This work is distributed as a Discussion Paper by the STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH

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1 This work is distributed as a Discussion Paper by the STANFORD INSTITUTE FOR ECONOMIC POLICY RESEARCH SIEPR Discussion Paper No A Spatial Theory of Trade By Esteban Rossi - Hansberg Stanford University June 2003 Stanford Institute for Economic Policy Research Stanford University Stanford, CA (650) The Stanford Institute for Economic Policy Research at Stanford University supports research bearing on economic and public policy issues. The SIEPR Discussion Paper Series reports on research and policy analysis conducted by researchers affiliated with the Institute. Working papers in this series reflect the views of the authors and not necessarily those of the Stanford Institute for Economic Policy Research or Stanford University.

2 A Spatial Theory of Trade Esteban Rossi-Hansberg Stanford University June 2003 Abstract The equilibrium relationship between trade and the spatial distribution of economic activity is fundamental to an analysis of the effect of trade policy and other trade frictions. We study this link using a trade model with a continuum of regions, transport costs, and agglomeration effects via production externalities. We analyze the equilibrium specialization and trade patterns for different levels of transport costs and show that the theory is consistent with evidence from estimations of the Gravity Equation. The effect of trade barriers is amplified by changes in specialization patterns thereby producing important border effects, as documented in the empirical literature. JEL Classification Code: F1, R0, R3 I would like to thank Robert E. Lucas Jr. for his help and advice on this proyect. Fernando Alvarez, Rudiger Dornbusch, Lars P. Hansen, Claudio Irigoyen, Boyan Jovanovic, Timothy Kehoe, Narayana Kocherlakota, Nancy Stokey, Alejandro Rodriguez, Mark Wright, Jaume Ventura and Mehmet Yorukoglu provided helpful comments. I also thank seminars participants at Chicago, Fuqua,ITAM,MIT,Minnesota,Stern,Stanford, the SED Meetings in New York, Rice and UCLA. erossi@stanford.edu, Ph: (650) Address: 579 Serra Mall, Stanford, CA, , USA. 1

3 1. Introduction Tradeisspatialbynature. Itallowsregionstospecializeintheproductionofasmall number of goods, while consumers and firms demand a much larger basket of products. The benefits that firms in a particular location may derive from specialization have to compensate them for the extra transport costs that result from exporting production and importing intermediate inputs. This trade-off results in a variety of possible patterns of production and trade flows. The paper presents a framework in which these patterns can be studied using a continuum of regions and the two other essential elements of any spatial model: transport costs and agglomeration effects. Without transport costs or agglomeration effects the distribution of economic activity would either be perfectly concentrated or evenly distributed. Trade frictions affect the incentives that firms face when choosing production locations and thereby the spatial distribution of economic activity and the implied trade flows. Trade theory has neglected these effects either because theories lack a spatial dimension or because the spatial dimension is introduced in a way that is not flexible enough to incorporate and analyze these frictions. One example is the effect of national borders on trade. National borders have been found to be an important obstacle for specialization. The empirical literature has found that national borders reduce regional trade substantially. That is, regions in the same country trade more between themselves than with regions in other countries, after controlling for distance and output levels. What makes borders have this important effect on trade flows and specialization patterns? Transport costs play an important role in determining specialization patterns. If transport costs are high, firms will locate near their customers even though they may gain from locating near other producers in the same sector, for example in the presence of knowledge spillovers. Nevertheless, transport costs can not explain the importance of national borders. The reason is that national borders matter even after controlling for distance, and transport costs clearly depend on distance. 2

4 Other trade barriers like import taxes may play an important role in explaining the role of borders. In all models of international trade, barriers imply a reduction in international trade flows. Nonetheless, the effect of borders has been found to be so important that traditional explanations using import taxes or other tariffs would require either unreasonably high tariffs or very high elasticities of trade with respect to these barriers. Clearly, high tariffs can not be part of the explanation, in particular for the European Union or North America, since we have seen important liberalization of trade in the form of free trade agreements 1. What about a high elasticity of trade with respect to tariffs? This paper proposes a mechanism that yields this large elasticity. The idea is that tariffs imply a discontinuity in the relative price at the border that changes the specialization pattern in all countries. Agglomeration effects and transport costs will amplify this effect in equilibrium. The first one because new or bigger clusters of firms will change the productivity of nearby firms. The second because these new clusters supply goods mostly for the domestic market, consequently increasing regional trade and reducing international trade. Hence, the elasticity of trade will be larger than in standard models, thereby augmenting the importance of borders in the presence of tariffs. The model is consistent with two other stylized facts in international trade. On one hand, several empirical studies have shown that trade flows decrease with distance. In our model, regions that are close to each other trade more than regions that are far apart. The model is consistent with this evidence as are many models with transport costs. Nevertheless, we will show that transport costs affect trade directly, and via changes in specialization patterns. On the other hand, relative prices of capital goods seem to be lower in developed countries than in underdeveloped countries. If we associate developed countries with countries that specialize in the production of 1 Notice, however, that other barriers like different standards, legal systems, languages or currencies, may have similar effects as deliberate trade policy. 3

5 intermediate goods, the model can explain why relative prices of intermediate goods are lower in these countries than in developing countries that produce final goods. The existing literature in international trade has paid little attention to space. Krugman (1991) pointed this out and presented a model of trade between two regions. Inhismodel,distanceisimportantsince itaffects the price of manufactured goods in the different regions. After this paper, several studies have tried to model space and to demonstrate how space, via transport costs, can explain several puzzles in international economics. Some examples are Obstfeld and Rogoff (2000), Eaton and Kortum (2002), Hummels (1999), Puga and Venables (1995, 1996), Krugman and Venables (1995b), and Krugman (1993, 1994 and 1999). In virtually all papers in this literature regions are modeled as points in space. This is clearly a simplifying assumption, but we believe that it is also an important one in the sense that it limits dramatically the set of possible equilibrium specialization patterns. Modeling countries as points in space results in transport costs having the same effect as any other trade barrier 2. In these models, it is difficult to study why regions specialize in the production of some goods if more than two regions are considered. In general, the theories are not designed to analyze production in different regions inside a country and, in the same framework, study international trade flows. One noteworthy exception is Krugman and Venables (1995a). The paper presents a model of spatial specialization between manufacturing and agricultural sectors with a continuum of locations. It assumes, however, that factors are geographically immobile. Factor mobility plays a crucial role in our framework since we want to analyze trade flows, and they will depend directly on the location of economic activity. International barriers on migration are important and we will consider them, but we relax the assumption of no regional or local mobility of labor. Even more important is that the paper does not study the effect of national tariffs orfrictions. We build a spatial model of trade that considers a continuum of regions in a line. 2 Setting aside general equilibrium effects resulting from the possible uses of tax revenue. 4

6 The model can be seen as a Ricardian model of comparative advantage in the spirit of Dornbusch, Fischer and Samuelson (1977), but the comparative advantage is endogenousandwilldependonthedensityofemploymentineachsectorateachlocation. Some elements of the framework are similar to the one developed in Lucas and Rossi- Hansberg (2002) to study land use in cities. There are two goods, a consumption good and an intermediate good that is used in the production of the final good. Therefore, regions will have to trade if they specialize in the production of one good. Two main forces drive the results in the model, transport costs and spatial production externalities. The external effects are specific to each sector, that is, firms are more productive if other firms in the same sector are located nearby. Higher external effects will concentrate production in each sector, while higher transport costs will result in several disconnected regions producing the same good or regions producing both goods. We want to study the location of firms producing each of these goods and analyze the trade flows and prices that this production map determines in equilibrium. Production externalities in each sector are assumed to be linear in the density of employment at other locations and to decrease exponentially with distance. Since employment at the different locations is determined in equilibrium, the production externality will be determined in equilibrium as well. The specific form of the production externalities assumed in this paper is arbitrary, however, we believe that it captures important elements of reality. Fujita and Thisse (2002) show how this particular specification of external effects can, for example, be justified as knowledge spillovers. If the production of ideas requires the interaction between people working in different firms at different locations, the closer are other producers, the more ideas will be generated and hence the more productive firms in that region. This makes distance to other producers in the same industry an important determinant of productivity. Dumais, Ellison and Glaeser (1997), Henderson (1988), Pyke, Becattini and Sengenberger (1990) and Saxenian (1994) provide evidence on this type of 5

7 externalities at the regional level. The remainder of the paper is organized as follows. The next subsection presents and discusses evidence on the stylized facts mentioned above. We set out the model in Section 2 and prove existence of an equilibrium in Section 3. In Section 4 we characterize the equilibrium and prove results related to the magnitude of relative prices in the different regions and distance as a determinant of trade. Section 5 introduces trade barriers and discusses their effect on trade flows. Up to this point the paper assumes perfect labor mobility, in Section 6 we relax this assumption and in Section 7 we present several numerical simulations of the model. Section 8 concludes. 1.1 Empirical evidence on several stylized facts Distance and borders as determinants of trade flows have been studied in the context of the Gravity equation. The standard specification of the Gravity equation with border effects is given by ln X ij = α 0 + α 1 ln M i + α 2 ln M j + α 3 ln Dist ij + α 4 B ij + ε ij, where X ij is some measure of trade flows between region i and region j, M i and M j are measures of economic activity in regions i and j respectively, Dist ij is distance between the regions and B ij is a dummy variable that takes the value of 0 if i and j are regions in different countries and 1 if they are regions in the same country. Bergstrand (1989), Sanso et all. (1993), Egger (1999) and Evenett and Keller (1998) all show that the estimate of α 3 is negative, significant and above 1. They use data for different sets of OECD countries and sometimes slightly different specifications. In particular, Egger (1999) estimates the log of exports to some specific countryusing relative factor endowments, the sum of the GDP of both countries, relative size of both countries, distance, and time and fixed effects. He uses data from 1985 to 1996 of OECD countries and estimates an elasticity of exports with respect to distance of The estimates are lower but higher than one if instead of OLS he uses a fixed or random effects model. 6

8 McCallum (1995),Wei (1996) and Helliwell (1998) using data for US-Canada trade and trade between OECD countries show that the estimate of α 4 is significant and ranges from 3 for Canada-US trade to 1.7 in some specifications of the model using OECD countries. Hence, regions in the same country trade between 20 and 5.4 times more with other regions in the same country than with regions in other countries. Anderson and van Wincoop (2003) show that it is important to control for the average trade barriers with respect to other countries. They control for them and estimate that provinces in Canada trade around 6 times more between themselves than with states in the US, after controlling for size, distance, and average barriers. The number for the US is much lower, states in the US trade about 25% more between themselves than with provinces in Canada. Even thou these numbers are not as spectacular as McCallum initial finding, they are still very large. Specially given NAFTA and standard estimates of import and export price elasticities of substitution (< 2.5, Goldstein andkhan(1985)) 3. Anderson and van Wincoop (2003) propose a theoretical model from which they derive the specification of the Gravity equation. Their theory, however, assumes constant production and so does not allow for potentially important supply side effects which our theory emphasizes. To my knowledge, there is no study that systematically tries to explain border effects using trade taxes and other barriers. The effective tax rate between the US and Canada as reported by Trefler (2001) was around 4% in This rate seem still 3 Wolf (2000) estimates the same regression for intra versus inter-state trade in the US and finds muchsmallerbordereffects (without controlling for average tariffs). He finds coefficients around 1.2, that imply a border effect of around 3. Hence, looking at explanations at the national level, that is looking at trade barriers that apply at national borders, seems to be an important line of research. However,thefactthatthesecoefficients are still significant implies that trade policy can not be the only element in an explanation of border effects. The mechanism proposed in this paper depends on trade barriers that change relative prices at the border. Different tax systems in different states in the US, or different industrial policies across states may create this discontinuity in relative prices (see Holmes (1998)). 7

9 important and able to create the changes in specialization that are essential to our explanation. Helliwell (1998) presents some evidence showing that the estimates of the border effect between the US and Canada have declined trough time. Nevertheless, a more complete study that analyzes how important are trade barriers to explain border effects is necessary and could provide evidence of the extent to which the mechanism we propose is important. Helliwell (1998) provides some evidence that border effects are important for relative prices (see also Engel and Rogers (1996)). Our mechanism has one extra empirical implication. It implies that trade barriers between countries that have a common border should have a larger effect. Hence, a border dummy that takes the value of zero if two regions have a common border and one if they do not would have to be significant and positive if added to the specification above (see Hanson, Mataloni and Slaughter (2002) for related evidence). We mentioned in the introduction that our model is consistent with equipment prices being higher in underdeveloped countries. Eaton and Kortum (2001) and Santos (2000) provide evidence on this fact. In 1996 data from the World Penn Tables implied that prices of equipment relative to prices of equipment in the US were: 1.27 for Canada, 1.12 for Japan, 1.26 for the European Union and 2.16 for Latin America. These numbers provide some evidence of the fact stated above. By contrast, the same is not true for structures, where the relative price in terms of the price in the US is 1.1 in Canada, 1.15 in Japan, 1 in Europe and 0.95 in Latin America. 2. The model The spatial structure of the model is given by a line from S to S. We interpret S as the northern (western) border and S as the southern (eastern) border of the region under study. Hence the sum of the length of all countries is 2S. Countries will be ordered sequentially and will be connected intervals in that line. For what follows in this section we will assume that there are no restrictions to trade or to the flow of labor. Hence, for the moment the specific location of borders is irrelevant. In 8

10 following sections we will analyze cases with these types of restrictions. There are two goods, a final and an intermediate good. Agents consume the final good only. The final good is produced using land, labor and the intermediate good. Production of the finalgoodperunitoflandatsomelocationr [ S, S] is given by x F (r) =g F (z F (r))f F (n F (r),c I (r)), where n F (r) is the density of workers in this sector at location r, c I (r) the demand for the intermediate good per unit of land at r, and z F (r) a production externality that depends on how many workers are employed at all locations in the final good sector (we will specify the construction of z F (r) below). The intermediate good is produced using land and labor. Production per unit of land of the intermediate good at location r is given by x I (r) =g I (z I (r))f I (n I (r)), where n I (r) is the density of workers in this sector at location r and z I (r) a production externality that depends on how many workers are employed at all locations in the intermediate good sector. There are sector specific production externalities. The external effect depends on the number of workers employed in the same sector at each location. It is supposed to be linear and to decay exponentially with distance at rate δ F for the final good sector and δ I for the intermediate good sector. Hence, in the final good sector, z F (r) = Z S S δ F e δf r s n F (s)θ(s)ds, where θ(r) is the proportion of land used at location r in the production of final goods. So 1 θ(r) 0 for all r [ S, S]. For the intermediate good the external effect is calculated similarly, z I (r) = Z S S δ I e δi r s n I (s)(1 θ(s))ds. 9

11 Agents consume only the final good, they do not consume land or the intermediate good. The utility of an agent living and working at location r is given by U(c F (r)), where c F (r) is the demand for the final good of a worker at location r. There is a perfectly elastic supply of workers at utility ū, so agents live and work at location r if U(c F (r)) ū. That is, there is free mobility of labor across regions and so the utility of each worker isthesameregardlessofthelocation. Notice that this implies that c F (r) = c F U 1 (ū), assuming that U is a strictly increasing function. Hence demand for the final good by workers is constant. It is costly to transport goods between locations. If one unit of the good i is transported from r to s, only 1 κ i r s e κi r s,i= F, I, reaches s. Namely, Iceberg transport costs where κ i is the transport cost per unit of distance in industry i. In what follows we will use the exponential approximation. Let p F (r) and p I (r) be the price of the final and intermediate goods respectively at location r. Then, an agent s wage at location r must be given by w F (r) =p F (r) c F and w F (r) =w I (r) w(r), 10

12 where the first equation is the agents budget constraint and the second guarantees that the supply of labor is positive at all locations. Hence, wages are proportional to the price of the final good. Landlords are assumed to consume what they earn from land rents at the location where they live and they consume final goods only. Hence, in final good producing regionsagentsthatownlandconsumepartoftheproduction. Inintermediategood producing regions landlords consume part of the imports of final goods. Landlords do not work. The maximization problem of a firm that produces the intermediate good at location r is given by max n pi (r)g I (z I (r))f I (n) w(r)n. (2.1) The first order condition with respect to n is p I (r)g I (z I (r))f I n I (n I (r)) = w(r) =p F (r) c F, where n I (r) is the density of workers of the intermediate good sector at location r that solves the problem above. The production of the intermediate good at location r is competitive so firms take all prices as given and earn zero profits. Maximized output minus labor costs is then the land bid rent by intermediate goods firms (R I (r)). So R I (r) =p I (r)g I (z I (r))f I (n I (r)) w(r)n I (r). Denote the relative price of intermediate goods by p(r) pi (r) p F (r), then the first order conditions defines the density of workers in the intermediate good sector (ˆn I (p(r),z I (r)) = n I (r)) asafunctionoftherelativepriceoftheintermediate good and productivity z I (r). r is Similarly, the maximization problem for firms in the final good sector at location max n,c pf (r)g F (z F (r))f F (n, c) w(r)n p I (r)c. (2.2) 11

13 So the first order conditions are g F (z F (r))f F n (n F (r),c I (r)) = c F and g F (z F (r))f F c (n F (r),c I (r)) = p(r) these conditions define ˆn F (p(r),z F (r)) = n F (r) and ĉ I (p(r),z F (r)) = c I (r) as the density of workers in the final good sector and the demand of the intermediate good as a function of the relative price of the intermediate good and productivity z I (r). Notice that the marginal product of labor in the final good sector is constant across locations. Again the final goods industry is supposed to be competitive at all locations and so firms earn zero profit. The land bid rent by final good firms (R F (r)/p F (r)) is given by R F (r) p F (r) = gf (z F (r))f F (n F (r),c I (r)) c F n F (r) p(r)c I (r). Let H F (r) be the excess supply of the final good accumulated between S and r. Then by definition H F ( S) =0and in order for the excess supply in the world to equal zero we need H F (S) =0. The evolution of this stock is dictated by the following differential equation H F (r) r = θ(r) x F (r) c F n F RF (r) p F (r) (1 θ(r)) c F n I (r)+ RI (r) p F (r) κ F H F (r) (2.3) wherewesubtractκ F H F (r) when H F (r) 0, and we add it when H F (r) < 0. At each location r we add to the stock of excess supply the total production minus total consumption of the final good in the final good sector, θ(r)[x F (r) c F n F R F (r)/p F (r)], and we subtract total consumption of the final good in the intermediate goods sector (1 θ(r)) c F n I (r) +R I (r)/p F (r). Wethenneedtotakeintoaccountthefactthat, for example if H F (r) is positive, as we increase r we need to transport some final goods further away from where they are produced. A fraction κ F 12 of these goods is

14 destroyed during transportation so we need to reduce the accumulated excess supply accordingly. If H F (r) is negative the intuition is similar. Using the definition of R F (r) and R I (r) we can simplify the equations to get H F (r) r = p(r) θ(r)c I (r) (1 θ(r))x I (r) (+)κ F H F (r). The construction of H I (r) parallels the one of H F (r) so H I (r) r =(1 θ(r))x I (r) θ(r)c I (r) κ I H I (r) (2.4) wherewesubtractκ I H I (r) if H I (r) 0 and we add the same term otherwise. At each location r weaddthetotalproductionofintermediategoodsinintermediate goods sectors, (1 θ(r))x I (r), subtract the number of intermediate goods used for production in final goods sectors, θ(r)c I (r), and adjust for extra transport costs, ±κ I H I (r). Free mobility of firms and no entry costs imply that land is assigned to its highest value. Namely, R F (r) > R I (r) implies θ(r) =1, (2.5) R F (r) = R I (r) implies θ(r) (0, 1) (2.6) and R F (r) <R I (r) implies θ(r) =0. (2.7) Combining this with the first order condition of the firm s problem, we know that θ(r) (0, 1) if x F (r)+ c F n I (r) n F (r) = p(r) x I (r)+c I (r). Therefore we can define the mixed relative price, p m (r), implicitly by p m (r) = ˆxF p m (r),z F (r) + c ˆn F I p m (r),z I (r) ˆn F p m (r),z F (r), [ˆx I (p m (r),z I (r)) + ĉ I (p m (r),z F (r))] 13

15 where ˆx F p m (r),z F (r) g F (z F (r))f F n F p m (r),z F (r),c I p m (r),z F (r), and ˆx I p m (r),z I (r) g I (z I (r))f I ˆn I p m (r),z I (r). The mixed relative price p m is the relative price that equalizes the value of land in both sectors. Notice that p m (r) depends on r only through the productivity functions. Hencewecandefine ˆp m (z F (r),z I (r)) p m (r). Given z F and z I the Envelope Theorem implies that RI (r) p F (r) p(r) = gi (z I (r))f I (n I (r)) > 0, and RF (r) p F (r) p(r) = ci (r) < 0, for all r [ S, S]. So (2.5)-(2.7) can be restated as p(r) < p m (r) implies θ(r) =1, p(r) = p m (r) implies θ(r) (0, 1) and p(r) >p m (r) implies θ(r) =0. The intuition for this is clear, if the relative price of intermediate goods is higher than a certain threshold, land is more valuable if used in the production of intermediate goods and vice versa. In equilibrium, prices have to satisfy no arbitrage conditions. Suppose that a firm located at r is shipping intermediate goods to location s. Then, given free mobility and no entry costs, by no arbitrage it has to be the case that p I (r) =e κi r s p I (s). 14

16 Suppose not, then it is profitable for firms to sell the goods at location r or produce them at location s. Hence the above condition has to be satisfied if intermediate goods are being transported from r to s. Same for the final good, if final goods are being transported from r to s it means, by no arbitrage, that p F (r) =e κf r s p F (s). Since the model is static, and so there is no credit, we know that if intermediate goods are transported from r to s, it must be the case that final goods are transported from s to r. That is, the current account is balanced at each location 4. Hence, we know that if intermediate goods are shipped from r to s, and if intermediate goods are shipped from s to r. p(r) =e (κi +κ F ) r s p(s) (2.8) p(r) =e (κi +κ F ) r s p(s) (2.9) Current account balance at each location is given, for all r [ S, S], by H F (r)+p(r)h I (r) =0. (2.10) That is, the value of the sum of excess supplies at each location is zero 5. The expression takes into account transport costs. An immediate implication is that for p(r) > 0, H F (r) =0is equivalent to H I (r) =0. If location r is a mixed area -an area where both goods are produced- it has to be thecasethatnogoodsareshippedfromortolocationr. The reason is that if not, 4 Even in a dynamic context with credit, locations can not borrow infinitely so a similar condition, in present value, would apply. 5 The condition can be reexpressed in terms of changes in the excess supplies as H F (r)/ r + p(r) H I (r)/ r + p(r)/ rh I (r) = 0. Since p(r)/ r = ± κ I + κ F p(r) we obtain that current account balance location by location is equivalent to H F (r)/ r + p(r) H I (r)/ r ± κ I p(r)h I (r) κ F H F (r) =0for all r [ S, S]. 15

17 prices would have to satisfy the wage no arbitrage conditions in (2.8) or (2.9) and p(r) =p m (r). This can happen but only for very particular circumstances, a set of Lebesgue measure zero. Hence, in mixed areas H I (r) =0and (1 θ(r))x I (r) θ(r)c I (r) =0, which implies by 2.10 that H F (r) =0and θ(r) x F (r) c F n F RF (r) (1 θ(r)) c F n I (r)+ RI (r) p F (r) p F (r) We still need to guarantee that it is not beneficial for other locations in the mixed area to buy goods from or sell goods to location r. Sowhenθ(r) (0, 1), prices have to satisfy =0. e (κi +κ F ) r s p(s) p(r) e (κi +κ F ) r s p(s). (2.11) The condition guarantees that in mixed areas prices do not compensate for transport costs and so in equilibrium these areas do not trade. Let M + be the space of non-negative and continuous functions. For the definition of equilibrium it is useful to define two operators T F : M + M + M + and T I : M + M + M + by and T F (z F,z I )(r) T I (z F,z I )(r) Z S S Z S S δ F e δf r s n F (s; z F,z I )θ(s; z F,z I )ds, δ I e δi r s n I (s; z F,z I )(1 θ(s; z F,z I ))ds. Notice that we are stressing in the notation that n F,n I and θ are functions of both productivity functions z F and z I. We are ready to define an equilibrium for this economy. Definition: An equilibrium is a set of functions {n F,n I,c F,c I,z F,z I,H F,H I, p F,p I,R I,R F,θ} such that 16

18 i) In all locations r [ S, S], U(c F (r)) = ū. ii) Firms solve problems (2.1) and (2.2). iii) The no arbitrage conditions (2.8), (2.9) and (2.11) are satisfied (e.g. free mobility and free entry). iv) Land is assigned to its highest value, so conditions (2.5)-(2.7) are satisfied. v) H F and H I evolve according to equations (2.3) and (2.4), H F ( S) =H I ( S) =H F (S) =H I (S) =0 and for all r [ S, S], and vi) For all r [ S, S], H F (r)+p(r)h I (r) =0. T F (z F,z I )(r) =z F (r) T I (z F,z I )(r) =z I (r). We could also see the model as a Ricardian model of comparative advantage and define equilibrium by conditions (i)-(v) given z I and z F. In this case the productivity of each sector at each location is given exogenously and we can determine the optimal distribution of production activities. This would be the case if, for example, natural resources determine the productivity of regions in both industries. In the next section we will show that such an equilibrium exists and is unique. 3. Existence of an equilibrium The construction of an equilibrium in this setup falls naturally into two steps. The first step is to find an allocation that satisfies conditions (i)-(v) in the definition of equilibrium given the productivity functions (z F,z I ). The second step is to find a pair of functions (z F,z I ) such that condition (vi) is satisfied. 17

19 Given a pair of productivity functions (z F,z I ), an allocation is uniquely determined by a relative price path p. Letp(r, π) be the relative price at location r associated with a price path starting at π, p( S, π) =π. At any location r, the relative price path has to grow at rate (κ F +κ I ) if H F (r) < 0, decline at rate (κ F +κ I ) if H F (r) > 0, and either grow (if p(r) p m (r)) ordecline(ifp(r) p m (r)) atrate(κ F + κ I ) or follow p m (p(r) =p m (r)) ifh F (r) =0. To construct an equilibrium relative price path, we can start with an initial relative price, π, and follow these rules to obtain a candidate equilibrium relative price path. Since every time H F (r) =0and p(r) =p m (r) the relative price path can evolve in three different ways, there may be many relative price paths associated with an initial relative price. Hence, each initial relative price path may be associated with many final stocks of excess supply of the final good ϕ F (π) H F (S, π). That is, ϕ F (π) is a correspondence. We are looking for a π such that 0 ϕ F (π). In the same way, one can define ϕ I (π) as the final stock of excess supply of intermediate goods given that the relative price path start at π. So ϕ I (π) H I (S, π). Clearly, in equilibrium, the initial relative price has to satisfy 0 ϕ I (π) also. Theorem 1 guarantees that there exists such a π and an associated price path so that the first five equilibrium conditions are satisfied. Theorem 2 uses Schauder fixed point theorem to prove that there exists a pair of functions (z F,z I ) that satisfy condition (iv). Many of the arguments in the proofs of the theorems parallel the arguments in Lucas and Rossi-Hansberg (2001). To prove both theorems we use the following assumptions. Assumption A: i) Both g F and g I are continuously differentiable and concave. ii) f F is continuously differentiable, strictly increasing in both arguments, strictly concave and fn F (the marginal product of labor) is strictly increasing in c I. 18

20 iii) f I is continuously differentiable, strictly increasing, and strictly concave. iv) There exists a pair of numbers 0 <ε< ε< such that for any x R + ε g F (x) εand and ε g I (x) ε. v) g F,g I,f F and f I satisfy, for all c I R +, g F (x)f F (x, c I ) g I (x)f I (x) lim =lim =0. x x x x vi) U is strictly increasing and strictly concave. Part (iii) of Assumption A implies that when the relative price of intermediate goods increases, the density of employment in the final good sector decreases. Part (iv) of Assumption A guarantees that the productivity of any firm is positive, and that even if the external effect is infinitely large, the productivity of firms is bounded. The magnitude of these bounds is arbitrary. We are ready to prove the existence theorems. Theorem 1: Under Assumption A, for any pair of positive and continuous productivity functions z F and z I there is an allocation that satisfies conditions (i)-(v). Any such allocation is associated with a uniquely determined relative price path p(r). Except for intervals on which p(r) coincideswiththemixedpathandwitheither(2.8) or (2.9), the allocation is uniquely determined. Proof: See Appendix 1. Theorem 2: Under Assumption A, there exists an equilibrium allocation. That is, there exists a pair of functions (z F,z I ) such that T F (z F,z I )(r) = z F (r) and T I (z F,z I )(r) =z I (r). Proof: SeeAppendix2. 19

21 Theorem 1 guarantees uniqueness of the equilibrium given a pair of productivity functions. Theorem 2, on the other hand, does not guarantee that the equilibrium is unique. This implies that the initial productivity functions will determine the equilibrium that is reached. One interpretation of this dependence is that history matters. The model is of course static and so taking about history is awkward, nevertheless if the adjustment in productivities takes time, this interpretation is natural. Location of firms in different sectors and their productivity at some point in time will determine the equilibrium allocation in the future. 4. Characterization of the equilibrium In order for an allocation to be an equilibrium, prices have to satisfy the no arbitrage conditions and land has to be assigned to its highest value. From these two elements we can construct a graph that will help us understand how land is assigned to its different uses. If we plot p m for all different locations, we know that if the relative price of intermediate goods is above this curve the region will produce the intermediate good, so θ =0. If the relative price is below this curve the area will produce the final good, θ = 1, and if it is equal it will produce both goods, θ (0, 1). We also know by (2.8), (2.9) and (2.11) that the relative price function has to grow at rate (κ F + κ I ), (κ F + κ I ) or be equal to p m. This construction implies that there are several equilibrium possibilities depending on κ F +κ I and p m. Noticethatatlocations where H F (r) =H I (r) =0, thereisakinkintherelativepricefunction. Thereason is that the shipments of goods change direction. Notice that in this construction relative prices of intermediate goods are higher in final good producing regions than in adjacent intermediate good producing regions. Thereby, confirming our claim that this model could explain the high capital goods prices in developing countries. Relative prices of intermediate goods may be low or high in regions that produce both goods. We formalize the result in the following proposition, the proof is included in Appendix 3. 20

22 Proposition 3: The relative price of intermediate goods is higher in regions such that θ =1than in adjacent regions such that θ =0. Defining bilateral trade between two location is cumbersome because firms in a particular location are indifferent about trading with several trading partners. Regions will trade -or will be indifferent about trading- with the closest region that produces the good that they do not produce. That is, prices will adjust to compensate for transport costs in regions such that H F > 0 or H F < 0. Therefore, locations are indifferent between trading with any other location in this region. If H F (r) =0, there is no trade between locations r 0 such that r 0 <rand locations r 00 such that r 00 >r. We define trade between two locations formally and prove the next proposition in Appendix3. Theideaisthattradebetween twolocations isdefined as the minimum between the two locations of the maximum amount of goods that flow trough them. The following result shows that distance reduces trade between location using the definition of trade described above. Proposition 4: If two locations produce different goods, the closer they are, the more they trade. 5. Trade barriers Assume that a final good producing country decides to impose a tax on the imports of the intermediate good from a specific country. Suppose, for example, that this country occupies locations [s 2,s 3 ], and that the country from which it imports intermediate goods occupies locations [s 1,s 2 ]. Then, the relative price of intermediate goods has to satisfy p(r) =e (κi +κ F ) r s p(s)+τ for r (s 2,s 3 ] and s [s 1,s 2 ), where τ is the tax per unit of the intermediate good. Other types of tariffs can be introduced in an analogous way. Versions of Theorem 1 and2canbeprovenwithminormodifications. The result of such a policy is that the country specialized in the production of final goods will produce more intermediate 21

23 goods and the other country will produce more final goods. This is a spatial version of the standard effect of tariffs. Figure 1 illustrates this example given productivity functions. Country 2 imposes a tax on the imports from Country 1. The solid line is the relative price of intermediate goods without taxes. Once the tax is imposed, the relative price of intermediate goods in Country 2 goes up at the boundary. This is illustrated by the solid thin line in the figure. The tax makes some locations in Country 2 switch and produce the intermediate good instead of the final good. This allocation can not be an equilibrium since at these prices the world produces too many intermediate goods. There is an excess supply of intermediate goods which leads to a decrease in the relative price of intermediate goods at all locations. The resulting relative price is the dashed line in the figure. Notice that because of the decrease in the price of intermediate goods, some locations in Country 1 will start producing final goods. Insert Figure 1 This example stresses two implications of the model. First, transport costs and trade barriers have potentially very different effects on the equilibrium allocation. Second, trade barriers will reduce trade, as expected, and increase the set of goods that each country produces. Notice that once we allow the productivity function to adjust, these effects will be even larger. The reason is that producing the final good, for example, in Country 1 will make Country 1 better at producing this good. The important conclusion out of this section is that trade barriers in the form of import or export taxes/subsidies imply discontinuities of the price function at borders. Without trade barriers of this sort, the relative price function is continuous. Different transport costs will imply a different evolution of the relative price function, but no discontinuities. Hence, the results of tariffs and changes in transport costs will in principle be different, except for special cases. Notice that if instead of having a continuum of regions we had considered several points in space (as most of the 22

24 Figure 1: The effect of an import tax with fixed productivities 3 Pm Tax P Country 1 Intermediate Country 2 F I Final I Country 3 Intermediate Location

25 previous literature) we could not make this distinction, so by assumption, as Obstfeld and Rogoff (2000) point out, the model would imply that tariffs and transport costs have exactly the same effect. We formalize the effect of trade barriers given productivity functions in the next proposition. Tariffs reduce bilateral trade and increase regional trade. The spatial version of the standard effect of tariffs. Proofs of the next two propositions are in Appendix 3. Proposition 5: Given a pair of productivity functions, trade frictions at the border, that do not imply trade reversals, weakly reduce trade between countries and weakly increase trade within countries. The presence of the amplification effect that helps us explain border effects can also be proven analytically. In particular, we want to establish that the standard implication described above is amplified in equilibrium, thereby increasing the elasticity of trade flows with respect to tariffs. We can formalize the result under some conditions. The first one is that production externalities decrease fast enough with distance. The condition is necessary since we want the local increase in production of the protected good to influence productivity more than the decrease in production in other countries. Given that most examples of regional externalities -like Silicon Valley or Route 128- are for relatively small regions, we do not believe that this is a very restrictive assumption. We also need no trade reversals. The reason is that countries may start exporting adifferent good after imposing the tariff, and they potentially could export large amounts. This, however, does not imply that the elasticity of trade barriers is small since in that case the effective tariff, or friction, becomes zero. The last condition comes out of the potential multiplicity of equilibria in the model. Starting from an equilibrium without tariffs we need to determine which of the potentially many equilibria we will reach. Proposition 6 presents the results. 23

26 Proposition 6: (Amplification effect) In equilibrium, trade frictions at the border weakly reduce trade between countries and weakly increase trade within countries more than the respective reduction and increase given the productivity functions if production externalities decrease fast enough with distance (high δ F and δ I ), frictions imply no trade reversals, and starting from an equilibrium with productivity functions z F,z I,thesequence of functions Tτ F n z F,z I and Tτ I n z F,z I, n =1, 2,..., converge. Where the operators Tτ F and Tτ I are defined for the economy with frictions. 6. The model without international labor mobility Probably the most natural specification of the model is to allow for labor mobility inside a country but not between countries. Migration laws restrict the free mobility of workers in almost all nations in the world (Europe is an example for which the version presented so far is better suited). In this subsection we will modify the model presented above so that workers can move freely inside a country, but not across countries. Free mobility of labor equalizes the utility of agents inside the same country. Hence, in this version of the model, utility levels will vary between but not within countries. In order to incorporate the no migration restriction we can proceed in two ways. We can define the utility levels of agents in each country and let the population size in each country be determined in equilibrium, or we could determine the population size in each country and let the utility level that agents in each country experience be determined in equilibrium. For the first case we need to specify a utility level for each country. Let the number of countries be given by an integer NC. Let the two borders of each country be determined by B(i) and B(i),fori =1, 2,..., NC. Where B(i) =B(i 1) by definition. Denote the utility level of agents in country i by ū(i), i =1, 2,..., NC. Then, condition i) in the definition of equilibrium becomes: 24

27 i )For all r B(i), B(i),i=1,..., NC, U(c F (r)) = ū(i); and U(c F (S)) = ū(nc). The rest of the analysis follows as in the previous sections. Notice that the modified definition of equilibrium implies that, for U strictly increasing, c F (r) is a step function with constant value inside each country and different value between countries. That is, real wages vary between but not within countries. For the second case, we need to determine the population of each country exogenously. Let Pop(i) be the population size in country i. Then Pop(i) = Z B(i) B(i) (θ(r)n F (r)+(1 θ(r))n I (r))dr, i =1,..., NC, is the equilibrium condition in country i s labor market. Notice also that in this case the utility of agents in country i, ū(i), will be determined in equilibrium. Condition i) in the definition of equilibrium then becomes: i )For all r B(i), B(i),i=1,..., NC, U(c F (r)) = ū(i); and U(c F (S)) = ū(nc). Population sizes in all countries are given by Pop(i) = Z B(i) B(i) (θ(r)n F (r)+(1 θ(r))n I (r))dr, i =1,..., NC. Again, all other features of the model presented in previous sections remain unchanged. Theorems 1 and 2 apply with slight modifications. Notice that the two 25

28 cases presented above (exogenous utilities or population sizes) are equivalent. That is, there is a one to one mapping between countries population sizes and countries utility levels 6. It is important to point out that since the function c F ( ) is not a continuous function, n F,n I,c I,x F and x I may not be continuous functions either. Hence, p m will have discontinuities at the country borders. Nevertheless the relative price function is still continuous in the absence of trade barriers. 6.1 Permanent productivity differences without international migration One of the problems of restricting international labor mobility is that countries with large populations and hence low real wages, will produce relatively more than other countries. The extra production, via larger employment densities, will increase the productivity of these countries in equilibrium. The reason is that the production externality depends on the number of workers employed in the same sector at nearby locations. This effect may end up making poor countries (low real wage) much more productive than rich countries (high real wage). This is an undesirable property of the model since in reality we observe permanent differences in the level of total factor productivity (TFP) between countries (See for example Parente and Prescott (2000)). The only determinant of productivity in this model is proximity to other producers in the same sector. Clearly, there are other country-specific factors that determine the level of TFP inside a country. Two important examples are capital market frictions and institutional differences. One simple way of adding country-wide differences in the level of TFP to our 6 To see this, remember that the first order conditions with respect to employment densities of the firms problems are given by g F (z F (r))fn F ( n(r)θ(r),ci (r)) = U 1 (ū(i)), and p(r)g I (z I (r))fn I (n I (r)) = U 1 (ū(i)), where r B(i), B(i). Hence, under Assumption A, a higher I utility level in country i implies a lower employment density in both industries. Of course, changes in the utility function will change the productivity and prices at different locations and so the result is more complex, but the direct effect is the one described above. 26

29 model is to make the level of the production function depend on the country in which production is located. Let production per unit of land of the intermediate and final good in country i be determined by x I (r) =A(i)g I (z I (r))f I (n I (r)), and x F (r) =A(i)g F (z F (r))f F (n F (r),c I (r)), for r B(i), B(i). Then, A(i) represents the permanent level of TFP in country i. So if country i is the rich country and country j thepoorone,wewouldexpect A(i) >A(j). Since, under Assumption A, the production functions exhibits diminishing returns with respect to the densities of workers and the number of intermediate goods per unit of land, differences in A across countries will imply discontinuities in p m. Nevertheless, the discontinuities will tend to cancel out (not completely except for very special cases) with the discontinuities created by the differences in population sizes. 7. Numerical examples We will illustrate the different equilibrium possibilities of the model with numerical examples. In all numerical examples we will use a Cobb-Douglas specification for the utility function and both production functions. In particular, we will use 7, x F (r) = z F (r) γf n F (r) αf c I (r) βf, x I (r) = z I (r) γi n I (r) αi, and U(c F (r)) = c F (r) β. 7 Notice that in order for part iv) of Assumption A to be satisfied, we need to h i use the following specification: x F (r) = min z F (r) γf + ε, ε n F (r) αf c I (r) βf and x I (r) = h i min z F (r) γf + ε, ε n I (r) αi. However, we can choose ε low enough and ε high enough so that the bounds do not play a role in the results. The parameter values choosen are such that all the other parts of Assumption A are satisfied. 27

30 In all numerical exercises presented below we will let γ F = γ I =0.04, α F =0.8, β F =0.1, α I =0.7 and δ F = δ I =5. These parameter values are arbitrary but allow us to illustrate several possible equilibrium allocations. All the results presented in the next section are for the case of perfect labor mobility using ū =1. Total population, however, does not change significantly in the different examples. 7.1 Specialization patterns for different levels of transport costs Transport cost parameters are of particular importance for the qualitative features of the equilibrium. If transport costs are very high, regions will not trade and the solution of the model is autarchy. Different regions will produce different amounts of the intermediate and final goods and population densities will vary across regions. If transport costs are low, regions will specialize completely. In the example presented in Figure 2 for κ F = κ I =0.005 thereisaregionatthecenterproducing the final good, and the rest of the world will produce the intermediate good. The intermediate good region at the left will trade with the closest half of the final good region (the region to the left of the kink in the relative price curve). There is no trade between this intermediate good region and the other half of the final goods region. Relative prices do not cover transport costs, so producers prefer to trade with regions that are closer. It is evident that regions trade with the closest region that produces the good that they do not produce and that relative prices of intermediate goods are higher in final good producing regions than in intermediate good areas. Insert Figure 2 Higher transport costs (κ F = κ I =0.1) implytwofinal good regions as shown in Figure 3. Notice that now we have four regions in which there is trade within the region but not between regions. These areas can be identified as the areas between the kinks in the relative price curve. These kinks correspond to the locations at which both H F and H I are equal to zero. Trade flows are much lower than in the previous 28