Fixed costs of trade and equilibria. The introduction of fixed costs of trade, orequivalently link formation, leads to subsets of stable and efficient equilibria. If one country paysfor the link formation, this is a similar setup as the symmetric connections model in Jackson andWolinsky (1996). For low fixed costs, countries that are indifferent between autarky and trade, willprefer to stay in autarky. This still leads to stable and efficient outcomes, but maximum efficiencyis now less than in the frictionless case, as W =
Pi
Ui
�P
i
Pj
fij
, where fij
indicates the cost oflink formation between i and j. For medium-sized fixed costs, the most efficient network is a starnetwork. The number of links the center pays for is determined by the utility it gets in return. Forhigh fixed costs, a star network is not stable, but non-empty networks exist, such that each nodehas at least two links.
Corollary 6. Complements and substitutes. Games with strategic substitutes and fixed costs oflink formation generally result in structures proposed in Conjecture 7. This is the case for hori-zontal varieties as proposed in this paper. Simple games with strategic substitutes can be analyzedwith maximal independent subsets (Jackson and Zenou (2014)). Games with strategic comple-ments can generate much more complex network structures, as is the case with intermediate goods.Characterization of equilibria in these networks is much more involved.
6 Towards a general framework
The baseline examples in this paper are stylistic and provide some intuition about the evolution ofequilibrium quantities when countries open up sequentially to trade, and whether network formationleads to stable and efficient outcomes. I now propose a more general model that allows for networkformation in an existing Ricardian framework of Eaton and Kortum (2002). I show that the impactof the network formation process is captured completely by changes in relative wages. From thisprobabilistic model, comparative statics for network formation and pairwise stability can be derivedin future work.
6.1 Setup
There are n countries in the world i 2 N = {1, ..., n}, each of size Li
. There is a continuum of goodson the unit interval, ! 2 (0, 1), where the set ⌦
t
denotes the set of goods available for consumptionin the network g
t
. Efficiency for good ! produced in i is given by zi!
=
1ai!
, which is the realization
18
of a draw from a random variable from a distribution Fi
(zi!
). zi!
is the inverse of the unit laborrequirement a
i!
. Labor is the only factor of production, paid in wages wit
(gt
). Labor is perfectlymobile across sectors, perfectly immobile across countries.
Firms compete in perfect competition, and the cost of producing one unit of output ! is givenby p
i!t
(gt
) =
wit(gt)zi!
. Exporting from i to j entails time-invariant iceberg trade costs ⌧ij
� 1 (with⌧ii
⌘ 1), such that the price for goods produced in i and served in j at time t is given by
pij!t
(gt
) =
wit
(gt
)
zi!
⌧ij
(14)
From perfect competition, prices for good ! in j settle as
pj!t
(g) = min{pij!t
(gt
); for all iin the same component} (15)
Note that the original Eaton and Kortum (2002) framework implicitly assumes that all countriesare in the same component, as all prices equalize over all countries i.
In each country j, there is a representative consumer, maximizing utilities over CES preferencesat time t:
Ujt
=
2
64ˆ
!2⌦jt(gt)
qt
(!, gt
)
��1� d!
3
75
���1
(16)
where ⌦
jt
(gt
) is the measure of goods available for consumption in j at time t, dependent on thenetwork g
t
, q(!, gt
) is the quantity consumed of good !, and � > 1 represents the elasticity ofsubstitution across goods, common across countries.
Technology is given by zi!
, and is drawn from a random variable that is Fréchet distributed:
Fi
(z) = Pr(Zi
z) = exp{�Ti
z�✓} (17)
where the location parameter Ti
> 0 represents an absolute technology parameter, common acrossfirms in i: a larger T
i
shifts the distribution to the right, implying that a country i can be betterin a range of goods, as a higher T
i
first-order stochastically dominates a lower Ti
. ✓ > 1, the shapeparameter of F
i
(·), represents a measure of comparative advantage, and defines the “width” of thedistribution: a higher ✓ is related to a narrower shape of the distribution. In general, lower valuesof ✓ generate more heterogeneity in productivities, leading to a higher probability that a firm is thecheapest supplier of a good !. I assume ✓ is equal for all countries i. From the law of large numbers,(17) is also the fraction of goods for which country i’s efficiency is less or equal than z. From (14),prices are also a realization of a draw from z: P
ijt
(gt
) =
wit(gt)Zi
⌧ij
, and so the lowest price for eachgood in j at time t is a realization of P
jt
(gt
) = min{Pijt
(gt
), for all iin the same component}.The Fréchet distribution is an extreme-value distribution, and can be most easily visualized as
the distribution of extreme draws from a Pareto distribution of technologies (see Eaton and Kor-tum (2012)): a lesser-known result of the central limit theorem states that the highest and lowest
19
observations in a sample drawn from any well-behaved distribution approaches an extreme-value dis-tribution (Eaton and Kortum (2012)). As the Ricardian model is expressed in terms of technologies,the counter cumulative distribution 1� F
i
(p) (which is Weibull distributed) describes the distribu-tion of the lowest prices. This provides a direct link between the distribution of productivities z
and prices p.Plugging P
ijt
(gt
) into (17): Pr(Pijt
p) = 1 � Fi
⇣wit(gt)p(gt)
⌧ij
⌘, makes explicit that the lowest
price in j will be less than p, unless each source’s price is greater than p. The ex ante distributionof prices for all goods presented by i to j, H
ijt
(gt
), is then given by
Hijt
(p, gt
) = 1� exp
(�T
i
✓wit
(gt
)
p(gt
)
⌧ij
◆�✓
)(18)
This expression is similar to Eaton and Kortum (2002), where I now explicitly allow for the networkstructure g
t
. A higher Ti
(higher absolute technology for i) is related to a higher probability for i
to be a lowest cost supplier, all other things equal. Similarly, a higher wage wit
(gt
) generates largerinput costs, decreasing the probability of being the cheapest supplier. Also, higher variable tradecosts ⌧
ij
for goods from i to j decrease the probability of providing the lowest price. Regardingthe network, I conjecture that a network which exhibits more trade, generates more internationalcompetition, driving down the probability that i provides the cheapest goods. This then generallydrives down prices for all goods.
Finally, the price the consumer pays, is given by the ex post distribution of prices, dictated bythe probability that i is the cheapest supplier of goods in j:
Hjt
(gt
) = Pr(Pjt
p) = 1�nY
i=1
[1�Hijt
(p, gt
)] (19)
This can also be written as
Hjt
(gt
) = 1� exp�
jt
(gt
)p(gt
)
✓ (20)
where �
jt
(gt
) ⌘P
n
i=1 Ti
(wit
(gt
)⌧ij
)
�✓ summarizes how (i) the states of technologies across theworld (T
i
), (ii) input costs across the world (wit
(gt
)) and (iii) trade costs (⌧ij
) govern prices inside j.�
jt
(gt
) is a general form of the equilibrium conditions in the baseline model in Section 3. Note thatTi
and ⌧ij
are exogenous, and that �
jt
(g) captures the network structure through wit
(gt
) and thesummation operator
Pn
i=1: the more countries there are in the network, the higher the probabilitythat consumers in country j pay a lower price, increasing global welfare. �
jt
(gt
) can also be viewedas a measure of “remoteness”: a large �
jt
(gt
) implies that j is close to all other countries in theworld, leading to a higher probability to receive lower prices for consumption. In autarky (letting⌧ij
! 1), �jt
(gt
) reduces to Tj
w�✓
jt
, country j’s own state of technology and input costs. In theother extreme of frictionless trade, ⌧
ij
= 1 for all i and j, and �
jt
(gt
) = �
t
(gt
) for all j. This impliesthat the law of one price holds for all goods and all countries in the same component, confirming
20
earlier results.Finally, the probability that i provides a good at the lowest price to j at time t is given by
⇡ijt
=
Ti
(wit
(gt
)⌧ij
)
�✓
�
jt
(gt
)
(21)
This results in a trade model where all the action is on the extensive margin: if i is the cheapestsupplier for good ! in j, i supplies that market. �
jt
(gt
) can then be seen as a general equilibriumimpact of third countries on the bilateral characteristics of i and j. �
jt
(gt
) is then also reminiscentto the multilateral resistance term in Anderson and van Wincoop (2003), where all the action isinstead on the intensive margin, and all countries export to all destinations from an Armingtonassumption.
From the law of large numbers, the probability that i exports to j coincides with the fraction ofgoods that i exports to j. Export values from i to j, X
ijt
, as a fraction of total imports of j, Xjt
,can then be written as
⇡ijt
=
Xijt
Xjt
=
Ti
(wit
(gt
)⌧ij
)
�✓
�
jt
(gt
)
(22)
6.2 Network formation, welfare and externalities
From market clearing conditions wit
(gt
)Li
=
Pn
i=1Xijt
and trade balance wjt
Lj
= Xjt
, I can writecounty i’s expenditure as
wit
Li
=
nX
j=1
wjt
Lj
Ti
(wit
(gt
)⌧ij
)
�✓
�
jt
(gt
)
(23)
Let ⇡iit
denote the fraction of goods sold at home, i.e. ⇡iit
= 1 denotes autarky. Rearranging (21)for ⇡
ijt
= ⇡iit
then:
wit
(gt
) =
✓⇡iit
(gt
)�
it
(gt
)
Ti
◆� 1✓
This leads to the gains from trade
wit
(gt
)
pit
(gt
)
=
�
✓✓ + 1� �
✓
◆� �11��
✓⇡iit
(gt
)
Ti
◆� 1✓
where �(·) represents the gamma function, from pit
(gt
) =
⇥�
�✓+1��
✓
�⇤ �11��
�
it
(gt
)
�1✓ (see Eaton
and Kortum (2002) for the link between the gamma function and the price index). Remarkably,all parameters except ⇡
ii
are exogenous, and hence all gains from trade are captured by ⇡ii
, theopenness to trade of country i. A larger openness to trade (through trade connections) leads tolarger welfare gains. In turn, openness to trade is determined by the network formation gameoutlined in this paper. This framework allows for comparative static analysis of changes in thenetwork structure of trade from strategic network formation. Future analysis has to characterizethe impact of the network formation game on global welfare and externalities that involve multiple
21
countries.
7 Conclusion
Most models of international trade compare two extreme equilibria: autarky and free trade. Inthis paper, I have explored the dynamics of international trade in a simple model of trade withdifferences in technology between countries. There are several real-world examples of sequentiallyand exogenously opening up to trade, such as the fall of the Berlin wall, changes in external policythat integrate China into the world market, Russian and Middle-East trade embargoes etc. Theanalysis of a simple example with three countries and two goods has generated some intuitions thatserve as a foundation for future research.
First, myopic network formation leads to stable and efficient trade networks. This is a strongresult: in the absence of trade policy, transfers, coalitions etc. the network always ends up ina free trade equilibrium, even if each individual improving path is not always the global welfaremaximizing choice at that stage. However, in my setting, there is a plethora of equilibria, which allexhibit the same general equilibrium quantities. This comes from the absence of variable and fixedcosts of trade. In the game of substitutes that I analyze, fixed costs probably lead to more star-likestructures and sparse networks, depending on the exact size of these fixed costs. However, as shownby Jackson and Wolinksy (1996), with fixed costs of network formation, there is a tension betweenstability and efficiency. It is possible to end up in stable but sub-optimal networks.
Second, the path to efficient networks is paved with negative externalities. Again, this is astrong result, and overlooked in classical trade theory. A country that is in a trade relationshipwith another country is worse off if its trading partner opens up to other trading partners. At thesame time, global welfare increases. This has important policy implications: the negotiation of anew trade agreement can benefit both partners, but it can hurt other countries that are not involvedin this agreement. Policy makers mostly only estimate the bilateral gains from this agreement, whilethe global impact of this agreement can largely offset these bilateral gains. I conjecture that thisresult is independent of trade costs, as relative prices change, depending on the amount of goodsexchanged bilaterally. Furthermore, for some ranges of increases in mutual gains, these externalitiescannot be offset by side transfers. It might even be the case that total negative externalities outweighthe benefits of mutual gains, in which case global welfare would actually decrease. A more completeanalysis is necessary to confirm or reject this potential result.
One possibly fruitful path in this direction is the dynamic analysis of the Eaton and Kortum(2002) model of international trade. Probabilistic models of networks completely line up with theconcept of inhomogeneous random graphs (i.e. models of random graphs that allow for heteroge-neous probabilities of link formation, based on node characteristics). A more complete analysisof this model in view of inhomogeneous random graphs is potentially rewarding, as graph theoryprovides theorems that help in characterizing the possible outcomes.
Third, the definitions of global welfare, mutual gains and network externalities can be written as
22
network potentials (e.g. Monderer and Shapley (1996) and Chakrabarti and Gilles (2007)). Networkpotentials are a very strong tool to answer the questions of existence and stability of networks, andwould present a clear first direction to extend the current analysis.
Finally, in empirical analysis it is hard to evaluate the gains from trade by comparing autarkyand free trade, as autarky is (almost) never observed. The sequential equilibria in this paper providea simple framework for the comparison of two “snapshots” of the world trade network. This makesit possible to evaluate gains from trade from observed networks, and not only hypothetical initialstates of the world. This links back to the sufficient statistics approach in Arkolakis, Costinot andRodriguez-Clare (2012).
References
[1] Arkolakis, C., Costinot, A. and Rodriguez-Clare, A. (2012). New Trade Models, Same OldGains?, American Economic Review, 102 (1): 94–130.
[2] Bernard, A.B., Eaton, J., Jensen, B. and Kortum, S. (2003). Plants and Productivity in Inter-national Trade, American Economic Review, 93: 1268–1290.
[3] Chakrabarti, S. and Gilles, R.P. (2007). Network Potentials, Review of Economic Design, 11(1): 13–52.
[4] Chandrasekhar, A. and Jackson, M. (2012). Tractable and Consistent Random Graph Models,mimeo.
[5] Chaney, T. (2014). The Network Structure of International Trade, American Economic Review,104 (11): 3600–3634.
[6] Costinot, A. (2009). An Elementary Theory of Comparative Advantage, Econometrica, 77 (4):1165–1192.
[7] Deltas, G. Desmet, K. and Facchini, G. (2012). Hub-and-Spoke Free Trade Areas: Theory andEvidence From Israel, Canadian Journal of Economics, 45 (3): 942–977.
[8] Dornbusch, R., Fischer, S. and Samuelson P. (1977). Comparative Advantage, Trade, andPayments in a Ricardian Model with a Continuum of Goods, American Economic Review, 67(5): 823–839.
[9] Eaton, J. and Kortum, S. (2002). Technology, Geography and Trade, Econometrica, 70 (5):1741–1779.
[10] Eaton, J. and Kortum, S. (2012). Putting Ricardo to Work, Journal of Economic Perspectives,26 (2): 65–90.
[11] Even-Dar, E., Kearns, M. and Suri, S. (2007). A Network Formation Game for Bipartite Ex-change Economies, Symposium on Discrete Algorithms (SODA).
23
[12] Freund, C. (2000). Different Paths to Free Trade: The Gains from Regionalism, QuarterlyJournal of Economics, 115 (4): 1317–41.
[13] Furusawa, T. and Konishi, H. (2007). Free Trade Networks, Japanese Economic Review, 72 (2):310–335.
[14] Goyal, S. and Joshi S. (2006). Bilateralism and Free-Trade, International Economic Review, 47(3): 749–778.
[15] Jackson, M. and Wolinksy, A. (1996). A Strategic Model of Social and Economic Networks,Journal of Economic Theory, 71 (1): 44–74.
[16] Jackson, M. and Rogers, B. (2007). Meeting Strangers and Friends of Friends: How Randomare Social Networks?, American Economic Review, 97 (3), 890–915.
[17] Jackson, M. and Zenou, Y. (2014). Games on Networks, mimeo.
[18] Jones, R. (1961). Comparative Advantage and the Theory of Tariffs: A Multi-Country Multi-Commodity Model, Review of Economic Studies, 28: 161–175.
[19] Kakade, S., Kearns, M. and L.E. Ortiz (2004). Graphical economics, Proceedings of the AnnualConference on Learning Theory (COLT) 23: 17–32.
[20] Manea, M. (2015). Models of Bilateral Trade in Networks, The Oxford Handbook of the Eco-nomics of Networks, Eds Bramoulle, Y. Galeotti, A. and Rogers, B.), Oxford University Press,forthcoming.
[21] McKenzie, L.W. (1954). On Equilibrium in Graham’s Model of World Trade and other Com-petitive Systems, Econometrica, 22: 147–161.
[22] Monderer, D. and Shapley, L.S (1996). Potential Games, Games and Economic Behavior, 14(1): 124–143.
24
Appendix
A. Global market clearing conditions
From global market clearingP
n
i=1 xikt(gt) =
Pn
i=1 �ikt(gt)Liaik
for all k, plug in optimal demandsxikt
(gt
) = ↵k
wit(gt)Li
pikt(gt). Then use the budget constraint
Pm
k=1 pikt(gt)yikt(gt) = wit
(gt
)Li
to arrive at
↵k
pkt
(gt
)
nX
i=1
mX
k=1
pkt
(gt
)
�ikt
(gt
)Li
aik
| {z }Global demand for k
=
nX
i=1
�ikt
(gt
)Li
aik
| {z }Global supply of k
(24)
B. General equilibrium quantities all possible networks
The following Tables present the general equilibrium quantities for all possible networks over theset of three countries with two goods.
Table 6: Autarky.
A
B
C
A B CAutarky bundles (x
ict
,xiwt
) (2, 8) (16, 4) (
35/2, 7/2)Relative prices (pc/pw) 4
1/4 1/5
Country welfare (U i
0) 4 8 7.83Global welfare (W0) 19.83
Table 7: A and B trade, C in autarky.
A
B
C
A B CRelative prices (pc/pw) 1/2 1/2 1/5
Production (0,16) (32,0) (
35/2, 7/2)Consumption (16,8) (16,8) (
35/2, 7/2)Net exports (-16,8) (16,-8) (0,0)
Country welfare (Uit
) 11.31 11.31 7.83Global welfare (W
t
) 30.45
25
Table 8: B and C trade, A in autarky.
A
B
C
A B CRelative prices (pc/pw) 4 8/35 8/35
Production (2,8) (0,8) (35, 0)Consumption (2,8) (35/2,4) (
Figure 3 shows all improving paths on the set of possible networks G. In every case, the resultingnetwork is stable and efficient. However, three stable networks are possible. These stable networksonly differ in the amount of re-exports that flow through the nodes.
27
Figure 3: Alternative network formations.
A
B C
c,w
c,w c,w
A A A
B B B
A A A
A
B B B
B
C C C
C C C
C
c,w
c,w c,w
w
c w c
w
c
w
c,w c
w
c,w c
w
c,w c
w
c,w c
D. Pareto optimal and efficient networks
Figure 4 shows the set of Pareto optimal networks (blue) and the set of efficient networks (red).
