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HIERARCHICAL TRADE.

I. INTRODUCTION

A. Motivation: Hierarchical Trade and Trade Hubs

Resource allocation mechanisms often have a hierarchical structure. The most prominent example is provided by international commodity trading such as oil, wheat, metals, and other raw materials: Large-scale trade takes place at the global price system while individual trade occurs at local price systems within the countries. Global prices may differ from the local prices if significant barriers to international goods arbitrage exist that drive a wedge between local and global prices. (1)

In two-stage trading, one or more agencies bridge the two stages by trading in global and domestic markets, thereby ensuring that excess demand/supply in domestic markets enters the demand/supply of global markets. While such agencies may simply aim at clearing domestic and global markets, two additional objectives can be important. First, an agency may promote the welfare of a nation when such an agency is public. Second, an agency may maximize its own profit when it is privately owned.

In the real world, such agencies--in the form of trading hubs or trading clusters--play an important role, and the two-stage trade occurs in the form of global/wholesale trade and domestic/retail trade (see, e.g., Lankhuizen, de Graaff, and deGroot 2015). (2)

There are several prominent examples for this. (3) First, Switzerland is famous for its commodity clusters, with trading houses and all services needed in commodity-trading business. It is assumed that in a variety of commodity areas, more than one third of international trading takes place in Switzerland, which connects global trade to trade in specific countries. Second, hubs for international commodity-trading have developed in other areas such as London, Singapore, and the United States, for instance.

B. Motivation: Research Question and Contribution

In this paper, we study the properties of hierarchical international trade incorporating two particular features: First, in each nation, the bridge between trading in global and local markets is formed by a trading hub as described above. Second, other market participants cannot exploit differences between local and global prices. There is a large literature on the reasons why such international goods arbitrage is limited. They often are summarized under trade costs. These costs can be divided up in a first category consisting of policy barriers (tariffs and nontariffs) and transportation costs, and a second category consisting of communication, contract enforcement, insurance, and legal and regulatory costs (see World Trade Organization 2008). We focus on the presence of the second category of trade costs, which can be reduced by trading hubs since there are strong economics of scale, as described above. For the sake of simplicity, we assume in the main body of the paper that a trading hub can completely eliminate such trading costs. In this sense, a trading hub becomes a local monopolist by being the only agent in a country that can reduce--or in the model, eliminate--the trade costs.

An example is the natural gas industry (see Barbe and Ricker 2015) in which trade costs of the second category are substantial (4) and could be reduced by a suitable organization of international trade.

Given the two main assumptions, we investigate five research questions related to international trade with trading hubs whose equilibria between (costless) free trade and autarky are called intermediate equilibria.

* Under which circumstances do intermediate equilibria exist and how can particular types be constructed when the trading hubs are solely interested in their budget balance?

* How is international trade shaped by additional objectives of trading hubs?

* How does bargaining between trading hubs affect international trade?

* Do profit-maximizing trading hubs promote free trade?

* Can hierarchical trade be related to outcomes of traditional trade theory?

By answering these questions, we contribute to the literature on the rate of trade costs and the limits to international goods arbitrage in four ways. First, the literature has focused on the nature and heterogeneity of international trade costs and on how much they reduce or shape international trade; see the survey of Anderson and van Wincoop (2004) and recent contributions by Atkeson and Burstein (2008), Arkolakis, Costinot, and Rodriguez-Clare (2012), and De Sousa, Mayer, and Zignago (2012). We examine international trade when trade costs can be reduced by trading hubs but not by other market participants. Second, it is well documented that transportation costs in international trade have dramatically fallen in the past few decades (see Glaeser and Kohlhase 2004; Hummels 2007), and trade costs of the second category have become a comparatively more important source of barriers to international trade. We investigate the impact of such trade costs.

Third, in the context of firms in different national markets, several authors have studied how price-to-market (e.g., Atkeson and Burstein 2008) and principal-agent relationships of original home manufacturers and foreign retailers (e.g., Maskus and Stahler 2014) can cause retail prices to differ significantly across countries. Persistent price differences can be observed even between countries at short geographical distances and without explicit trade barriers. We investigate how trading hubs can or may cause price differences in retail markets in countries. Of course, goods arbitrage sets in when local and global prices diverge too much. (5) Such considerations will be added in the course of the paper.

Fourth, we investigate how the control of international trade by one or several trading hubs--of the type discussed above, which can set global prices--shapes international trade. To sum up, the existence of trading hubs and their potential to reduce trade costs require a new type of international trade model, which the current paper aims to develop. It is a nonstandard international trade model complementing other new approaches to international trade in general equilibrium (see Neary 2009).

The model works as follows: Members of a group or a nation trade within the group at an internal price system. However, internal markets need not be cleared and the group can act as a trading block towards the rest of the world. The group's trade with the rest of the world occurs through an external trade agency at an external price system. Balancing the group's external trade budget and market-clearing across groups has to be done at the external price system. Internal and external prices can differ, since we assume that goods arbitrage across groups is limited.

In the first part of this paper, we assume that the external agency simply aims at clearing markets. In the second part of the paper, we explore what happens if price differentials are limited by international goods arbitrage and if the external trade agencies either promote the welfare of their countries or maximize their own profit.

C. Model and Results

We begin by devising a framework to study hierarchical market equilibria where groups trade at the global price system while individuals within a group trade at local price systems, and trade hubs link local and global trade and are only concerned about balancing their budget. We identify the free trade and the autarky equilibrium as polar cases. As a rule, free trade yields efficient allocation of commodities whereas autarky leads to inefficient allocation. Thus the question arises if there is the possibility of an intermediate outcome that Pareto dominates an autarky equilibrium allocation, but is itself Pareto dominated by some other feasible allocation, though not necessarily by an equilibrium allocation.

Then, we aim for existence results for intermediate equilibria and study their properties. A first finding indicates that intermediate equilibria between free trade and autarky need not exist for low-dimensional commodity spaces, even under standard assumptions that guarantee competitive equilibria within each group of the economy. In particular, we show that no other market equilibria than free trade and autarky exist if the commodity space is two-dimensional: The world is divided into a free trade zone and an autarky zone. An example demonstrates that nontrivial intermediate equilibria exist for three-dimensional commodity spaces.

We then provide two existence proofs for intermediate equilibria in higher dimensions. (6) In summary, nontrivial intermediate equilibria exist for three-dimensional or higher-dimensional commodity spaces. Since extensive empirical work suggests that international trade is best understood by considering many commodities--starting from the work of Learner and Levinsohn (1995)--one would expect such intermediate equilibria to exist.

In the second part of the paper, we outline plausible refinements of the notion of intermediate equilibrium. Specifically, we explore whether particular types of equilibria can be rationalized by specifying further objectives for the external agency. We outline several conceivable ways to formulate such objectives, as for instance, maximization of the welfare of one country or profit maximization. The refined equilibrium notion opens up a variety of interesting applications, relating the power to control the external agency to trade flows and to the welfare of countries. This analysis yields three insights:

* If a single country can control global trade, a trading hub can achieve substantial welfare gains over and above the level achievable under free trade. In our example, trade flows gravitate towards the country with control of global trade.

* If several countries could control global trade, trade flows become more even. If all countries share control of global trade and bargaining power is the same, we obtain free trade outcomes.

* If external agencies are profit-oriented, we do not necessarily obtain free trade. Large deviations from free trade could occur even if the external agency obtains a small fee from each unit of trade.

Overall, we stress that differences between global and domestic prices are determined endogenously in equilibrium. We assume that these price distortions do not trigger international goods arbitrage, at least within certain boundaries. (7) In Section VII, we will examine how limits on price differentials between global and domestic markets that are caused by international goods arbitrage will impact our results.

Finally, we will establish a link between traditional trade theory and hierarchical trade. Since the organization of markets is different under hierarchical trade, it cannot be compared directly to traditional trade theory. However, we show that a two-stage equilibrium allocation can be replicated by a one-stage market equilibrium if we introduce particular (hypothetical) tariffs in the latter but no lump sum taxes. The reverse is not true, however.

Following the foregoing agenda, the paper is organized as follows. In the next section we relate our analysis to relevant branches in economic theory. In Sections III and IV, we introduce the formal framework. In particular, we define two-stage market equilibria. Free trade and autarky are identified as the polar cases. In Section V we show that no intermediate equilibrium exists if the dimension of the commodity space is two. In Section VI, we establish existence of intermediate equilibria by using two different methods of constructing intermediate equilibria. We also discuss the kinds of indeterminacy of intermediate equilibria. In Section VII, we analyze an example with three consumers and three commodities and calculate explicitly a two-stage market equilibrium, which is intermediate in the sense that the equilibrium allocation Pareto dominates the initial endowment allocation (autarky in this case). We also outline plausible refinements of the notion of intermediate equilibrium that rationalize particular types of equilibria by specifying decision rules for the external agency. In Section VIII, we discuss the relationship between trade in single and in dual market places. Section IX concludes.

II. RELATION TO THE FURTHER LITERATURE

A. Traditional Trade Theory

In this paper, we develop a hierarchical model of international trade. The standard trade literature (e.g., Bell 2006) assumes that all nations and their constituents trade in a single market place, though each nation may face a different price system due to tariffs or subsidies. We view trade within and trade among groups as activities that take place in two different market places linked by an external agency. We shall demonstrate that different market places for global and local trade and, consequently, double budget constraints play an important role for equilibrium existence, the occurrence of wedges between local and global prices, and welfare considerations.

As an example consider the standard setting of conventional trade theory with a two-dimensional commodity space. Here we observe a stark contrast between single and double budget constraints. For international trade with a single budget constraint, Yun (1995) obtains a continuous path from a distortionary equilibrium to the optimum. For two-stage market equilibria, we obtain a dichotomy, either autarky or free trade, and consequently the absence of a continuous path between the two.

In general, two-stage market equilibria cannot be associated with equilibria in one market place. For three or higher dimensional commodity spaces, we will, however, establish in Section VIII a link between two-stage market equilibria and trade in a single market place with hypothetical tariffs and subsidies and balanced government budgets.

The existence of dual market places and the occurrence of price distortions is reminiscent of several non-Walrasian market theories. We now discuss the parallels and differences between our model and strategic market games, fixed price equilibria and incomplete markets.

Our paper is complementary to the literature on theories on how to explain persistence of substantial variations of retail prices across countries and, in particular, across countries in the European Union. Why retailers cause little arbitrage was set out in important papers by Maskus and Chen (2004), Chen and Maskus (2005), and Ganslandt and Maskus (2007) in the context of vertical control, and by Maskus and Stahler (2014) in the context of principal-agent relationships of original home manufacturers and foreign retailers. These papers show that parallel trade fails to eliminate retail price differentials. Lutz (2004) identifies arbitrage barriers in the car market as a main reason why prices differ across countries. We start from the assumption that within some ranges, retail price differentials across countries are not eliminated.

B. Strategic Market Games

Koutsougeras (2003) develops a strategic market game where a commodity can be traded in multiple "trading posts." He obtains instances of violation of "the law of one price," even though all traders can trade in all places. Attempts to take advantage of the apparent arbitrage opportunities are to no avail, because traders' actions impact on prices in such a way that the attempts make the opportunities disappear. Since the traders are aware of this effect, the price discrepancies are sustainable in equilibrium. In our model, consumers are price-takers and confined to their own market place, a very different non-Walrasian environment. The common feature is the existence of multiple market places and wedges between prices in different locations.

C. Fixed Price Equilibria

Dreze (1975) shows that under standard assumptions, a pure exchange economy with price rigidities has an equilibrium with uniform rationing of net trades. In particular, for every fixed strictly positive price vector, there exists an equilibrium with uniform rationing of net trades. Herings (1998) provides conditions so that for every fixed strictly positive price vector, there exists a continuum of equilibria with quantity rationing. Hence there exists multiplicity of equilibria with respect to equilibrium prices and multiplicity given equilibrium prices. This indeterminacy prevails even if all consumers face the same price system and the same rationing scheme. Citanna et al. (2001) extend the analysis to economies with production and find that a continuum of supply constrained equilibria can exist at Walrasian prices. In contrast, multiplicity of equilibria in our model is driven by price wedges between local markets on the one hand and local markets and the global market on the other hand.

D. Incomplete Markets

It is well known from the study of allocation under uncertainty and a comparison of complete versus incomplete markets that the difference between single and multiple budget constraints is a significant one. Like Debreu's (1970) result in the case of complete markets, Geanakoplos and Polemarchakis (1986) show generic existence and finiteness of competitive equilibria for finite pure exchange economies with finitely many states of nature and real numeraire assets. In stark contrast, Geanakoplos and Mas-Colell (1989) find that with finitely many states of nature and financial assets, there tends to exist a continuum of equilibria exhibiting real indeterminacy. Two equilibria may only differ in the "price levels" in various states, yet yield different allocations. In our model, the analogue of the latter cannot happen: If all local equilibrium price systems are collinear, then the equilibrium allocation also arises as a free trade equilibrium allocation. As a rule, there are only finitely many such allocations--and they are Pareto optimal. In the incomplete market context, the transferability of wealth across states proves crucial. In our framework, markets are complete from an individual's perspective in that the individual can trade all commodities (assets) and is not concerned about wealth transfers across markets. Markets are also complete, yet distorted vis-a-vis the global market, in inter-group trade.

III. TWO-STAGE MARKET ALLOCATIONS

We consider a model of a finite pure exchange economy where commodities and consumer characteristics are standard. The distinguishing feature of the model is the allocation mechanism, a two-stage hierarchy of markets.

A. Commodities, Consumers, and Allocations

There exists a finite number l [greater than or equal to] 1 of different commodities. Thus the commodity space is [R.sup.l]. Each commodity is a private good.

There is a finite population of consumers or individuals, represented by a set I. A generic consumer is denoted by i or j. Each consumer i [member of] I has consumption set [X.sub.i] = [R.sup.l]. The endowment of i is a commodity bundle [w.sub.i] [member of] [R.sup.l.sub.++]. For a given price system p [member of] [R.sup.l], [B.sub.i](p) = {[x.sub.i] [member of] [X.sub.i]| p x [x.sub.i] [less than or equal to] p x [w.sub.i]} denotes i's budget set. Individual i has continuous, strictly convex, and strictly monotonic preferences on [X.sub.i] represented by a utility function [U.sub.i]: [X.sub.i] [right arrow] R.

An allocation of commodities assumes the form x = [([x.sub.i]).sub.i[member of]I] and belongs to the allocation space X [equivalent to] [[PI].sub.j[member of]I] [X.sub.j]. In x [member of] X, the consumption bundle [x.sub.i] [member of] [X.sub.i] is assigned to individual i [member of] I.

B. Groups and Two-Stage Markets

The population I is partitioned into groups or nations, that is, there exists a partition P of I into nonempty subsets. P has generic elements h and consists of H groups frequently labeled h = 1, ..., H. For each group h [member of] P, set [X.sub.h] = [[PI].sub.i[member of]h] [X.sub.i], the consumption set for group h. [X.sub.h] has generic elements [x.sub.h] = [([x.sub.i]).sub.i[member of]h]. If x [member of] X is a commodity allocation, then consumption for group h is [x.sub.h] = [([x.sub.i]).sub.i[member of]h], the restriction of x = [([x.sub.i]).sub.i[member of]I] to h. Thus group h attains the group consumption [x.sub.h] [member of] [X.sub.h]. We set [w.sub.h] [equivalent to][[SIGMA].sub.i[member of]h], the social endowment of group h.

Next we define two-stage market equilibria. The definition is based on the given partition P of the consumer population I into groups. For i [member of] I, P(i) denotes the group to which individual i belongs.

Definition: A two-stage market equilibrium is a tuple (p;[([q.sub.h]).sub.h[member of]P];x) with p [member of] [R.sup.l.sub.+] being an external (global, world) price system, [q.sub.h] [member of] [R.sup.l.sub.+] being an internal (local, domestic) price system for each h [member of] P, and x being an allocation of commodities to consumers, such that

(1) [x.sub.i] [member of] arg max {[U.sub.i] ([y.sub.i]) | [y.sub.i] [member of] [B.sub.i] ([q.sub.h])}

for each individual i [member of] I and household (group, country) h = P(i);

(2) [mathematical expression not reproducible]

(3) [mathematical expression not reproducible]

C. Discussion of the Equilibrium Concept

Central Ideas. Two empirical facts discussed in the introduction motivate the definition and analysis of two-stage equilibrium. First, there are limits to international goods arbitrage and differences between the prices of tradeable goods in different locations can persist. Second, trade is organized in two stages: global trade and domestic trade linked by an external agency or trading hub.

Accordingly, individuals can only trade freely within their group (household, country) h, taking the internal price system qh as given. This condition is formalized as Equation (1). Under our assumptions on preferences, individual budget constraints are binding and Walras' Law holds group by group:

(4) [q.sub.h][Z.sub.h] = 0

where [z.sub.h] [equivalent to] [[SIGMA].sub.i[member of]h][x.sub.i] - [[SIGMA].sub.i[member of]h] [w.sub.i] is the group's aggregate excess demand. The fact that individuals can only trade within their respective group does not necessarily mean that the group's internal market has to be cleared. Rather the group h as a trading block can have a nonzero net trade [z.sub.h], with the rest of the world. In external trade, the group takes the external price system p as given and is subject to an external budget constraint. This condition is reflected in Equation (2). Because preferences are strictly monotonic, inequality Equation (2) holds as an equality.

Finally, Equation (3) is the formal expression of the global market-clearing condition. Conditions (2) and (3) imply for each group h balancing of its external trade account:

(5) p[z.sub.h] = 0.

Note that the equilibrium allocation of a group has to satisfy two budget constraints: the budget constraint with respect to the local prices that enter the individual budgets, giving rise to Equation (4), and a second budget constraint with respect to global prices, giving rise to Equation (5). This reflects our view of trade in two different market places.

Procedure. We will proceed as follows. In the next sections, we will assume that the sole objective of the external agency is market-clearing. This will allow us to discuss the set of possible two-stage equilibria. In Section VII, we impose further objectives for the external agency (e.g., welfare maximization of specific countries or profit maximization) which reduces the set of two-stage equilibria and provides interesting applications. Moreover, we impose limits on price differentials between domestic and global prices to account for limited international goods arbitrage.

We note that rather general assumptions guarantee the existence of Walrasian equilibria for an economy. Assuming strictly positive endowments and monotonic, strictly convex, and continuous preferences for each individual consumer suffices. Ensuring equilibrium existence facilitates the discussion of efficiency of two-stage market equilibria. Moreover, the assumptions on preferences imply that all individuals and groups exhaust their budgets and hence the budget balancing conditions (4) and (5) hold. Finally, the assumptions guarantee that [q.sub.h] [much greater than] 0 for each h [member of] P.

IV. FREE TRADE AND AUTARKY

To begin with, we can state the following proposition for nontrivial P, that is, 1 < [absolute value of P] < [absolute value of I];

PROPOSITION 1. For generic consumer characteristics, there are at least two two-stage market equilibria.

Proof. For a proof, we construct a "free trade equilibrium" which is Pareto optimal and an "autarky equilibrium" which is not Pareto optimal.

Free Trade Equilibrium: Under our standard assumptions, there exists a Walrasian equilibrium ([??]; [??]) for the entire economy. Set [[??].sub.h] = [??] for h [member of] P, and [??] = [([[??].sub.i]).sub.i[member of]I]. Then ([mathematical expression not reproducible]) is a two-stage market equilibrium and x is a Pareto-optimal allocation corresponding to a "free trade equilibrium."

Autarky Equilibrium: Also under standard assumptions, there exists a Walrasian equilibrium ([q*.sub.h];[x*.sub.h]) for the sub-economy formed by the members of any group h [member of] P. Now fix a family ([q*.sub.h], [x*.sub.h]), h [member of] P, of such "local" equilibria, choose an arbitrary p* [member of] [R.sup.l] and set x* = [([x*.sub.i]).sub.i[member of]I]. Then (p*; [([q*.sub.h]).sub.h[member of]P]; x*) constitutes a two-stage market equilibrium. It is an "autarky equilibrium" where each group has zero external trade since [x*.sub.h] = [w.sub.h]. As a rule, the internal equilibrium price systems [q*.sub.h], h [member of] P, are not collinear and the equilibrium allocation x* is not Pareto-optimal. We note that p* can be chosen arbitrarily as there is no global trade.

The existence of these two particular equilibria establishes the claim of the proposition.

V. THE TWO-DIMENSIONAL CASE

After having established two distinct types of equilibria--a free trade equilibrium that is Pareto-optimal and an autarky equilibrium--an obvious question is whether there is room for intermediate degrees of equilibrium inefficiencies and how their properties differ from trade with price distortions in a single market place. As will become clear, the existence and nature of intermediate equilibria depends crucially on the dimension of the commodity space. We first examine exchange economies with two goods that are reminiscent of most of the classical international trade models.

PROPOSITION 2. Suppose l = 2. Then, at a two-stage market equilibrium with p [much greater than] 0, the world is divided into an autarkic trade zone and a free trade zone. One of the zones may be empty.

Proof. Because of our assumptions on preferences, a two-stage market equilibrium requires

[mathematical expression not reproducible]

Hence, we also have

[mathematical expression not reproducible]

If [[SIGMA].sub.i[member of]h][x.sub.i] [not equal to] [[SIGMA].sub.i[member of]h] [w.sub.i], then both [q.sub.h] and p are orthogonal to [[SIGMA].sub.i[member of]h] ([x.sub.i] - [w.sub.i]) [not equal to] hence they are collinear. But since p [much greater than] 0 and [q.sub.h] > 0, this implies that the two price systems are identical up to normalization. Therefore with prices restricted to the unit simplex, [[SIGMA].sub.i[member of]h][x.sub.i] = [[SIGMA].sub.i[member of]h] [w.sub.i] (autarky) or [q.sub.h] = p (free trade) prevails for each country h.

Thus, with a two-dimensional commodity space, no intermediate equilibria exist, except possibly equilibria that divide the world into an autarkic and a free trade zone. The basic geometric intuition is as follows: A group's aggregate consumption has to lie on two budget lines through the group's social endowment bundle, the line given by local prices and the line given by global prices. This can only happen if the two lines coincide--that is, the two price systems are collinear--or if the group's aggregate consumption is located at the intersection of the two lines--which means it is the group's social endowment bundle. This observation also suggests that nontrivial intermediate equilibria might exist with at least three commodities, since then the intersection of two budget hyper-planes through the same endowment bundle has at least dimension 1. This will be taken up in the following section.

We note that the above result is in contrast to classical international trade theory, where the existence of competitive equilibria is guaranteed under general conditions (see, e.g., Dixit and Norman 1980) even when significant, but not totally prohibitive, barriers to international trade exist in the form of tariffs and nontariff restrictions.

The contrast between single and double budget constraints is further accentuated by the existence and nonexistence, respectively, of a continuous path from autarky to free trade. Ideally, one would like to trace a continuous path from an inefficient, distorted equilibrium (e.g., autarky) to the optimal free trade equilibrium, perhaps even continuously improving welfare along the path. With single budget constraints, the important work of Yun (1995) shows that, indeed, a unique path can be constructed, beginning at a distortion equilibrium, going through proportional changes of price distortions, and ending at the targeted optimum. With double budget constraints, the nonexistence of intermediate equilibria rules out any path between the two polar outcomes.

VI. EXISTENCE AND INDETERMINACY OF INTERMEDIATE EQUILIBRIA

In this section we establish existence of intermediate equilibrium allocations for three-or higher dimensional commodity spaces. The propositions also demonstrate how one can construct special classes of intermediate equilibria. At the end of this section, we illustrate what types of indeterminacy of intermediate equilibria can arise.

A. Existence

In order to avoid pathological cases we assume throughout this subsection that autarky and free trade differ and that under free trade, each group has a nonzero net trade and each individual attains a strictly positive consumption bundle.

PROPOSITION 3. Suppose l [greater than or equal to] 4 and 2 < H < l. Then an intermediate two-stage equilibrium exists for generic consumer characteristics.

The proof of the Proposition is given in the Appendix. The intuition for the proof of the Proposition runs as follows: The set of individuals is divided into two groups and an intermediate equilibrium is constructed by local prices that are equal to the market clearing prices when each group is considered separately as a subeconomy. Since market clearing prevails for both subeconomies, overall market clearing is also guaranteed.

Proposition 3 can easily be extended to type economies. A type economy in our context is defined as follows. Two groups are of the same type if their excess demand function is identical. The most natural case occurs when both groups contain the same number of individuals and each individual in one group has an identical counterpart with respect to endowments and preferences in the other group. We obtain:

PROPOSITION 4. Suppose l [greater than or equal to] 3, and M types of groups with 2 [less than or equal to] M < l. Then, generically, an intermediate two-stage equilibrium exists.

The proof of Proposition 4 is given in the Appendix. The proof of Proposition 4 follows a similar albeit slightly modified logic as Proposition 3.

Given that one may distinguish between hundreds if not thousands of commodities, the conditions H < l and M < l are quite plausible. Still, it is not always necessary to impose H < l or M < l in order to establish generic existence of intermediate equilibria:

PROPOSITIONS. Suppose l > 2 and that all groups are singletons. Suppose further that for each i [member of] I, interior consumption bundles are preferred to boundary ones, the utility function is concave and differentiable in the interior of [X.sub.i], and [w.sub.i] belongs to the interior of [X.sub.i]. Then, generically, intermediate equilibria exist.

The proof of Proposition 5 is given in the Appendix. The equilibria in Proposition 5 are constructed by assuming that individuals are limited in making trades in at least one commodity. Then, each group has nonzero net trades with the rest of the world but those trades differ from free trade. Notice that in the intermediate two-stage market equilibrium constructed in the proof, all commodities are tradeable, despite the fact that the construction is based on an artificial economy E where every group h cannot trade a specific commodity [k.sub.h]. Finally, we note that with the above construction method, a variety of intermediate equilibria can be constructed.

In traditional international trade theory, the central results such as the law of comparative advantage and Heckscher-Ohlin theorems are sensitive to dimensionality and survive only as correlations or in an average sense in higher dimensions. Existence of equilibria including distortions is, however, not sensitive to dimensionality (see, e.g., Ethier 1984). For trade in two market places, dimensionality is decisive.

We have phrased the central existence theorem in terms of one-person groups. However, the existence result can be extended to groups containing an arbitrary number of consumers, as long as there exists a representative consumer for each group, that is, the aggregate demand function of each group is generated by the demand function of its representative consumer. Then we can apply the same arguments as above to establish the existence of intermediate equilibria for groups with an arbitrary number of individuals.

B. Indeterminacy

As is well known, a finite pure exchange economy can have multiple Walrasian equilibrium allocations. But as a rule, the economy is regular and has a finite number of equilibrium allocations. See for instance Propositions 17.D.5 and 17.D.2 in Mas-Colell, Whinston, and Green (1995). In our model, the possibility of two kinds of indeterminacy of intermediate equilibria exists. First, there is the possibility of significant nominal or price indeterminacy. The world market price systems supporting a particular equilibrium allocation may span a multidimensional subspace of the commodity space, an indeterminacy that is not eliminated by price normalization. Second, there can be a continuum of equilibrium allocations and utilities. We briefly illustrate both kinds of indeterminacy.

Nominal Indeterminacy. Suppose 2 < H < l - 1 in Proposition 3. Then in the proof of Proposition 3, one can choose p [not equal to] 0 in [[intersection].sub.h] [c.sup.[perpendicular to].sub.h] and the latter has at least dimension 2. The orthogonal complement of the excess demand vector [c.sup.[perpendicular to].sub.h] has been defined in the proof of Proposition 3 in the Appendix.

Real Indeterminacy. Rewrite Equations (A1)-(A6) as

F([q.sup.2.sub.1], [q.sup.3.sub.1], [q.sup.1.sub.2], [q.sup.1.sub.3], [q.sup.3.sub.2], [q.sup.2.sub.3], [p.sub.2], [p.sub.3]) = 0

where F: [R.sup.8] [right arrow] [R.sup.6]. Then at the intermediate equilibrium values ([q.sup.2.sub.1], [q.sup.3.sub.1], [q.sup.1.sub.2], [q.sup.1.sub.3], [q.sup.3.sub.2], [q.sup.2.sub.3], [p.sub.2], [p.sub.3]) = (2,3; 2,2/3,2/3,3/5,5/2,7/3),

[mathematical expression not reproducible]

Since this sub-Jacobian matrix has full rank and F is continuously differentiate, the implicit function theorem applies at [y*.sub.1] = ([q.sup.2.sub.1], [q.sup.3.sub.1]) = (2,3) and [y*.sub.2] = ([q.sup.1.sub.2], [q.sup.1.sub.3], [q.sup.3.sub.2], [q.sup.2.sub.3], [p.sub.2], [p.sub.3]) = (2,2/3,2/3,3/5,5/2,7/3): There exist open neighborhoods [U.sub.1] of [y*.sub.1] and [U.sub.2] of [y*.sub.2] and a continuously differentiate mapping [phi] : [U.sub.1] [right arrow] [U.sub.2] such that for every [y.sub.1] [member of] [U.sub.1], [y.sub.2] = [phi][(.sub.y1]) is the unique element of [U.sub.2] satisfying F([y.sub.1], [y.sub.2]) = 0. Therefore, every price system [q.sub.1] = (1, [y.sub.1]) with [Y.sub.1] [member of] [U.sub.1] is part of a two-stage market equilibrium price system in our main example. Different [y.sub.1] yield a different local price system [q.sub.1] and different equilibrium consumption for consumer 1 (group 1, country 1). For [y.sub.1] sufficiently close to [y*.sub.1], the corresponding two-stage market equilibrium is an intermediate equilibrium. Finally, choosing [y.sub.1] = [lambda] * [y*.sub.1], with [lambda] varying in a sufficiently small neighborhood of 1, generates a continuum of equilibrium utility levels for consumer 1 (group 1, country 1).

The real indeterminacy could be generalized significantly. In any setting with at least three commodities and three groups, the following conditions ensure the presence of real indeterminacy. Strictly monotone and concave utility function of consumers, positive endowments of each group in some commodities, and positive trade of all commodities in an intermediate equilibrium generates nondegenerate market clearing and budget balance conditions with more unknown variables than equations. This ensures the application of the implicit function theorem and the presence of real indeterminacy.

VII. EQUILIBRIUM REFINEMENTS

Until now we have not specified any objective of the external agency other than market-clearing. This can leave several degrees of freedom for market-clearing price systems and can generate significant indeterminacy of intermediate equilibria when commodity spaces are higher dimensional. In Section VI.B, we found the possibility of price indeterminacy that cannot be eliminated by price normalization and the possibility of real indeterminacy that can produce a continuum of potential equilibrium utility levels.

In this section we outline plausible refinements of the notion of intermediate equilibrium that rationalize particular types of equilibria and reduce or eliminate indeterminacy of intermediate equilibria. Our main emphasis lies on the effects of control of the external agency by a single group which in the following is identical to a country. We will also outline some further extensions.

We will proceed as follows. In the next subsection, we introduce an example that is used to show that intermediate equilibria exist. Then, for every possible equilibrium refinement we use the example to illustrate the consequences of such a refinement.

A. An Example

In this subsection we illustrate intermediate equilibria in which external agencies only balance their budget by a simple example. Because of Proposition 2 there must be at least three commodities. While it is possible to construct intermediate equilibria with only two groups, we proceed with three groups as the example will be used throughout this section and is flexible enough to incorporate different objectives of the external agency. Suppose l = 3, [absolute value of I] = H = 3. Thus, each group contains exactly one individual. For every individual i the utility function is given by

(6)

[U.sub.i] = [U.sub.h] = [1/3] ln ([x.sup.1.sub.h]) + [1/3] ln ([x.sup.2.sub.h]) + [1/3] ln ([x.sup.3.sub.h]), I = h = 1, 2, 3,

where [x.sup.k.sub.h] denotes the consumption of the k-th good by individual h. The endowments are given by

[w.sub.1] = (1,0,0), [w.sub.2] = (0,1,0), [w.sub.3] = (0,0,1).

The autarky solution leaves every individual with his endowments. We note that autarky is a particularly bad outcome for consumers as [mathematical expression not reproducible]. Due to symmetry, it is obvious that free trade is characterized by [p.sup.1] = [p.sup.2] = [p.sup.3] = 1 and

[x.sub.1] = ([1/3], [1/3], [1/3]), [x.sub.2] = ([1/3], [1/3], [1/3]), [x.sub.3] = ([1/3], [1/3], [1/3]).

To calculate an intermediate equilibrium, we normalize prices so that [p.sup.1] = 1, [q.sup.1.sub.1] = 1, [q.sup.2.sub.2] = 1, [q.sup.3.sub.3] = 1. [q.sup.k.sub.h] denotes the local price for good k in group h.

FACT 1. There exists an intermediate equilibrium characterized by

p = (1, [7/5], [5/2]),

[q.sub.1] = (1, 2, 3),

[q.sub.2] = (2, 1, [3/5])

[q.sub.3] = ([2/3], [2/3], 1)

and the allocation

[x.sub.1] = ([1/3], [1/6], [1/9]), [x.sub.2] = ([1/6], [1/3], [5/9]), [x.sub.3] = ([1/2], [1/2], [1/3]).

Proof. See Appendix.

This allocation differs from autarky and free trade. Thus, the allocation together with the global and local prices constitute an intermediate equilibrium and all individuals are better off than under autarky. Note that the third individual is better off in the intermediate equilibrium than under free trade. However, the allocation in the intermediate equilibrium is not Pareto-efficient. Individuals would like to trade again in one market place, starting from the allocations they received in the intermediate equilibrium.

For later use we state the system of equations that describe market clearing and budget balance for each external agency derived in the proof of Fact 1 which in turn determines the remaining global and local prices beyond the price normalization

(7) [q.sup.1.sub.2] + [q.sup.1.sub.3] = 2[q.sup.1.sub.2][q.sup.1.sub.3]

(8) [q.sup.2.sub.1] + [q.sup.2.sub.3] = 2[q.sup.2.sub.1][q.sup.2.sub.3]

(9) [q.sup.3.sub.1] + [q.sup.3.sub.2] = 2[q.sup.3.sub.1][q.sup.3.sub.2]

(10) [p.sub.2][q.sup.3.sub.1] +[ .sub.p3][q.sup.2.sub.1] = 2[q.sup.2.sub.1][q.sup.3.sub.1]

(11) [q.sup.3.sub.2] + [p.sub.3][q.sup.1.sub.2] = 2[p.sub.2][q.sup.3.sub.2][q.sup.1.sub.2]

(12) [q.sup.2.sub.3] + [p.sub.2][q.sup.1.sub.3] = 2[p.sub.3][q.sup.1.sub.3][q.sup.2.sub.3].

B. Different Bargaining Schemes

Control of External Agency by a Single Country. In practice, the operations of external agencies are often located in particular countries. For instance, Switzerland is famous for hosting platforms for global commodity-trading while platforms for agricultural products are operating in the United States. To capture such aspects we denote by [h.sup.c] the country that controls the external agency. The external agency sets global prices. A plausible equilibrium refinement is thus as follows:

Definition: A two-stage market equilibrium with control of global prices by the external agency by country [h.sup.c] is a tuple (p;[([q.sub.h]).sub.h[member of]P];x) such that:

(13) ((p;[([q.sub.h]).sub.h[member of]P];x)

is a two-stage market equilibrium. Moreover, there exists no other two-stage market equilibrium ([mathematical expression not reproducible]) with

(14) [U.sub.i] ([[??].sub.i]) [greater than or equal to] [U.sub.i] ([x.sub.i]) [for all]i [member of] [h.sup.c],

(15) [U.sub.j]([[??].sub.j]) > [U.sub.j] ([x.sub.j]) for some j [member of] [h.sup.c].

We note that the external agency acts as the monopolist regarding global prices, but cannot influence local prices directly--only indirectly through its choice of global prices. Essentially, we require then that the external agency controlled by country [h.sup.c] will select global prices such that the ensuing allocation is optimal for group [h.sup.c] given the hierarchical nature of trade. Optimality for group [h.sup.c] means that there is no other two-stage equilibrium that is Pareto superior for the individuals in group [h.sup.c].

Equilibrium refinements can be too stringent at times. In our context, there exists a free trade equilibrium under standard assumptions and, consequently, there exists at least one two-stage market equilibrium. Let us normalize prices so that the price systems [mathematical expression not reproducible] belong to the unit price simplex in[ .sup.Rl]. Suppose that the set of price systems [mathematical expression not reproducible] which are part of a two-stage market equilibrium is a compact subset [mathematical expression not reproducible] of the unit price simplex in [R.sup.l]. Further assume that each consumer i [member of] [h.sup.c] has a continuous indirect utility function [mathematical expression not reproducible]. Finally, assume that group [h.sup.c] aims at maximizing a utilitarian social welfare function of the form [mathematical expression not reproducible]. (8) Then there exists a [mathematical expression not reproducible] that maximizes [mathematical expression not reproducible]. By definition, [mathematical expression not reproducible] is associated with a two-stage market equilibrium ((p*[([q.sub.h]).sub.h[member of]P]; x) where [mathematical expression not reproducible] and country [h.sup.c] is in control of the external agency.

Two qualifications are warranted: First, the equilibrium ((p*[([q.sub.h]).sub.h[member of]P]; x) may be a free trade equilibrium if there is no two-stage market equilibrium at which group [h.sup.c] fares better than under free trade. In such cases, it is not detrimental with regard to Pareto optimality that one country controls trade. In general, we can examine the question, which country or group of countries should control trade if such trade has to be managed. This will be illustrated in the final subsection of VII.B. Second, setting the global price system equal to p* may bring about the desired equilibrium ((p*[([q.sub.h]).sub.h[member of]P]; x), but there may exist other two-stage market equilibria that are consistent with the global price system p* and are less desirable for group [h.sup.c].

While equilibrium refinements tend to reduce indeterminacy, they do not a priori yield uniqueness. We next illustrate, however, how two-stage market equilibria with control by a single group can indeed yield uniqueness of equilibria. Moreover, the ensuing equilibrium differs from free trade.

We use our main example from Section VII.A and assume that one country, say country 3, controls the external agency. We observe:

FACT 2. Suppose that a two-stage equilibrium with control by country 3 exists. Then it differs from autarky and free trade.

The fact follows from the observation made in Section V. By selecting [p.sub.2] = [7/3] and [p.sub.3] = [5/2] country 3 induces a two-stage market equilibrium with [q.sup.2.sub.1] = 2, [q.sup.3.sub.1] = 3, [q.sup.1.sub.2] = 2, [q.sup.3.sub.2] = [3/5], [q.sup.1.sub.3] = [2/3], [q.sup.2.sub.3] = [2/3] and the equilibrium allocation

[x.sub.1] = ([1/3], [1/6], [1/9]),

[x.sub.2] = ([1/6], [1/3], [5/9]),

[x.sub.3] = ([1/2], [1/2], [1/3]).

Hence, in this two-stage market equilibrium group 3 is strictly better off than in autarky and free trade. As the intermediate equilibrium with control by the third country has to generate at least the same utility for the third group as in this arbitrarily chosen example, it must differ from free trade and autarky which proves Fact 2.

We next show how the equilibrium refinement yields uniqueness--up to price normalization--in our main example. Using the equilibrium conditions from Section V, the problem of the external agency controlled by country 3 can be formulated as follows:

[mathematical expression not reproducible]

We obtain:

FACT 3. There exists a unique two-stage equilibrium with control by country 3. It is given by the prices

p = (1,1,2), [q.sub.1] = (1, [infinity], 1), [q.sub.2] = ([infinity], 1, 1), [q.sub.3] = ([1/2], [1/2], 1),

and the allocation

[x.sub.1] = ([1/3], 0, [1/3]), [x.sub.2] = (0, [1/3], [1/3]), [x.sub.3] = ([2/3], [2/3], [1/3]).

The proof of Fact 3 is given in the Appendix. We note that the third country, which controls the external agency, can obtain much higher consumption and utility through its judicious choice of the global prices that make endowments of the controlling country particularly valuable.

We observe that two of the local equilibrium prices are infinite. Hence, the derived two-stage market equilibrium can at most be understood as the limit of a sequence of equilibria constructed as follows: Assume a finite upper bound on local and global prices and allow the upper bound to become larger and larger. In the final subsection of VII.C we derive two-stage market equilibria with control of group 3 when there are limits on the price differential between local and global goods. For a given limit on price differentials we will obtain a unique two-stage market equilibrium with finite prices.

Control by Many Countries. We could allow that an arbitrary set of countries controls the external agency. We denote by [H.sup.c] the set of groups controlling the external agency and by [I.sup.c] the union of those groups.

The second property of the equilibrium definition in Section VII. A has to be replaced by the following property:

There exists no other two-stage market equilibrium ([mathematical expression not reproducible]) s.t.

[U.sub.i] ([[??].sub.i]) [greater than or equal to] [U.sub.i] ([x.sub.i]) [for all]i [member of] [I.sup.c]; [U.sub.j] ([[??].sub.j]) > [U.sub.j] ([x.sub.j]) for some j [member of] [I.sup.c].

The interpretation is that no other two-stage market equilibrium exists which would be a Pareto improvement for the set of individuals in groups that control the external agency. The following fact is obvious:

FACT 4. The unique two-stage market equilibrium with control of the external agency by group 3 is also a two-stage market equilibrium with control of the external agency by [H.sup.c] if group 3 belongs to [H.sup.c].

Several remarks are in order. First, if for instance groups 1 and 2 control the external agency, the two-stage equilibrium with control by country 3 is not an equilibrium anymore as already free trade would be better for groups 1 and 2. Second, even if all countries control the external agency free trade may not necessarily emerge as the equilibrium with control by group 3 continues to be an equilibrium. However, the first welfare theorem implies that free trade constitutes a two-stage equilibrium when all countries jointly control the external agency.

We could further specify a detailed decision process regarding the selection of global prices when several countries control the external agency. Again we can illustrate this possibility for the leading example from Section VII.A, for example, by assuming that groups 1 and 2 control the external agency and arrive at global prices via the Nash bargaining solution. Let us also assume that the bargaining power is the same among groups. We obtain for this case:

FACT 5. Suppose there exists a lower bound [epsilon] > 0 for the relative global market price of good 3 in any two-stage market equilibrium. Then there exists a unique two-stage market equilibrium with control of the external agency and symmetric Nash bargaining by groups 1 and 2. This equilibrium differs from autarky and free-trade and is given by the prices

p = (1,1, [epsilon]),

[q.sub.1] = (1, 1/2-[epsilon], 1), [q.sub.2] = (1/2-[epsilon], 1, 1), [q.sub.3] = (1/[epsilon], 1/[epsilon], 1),

and the allocation

[x.sub.1] = ([1/3], 2-[epsilon]/3, [1/3]), [x.sub.2] = (2-[epsilon]/3, [1/3], [1/3]), [x.sub.3] = ([epsilon]/3, [epsilon]/3, [1/3]).

The proof of Fact 5 is given in the Appendix. Now for [epsilon] < 1, country 1 and country 2--which control the external agency--can achieve higher consumption and higher utility by selecting optimal global prices that makes their excess supply in this market particularly valuable. The third country is harmed and for small [epsilon] suffers considerably from the control of the other countries.

Also note that as a by-product of Fact 5, we obtain a family of intermediate equilibria parametrized by [epsilon] [member of] (0, 1) such that the equilibrium welfare of group 3 increases and that of groups 1 and 2 decreases as e increases.

Shared Control by All Countries and Free Trade. We next investigate the situation when all trading hubs share control of global prices and where all of them have equal bargaining power.

To construct the two-stage equilibrium with control by all groups 1, 2, and 3, we denote by B the set of global prices such that for every p [member of] B, there exists a two-stage market equilibrium ((p;[([q.sub.h]).sub.h[member of]P]; x). The problem facing the stakeholders of the external agencies is to maximize the Nash product

[mathematical expression not reproducible]

Again using the symmetry of country I and country 2 yields--as in the proof of Fact 5--that the first two groups obtain the consumption levels ([1/3], 2-[p.sub.3]/3, [1/3]) and (2-[p.sub.3]/3,[1/3],[1/3]), respectively, for given price[ .sub.p3]. The third group will consume ([p.sub.3]/3, [p.sub.3]/3, [1/3]) The problem thus can be reduced to finding [p.sub.3] such that the Nash product

[{-[1/3]ln([1/3]) + [1/3]ln(2-[p.sub.3]/3)}.sup.[2/3]] [{[2/3]ln([p.sub.3]/3) - [2/3]ln([1/3])}.sup.[1/3]]

is maximized. The first factor captures the excess utility of group 1 and group 2, which obtain a combined bargaining weight 2/3, and the second factor represents group 3. We observe that this expression is maximized for [p.sub.3] = 1 as otherwise one factor would become negative. Hence, we obtain the free trade equilibrium in which each group consumes the bundle ([1/3], [1/3], [1/3]). We thus obtain:

FACT 6. If all groups share control of global trade through their external agency with equal bargaining power, we obtain free trade as the unique equilibrium.

One could consider an alternative bargaining process in which the default point of groups is lower than assumed so far and bargaining power is heterogeneous across countries. By going through the same exercise as before one can show that free trade only occurs if default utility levels and bargaining power are the same across groups.

C. Limits on Price Differentials

Exogenous Limits. The set of equilibria could also be constrained by limits on the price differentials between local and global prices. For instance we could assume that all local and global prices have to satisfy

[absolute value of [q.sup.k.sub.h] - [p.sup.k]]<[DELTA] [for all]k, [for all]h

for some [DELTA] > 0. This would reflect the fact that goods arbitrage sets in when local and global prices diverge too much. Goods arbitrage constraints can be combined with the refinements of the equilibrium notion when some groups control the external agency.

As an illustration, we consider again the example in which group 3 controls the external agency. Let us assume [DELTA] = 1. Then the two constraints

[absolute value of [q.sup.1.sub.2] - [p.sub.1]] [less than or equal to] 1, [absolute value of [q.sup.2.sub.1] [p.sub.2]] [less than or equal to] 1

and

[q.sup.2.sub.1] = [p.sub.2]/2 - [p.sub.3], [q.sup.1.sub.2] = 1/2[p.sub.2] - [p.sub.3]

imply [p.sub.3] [less than or equal to] 2[p.sub.2] - [1/2] and [p.sub.3](l + [p.sub.2]) [less than or equal to] 2 + [p.sub.2].

We calculate the maximal value of [p.sub.3] that fulfills both constraints. Using [p.sub.2] = [p.sub.3]/2 + [1/4] from the first constraint yields 2[p.sub.3.sup.2] + 3[p.sub.3] - 9 [less than or equal to] 0. The maximal value of [p.sub.3] satisfying this constraint is [p.sub.3] = [3/2]. This implies [p.sub.2] = 1, [q.sup.2.sub.1] = 2, [q.sup.1.sub.2] = 2. The entire equilibrium is described by the price system

p = (1,1,[3/2]),

[q.sub.1] = (1,2,1),

[q.sub.2] = (2,1,1),

[q.sub.3] = ([2/3],[2/3],1),

and the allocation

[x.sub.1] = ([1/3], [1/6], [1/3]),

[x.sub.2] = ([1/6], [1/3], [1/3]),

[x.sub.3] = ([1/2], [1/2], [1/3]).

We observe that constraints from goods arbitrage limit the utility group 3 can achieve by controlling the external agency.

As a second illustration, let again [DELTA] = 1. Imposing the conditions [absolute value of [q.sup.1.sub.2] - [p.sub.1]] [less than or equal to] 1, [absolute value of [q.sup.1.sub.3] - [p.sub.1]] [less than or equal to] 1 in the following subsection yields [epsilon] [greater than or equal to] 1/2. Hence constraints from goods arbitrage limit the Nash-bargained utility level groups 1 and 2 can achieve when they control the external agency.

Endogenous Limits. Instead of taking price differentials between local and global prices to be exogenous, one could take another approach. Suppose that all groups have to achieve a minimal utility, say [U.bar] that is finite and below ln([1/3]) Such minimal utility levels restrict price wedges between local and global prices in the following way. The minimal utility constraint imposes an upper bound on price differentials between local and global price. We illustrate this property within the situation explored in Fact 5. The minimal utility [U.bar] translates into a lower bound [[epsilon].bar] on the good 3 in the two-stage market equilibrium described in Fact 5, through the follow condition for country 3:

[2/3]ln ([[epsilon].bar]/3) + [1/3]ln([1/3]) = [U.bar]

which yields

[[epsilon].bar] = exp([U.bar] + ln(3)).

We note that for 0 < [epsilon] < 1, the utility of country 3 is minimal. (9) This, in turn, implies that the largest price differential between local and global prices is given by 1/[[epsilon].bar] - 1.

D. Profit-Oriented External Agencies

A totally different approach is to consider the external agency as an institution that demands a fixed percentage f (0 < f < 1) from each trade and maximizes profits arising from external trade. We take f as exogenously given in this subsection. However, one might imagine that f is the level of the fee above which a second agency would enter and would offer a trading platform. The equilibrium notion can be adapted to a profit-oriented agency by substituting condition (5) by

(5')

[l.summation over (k=1)][p.sub.k][max {[z.sup.k.sub.h], 0} (1+f)+min {[z.sup.k.sub.h], 0}(1 - f)] [less than or equal to] 0 for each h [member of] P,

where [z.sup.k.sub.h] is the excess demand of group h for commodity k.

The objective of the trade agency is to maximize profits, as denoted by [pi]. Its problem is given by

[mathematical expression not reproducible]

subject to

(16) [x.sub.i] [member of] arg max {[U.sub.i] ([y.sub.i]) |[y.sub.i] [member of] [B.sub.i] ([q.sub.h])} with (5') and

(17)

[mathematical expression not reproducible]

In this specification, the fee at rate f is levied twice, both on positive and negative intergroup excess demands. As an example, we consider the variant of the model with limits on goods arbitrage. Specifically, we assume

(18) [absolute value of [q.sup.k.sub.h] - [p.sub.k]] [less than or equal to] 1 [for all]h, [for all]k.

We obtain:

FACT 7. Intermediate equilibria with profit-maximizing trade agencies can induce large deviations from free trade outcomes. For instance, for f=[epsilon] and arbitrarily small [epsilon]>0, an equilibrium exists with the allocation

[x.sub.1] = ([1/3], [1/6], 3-8[epsilon]/6),

[x.sub.2] = (3-8[epsilon]/6, [1/3], [1/6]),

[x.sub.3] = ([1/6], 3-8[epsilon]/6, [1/3]).

Proof. See Appendix.

The important insight of this example is that large deviations from free trade occur even is f is arbitrarily small. In fact, for f [less than or equal to][1/5], the equilibrium is independent of the size of f and exists for f arbitrarily small. The example is meant to illustrate one possible approach, but by no means exhausts all possibilities. Partly for that reason, two qualifying remarks are warranted:

(i) With a profit-oriented agency, not only is Equation (5) replaced by Equation (5'), but also the market-clearing condition (3) is replaced by Equation (17). This begs the question to what extent consuming the bundle [mathematical expression not reproducible] is optimal for the agency and its constituents, given the maximal profit [pi]. In the example, that choice can be rationalized with the utility function [mathematical expression not reproducible]. In general, the agency's objective could be a more intricate issue. (10)

(ii) With limits on goods arbitrage in the form of Equation (18), price normalization can have real effects. For instance, if one normalizes prices so that [q.sup.1.sub.1] = [q.sup.2.sub.2] = [q.sup.3.sub.3] = [1/2] while Equation (18) is still imposed, the equilibrium allocation changes.

We conclude that there are a variety of different and interesting approaches to model the external agency which can be pursued further.

VIII. HIERARCHICAL TRADE VERSUS TRADE IN ONE MARKET PLACE

Hierarchical trade differs from traditional trade theory in terms of the organization of markets. Hence, they cannot be compared directly. However, in one direction it is possible to establish a relationship--if we hypothetically impose tariffs. Let us denote the price distortion for commodity k in group h, for a given intermediate equilibrium ((p;[([q.sub.h]).sub.h[member of]P];x) by [[tau].sup.k.sub.h]. Thus:

[[tau].sup.k.sub.h] = [q.sup.k.sub.h]/[p.sup.k] - 1.

Moreover, let [[tau].sub.h] be the price distortion vector of group h. Let us now assume that all individuals trade in one market place. Moreover, assume hypothetically that each government imposes tariffs (or export subsidies) equal to [[tau].sub.h]. Let (([p.sup.n][([[tau].sub.h]).sub.h[member of]P];y) denote a competitive equilibrium in which each nation trades in a single market place, where y is the equilibrium allocation, [p.sup.n] is the world market price vector, and where group h faces the tariff vector [[tau].sub.h] and thus the price vector [p.sup.n] (1 + [[tau].sub.h]). Then the following proposition holds:

PROPOSITION 6. Suppose ((p;[([q.sub.h]).sub.h[member of]P];x) is a two-stage market equilibrium. Then there is a single market equilibrium with tariffs [[tau].sub.h], [p.sup.n] = p, y = x, and zero lump-sum transfers.

The proof is straightforward and carried out in a precursor of this paper (Gersbach and Haller 2007). We stress that this relationship is of a hypothetical nature as there are no governmentally imposed tariffs in the two-stage market equilibria. (11) Hence, the relationship requires that government budgets are balanced in the hypothetical one-stage market equilibrium, which is expressed by zero lump-sum transfers. Also as demonstrated in Gersbach and Haller (2007), the situation is in general different if we start with a single market equilibrium. Given a set of price distortions for a proper subset of groups, there are three possible cases. First, no corresponding two-stage-market equilibrium may exist. Second, wedges between local and global prices for the other groups may arise and thus the allocation and distortions in the two-stage market equilibrium will differ in general from those in the one-stage market equilibrium. Third, if lump-sum transfers are zero in the single-market equilibrium, then there exists a two-stage market equilibrium with the same allocation and distortions as the single market equilibrium.

IX. CONCLUSION

We have developed a simple model of trade in different market places. There remain several interesting open questions. First, how many intermediate equilibria are there in general? Specifically, if the number of commodities exceeds two, are there sufficient conditions for the existence of a continuous path from autarky to free trade? Second, how is the analysis affected by the incorporation of producers? Whereas the introduction of local producers seems to cause mainly notational complications, the modeling of multinational corporations constitutes a much more formidable challenge. Moreover, as already outlined in Gersbach and Haller (2007), several variations of the basic frameworks can be pursued, such as allowing for explicit tariffs, trade surpluses or deficits, for instance.

APPENDIX

A.1 Proof of Fact 1

The corresponding excess demand vector amounts to

[z.sub.1] = (-[2/3], 1/3[q.sup.2.sub.1], 1/3[q.sup.3.sub.1]),

[z.sub.2] = (1/3[q.sup.1.sub.2], [2/3], 1/3[q.sup.3.sub.2]),

[z.sub.3] = (1/3[q.sup.1.sub.3],1/3[q.sup.2.sub.3], -[2/3]).

Market clearing and the global budget constraint yield:

(Al) [q.sup.1.sub.2] + [q.sup.1.sub.3] = 2[q.sup.1.sub.2][q.sup.1.sub.3]

(A2) [q.sup.2.sub.1] + [q.sup.2.sub.3] = 2[q.sup.2.sub.1][q.sup.2.sub.3]

(A3) [q.sup.3.sub.1] + [q.sup.3.sub.2] = 2[q.sup.3.sub.1][q.sup.3.sub.2]

(A4) [p.sub.2] [q.sup.3.sub.1] + [p.sub.3][q.sup.2.sub.1] = 2[q.sup.2.sub.1][q.sup.3.sub.1]

(A5) [q.sup.3.sub.2] + [p.sub.3][q.sup.1.sub.2] = 2[p.sub.2][q.sup.3.sub.2][q.sup.1.sub.2]

(A6) [q.sup.2.sub.3] + [p.sub.2][q.sup.1.sub.3] = 2[p.sub.3][q.sup.1.sub.3][q.sup.2.sub.3].

Let us choose [q.sup.2.sub.1] = 2, [q.sup.3.sub.1] = 3. Solving the system of equations immediately yields [q.sup.2.sub.3] = [2/3], [q.sup.3.sub.2] = [3/5] and the remaining equations as:

[q.sup.1.sub.2] + [q.sup.1.sub.3] = 2[q.sup.1.sub.2][q.sup.1.sub.3] 3[p.sub.2] + 2[p.sub.3] = 12 3 + 5[p.sub.3][q.sup.1.sub.2] = 6[p.sub.2][q.sup.1.sub.2] 2 + 3[p.sub.2][q.sup.1.sub.3] = 4[p.sub.3] [q.sup.1.sub.3] [p.sub.3] = [5/2], [p.sub.2] [7/3], [q.sup.1.sub.3] = [2/3], [q.sup.1.sub.2] = 2 is a solution of this reduced system. The resulting intermediate equilibrium allocation is:

[x.sub.1] = ([1/3], [1/6], [1/9]), [x.sub.2] = ([1/6], [1/3], [5/9]), [x.sub.3] = ([1/2], [1/2], [1/3]).

A.2 Proof of Proposition 3

Let [x.sub.h] ([q.sub.h]) = [[SIGMA].sub.i[member of]h][x.sub.i] ([q.sub.h]) for h [member of] P, [q.sub.h] [member of] [R.sup.l.sub.+ denote the aggregate demand vector of group h at prices [q.sub.h]. Finally, denote the excess demand of group h at prices [q.sub.h] by [z.sub.h]([q.sub.h]) = [x.sub.h]([q.sub.h]) - [w.sub.h]. Let us divide the set of groups P into two nonempty distinct subsets [P.sup.1] and [P.sup.2] (P = [P.sup.1] [union] [P.sup.2]). Consider two pure exchange economies [E.sup.1] and [E.sup.2]. In [E.sup.1] ([E.sup.2]) consumers belong to groups in [P.sup.1] ([P.sup.2]), respectively. Take two corresponding equilibria with price vectors [p.sup.1] and [p.sup.2] and allocations denoted by [mathematical expression not reproducible]. Generically, [p.sup.1] or [p.sup.2] differs from both autarky and full trade prices. Consider for each group h the orthogonal complement of the equilibrium excess demand vector,

[c.sup.[perpendicular to].sub.h] = {[y.sub.h] [member of] [R.sup.l] |[y.sub.h] * [z.sub.h] ([p.sup.1]) = 0 if h [member of] [P.sup.1],[ .sub.yh] * [z.sub.h]([p.sup.2]) = 0 if h [member of] [P.sup.2], resp.}.

In general, [c.sup.[perpendicular to].sub.h] has dimension l - 1 or dimension l. Consider the intersection [[intersection].sub.h][c.sup.[perpendicular to].sub.h] taken over all groups h in P. This intersection has at least dimension 1 since we have at most l - 1 orthogonal complements (as H < l) and one intersection operation reduces the dimension at most by one. Now take any vector p [not equal to] 0 in [[intersection].sub.h] [c.sup.[perpendicular to].sub.h]. We claim that [mathematical expression not reproducible] is a two-stage market equilibrium, [p.sup.1] and [p.sub.2] generate local price systems for the corresponding groups. Moreover, ([mathematical expression not reproducible]) is an allocation of commodities. By construction, we have p * ([[SIGMA].sub.i[member of]h][x.sub.i] - [[SIGMA].sub.i[member of]h][w.sub.i]) = 0. Since market-clearing prevails for both sub-economies [E.sup.1] and [E.sup.2], overall market-clearing obtains.

A.3 Proof of Proposition 4

We can apply the same construction as in Proposition 3 with one additional consideration. When dividing the economy into two nonempty subsets, all groups of the same type have to be put into one exchange economy. Then, we have at most l - 1 different orthogonal complements of the equilibrium consumption vector, since each group of the same type has the same equilibrium excess demand vector. Again, by considering the intersection of all complements, we can find the global prices.

A.4 Proof of Proposition 5

Here we identify individual i with group {i) and, accordingly, label both individuals and groups by h = 1, ..., H.

Consider the following exchange economy, denoted by E. Individual h is allowed to trade except in one arbitrarily chosen commodity [k.sub.h]. Consider a corresponding equilibrium of E, denoted (p, x*). For each individual h, let

[q.sub.h] = grad [U.sub.h] ([x*.sub.h]).

Then, [q.sub.h] is a supporting price system for group (individual) h at [x*.sub.h] [much greater than] 0. We claim that (p;[([q.sub.h]).sub.h[member of]P];x*) is an intermediate two-stage equilibrium. We first observe that p and [q.sub.h] typically differ. Since generically, each group has a nonzero net trade under free trade, we can choose the commodity in which group h is not allowed to trade in E, so that the excess demand [x*.sub.h] - [w.sub.h] differs from that under free trade. Hence, the allocation under free trade is different from x*. Clearly, x* is a feasible allocation of commodities. Since l > 2 and every group is only restricted in trading of one commodity, it follows that, as a rule, groups have nonzero net trades with the rest of the world in the exchange economy E. Thus the allocation x* also differs from autarky. The incorporation of the nontradeable commodity into the budget constraint does not matter in E and, therefore, we have p * [z*.sub.h]= 0 for the group's excess demand [z*.sub.h] [[SIGMA].sub.i[member of]h]= ([x*.sub.i] - [w.sub.i]).

In the next step we show that [q.sub.h][z*.sub.h] = 0. Suppose that group h is not allowed to trade in commodity [k.sub.h] [member of] (1, ..., l} in the exchange economy E. Because of [x*.sub.h] [much greater than] 0 and the hypothesized properties of the utility functions, there exists a scalar [[lambda].sub.h] > 0 such that

(A7) [partial derivative][U.sub.h]([x*.sub.h])/[partial derivative][x.sup.k.sub.h] = [[lambda].sub.h][p.sup.k] for k [not equal to] [k.sub.h].

Equation (A7) characterizes the first-order conditions for an interior competitive equilibrium in E. Furthermore,

(A8) [partial derivative][U.sub.h]([x*.sub.h])/ [partial derivative][x.sup.k.sub.h] = [q.sup.k.sub.h] for all k.

Hence, we obtain:

[mathematical expression not reproducible]

The first term is zero because group h did not trade in commodity [k.sub.h]. The second term is zero since this represents the budget constraint of group h in the exchange economy E. Finally, market-clearing is guaranteed since all groups participate in the exchange economy E. Thus ((p, [([q.sub.h]).sub.h[member of]P];x*) is an intermediate equilibrium, that is a two-stage market equilibrium with the desired properties: no group enjoys (quasi-)free trade or autarky.

A.5 Proof of Fact 3

To prove Fact 3 we construct an intermediate equilibrium with intuitive properties which will turn out to be the unique two-stage equilibrium with control by country 3.

Step 1: Suppose that country 3 consumes the same amount of good 1 and good 2 in the two-stage equilibrium in which it controls the external agency. (12) Using [q.sup.1.sub.3] = [q.sup.2.sub.3] in Equation (A6) yields

[q.sup.1.sub.3] = [q.sup.2.sub.3] = 1 + [p.sub.2]/2[p.sub.3].

Hence the remaining problem is

[mathematical expression not reproducible]

which is equivalent to

[mathematical expression not reproducible]

Step 2: Suppose that group 1 and group 2 are treated symmetrically regarding consumption of the third good, that is, [q.sup.3.sub.1] = [q.sup.3.sub.2]. From Equation (A3) we obtain 2[q.sup.3.sub.1] = 2[([q.sup.3.sub.1]).sup.2] and [q.sup.3.sub.1] = [q.sup.3.sub.2] = 1. From Equation (A4) with [q.sup.3.sub.1] = 1 follows

[p.sub.2] + [q.sup.2.sub.1][p.sub.3] = 2[q.sup.2.sub.1] and [q.sup.2.sub.1] = [p.sub.2]/2 - [p.sub.3].

This yields the constraint: [p.sub.3] [less than or equal to] 2. Step 3: From (A5) with [q.sup.3.sub.2] = 1 follows

1 + [q.sup.1.sub.2][p.sub.3] = 2[q.sup..sub.2][p.sub.2] and [q.sup.1.sub.2] = 1/2[p.sub.2] - [p.sub.3].

This yields the constraint: [p.sub.3] < 2[p.sub.2].

Step 4: The remaining problem is

[mathematical expression not reproducible]

The second constraint will hold as an equality as the objective function is monotonically increasing (decreasing) in [p.sub.3] ([p.sub.2]), respectively. Hence, we are left with

[mathematical expression not reproducible]

As the objective function is monotonically increasing in [p.sub.3] we obtain [p.sub.3] = 2. As a consequence we obtain equilibrium prices

p = (1,1,2), [q.sub.1] = (1, [infinity], 1), [q.sub.2] = ([infinity], 1, 1), [q.sub.3] = ([1/2], [1/2], 1),

and allocation

[x.sub.1] = ([1/3], 0, [1/3]), [x.sub.2] = (0, [1/3], [1/3]), [x.sub.3] = ([2/3], [2/3], [1/3]).

Step 5: We note that, indeed, the solution above is the optimal intermediate equilibrium for group 3. At any system of internal prices individuals in group 1, group 2, and group 3 will supply 2/3 of their endowments to local markets. In the intermediate equilibrium we just constructed, group 3 receives all commodities offered by the first and second group, respectively. Hence given the hierarchical nature of trade the utility of group 3 cannot be improved further as no more commodities are traded in the markets.

A.6 Proof of Fact 5

Step 1: To construct the two-stage equilibrium with control by groups 1 and 2, we denote by B the set of global prices such that for every p [member of] B, there exists a two-stage market equilibrium ((p;[([q.sub.h]).sub.h[member of]P];x). Assuming that free trade is the default option, the problem facing the stakeholders of the external agency is maximization of the Nash product

[mathematical expression not reproducible]

As we shall see this procedure selects a two-stage equilibrium in which countries 1 and 2 obtain the same utility.

Step 2: We assume that group 1 and group 2 are treated symmetrically as they have the same bargaining power: symmetry is expressed by

[p.sub.2] = [p.sub.1] = 1, [q.sup.3.sub.1] = [q.sup.3.sub.2].

The property [q.sup.3.sub.1] = [q.sup.3.sub.2] and Equation (A3) yield [q.sup.3.sub.1] = 2[([q.sup.3.sub.2]).sup.2] which leads to [q.sup.3.sub.1] = [q.sup.3.sub.2] = 1. From Equation (A4) we obtain 1 + [p.sub.3][q.sup.2.sub.1] = 2[q.sup.2.sub.1] and hence

[q.sup.2.sub.1] = 1/2-[p.sub.3]

which yields the constraint [p.sub.3] [less than or equal to] 2. From Equation (A5) we obtain 1 + [p.sub.3][q.sup.1.sub.2] = 2[q.sup.1.sub.2] which yields

[q.sup.1.sub.2] = 1/2-[p.sub.3] = [q.sup.2.sub.1].

Step 3: The remaining problem is given by

[mathematical expression not reproducible]

as the objective functions of group 1 and group 2 are identical as functions of[ .sub.p3]. We assume that the smallest possible price [p.sub.3] is [epsilon] where [epsilon] > 0 can be arbitrarily small. Then, groups 1 and 2 will choose [p.sub.3] = e to maximize their common objective function. Given [p.sub.3] = [epsilon] we obtain

[q.sup.1.sub.2] = [q.sup.2.sub.1] = 1/2-[epsilon].

From Equation (A1) we get

[q.sup.1.sub.3] = [q.sup.1.sub.2]/2[q.sup.1.sub.2] - 1 = 1/[epsilon].

From Equation (A2) we get

[q.sup.2.sub.3] = [q.sup.2.sub.1]/2[q.sup.2.sub.1] - 1 = 1/[epsilon].

Finally, Equation (A6) reads

2 1/[epsilon] = 2[(1/[epsilon]).sup.2] [epsilon]

and thus holds.

Step 4: To sum up, the two-stage equilibrium with control by groups 1 and 2 is characterized by

p = (1,1,[epsilon]),

[q.sub.1] = (1, 1/2-[epsilon], 1),

[q.sub.2] = (1/2-[epsilon], 1,1),

[q.sub.3] = (1/[epsilon],1/[epsilon],1),

and the allocation

[x.sub.1] = ([1/3], 2-[epsilon]/3, [1/3]),

[x.sub.2] = (2-[epsilon]/3, [1/3], [1/3]),

[x.sub.3] = ([epsilon]/3, [epsilon]/3, [1/3]).

A.7 Proof of Fact 7.

We normalize prices so that [q.sup.1.sub.1] = 1, [q.sup.2.sub.2] = 1, [q.sup.3.sub.3] = 1. The corresponding excess demand vectors amount to

(A9) [z.sub.1] = (-[2/3], 1/3[q.sup.2.sub.1], 1/3[q.sup.3.sub.1]),

(A10) [z.sub.2] = (1/3[q.sup.1.sub.2], [2/3], 1/3[q.sup.3.sub.2]),

(A11) [z.sub.3] = (1/3[q.sup.1.sub.3], 1/3[q.sup.2.sub.3], -[2/3]).

Market clearing and the global budget constraint lead to modified Equations (A1)-(A6):

533

(A12) [q.sup.1.sub.2] + [q.sup.1.sub.3] = 2 1 - f/1 + f [q.sup.1.sub.2][q.sup.1.sub.3]

(A13) [q.sup.2.sub.1] + [q.sup.2.sub.3] = 2 1 - f/1 + f [q.sup.2.sub.1][q.sup.2.sub.3]

(A14) [q.sup.3.sub.1] + [q.sup.3.sub.2] = 2 1 - f/1 + f [q.sup.3.sub.1][q.sup.3.sub.2]

(A15) [p.sub.2][q.sup.3.sub.1] + [p.sub.3][q.sup.2.sub.1] = 2 1 - f/1 + f [p.sub.1[q.sup.2.sub.1][q.sup.3.sub.1]

(A16) [p.sub.1][q.sup.3.sub.2] + [p.sub.3][q.sup.1.sub.2] = 1 - f/1 + f [p.sub.2][q.sup.3.sub.2][q.sup.1.sub.2]

(A17) [p.sub.1][q.sup.2.sub.3] + [p.sub.2][q.sup.1.sub.3] = 2 1 - f/1 + f [p.sub.3][q.sup.2.sub.3][q.sup.1.sub.3]

Let us denote 1-f/1+f by A. Solving the above system of equations with respect to [q.sup.1.sub.2], [q.sup.1.sub.3], [q.sup.2.sub.1], [q.sup.2.sub.3], [q.sup.3.sub.1], [q.sup.3.sub.2] yields

[mathematical expression not reproducible]

with [q.sup.3.sub.2] as a parameter, as Walras' law applies and only five prices can be determined by the six Equations (A12)-(A17).

It remains to determine [q.sup.3.sub.2], [p.sub.1], [p.sub.2] and [p.sub.3]. Substituting the obtained group prices in the objective function of the external agency leads to

(A18) [pi] = [2/3]/f (1 + 1-f/1 + f)([p.sub.1] + [p.sub.2] + [p.sub.3])

Note that [pi] is independent of [q.sup.3.sub.2] and strictly increasing with the global prices. As [q.sup.1.sub.1] = [q.sup.2.sub.2] = [q.sup.3.sub.2] = 1 the global prices [p.sub.l], [p.sub.2] and [p.sub.3] can be at most 2. If we find a solution, that is, an equilibrium, with [p.sub.1] = [p.sub.2] = [p.sub.3] = 2 =: [??], we can conclude that the global prices and the associated equilibrium maximize the profit of the external agency. We therefore seek a symmetrical solution to the problem and set accordingly [??] = [p.sub.1] = [p.sub.2] = [p.sub.3]. Hence, we look for a maximal [??] at which the goods arbitrage condition becomes binding. The solution structure for the group prices becomes:

(A19) [q.sup.1.sub.2] = [q.sup.3.sub.2]/2A[q.sup.3.sub.2] - 1

(A20) [q.sup.1.sub.3] = [q.sup.3.sub.2]

(A21) [q.sup.2.sub.1] = [q.sup.3.sub.2]

(A22) [q.sup.2.sub.3] = [q.sup.3.sub.1]/2A[q.sup.3.sub.2] - 1 = [q.sup.1.sub.2]

(A23) [q.sup.3.sub.1] = [q.sup.3.sub.2]/-1 + 2A[q.sup.3.sub.2] = [q.sup.1.sub.2]

still with [q.sup.3.sub.2] as parameter. Note that all group prices are independent of [??].

We attempt to find an equilibrium with [??] = 2. By setting further [q.sup.3.sub.1] = 1, all prices are determined. We start by considering group price [q.sup.1.sub.2]. It follows

[q.sup.1.sub.2] = 1/2A - 1.

For the price [q.sup.1.sub.1] to be defined it must hold A > 1/2. To fulfill the goods arbitrage condition, it must hold that:

1 [less than or equal to] [q.sup.1.sub.2] [less than or equal to] 3

The left side of the inequality is fulfilled when A [less than or equal to] 1, which is always the case as f > 0. From [q.sup.1.sub.2] [less than or equal to] 3 it follows that A [greater than or equal to] 2/3 and f [less than or equal to] 1/5. It is readily verified that the choice of [q.sup.2.sub.3] = 1 produces the maximal lower bound on A among the feasible values 1 [less than or equal to] [q.sup.3.sub.2] [less than or equal to] 3 and ensures that the equilibrium exists for low values off. We note that for f < 1/5 and [q.sup.3.sub.2] = 1, all prices satisfy the goods arbitrage condition. An alternative equilibrium can be constructed by setting [q.sup.3.sub.2] = 2 which implies [q.sup.1.sub.3] = [q.sup.2.sub.1] = 2 and [q.sup.2.sub.3] = [q.sup.1.sub.2] = [q.sup.3.sub.1] = 2/4(1 2[epsilon])-1 = 2/3-8[epsilon]. This yields the allocation

[x.sub.1] = ([1/3], [1/6], 3-8[epsilon]/6), [x.sub.2] = (3-8[epsilon]/6, [1/3], [1/6],) [x.sub.3] = ([1/6], 3-8[epsilon]/6,[1/3]).

ABBREVIATION

MIP: Macroeconomics: Innovation and Policy

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HANS GERSBACH and HANS HALLER *

* A precursor of this paper appeared in 2007 as discussion paper entitled "Hierarchical Trade and Endogenous Price Distortions," CER-ETH Working Paper 07/72. We are grateful to Maggie Chen and two referees for beneficial critique and suggestions. We thank Clive Bell, Lucas Bretschger, Jurgen Eichberger, Switgard Feuerstein, Timo Goschl, David Orden, Till Requate, Tom Rutherford, Simone Valente, and seminar participants in Heidelberg, Blacksburg, and Zurich for insightful comments. Hans Haller gratefully acknowledges the hospitality and support of ETH Zurich, Chair of Macroeconomics: Innovation and Policy (MIP).

Gersbach: Professor, CER-ETH--Center of Economic Research, at ETH Zurich and CEPR, 8092, Zurich, Switzerland. Phone +41 44 632 8280, Fax +41 44 632 1830, E-mail hgersbach@ ethz.ch

Haller: Professor, Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0316, Phone 540 231 7591, Fax 540 231 5097, E-mail haller@vt.edu

doi: 10.1111/ecin.12513

(1.) For empirical evidence, see, for example, European Central Bank (2002), Lankhuizen, de Graaff, and de Groot (2015), or Engel and Rogers (1996).

(2.) In practice, such agencies provide a variety of services besides matching suppliers and customers, for example, brokerage services, organization of logistics, and contract design.

(3.) See for example, State Secretariat for Economic Affairs SECO, Switzerland (2013).

(4.) Trade costs of the first category are also present, which could be lowered through new storage and transportation techniques (see Barbe and Ricker 2015).

(5.) Empirical evidence and explanations are discussed in the literature reviewed below.

(6.) We also show how one can construct special classes of intermediate equilibria. First, if the number of economic groups is three at least, and is smaller than the number of goods traded, an intermediate equilibrium can be constructed by dividing the economy into two distinct exchange economies. Second, if the commodity space is at least threedimensional and the market demand of every group can be generated as the demand function of its representative consumer, it can be shown that intermediate equilibria exist and that they can be constructed by restricting the trade of a particular good within each group.

(7.) Moreover, the trade barriers are not imposed by the government and are intangible by nature.

(8.) In other words, the behavior of country [h.sup.c] based on indirect utilities can be derived from a representative consumer with utility function equal to [mathematical expression not reproducible].

(9.) A symmetrical case can be constructed for [epsilon] > 1 when the utilities for country 1 and 2 become minimal.

(10.) If the constituents of the agency have a concave utility function, the profits of the agency will be used to buy a bundle of commodities that maximizes the utility. The market-clearing condition (All) needs to be adjusted accordingly.

(11.) Yet it is also possible to impose tariffs in two-stage market equilibria. Such extensions are left for future research.

(12.) As and [q.sup.1.sub.3] and [q.sup.2.sub.3] are treated symmetrically in the objective function and in the constraints, and the objective function is logarithmic, the property [q.sup.1.sub.3] = [q.sup.2.sub.3] also follows formally from the maximization problem.
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Author:Gersbach, Hans; Haller, Hans
Publication:Economic Inquiry
Article Type:Abstract
Geographic Code:1USA
Date:Apr 1, 2018
Words:15066
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